Conformal Geometry of Discrete Groups and Manifolds (Degruyter Expositions in Mathematics)

Conformal Geometry of Discrete Groups and Manifolds by

Boris N. Apanasov

W DE

G Walter de Gruyter - Berlin New York 2000

Author Rotis N. Apanasov 176partment of Mathematics University Oklahoma 601 Elm Avenue, PHSC 423 Norman, OK 73019-0315 USA [email protected]

Mathematics Subject Classification 2000: 51-01, 51-02, 57-01, 57-02; 11Fxx, 20-XX, 22E40, 30C65, 30F40, 32Gxx, 52C25, 53Axx, 53Cxx, 55Nxx, 55Rxx, 57Mxx, 57Nxx Key words:

Geometric Structures on Manifolds, Discrete Groups, Conformal Geometry, Hyperbolic Geometry, Deformations of Structures, Teichmnller Spaces ® Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability.

Library of Congress - Cataloging-in-Publication Data Apanasov, B. N. (Boris Nikolaevich) Conformal geomety of discrete groups and manifolds / by Boris N. Apanasov.

p. cm. - (De Gruyter expositions in mathematics, ISSN 0938-6572; 32) Includes bibliographical references and index. ISBN 3110144042 (alk, paper)

1. Conformal geometry. 2. Discrete groups. (Mathematics) I. Title. II. Series. QA609 .A63 2000 516.3'5-dc2l

3. Manifolds

00-043017

Die Deutsche Bibliothek - Cataloging-in-Publication Data Apanasov, Boris N.: Conformal geometry of discrete groups and manifolds / by Boris N. Apanasov. - Berlin ; New York : de Gruyter, 2000 (De Gruyter expositions in mathematics ; 32) ISBN 3-11-014404-2

@C Copyright 2000 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany.

All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage or retrieval system, without permission in writing from the publisher. Typesetting using the author's TEX files: I. Zimmermann, Freiburg. Printing: WB-Druck GmbH & Co., Rieden/Allgau. Binding: Liideritz & Bauer-GmbH, Berlin. Cover design: Thomas Bonnie, Hamburg.

To Tatiana, Anton and Nikolay

Preface

One of the most beautiful results in classical complex analysis that has a great appeal to geometry is the solution by F Klein and H. Poincare of the uniformization problem for multivalent analytic functions and (consequently) Riemann surfaces. It states that each conformal structure on a Riemann surface is induced by one of the three classical geometries: Euclidean, spherical or hyperbolic (Lobachevskian), that is this structure

is represented by a Riemannian metric on the surface which has a constant (zero, positive or negative) curvature. According to Felix Klein's Erlangen program of 1872, geometry is the study of

the properties of a space which are invariant under a group of transformations. A geometry in Klein's sense is thus a pair (X, G) where X is a manifold and G is a Lie group transitively acting on X. Due to the Klein-Poincare geometrization theorem, the Euclidean, spherical and hyperbolic geometries are the most important ones in dimension two. However, they are all particular cases of the more general conformal geometry, that is the (S2, M&(2))-geometry, where Mob(n) is the group of conformal (Mobius) transformations of the n-dimensional sphere S" . This is not a Riemannian geometry. A conformal structure on a manifold M is the same as a conformal class of Riemannian metrics, each locally conformally equivalent to a flat metric. In dimension three, due to Thurston's geometrization, many 3-manifolds admit conformal structures, although relatively simple ones might not (among the eight possible 3-geometries, nontrivial closed solvable and nilpotent manifolds are examples of this). Generally, conformal geometries naturally appear at infinity for negatively curved Riemannian

geometries. Moreover, due to M. Gromov's [5, 6] geometric approach to infinite groups, conformal geometry invents new fruitful methods in combinatorial group theory.

The main goal of our book is to present the first systematic study of conformal geometry of n-manifolds, as well as its Riemannian counterparts (in particular, hyperbolic geometry). A unifying theme is the discrete holonomy groups of the corresponding geometric structures, which also involves algebra and dynamics. However, we do not pay much attention to 2-dimensional geometries covered by many classical and recent books (see, for example, Casson-Bleiler [1], Beardon [4], Ford [1], Kra [3], Maskit [12]). Also, this book does not cover conformal geometries that appear at

viii

Preface

infinity for noncompact symmetric spaces with variable sectional curvature. Nevertheless we indicate some relationship to those geometries and provide the necessary references. Regarding hyperbolic geometry, some recent books may be useful for the reader as a source of alternative approaches and references: Benedetti-Petronio [1], Ratcliffe [1] (with rich historical notes) and Apanasov [36]. We have minimized the unavoidable overlap in the covered results on hyperbolic manifolds by using those books, especially the last one (which is an English edition of the 1983 Russian book Apanasov [15]), as a source for preliminary results and constructions. Additionally, the enormous expansion in journal literature on conformal geometry of manifolds and the new important results obtained in the last decades years have allowed us to point out new connections and perspectives in this field and to illustrate various aspects of the theory. In addition to formal proofs we also indicate some intuitive approaches, which emphasize the ideas behind the constructions. This is complemented by a large number of concrete examples (continuing the book by S. Krushkal', B. Apanasov and N. Gusevskii [3]) and figures which both use and support the reader's geometric imagination and make the matter more transparent. We have tried to make the book as complete as possible, although the choice of topics obviously reflects our personal preferences. Our interest in this area started in the "golden years" of mathematics in Novosibirsk Akademgorodok in the seventies and eighties, when the author worked in a remarkable geometry/topology group that included A. D. Aleksandrov, P. P. Belinskii, V. Goldstein, N. Gusevskii, S. Krushkal', V. Marenich, A. Mednykh, I. Nikolaev, Yu. G. Reshetnyak, V. Toponogov and several doctoral students: D. Derevnin, V. Chueshev, M. Kapovich, E. Klimenko, G. Lyan, L. Potyagailo, A. Tetenov, A. Vesnin, S. Vodopyanov, I. Zhuravlev and others. This provided a perfect environment for our research and for advanced graduate courses we taught at the Novosibirsk State University and at the Sobolev Institute of Mathematics in the Academy of Sciences. These courses were continued in graduate courses we taught at the Universitat Autonoma de Barcelona (Spain) and at the University of Oklahoma in Norman, USA. This book is based on those courses, and it should be accessible to advanced graduate students in either mathematics or theoretical physics. In particular, the first three chapters (which make the book self-contained) are addressed to those graduate students who are approaching the subject for the first time. These chapters may be used as a text for a graduate class. The book quickly introduces these students to up-to-date problems. To the second type of readers, mature mathematicians working in other fields and theoretical physicists, this book gives new knowledge and understanding of conformal geometry on manifolds and the conformal action of fundamental groups. To the experts, the book presents some new material published for the first time. It is our deep pleasure to thank our colleagues and friends with whom we discussed the subject for a long time. In addition to those mentioned above, a debt of gratitude for a series of valuable remarks is owed to Francis Bonahon, Dubravko Ivangic, Anatoly Fomenko, Michael Gromov, Yoshinobu Kamishima, Ann Chi Kim, Ravi

Preface

ix

Kulkarni, Kyung Bai Lee, Gregory Margulis, Sergei Matveev, Alexander Mednykh, Darryl McCullough, Jose-Maria Montesinos Amilibia, Walter Neumann, Vladislav Nikulin, Ken'ichi Ohshika, Masahico Saito, Larry Siebenmann, Alexander Starkov, Bill Thurston, Andrei Vesnin, Ernest B. Vinberg, Oleg Viro and Chengbo Yue. Our special thanks are due to Nikolai Saveliev for discussions and help during the writing this book. Finally, we are deeply indebted to the Walter de Gruyter Publishing Company, especially to Dr. Manfred Karbe, for encouraging us to write this book. Norman, Spring 2000

Boris Apanasov

Table of Contents

Preface .................................................................. vii

1. Geometric Structures .................................................... 1

........................................ ..................................... 4 ............................................ 11

1.1. 1.2. 1.3.

(X, G)-structures on manifolds Conformal geometry on the sphere The hyperbolic space IHI'

1.4.

Lie subgroups of the MSbius group ................................... 19

1.5. 1.6. 1.7.

Structure developments and holonomy homomorphisms ................. 22 The eight 3-dimensional geometries

1

................................... 25

Four-dimensional geometries ......................................... 31

1.8. Geometry of orbifolds ............................................... 33 Notes ....................................................................39

2. Discontinuous Groups of Homeomorphisms

............................. 40

2.1.

Convergence groups ................................................. 40

2.2. 2.3. 2.4. 2.5.

Group action on the discontinuity set ................................. 54 Fundamental domains

............................................... 62

Convex polyhedra and reflection groups ............................... 77 Discrete group action on the limit set ................................. 86

Notes ...................................................................100

3. Basics of Hyperbolic Groups and Manifolds 3.1. 3.2.

............................ 101

Margulis's Lemma and splittings of hyperbolic manifolds .............. 101

Injectivity radius of hyperbolic manifolds ............................. 107

3.3.

Thin cusp submanifolds ............................................. 111

3.4.

Precisely invariant horoballs ......................................... 117

3.5.

Group action on the set of horoballs .................................. 121

3.6.

Convex hull constructions ........................................... 123

3.7.

Tessellations of manifolds by ideal hyperbolic polyhedra ............... 129

xii 3.8.

Table of Contents Hyperbolic arithmetics .............................................. 130

3.9. Arithmetic groups generated by reflections ............................ 133 3.10. Non-arithmetic groups of Gromov and Piatetski-Shapiro ............... 140 3.11. Fibonacci manifolds ............................................... 144

Notes ...................................................................151 4. Geometrical Finiteness ................................................ 153

4.1. 4.2. 4.3. 4.4.

Classical finiteness for planar Kleinian groups ......................... 153 Geometrical finiteness in higher dimensions .......................... 156 Equivalent definitions of geometrical finiteness ........................ 166

4.5.

Geometry of tessellations ........................................... 178

4.6. 4.7.

Cayley graphs and geometric isomorphisms of discrete groups ......... 191 Geometrical finiteness for discontinuity set components ............... 204

Geometrically finite ends and coverings .............................. 175

Notes ...................................................................215 5. Kleinian Manifolds ....................................................217

5.1. 5.2. 5.3. 5.4.

Basic topology related to Kleinian manifolds ..........................217 Topological aspects of combination theorems ......................... 225 Universal groups and Poincare Conjecture ............................ 241

5.5. 5.6.

Kleinian n-manifolds and hyperbolic cobordisms Finiteness problems for Kleinian n-manifolds .........................

6.1. 6.2. 6.3. 6.4. 6.5. 6.6. 6.7. 6.8.

Classical uniformization ............................................290

Ends of Kleinian manifolds, their compactification

and Ahlfors's Conjecture ........................................... 244

...................... 256

276 Notes ....................................................................288 6. Uniformization ....................................................... 289

.................................. 291

Modem concepts of uniformization Hyperbolization of manifolds and hyperbolic volumes ................. 308 Uniformizable conformal structures Conformal uniformization of "flat" connected sums ................... 335 Conformal uniformization of Seifert manifolds ....................... 342 Torus sums of conformal structures Canonical Riemannian metric on conformal manifolds ................. 355

.................................. 324

.................................. 350 Notes ....................................................................377 7. Theory of Deformations ............................................... 379

................................ 379

7.1.

Deformations of geometric structures

7.2.

Rigidity of hyperbolic structures .................................... 392

7.3. 7.4.

Quasi-Fuchsian structures: bendings Quasi-Fuchsian structures: cone deformations

................................. 408 ........................ 425

Table of Contents

7.5. 7.6.

xiii

Bendings along surfaces with boundaries .............................434 Global properties of deformation spaces .............................. 441

Notes ....................................................................457 Bibliography ............................................................ 461

Index ...................................................................513

Chapter 1

Geometric Structures

In addition to purely topological methods in the study of manifolds, last two decades results and especially Thurston's work [1-8] have shown that geometry also plays an important role in low-dimensional topology. The basic aim of this chapter is to introduce (following Thurston [1]) the important concept of geometric (Riemannian and sub-Riemannian) structures on manifolds and its generalization for manifolds with singularities (so-called orbifolds). We will also discuss various `nice' geometries which arise in dimensions three and four, as well as the relationship between geometric and topological properties of manifolds carrying those geometries. This discussion will be continued in Chapter VI.

§1. (X, G)-structures on manifolds Let M be a topological Hausdorff space with a countable basis. One calls M a topological n-manifold (n-dimensional manifold) if each point x E M has a neighborhood U homeomorphic to the Euclidean space I[8n = gp(U). If in this condition, the homeomorphic image p(U) may be either the Euclidean space RI or its closed half-space {x E Il8" : xn, > 0}, we arrive at the definition of a manifold with boundary. Here the set of points x E M not having neighborhoods homeomorphic to the Euclidean space forms the boundary 8M, which is a manifold of dimension (n - 1). A compact manifold without boundary is called closed. In such a way, a topological n-manifold is specified by its atlas {(Ui, O)}jEZ consisting of local charts cpi: Ui -* Rn where, for Ui fl Uj are homeomorphisms defined on subdomains 0, changes of charts in R. M is called a smooth manifold if, for any charts (U1, Cpl) and (U2, cP2) on M in the chosen atlas with Ui fl u2 ; 0, the changes of charts Cpl o o2 1 are smooth.

Now let M be a smooth manifold. For each x E M, we denote by Tx (M) the tangent space of the manifold M at x (an n-dimensional topological vector space). TX (M) and the natural projection p: T (M) -* M Consider the set T (M) = such that p(x) = T. (M). The triple {T (M), M, p} is called a tangent fiber bundle of the manifold M, while the set p -1(x) is called a fiber in the fiber bundle over x E M.

1. Geometric Structures

2

A Riemannian manifold is said to be a pair (M, g) consisting of a smooth manifold R, which is a positive-definite quadratic form M and a smooth mapping g: T (M) in each fiber Tx (M). The Riemannian manifold (M, g) is a metric space whose metric is generated by those quadratic forms. That is, for points yo, yl c M,

p(Yo, Yl) = inf length(y) = inf f 1 gY(t)(Y'(t))dt,

(1.1)

where the lower bound is taken over all continuously differentiable curves y in M joining points yo and yl, i.e., y (0) = yo and y (1) = yl. The function p: M x M -j R has all the properties of a metric. A curve y c M is described as being the shortest if its length is the least among all curves with the same ends. It is clear from (1.1) that the curve y with ends yo and yl will be the shortest if and only if its length is p (yo, yl). A curve y C M is called a geodesic if each of its points has a neighborhood such that each arc of the curve y in this neighborhood is the shortest one. Considering three kinds of curvatures for Riemannian manifolds (sectional curvature, Ricci curvature and scalar curvature), we will be mainly concerned with the sectional curvature which has a geometric description making it easier to handle. One may define the sectional curvature in a point p of a Riemannian manifold M with respect to a 2-subspace V C TpM as the curvature at p of the oriented Riemann surface obtained as the image of a small neighborhood U(0) C V under the exponential mapping.

There is another way to add more structure to a manifold. Namely, instead of considering an atlas on M, one can think about M as if it were composed of pieces of IRn, glued together by the homeomorphisms gij = (pi to, 1. Denoting this set of homeomorphisms by 9,, we see that it should satisfy the following obvious conditions which transform 9, to a pseudo-group of local homeomorphisms between open sets

in R': 1)

a restriction go of any element g E $. to an open set in its domain is an element of

2)

a composition g o h of any two elements g, h E 9, (if defined) is an element of

3)

the inverse element for g E 9, is an element of 9,; and

4) if D = Ui Di C IRn and g: D -> D' is a local homeomorphism such that

gi =gIDi E 9foralli,then g E 9,. Manifolds M, obtained by gluing together pieces of 1R by means of local homeomorphisms from a pseudo-group 9, are called 9-manifolds. Specifying a pseudo-group 9, in this definition, one can add more structure on a 9,-manifold M. For example, we arrive at the notion of a PL-manifold if $, is the pseudo-group of local piecewise-linear homeomorphisms in IRn, and at the notion of a CT-manifold (a C'-smooth manifold, for r > 1) if 9, is the pseudo-group of local

1.1. (X, G)-structures on manifolds

3

C'-diffeomorphisms in R. Furthermore, a very important class of g-manifolds can be obtained by using the following pseudo-groups g. Let us fix a manifold X and a group G of self-homeomorphisms of X and consider a pseudo-group 9 consisting of all restrictions of elements of G to open subsets in X. A manifold M with such 9,-structure is represented as composed of pieces of X by means of elements of the pseudo-group g associated with the group G. We call such g-manifolds (X, G)-manifolds or manifolds modeled on (X, G)-geometry. Here, we use the term "geometry" in the sense of F. Klein, meaning by geometry

of the pair (X, G) those properties of X that are left invariant under the group G action. Sometimes, this "invariant" approach to studying geometry is equivalent to the classical approach which can be used to study Euclidean and non-Euclidean geometries as well as to the Riemannian geometry approach (see Theorem 1.14).

Example 1.1 (Affine torus). Let X = R' and G = Aff (W) be the group of affine transformations in RI. Then a (Rn, Aff(Rn))-manifold is an affine n-manifold. In particular, an affine structure may be defined on two-dimensional torus T2 = S1 X S1 by gluing the opposite sides of a quadrilateral which needs not to be a parallelogram. This gluing is performed by means of two affine transformations in the plane (see Figure 1), i.e., by elements of the affine pseudo-group, which are restrictions of these affine transformations on corresponding open subsets in R2 containing sides of the quadrilateral. However, we shall see in what follows that, for a non-parallelogram case, a torus with such structure is not a complete affine manifold.

Example 1.2 (Euclidean torus). One can introduce a Euclidean structure on n-dimenx S1 whose fundamental group is a free Abelian group sional torus Tn = S1 x of rank n. In fact, this torus Tn may be obtained by gluing the opposite sides of an n-dimensional parallelepiped, (Figure 2). The gluing mappings may be chosen to be Euclidean translations, and thus a complete Euclidean structure is introduced on the torus Tn.

Figure 1

Figure 2

1. Geometric Structures

4

Example 1.3 (Torus shell). Considering the extension of the gluing mappings used for 2-torus from the plane R2 = {x E R3 : X3 = 0} to R3 as Euclidean isometries, we introduce a Euclidean structure on the interior of a torus shell M = T2 x (0, 1). Indeed, this open manifold is homeomorphic to the quotient space 1[83 / G where the (discrete) group G of Euclidean isometries of 1[83 is generated by the gluing translations

xHx+elandxI-rx+e2. Example 1.4 ("Thickened" Klein bottle). We define "thickened" Klein bottle as the compact 3-manifold M = (T2 x [0, 1])/7L2, where the group Z2 acts on T2 as the deck transformation group for torus covering the Klein bottle and flips the segment [0, 1]. In the interior of a thickened Klein bottle M, we may introduce a Euclidean structure. Indeed, int(M) is homeomorphic to the quotient space R3/G, where the (discrete) group G is generated by two Euclidean isometries, x -+ x + el and x =

(-xl, x2 + 1, -x3), which glue the opposite sides of the column (xl, x2, x3) {x E 1[83: lxl I < 1/2, 1x21 < 1/2}. Similarly to the Euclidean manifolds, one obtains spherical, or elliptic, n-manifolds if, in the definition of a (X, G)-manifold, considers X = Sn and G the isometry group of the n-sphere S". Examples of such manifolds are 3-dimensional lens spaces L(p, q) and the real projective n-space 1181". Also, considering X as the hyperbolic n-space En and G as the group IsomIHF of all isometries of EP', one arrives at the definition of a hyperbolic manifold, a Riemannian manifold with a complete metric of constant negative sectional curvature. All these (Riemannian) geometries are particular cases of the so-called conformal geometry which we define in the next section.

§2. Conformal geometry on the sphere 1. Spaces of constant positive curvature. A simplest possible geometry is the geometry of those homogeneous Riemannian manifolds which look the same from a viewpoint of any observer, regardless to his position on the manifold and the direction

he looks in, i.e., geometry of spaces of constant sectional curvature. These spaces are local models for some classes of manifolds we shall deal with in what follows. For zero curvature, we get the Euclidean geometry; for negative curvature, we have the hyperbolic (Lobachevsky) geometry which we shall consider in the next section; for positive curvature k, we get the spherical geometry whose model is the sphere S"(0, 1//) of radius 1/.,Ik- in the Euclidean space I[81+1 The distance between points on the sphere is defined as the length of the shortest arc on the great circle joining these points. The great circles act as straight lines in this geometry. However, the analogy with Euclidean straight lines is disrupted by the fact that the great circles intersect in two diametrically opposite points. It is convenient to identify these points. The distance on the quotient space is given by means of the

shortest arc. The resulting new space of constant positive curvature k is called the elliptic space. If we consider straight lines passing through pairs of the diametrically

1.2. Conformal geometry on the sphere

5

opposite points on the sphere S"(0, 1 /N/T) instead of these points themselves, we come to the concept of the elliptic space as the projective space R]P", i.e., the space of all straight lines in R1+1 passing through the origin. The projective space ]RIP can also be obtained from the n-dimensional ball having its center at the origin by identifying diametrically opposite points on its boundary, the sphere Sr-1. In the space 1I8P' we can introduce homogeneous coordinates by noting that the coordinates (xo, xl, ..., x") of different points on a straight line through the origin of R"+1 differ only by a common proportionality factor. A distance p in III" between two straight lines can be defined as follows. Suppose that one straight line passes through a point x E R"+1\{0} and the other through a point y E Ilk"+1\{0}, then the distance p between them is such that

cost kp =

(x, y) 2 (x, x) (Y, Y)

0 < kp < 7r/2,

(1.2)

where (x, y) denotes the dot product in Rn+1 Isometries of the spherical space are defined by orthogonal transformations in IIS1+1, and the same applies to the elliptic space and the projective space RIID".

2. Quasiconformal mappings. Let M and M' be metric spaces with metrics p and p', respectively, and let f : M -- M' be a homeomorphism. For each point y E M, we define a linear dilatation K (f, y) of the homeomorphism f :

(f (x), f (Y)) r-- oinfP(x,Y)=r p' (f W, f(Y))

K(f, y) = lim SupP(x,Y)=r ,O

(1.3)

The homeomorphism f is called quasiconformal in M if K (f, y) < oo for any y E M. If K (f, y) is bounded in M by a certain constant K, 1 < K < oo, then f is called K-quasiconformal, and the number K(f) = essyem sup K(f, y) is called the maximal quasiconformality coefficient. This definition of quasiconformality is intrinsic, i.e., is in terms of the metric on the space. When we have to deal with homeomorphisms of manifolds or domains in Euclidean space, we can also use many other equivalent definitions of quasiconformality such as those due to Caraman [1], Reshetnyak [2], and Vaisala [1]. In particular, if M and M' are domains in W1, this definition is equivalent to f being a homeomorphism from the Sobolev class Wn (M), for which there exists a constant C < oo such that

1grad f(x)J"
x E M,

(1.4)

where Df (x) is the derivative of the mapping f at the point x (Jacobi matrix).

It is evident that 1 < K (f) < oo. If K (f) = 1, or, equivalently, if (1.4) is an equality with C = 1, then the mapping f is called conformal in M. In that case (see Gehring [1] and Reshetnyak [2]):

1. Geometric Structures

6 1)

for any pointy there exist an orthogonal matrix A and a number (linear distortion) If '(y) I > 0,

If'(Y)I =x-+y lira p'(f(x), f(Y))/P(x,Y)

(1.5)

such that Df (y) = I f'(y)I A; 2)

the mapping f preserves the angles between the curves passing through a pointy.

The linear dilatation K (f, x) is invariant under the composition of f with a conD2 and f : D2 -* D3 be a conformal and a formal mapping. That is, let h: D1 quasiconformal homeomorphism respectively in domains D1, D2, and D3 in IR'2, then

K (f h, x) = K (f, h (x)). Since a quasiconformal homeomorphism f : D -+ D' is differentiable almost everywhere, we have a matrix-valued function D f defined almost

everywhere in D, while almost everywhere det Df (x) 0 0. A more exact measure of quasiconformality of a mapping f is provided by the following symmetrized and normalized Jacobi matrix (see Ahlfors [3, 5] and Reshetnyak [2]):

M(f, x) = (det Df

(x))-21n

. Df (x)T Df (x)

,

(1.6)

where A T denotes the transpose of a matrix A. It follows directly from the definition of

quasiconformality that the equation M(f, x) = 1 for almost all x E D is a necessary and sufficient condition for 1-quasiconformality, and consequently (Reshetnyak [1]) for the conformality of the homeomorphism f. Due to a result by Pavel Belinskii [1, 2], families of quasiconformal homeomorphic automorphisms of the sphere have a remarkable compactness property. In fact, normalizing these automorphisms, one can make such family compact. Following Gehring-Martin [1], we formulate this property as Theorem 1.5. For any infinite family G of K-quasiconformal self-homeomorphisms of the n-sphere Sn, there exists an infinite sequence {gi } of distinct elements gi E G such that one of the following holds:

(i) There exists a K-quasiconformal self-homeomorphism of Sn such that

lim gi = g and

lim gi1 = g-1

i-too

uniformly in Sn.

(ii) There exist points xo, yo E Sn such that lien gi (x) = Yo

i moo

and

lim g, (x) = xo

2-400

uniformly in compact subsets of Sn\{xo} and Sn\{yo}, respectively.

1.2. Conformal geometry on the sphere

7

3. Conformal structures. One can define a smooth conformal structure on a smooth n-manifold M as a smooth family of conformal structures on the tangent spaces Tp (M), which smoothly depend on p E M. Here, by conformal structure on (a vector space) Tp (M), we mean a conformal class of inner products g; on TIM. Two inner products,

gt and 92, are called conformally equivalent (i.e., belong to one conformal class) if gl = X92 for some .l > 0. Thus a conformal structure on M amounts to a covering {U1 h EI of M by open subsets U; C M each of which possesses a smooth Riemannian metric gi such that there are smooth positive functions on non-empty intersections UZ fl Uj, ff3 : Ul fl U3 -> R+, with g7 = fjj gj. Choosing a smooth partition of unity {,/ri } subordinate to Ui, we see that g = E,/ri g; is a Riemannian metric such that the conformal structure on M is ambient to it. We call a conformal structure on M integrable if each p E M has a coordinate neighborhood U with coordinates {xl, ... , xn } such that the frame } is an orthogonal basis of TpM with vectors of equal lengths (with respect to one, and hence any, compatible Riemannian metric). In terms of these coordinate systems {xl, ... , xn} on M, a compatible Riemannian metric on M has the form a.(p)Edx? where X(p) is a smooth JR+-valued function, and the manifold M with an integrable conformal structure may also be thought of as a locally conformally Euclidean manifold (shortly, a conformally fiat manifold). fax.

4. Stereographic projection. To introduce a metric on the compactified space Ilk" = Rn U {oo}, we consider a construction known as the stereographic projection. We Rn+i assume that Rn is embedded in Rn+1 in the form of the extended n -plane {x E

xn+1 = 0}. It can be mapped one-to-one onto the sphere Sn = Sn(0, 1) = {x E Rn+1 : x I = 11 by making each point x = (x, 0) of the plane r C Rn+1 correspond to the unique point r(x) of intersection of the straight line passing through x and en+t = (0, ... , 0, 1) with the sphere Sn (0, 1), see Figure 3.

Figure 3

1. Geometric Structures

8

This mapping jr: Ifgn -+ S' (0, 1) is called the stereographic projection. It can be where t E R is the number such that (en+1 I n (x) I= 1. Explicitly, for x 0 oo,

written in the form r(z) = i + t

_ rr(x)=

2xn

2x1

[IxI2+1,..., Ix12+l'

IxI2-1 Ix12+1]

(1.9)

and by definition, n(oo) = en+1 We use the one-to-one mapping in (1.9) to transfer the metric from the sphere Sn (0, 1) (chord length) to Rn, where we get a spherical metric s (x, y):

s(x,Y)=I7t(x)-' )I =

21x-Y1 (1+1x12)1/2.(1+Iy12)1/ 2

'

(1+1X11)1/2

if x, y,{ 00

if y=00.

(1.10)

It is readily seen that the topology thereby induced in R' coincides with the topology induced by the Euclidean metric. Therefore, functions continuous in R'1 are also

continuous in 9n. It is also readily seen that the mapping n is continuous in the Euclidean metric in R. The stereographic projection is a real-analytic conformal homeomorphism. A direct application of this is the fact that the n-sphere Sn with its standard induced metric is conformally flat. So we may identify Sn with Kn. 5. Reflections in spheres and planes. Given a hypersphere Sn-1 (a, r) = {x E Rn : Ix - a I = r }, one defines the reflection (inversion) x in I[Sn with respect to this sphere:

x(x) = a + (r/lx - aI)2(x - a) = a -I- r2(x - a)*,

(1.11)

where x* denotes the inversion with respect to Sn-1(0, 1), x* = x/Ix12. Given a plane P = P (a, r) = {x E IRn : (x, a) = r}, one defines the reflection in I[8n with respect to P as

R(x) =x-2((x,a)-r) a*,

(1.12)

where (*, *) is the standard scalar product in R", or in matrix form, (1.13)

where the matrix Qa consists of the elements a; as / I a 12, see Figure 4.

1.2. Conformal geometry on the sphere

9

Aa

P (a, k)

(a, y) = k 1*1

111 / R(x)

P (a, 0) 0

Figure 4

We see from (1.11)-(1.13) that the mappings 1(x) and R(x) are conformal homeomorphisms of lib (in the spherical metric (1.10)). The homeomorphisms of lib" that are compositions of a finite number of reflections (with respect to hyperspheres or hyperplanes in 118n) are called Mobius transformations

in Rn. The set of all Mobius transformations, Mob(n), forms a multiplicative group, which is called the Mobius group. A subgroup of this group consisting of orientationpreserving automorphisms is denoted by Mob+(n). It is readily seen that compositions of reflections with respect to hyperplanes are Euclidean isometries. The converse is also true: each Euclidean isometry of lib" is a composition of at most (n + 2) reflections with respect to hyperplanes. In dimensions n > 3, Mbbius transformations exhaust all conformal mappings. This can be easily seen from the following Liouville theorem, which is based on the observation that the image f (Bn) of a round n-ball Bn, n > 3, under a conformal map f is still a round ball in the sphere Sn.

Theorem 1.6. Let U, V be open connected subsets of Sn = Rn, n > 3, and let f : U -3 V be a conformal map. Then f is the restriction of a Mobius transformation f E MSb(n) which is uniquely determined by f.

6. Isometric spheres of Mobius transformations. Since each Euclidean isometry in Iign is a composition of (n + 2) reflections in hyperplanes, one can see that, for each

similarity in RI (i.e., a transformation g E Mob(n) with constant Ig'(x)1), there are an orthogonal transformation A E 0(n), a vector a E Rn and a positive number r > 0 such that g(x) = rA(x) + a. For any other Mobius transformations g (different from similarities) we can define the so-called isometric sphere I (g) as the set

I(g)={xER°:Ig'(x)I=1}.

1. Geometric Structures

10

In fact, this set is a hypersphere in R. To prove this, let us consider a point q E IR" such that g(q) = oo, and let x be the reflection with respect to the sphere Sn-1(q, 1).

Then we can write g o 1(x) = r A(x) + a, where r E R, a E R', and A is an orthogonal transformation. Hence the (n - 1)-sphere {y E 1R' : I det Dx (I (y)) I = r} = x ({x E IIS"

:

I det DI (x) I = r})

={xeW2:IdetDZ(x)I=1/r} is in fact the isometric sphere I(g). It is readily seen that the image of the isometric sphere I (g) under the mapping g is the isometric sphere for the inverse mapping g-1. Indeed, as the image g (I(g)) is again a certain sphere S where (local) lengths are not altered by the mapping g-1 (chain rule), we have by the definition that S is the isometric sphere for g-1. Moreover, I (g-1) has the same radius as I (g) because g does not change distances on I (g). These facts imply:

Proposition 1.7. Any Mobius transformation g E Mob(n) different from a similarity can be represented as a composition of the inversion x with respect to the isometric sphere I (g), a reflection R with respect to a plane P such that I (g-1) = R (I (g)), and (possibly) an orthogonal transformation U with the origin at the center q' of the isometric sphere I (g-1) for the inverse transformation g-1.

Using the representation g = U o R o x E Mob(n), g(oo) # oo, and representing the Euclidean isometry U o R in the form UR(x) = Ug(x) + a with Ug E O(n) and a = g(oo) - Ug (a), we have

g(x) = Ug (Z (x) + g(oo) - Ug(q))

= Ug(r2(x -q)/Ix - q12 + q) + g(oo) - Ug(q) = r2Ug ((x

- q)/Ix - q12) +g(oo),

where r is the common radius of spheres I (g) and I (g-1). This and Proposition 1.7 prove the following result.

Corollary 1.8. Every Mobius transformation g E M6b(n) can be written as g(x) _ r2UgV (x) + q' where Ug is an orthogonal transformation, r > 0, q' E 1W', and the mapping V (x) is of the form

V(x)

(x-q)/Ix-q12, if q'=g(oo) X,

oo;

if g (oc) = oo

Lemma 1.9. Let 1(x) = x/ Ix 12, r > 0 and x, a E R"\{O}. Then

z(r2x(x)+a) =

Ia12Rax

( x+ JaJ2 r2a) +1(a),

(1.14)

1.3. The hyperbolic space HI

11

where RQ is the reflection with respect to the subspace in llBn orthogonal to the vector a.

Proof. Let L C W' be the line that passes through the origin and the point a E R'1, and gj and g, be the transformations in the left- and right-hand sides of (1.14), respectively. Since L is invariant with respect to both gi and g, each 2-plane P containing the line L is also invariant for both g,, and g,-. Introducing the standard complex structure

on such a plane P so that it becomes the complex plane C with L as the real axis {z E C : Im z = 0}), we can easily show that

F

gi(co) = gy(oo)

and

-

2 Ja12

= ge l(oo) = g, '(oo)

are the centers of isometric spheres I (ge 1) = I (gr 1) and I (ge) = I (g,), correspondingly. These isometric spheres have the same radius equal r/!a1. Finally, for both gl and g,., we have no rotations around L but only a reflection with respect to a plane orthogonal to L.

§3. The hyperbolic space lHi'Z In this section we describe geometry (geodesics, isometries, distance functions, etc.) of the so-called hyperbolic space W which is a simply connected homogeneous Riemannian space of negative constant sectional curvature. To define a linear model of such a space of constant negative curvature k, we consider a bilinear form [x, y] = -xoyo +xl y1 + +xn yn in Rn+1. The corresponding metric space with length differential ds2 = -dxo+dxi + +dxn is called Minkowski space and denoted by M''+1 Let V be the set of straight lines in Mn+1 passing through

the origin and lying within the light cone L = {x E Mn+1 : [x, x] = 0}. For two straight lines in V, one passing through a point y E M1+1 and the other through a point x E Mn+1 we define a distance p between them induced by the form [x, y] such that y]2

cosh2 kp

=

[x, [x, x] [y, y]

,

(1.15)

which is analogous to (1.2).Since, for points x and y lying within the light cone L, [x, y]2/[x, x] . [y, y] ? 1, we see that p defined by (1.15) is a real non-negative number that is zero only if the lines coincide. It is readily seen that the so-defined function p([x], [y]) satisfies the usual distance (symmetry and triangle) axioms. The resulting metric space (Vn, p) is called the linear model of the n-dimensional hyperbolic (Lobachevsky) space. The quantity k, k < 0, is the sectional curvature of this space. The straight lines lying on the light cone L are points of the (n - 1)dimensional sphere at infinity of the hyperbolic space.

1. Geometric Structures

12

The group of linear transformations of Minkowski space M"+1 preserving the form [x, y] is the Lie group O(n, 1) of Lorentz transformations. The isometries of the hyperbolic space (V", p) are defined by linear transformations in Mn+1 preserving the

form [x, x] and each half-cone L+ = {x E L : xo > 01 and L- = {x E L : xo < 0). Such transformations form the index two subgroup O+(n, 1) of O(n, 1). This isometry group 0+ (n, 1) has two components where the unity component, SO+(n, 1) = 0+ (n, 1) fl SL(n + 1, R), consists of orientation-preserving isometries of the hyperbolic n-space. Such transformations are also characterized by the property that they Both groups O+(n, 1) and SO+(n, 1) are natupreserve the orientation of rally endowed with a Lie group structure and act transitively on the cone L+; their stabilizers of a point p E L+ are non-compact. Mn+1.

1. Hyperboloid model. Let us consider the upper fold of the hyperboloid

H" = {x E I09I"+1 : [x, x] = -1, xo > 01,

(1.16)

naturally associated with the above bilinear form of signature (n, 1), i.e., a sphere of imaginary radius i. Since this surface intersects each straight line lying in V,, in a unique point, we have the mapping V, --->- H" which allows us to introduce a metric in H" with the distance function p (x, y) given by Cosh kp

= [x, y] .

(1.17)

This metric is induced by the form [x, x] (with the arc differential k2ds2 = [dx, dx]) which is positive-definite on the tangent space Tx H" because there we have

(1.18)

The Riemannian space H" has a constant negative sectional curvature k and its isometries are again Lorentz transformations in M"+1 that preserve H", i.e., restrictions of elements of O+(n,1) on H". From (1.18) one obtains that straight lines in R"+1 passing through the origin intersect the hyperboloid H" in right angles (in the pseudo-metric induced by [x, x]), and consequently the same applies to (m + 1)-planes in l[8"+1 passing through the origin. This means that these intersections are totally geodesic m-planes in H", m > 1. 2. Projective (Klein) model. Let q: H" --* IRIP" be the restriction of the canonical projection l[8"+1 --

(xo, xl,

.... xn) --> (x1/xo, x2/xo,

... , xn/xo) ,

(1.19)

on the hyperboloid H". It is clear that q is a diffeomorphism of H" onto an open subset of TRIP" (actually, the unit n-ball in the plane {x E R"+1 : xo = 1}) which can be endowed with the pull-back metric with respect to q-1. This metric space is called the projective (Klein) model of the hyperbolic space H.

1.3. The hyperbolic space HI

13

As linear transformations of straight lines in Vn correspond to projective transformations of the n-plane {x : xp = 1} C IR'+1, the isometries of this hyperbolic space model are projective transformations of RIP" that preserve the interior of the ball Bn(0, 1). As the mapping q in (1.19) projects the geodesics in HI to chords in BI (0, 1), these chords are geodesics in this projective model. From (1.15) we get that the distance p (x, y) in this projective model of H between points x, y E Bn (0, 1) is defined by

p(x,Y)=2kth

Ix-al-lY-fill Ix -

aIJ'

(1.20)

where a and 8 are intersection points of the sphere Sn-1 (0, -1/k) and the chord passing through the points x and y, and the point x lies within the segment [a, y] of this chord. We see from (1.20) that p (x, y) tends to infinity if either x or y tends to the sphere Sn-1(0, 1) at infinity. As regards the angles between geodesic planes in this model, we note the following. If P C Bn (0, 1) is a geodesic plane intersecting Si-1 (0, 1) in some sphere S,-2, then there exists a point x E Rn such that all tangent planes to the ball Bn (0, 1) at points on the sphere Sn-2 contain it. Then all the geodesics in the hyperbolic space (chords in Bn (0, 1)) being the segments of straight lines passing through this point x intersect the geodesic plane P in right angles.

3. Hemisphere model. The set Sn (0, 1) = {x E Il8n+1 : Ix I = 1 , xn+1 < 0} gives us a fourth model for the hyperbolic space HI. To introduce a hyperbolic metric into it, we consider a mapping p: B'(0, 1) S" (0, 1) such that its inverse mapping p-1 is the restriction to S. (0, 1) of the orthogonal projection of Il8n+1 to the plane {x E IISn+1 : xn+1 = O} = IRn, n+1

p-1(x1, ,xn,xn+l)=(xl, ..,xn);

Ex? = 1.

(1.21)

i=1

We provide Sn (0, 1) with the pull-back metric induced (from the Klein model in B'(0, 1)) by this projection and transform this hemisphere into a hyperbolic space model. Here geodesics are circles in S!.(0, 1) orthogonal to the boundary sphere as. = Sn-1 (0, 1) C ]E8n (they are projections of chord-geodesics in Bn(O, 1)). The boundary (n -1)-sphere aSn is the sphere at infinity in this model. Isometries in this model are the Mobius transformations in II8n+1 that preserve Sn (0, 1) because they should convert all k-dimensional hemispheres orthogonal to the sphere Sn-1(0, 1) into hemispheres having the same property. Since Mobius transformations preserve angles between curves, angles in this model of the hyperbolic space IEIIn coincide with those measured in the Euclidean metric.

4. Poincare ball and half-space (conformal) models. The inverse of the stereographic projection rc : Rn -- S'(0, 1) in (1.9), 7r-1(x1, X2, ... , xn+1) _ (XI/(1 - xn+1), ... , xn/(1- xn+1)) ,

(1.22)

1. Geometric Structures

14

maps the hemisphere S"(0, 1) onto the unit ball n-1 (Sn (0, 1)) = Bn(0, 1) C Rn. Since the stereographic projection is a Mobius transformation, it converts round spheres into round spheres (planes) and preserves angles. So the images of hyperbolic geodesics in the model S"(0, 1) under .7r-1 are arcs of Euclidean circles (straight lines) in Bn (0, 1) orthogonal to the boundary sphere Sn-1(0, 1) = aBn (0, 1), which become geodesics for the pull-back metric in B'(0, 1) with respect to n. One can show that this metric has the arc differential

k2 ds2 =

4Idx12 (1.23)

(1 - Ix2l)2

and its distance function p(x, y) is defined by

p(x, y) = k In [(1 - h)/(1 + h)] ,

(1.24)

where

The unit ball B'(0, 1) with this metric is the Poincare ball model of H n (with the sectional curvature k < 0 and the above geodesics). Due to conformality of Tr, its isometries are Mobius automorphisms of the ball, as in the case of S! (0, 1). To convert the half-space R+ = {x E JRn : x, > 0} into a model of the hyperbolic space IH[n, we may define a Riemannian metric of constant sectional curvature k < 0 in 1R as the pull-back metric with respect to one of the following diffeomorphisms. One (conformal) diffeomorphism, 7r -I o U: S" (0, 1) R , is the composition of the rotation U in Jn+l U(xt, ... , xn, xn+l) _ (xl, , -xn+1, xn), and the inverse n-t to t h e stereographic projection. Another s u c h diffeomorphism is a Mobius trans-

formation h E Mob(n) which maps the unit ball B" (O, 1) to 1R .Both mappings induce the same pull-back metric in l[8+ with curvature k < 0 and the arc differential

k2ds2 = IdxJ2/xn .

(1.25)

Geodesics in this Poincare half-space model of H" are semicircles (rays) in R+,

orthogonal to the boundary plane Rn-1 = {x E 1R' : xn = 0} lying at infinity. The isometries of this model of HI are Mobius automorphisms of the half-space. So Isom(R+, p) is the group Mob(J+). It is useful to show directly that Mobius automorphisms of the half-space are

hyperbolic isometries. For this let us remark that, for a domain X C S' with the differential p(x)dx of ametric pin X, aMobius automorphism f E Mob(X) preserves p if and only if (1.26) If'(x)I =P(x)/P(f(x)) , X EX.

Due to this observation, it is enough to prove that the linear distortion If '(x) I of a transformation f E Mob(R+) is in fact [f (x)]n /xn, where [f (x)]n is the n-th coordinate of the point f (x). Then it would imply that, up to a constant factor, any Mob-invariant differential p(x)dx has p(x) = 1/xn; see (1.25)).

1.3. The hyperbolic space W

15

In order to evaluate I f (x) 1, let us find I f (x) - f (y) I / Ix - y I where x, z is the inversion with respect to a sphere S(a, r) then 1_1(x)

- x(y)I = I

r2

A

a-xix a- yl I

y E R. If (1.27)

which follows from the similarity of triangles with vertices a, x, and y and correspondingly a, 1(y), and 1(x); see Figure 5.

Figure 5

At the same time, for an = 0 and x E IIB+, the n-th coordinate [.r (x)]n of a point

1(x) is r2xn/Ix - a12, so

Ix(x)-x(Y)12

Ix-h12

[x(x)]n - IX(Y)1n

xn . Yn

(1.28)

Since the expression in (1.28) is also invariant for any reflection R with respect to a plane orthogonal to alf8+ and due to Proposition 1.12, we have for any f E Mob(R+) that

If(x)-f(y)12

Ix-y12

If (x)In . If (Y)1n

xn . Y-

and, consequently,

1 f'(x)I = Jim Y-+x

If(x) - f(Y)I = [f(x)]n Ix - Y1

xn

This completes the proof that f is an isometry of (R+, p). Now let us notice that the action of the group Isom lH[n of hyperbolic isometries in

the half-space model R+ is in fact the conformal action of the group Mob(n - 1) of MSbius automorphisms of the boundary Rn-1 of the half-space.

1. Geometric Structures

16

Actually, considering I8n-1 as the extended hyperplane {x E R" : x" = 0} in R", we can (uniquely) extend the action of g E Mbb(n - 1) to its conformal action in I8" preserving the half-space IIB+. This extension is based on Corollary 1.8 and on the extension of each reflection R E Mob(n - 1) with respect to a sphere/plane Sn-2 C R!'-1 to the reflection R E Mob(I8+) with respect to the sphere/plane Sn-1 C I8" which orthogonally intersects the hyperplane RI-1 along Sn-2. Also we may describe Isom IHI" as the group Mob (Bn) of Mobius automorphisms of the unit ball B" = B" (0, 1) = {x E IIg" : IxI < 1}. Since the stereographic projection 7r:II8n-1

-+ S"-1(0, 1) in (1.9) is the restriction of a Mobius transformation h E

Mob+ (n) which maps 1Il+ onto B" (0, 1), we have that Mob (B") = h Mob (R+) . h-1.

On the other hand, one may parametrize Mob(B") as Bn x 0(n). Indeed, for a given g E Mob(B"), let a = g-1(0). Then, assuming a y-E 0 (i.e., g(oo) ¢ oo), consider the (n - 1)-sphere Sn-1(a*, r) centered at a* = a/lal2 with the radius r > 0 such that it is orthogonal to the boundary sphere 8B" = Sn-1(0, 1), i.e., r = Ia*'2 - 1 = 1 - 1a12/Ial. Applying (1.11) and (1.12), we may define a canonical transformation

9.W = (1 - 2Qa) Za*(X) = (1 - 2Qa) - (a* +

ix*I2

a*I2

\\

(x - a*))

(1.29)

which is the composition of the reflection xa*(x) in the sphere Sn-1(a*, r) and the reflection (1 - 2Qa) in the plane P(a, 0) orthogonal to a and passing through the origin. Since ga(a) = 0 and ga (a*) = oc, the composition ga g-1 fixes the origin.

So gag-1 E O(n) and g = U ga for some orthogonal transformation U E O(n). Therefore we have Isom]HIn = Mob(I8+)

Mob(n - 1) - B" x O(n)

.

(1.30)

5. Busemann function and horospheres in 1EI1. Informally speaking, a horosphere in the hyperbolic space ]HI" is a limit case of a sphere S(y, r) = {x E IEIn : p (x, y) = r}

of radius r > 0, which may be defined as a (n - 1)-submanifold in 1W' orthogonal to the family of geodesics passing through the point y. In this limit case, changing the radius correspondingly, we drive centers of spheres to a point in the sphere at infinity 81EI". So a horosphere Sy C H" with center y E alEln may be defined as an (n -1)-manifold orthogonal to the family of geodesics y: R -± H" in H", having y

as their endpoint, limt,+0 y(t) = y. We call such geodesics asymptotic: for any two of them, y1 and y2, there exists a constant C > 0 such that p (Y1 (t), y2 (t)) < C

fort > 0.

Another (equivalent to the previous) definition of a horosphere may be obtained by replacing the distance function x -a p(x, y) in case of a sphere S(y, r) c H" by the Busemann function in the case of a horosphere centered at y E alln. Namely, for each point y E allln and each geodesic y c H", y(+oo) = y, we may define a function fy : 1H" -+ III such that, for all x E H", fy (x) = r-lii(p

(x, Y (t)) - t) .

1.3. The hyperbolic space 1FF

17

One can prove the following properties of the function fy: 1. fy is uniformly continuous and convex; in particular, I .fy (XI) - fy (x2)1

P(xl, x2)

2. If a geodesic $ is asymptotic to another geodesic y, then

fyWs))-.fy(f(t))=t-s; in particular, fy has no minima. 3. If a geodesic P is asymptotic to the geodesic y, then fy - fy = const. These properties show the correctness of the definition of the Busemann function fy (x) for a point y E a1HI' as the function fy (x), where y is some geodesic ending at the point y. Now we can define a horosphere Sy C H' centered at some y E dW and passing through a point z E 1HIn as Sy,Z = {x E 1HIn : fy (x) = fy (z)}. This horosphere bounds the horoball By,Z = {x E Hn : fy (x) < fy (z)}. It is essential to note here that, in the Poincare (ball or half-space) model of the hyperbolic space 1111", horospheres Sy are represented by Euclidean spheres tangent to the sphere a1H[n at points y. In particular, in the half-space model, the horospheres centered at oo are represented by Euclidean

planes S00,t={xE1E8+:x, =t},t>0. One can see straightaway from the latter that the restriction of the hyperbolic metric in R+ to a horosphere Soo C 1H[n converts this horosphere to the Euclidean space 1I8n-1

because, for all x1, x2 E Sw,t, we have p (xI, x2) = Ixl -x21/t. Geodesic perspective from the point oo at infinity aHn defines conformal maps S00,t -+ S.,, between the various horospheres centered at oo. Furthermore, these maps easily extend conformally to the one-point compactifications of these horooo, the Riemannian metrics on spheres, Sm,t U fool - Sn-1. In the limit, as t horospheres fail to converge but the conformal structure remains fixed. In this way, the hyperbolic geometry induces the conformal geometry on the sphere at infinity, a1H1', naturally identified (via the stereographic projection (1.9)) with the one-point compactification of the Euclidean space.

6. Examples of hyperbolic manifolds. Here we describe some examples of manifolds which have (HI, Isom that is complete Riemannian manifolds with constant sectional curvature k = -1. Example 1.10 (Gieseking manifold). Let us consider a regular ideal hyperbolic simplex 0 C 1HI3, i.e. a simplex with vertices at infinity and all dihedral angles n/3, and define orientations of its sides A, B, A' and B' as shown in Figure 6. For each two triples (p1, P2, p3) and (pi, p2, p3) of points in the circle aIH[2 S1 at infinity of the hyperbolic plane H2, there exists an isometry of 1H12 that maps one triple to the other. Hence we can find orientation reversing isometries of H3 which A' and B -* B', of the simplex A as indicated in Figure 6. It identify the sides, A

1. Geometric Structures

18

is easy to check that, after performing these isometric gluings, all the (oriented) edges of A are equivalent, and we obtain a non-oriented complete hyperbolic manifold. This manifold is non-compact since it has one end. Its volume is however finite and equal to the hyperbolic volume of A, V3 = 1.0149... . A

B

Figure 6

Figure 7

Example 1.11 (Figure eight knot complement). Let K be the so-called figure-eight knot in S3 shown in Figure 7. Its complement M = S3 \ K has the fundamental

group iri(M) = {x, y : xyx-1yx = yxy-ixy}. In order to introduce a complete hyperbolic structure on M, one can observe that M is homeomorphic to the manifold

obtained by gluing sides A H A', B H B', C H C' and D H D' of two simplices with deleted vertices shown in Figure 8. To see that, let us consider a 2-complex obtained from the knot K by adding two oriented segments and four films spanned by resulting 1-complex, see Figure 9. This 2-complex splits S3 into two topological balls bounded by our 4 films. So, contracting the non-oriented segments of the 1-complex (corresponding to arcs of the knot K) to points, we transform these two closed balls into two 3-simplices with deleted vertices shown in Figure 8.

Figure 8

Figure 9

1.4. Lie subgroups of the Mobius group

19

Now, as for the Gieseking manifold, we realize those 3-simplices as regular ideal simplices in the hyperbolic 3-space H3 and consider orientation preserving isometries of l l which identify sides of those simplices as indicated in Figure 8. Since exactly six dihedral angles (= x/3) meet at each edge of the obtained 3-complex, we see that this 3-complex is a 3-manifold locally isometric to H3 (i.e., a hyperbolic manifold) and homeomorphic to S3\K.

One can observe that this non-compact (finite volume) oriented hyperbolic 3manifold is the two-fold covering of the (non-oriented) Gieseking manifold. In other sections, we will discuss another hyperbolic structures, in particular, on the Borromean and Whitehead link complements in the 3-sphere S3. We remark that all considered in this section examples of Euclidean, spherical and hyperbolic manifolds simultaneously are examples of conformal structures on manifolds, i.e. (X, G)structures modeled on the (sub-Riemannian) conformal geometry (Sn, Mob(n)). Indeed, we can consider Isom Rn, Isom Sn-1 C Isom R' and Isom HI as subgroups of the Mobius group Mob(n) which preserve Rn C Rn - Sn, Sn-1 C W' and the round ball Bn(0, 1), correspondingly.

§4. Lie subgroups of the Mobius group Since the action of Mob(n) in the closed half-space 1R8+ 1 is conformally conjugate to the action of M6b(Bn+1) in the closed round (n + 1)-ball Bn+1, we may apply Browder's fixed point theorem in order to classify cyclic subgroups (g) C Mob(n). Actually, we call a transformation g E Mob(n) (i) parabolic if it has a unique fixed point in IlSn+ 1 that lies in ][8n = 8Rn+1;

(ii) loxodromic if it has precisely two fixed points in R+ 1 that lie in Rn; (iii) elliptic if it has a fixed point in R'+1

These classes exhaust Mob(n). In fact, if g E Mob(n) has three fixed points pi, p2, and p3 in Kn, i.e., is neither loxodromic nor parabolic, then all points on the circle

S1 C Rn passing through these points are fixed by g. Therefore, all points on the two-dimensional sphere S2 C Kn+1 orthogonal to the n-plane {x E Il81+1 : xn+1 = 0} = ]Rn and intersecting it in the circle S1 will be fixed by the transformation g, hence g is an elliptic transformation.

It is clear that a Mobius transformation g E Mob(n) and its conjugation ygy-1 E Mob(n) by Y E Mob(n) are of the same type. In particular, a parabolic transformation g E Mob(n) is conjugate to a transformation ygy-1(x) = A (x) + a, where the vector a E IRBn \ {0} is not orthogonal to any eigensubspace of the orthogonal transformation A.

For this purpose, it is sufficient to take as y E Mob(n) a transformation that converts the unique fixed point p, g(p) = p, to oo. Then the transformation ygy-1 is obliged to be a Euclidean isometry, since otherwise the method of contracting mappings can

20

1. Geometric Structures

be used to show that there is a fixed point for ygy-1 in 1l' different from oo. If, on the other hand, the vector a E R'\{0} is orthogonal to an eigensubspace of A, it is clear that g is obliged to be an elliptic transformation. If A = id, the parabolic transformation g is called purely parabolic. Let a loxodromic transformation g E Mob(n) have (precisely) two fixed points p, q E Rn. Then, taking a transformation y e Mob(n), y (p) = 0 and y (q) = oc, we conjugate g to the transformation ygy-1(x) = r - A(x) where r > 0, r 1, and A is an orthogonal transformation. If A = id, then g is called a hyperbolic transformation. Rn is conjuFinally, an elliptic transformation g E Mob(n) having fixed points in gate to a transformation of the form A(x) + a, where the vector a E I[ln is orthogonal to an eigensubspace of the orthogonal transformation A. This follows from the fact that, in this case, a sphere Sk C I[hn+1 of a minimal dimension k, 1 < k <_ n -1, that is orthogonal to Rn and contains all the fixed points for g in R1 and a fixed point for g in +1, will consist of fixed points for g. As a conjugating transformation y E Mob(n), we take one that converts one point from Sk n Rn to co and another point from Sc fl r ygy-1 E O(n) is an orthogonal transformation. to the origin. Then In dimension n = 2m + 1 (and only in that case), there are elliptic transformations g E Mob(n) not having fixed points in Kn. In particular, one can easily check that a (g-1)] transformation g(x) = UoRox (x) from Proposition 1.7, with dim [1(g) fl I = n - 2 and U ; id a rotation around the straight line passing through the centers of the I[82m+1. isometric spheres I (g) and I (g1), is elliptic and has no fixed point in Finally, let us remark that, for any g E Mob+(n), its fixed points lie on a certain (n - 2)-dimensional sphere (plane) in lRn, and g may be uniquely determined by its values at n + 1 points in IIBn not lying on any (n - 2)-dimensional sphere (plane).

In order to classify (non-cyclic) Lie subgroups of the Mobius group, we may identify Mob(n) with the isometry group of the hyperbolic (n + 1)-space 1n+1 due to (1.30). For a hyperbolic plane L C 11n+1, dim L < n, let us denote by K (L) the subgroup of Mob(n) IsomHn+1 that leaves L pointwise fixed. Let .M(L) be the subgroup of Mob(n) = Isom Hn+1 consisting of isometries of L in the metric induced by IEIIn+1 and let .M+ (L) be its orientation-preserving subgroup. Then the complete stabilizer subgroup for L is the direct product X (L) x At (L). In these terms, we now formulate Greenberg's classification of Lie subgroups of Mob(n), see [1]. Theorem 1.12. Let G be a connected Lie subgroup of the group Mob(n) of all Mobius transformations in Rn identified with the sphere at infinity of the hyperbolic space 1HIn+1 Then one of the following assertions applies:

1) the elements of G have a common fixed point in En+1 while the group G itself is conjugate to a Lie subgroup of the orthogonal group O(n + 1); 2) the elements of G have a common fixed point in iR' = d1E11n+1, while G itself is conjugate to a subgroup of the similarity group in Euclidean space Rn;

21

1.4. Lie subgroups of the Mobius group

3) there exists a hyperbolic straight line 1 C 1n+1 such that G = I' x ,M(l), where r C X (l), and M(l) is a one-parameter loxodromic group with the axis 1;

4) there exists a hyperbolic plane L C 1n+1 2 < dim L < n, such that G = r x A+(L), where r C X (L); 5) G = Mob.+(n). As another application of (1.30), we can obtain alinear representation of the Mobius group. As we have seen, in the half-space model of the hyperbolic space Hn, the action of the group Isom Hn is the conformal extension of the Mob(n -1)-action in the (n -1)sphere 8ffffn = ff8n-t At the same time, in the hyperboloid model of ]H[n, Isom IH[n

acts as the group 0°(n, 1) of Lorentz transformations. Considering the composition of isometries between the models Hn, B'(0, 1), Sn (0, 1) and R+ of ]HI' described in the previous section, we finally get an isometry cp: Hn --* R+ of the hyperboloid and half-space models of ffffn such that

(p(xo, x1, ... , xn) _ (xl/(xo - xn), ... xn-1/(xo - xn) , 1/(x0 - xn)) (p-t(x1, ... , xn) = (IxI2 + 1, 2x1, ... , 2xn-1, IxI2 - 1)/2xn . (1.31) The isometry cp conjugates the isometry groups of those models. Therefore, up to the identification of Mobius transformations g E Mob(n - 1) with their actions in the half-space ]IB+, we have the isomorphism

cp,,: Mob(n - 1) --> 0°(n, 1)/(fl}, Vg = p-1gCp,

g E Mob(n - 1). (1.32)

As each Mobius transformation g E Mob(n - 1) is a composition of a Euclidean translation x + v, v E Rn-1, an orthogonal transformation U E O(n - 1),

a homothety gr (x) = r x, r > 0 and, maybe, the inversion 1(x) = x/IxI2 (see Corollary 1.8), it is sufficient to know which matrices from O(n, 1) correspond to these four transformations. Some computations using (1.32) show that

0

1

0

*U= O U O 0

0 (IV 12

O

1

+ 2)/2 v1 ... Vn-1

-1 -IvI2/2 -V2

co*gv =

vn-1 Iv12/2

0..

(1.33)

-Vn-1

1

v1 ... vn-t

(2 -IvI2)/2

(r2 + 1)/2r 0

(r2

),

1

V1

1

0

1

0*x =

- 1)/2r

0

.

0

(r2

1)/2r

1. Geometric Structures

22

oo, we have g(x) = Due to Corollary 1.8, for any g E MSb(n - 1), g(oo) of the isometric spheres I (g) and r U Z (x - q) + q' where q and q' are the centers > 0. Hence we obtain I (g-1) of radius cP*g = (tij)0
(1.34)

In particular, the first colunm of the (n + 1) x (n + 1)-matrix cp*g E O(n, 1) has the form: too = [(Ig12 + 1) (Iq'12 + 1) + r2(U(q'), q)] /2r

do = (U(q'), ei) - (q, ei)(I q'12 + 1)/r , tno = [(1

1
- IRI2)(Iq'12 + 1) - r2 + 2r(U(q'), q)]/2r

(1.35)

,

where (*, *) is the dot product in Rn and ei is the i-th coordinate vector. Conversely, if we know the matrix cog = (tii) E O(n, 1), then the corresponding Mobius transformation g e Mob(lR f) acts in accordance with the formulae y = g(x) _ (Y1,...,Yn),

++n

Yi=

2

Enm-=1

(tio + tin) L.m=1 xm + 2 1 timxm + tip + tin n-1 to En 2 m=1 xm + 2 Em=1(tOm - tnm)xm + tl

,

1
2xn Yn

to

(1.36)

Em=1 xm + 2 EM= 1(tom - tnm)xm + ti

where

to =too+tOn-tno-inn,

ti =too+tnn -ton -tn0.

As a direct application of this linear representation of the Mobius group Mob(n) and the Selberg Lemma [1] that any finitely generated matrix group contains a normal torsion-free subgroup of finite index, we obtain:

Theorem 1.13. In any finitely generated group G C Mob(n), there exists a normal torsion free subgroup Go C G having finite index [G : Go] in G. Here and subsequently, the index [G : Go] for a normal subgroup Go in a group G means the cardinality of the quotient G/ Go.

§5. Structure developments and holonomy homomorphisms Let M be a manifold modeled on some (X, G)-geometry and {(Ui, cpi)}ieg' a corre-

sponding atlas where cpi : Ui -± X are charts. Then, for Ui n Uj # 0, changes of charts yi j are defined as restrictions to open sets of X of elements of the group G,

yij:xt-+cpiocpj 1 (x),

xEcoj(UinUj).

(1.37)

1.5. Structure developments and holonomy homomorphisms

23

Let us further assume that our (X, G)-geometry satisfies the following identity condition:

(IC) If two diffeomorphisms from G coincide on a non-empty open subset in X, then they identically coincide on X. First of all, any real analytic (X, G)-geometry satisfies this identity condition. Moreover, as we will show in Corollary 2.9, this condition holds for any discrete convergence group action on the n-sphere. In particular, it is the case of the fundamental group of a strictly negatively curved manifold because this group acts on the sphere at infinity of the universal covering space as a convergence group, see Martin [4] and Martin-Scora [1]. Now we fix some chart (U1, cpl) from the (X, G)-atlas on the manifold M and consider paths a in M with the origin at some xo E U1. Using (1.37), we can define direct (analytic) extensions of cpl to all Uj intersecting the fixed chart U1. Moreover, due to the identity condition (IC), each element of the group G is completely defined by its values on any open set from X, so we can assume that (1.37) defines a locally constant mapping of the manifold X to the group G (constant on any intersection cpj (Ui fl Uj)). Using this fact and induction on the length of chains of charts covering a given path a C M, we obtain the (analytic) extension of cpl along a to any chart Uj intersecting a, and this extension of Cpl has the form g o cpo for some g E G. This shows that there exists a global (analytic) extension of (pl well defined on the universal covering manifold M. To see that, we can represent the universal covering M as the set of pairs (x, [a]), where x E M and [a] is a homotopy class of paths a c M joining the points xo and x. Let us denote by d the defined extension of a chart Bpi: Ut --). X,

d: M -+ X,

(1.38)

and call it a development (map) of a given (X, G) -structure in M. As we saw above, this development d is a local (X, G)-diffeomorphism defined modulo composition with an element of the group G. Also we see that (1.38) induces a (X, G)-structure in the universal covering manifold M. Due to uniqueness of the development d (up to composition with g G), we see that each deck transformation Ta E G (M, M) of the universal covering M corresponds

to a unique (up to conjugation by g E G) element ga E G such that gad = dTa. Therefore we have a homomorphism

d,,: rrl(M) - G(M, M) -* G,

d*(Ta)d = dTa

,

(1.39)

which is well defined up to conjugation by an element of the group G. It is called a holonomy (homomorphism) of a given (X, G)-structure in M. In general, a holonomy of a (X, G) -manifold M does not determine the corresponding (X, G)-structure. For instance, one can show that the holonomy corresponding to an affine structure on 2-torus T2 (constructed in Example 1.1) may correspond to

24

1. Geometric Structures

another affine structures constructed in a similar way. However, there is an important special case when a holonomy defines the corresponding (X, G)-structure on a manifold M. This class contains complete (X, G)-manifolds M which, by the definition, are (X, G)-manifolds so that (M, d) are coverings over X, i.e., corresponding developments (1.38) are covering maps. For a simply connected space X, any complete (X, G)-manifold M can easily be reconstructed as the quotient manifold X/ r where r = d,, (rr1(M)) is the holonomy group of a given (X, G)-structure. As we will see in Chapter VI, the simply-connectedness condition for X maybe dropped if the r-action on X is discrete. We now prove a sufficient (X, G) -completeness condition.

Theorem 1.14. Let (X, G) be a geometry satisfying the identity condition (IC) and having a transitive action of the group G on X with compact isotropy subgroups

G. = {g E G : g(x) = x}, x E X. Then there exists a G-invariant Riemannian metric on X, and any closed (X, G) -manifold is (X, G)-complete.

Proof Let Tx (X) be the tangent space of the manifold X at a point x and let Q be a positively-definite quadratic form on Tx (X). We consider the set {g,k(Q), g E Gx} and use Haar's measure in the compact group Gx to average the quadratic form Q in such a way that it becomes invariant with respect to G. To save notations, we denote

this Gx-invariant form again by Q. Then, in the tangent space Ty(X) at any point y E X, we can define a quadratic form Qy as the image of that Gx-invariant form Q with respect to a diffeomorphism g E G, g(x) = y, i.e. we put Qy = g,(Q). It is clear that this definition does not depend on the choice of g E G that maps x to y. Hence it defines a complete G-invariant Riemannian metric on X as well as induces a Riemannian metric on any (X, G)-manifold, which is invariant with respect to any (X, G)-mapping. Now, for a closed (X, G)-manifold M, we can find a number s > 0 such that any metric ball B(y, s) C M (in the Riemannian G-metric on M) is convex and may be contracted to the point y. Using this choice of e > 0, we see that the development preimage d-1 (B(x, e/2)) of any ball B(x, s/2) C X is a disjoint union of sets that are homeomorphically mapped by d to the ball B(x, s/2). So the pair (M, d) is a covering over X, and M is a complete (X, G)-manifold. We remark that, for an n-dimensional (X, G)-geometry whose isotropy subgroups are 0(n), the obtained in Theorem 1.14 G-invariant metric on X is unique up to a scalar multiple. However, in general, there are many G-invariant metrics on X and they may give X very different properties. An interesting and relevant example is to study such G-invariant metrics on a Lie group G itself, i.e., consider (G, G)-geometries. A reader

may find a complete discussion of the possibilities in the case of 3-dimensional Lie groups in Milnor [3]. We complete this section with investigating a relationship between (X, G)-completeness of a (X, G)-manifold M and its completeness as a metric space (with a G-invariant Riemannian metric).

1.6. The eight 3-dimensional geometries

25

Theorem 1.15. Let an (X, G) -geometry satisfy the conditions of Theorem 1.14. Then for a (X, G) -manifold M, the following assertions are equivalent: (1) M is a complete (X, G)-manifold; (2) M is complete as a metric space,

(3) there exists an e > 0 such that any closed metric ball B(x, e) C M is compact;

(4) for any k > 0, any closed ball B(x, k) c M is compact; and (5) there exists a family Al }, ER of compact sets that exhausts M and is such that, for any $ > 0, Ma+p contains a neighborhood of radius P about Ma. Proof. (2) (1). Since the development d: M -* X of the (X, G)-structure on M is a local homeomorphismthe pair (M, d) is a covering over X if any path y : [0, 1] X can be lifted to a path in M. Let us assume that Io C [0, 1] is a maximal subinterval such

that the path yo: Io -± X, yo = Y 110, can be lifted to M. Then Io is simultaneously a closed and an open subset in [0, 1] because d is a local homeomorphism and M (together with M) is a complete metric space. This shows that Io = [0, 1] and hence proves the (X, g)-completeness of the manifold M. One can use standard elementary arguments to prove the following assertions: (2) (3) # (4) q (5). So we will complete the proof by showing that (1) = (3). To prove that (1) = (3), we first note that the universal covering space X of the manifold X satisfies (2), (3), (4) and (5). We obtain this fact by a direct application of Theorem 1.14 because the pair (X, G) satisfies corresponding conditions on transitivity action and compactness of isotropy subgroups in G. Here G is a group of all (X, G)-diffeomorphisms of the universal covering space X which are lifts of elements of G. Therefore, there exists a G-invariant complete Riemannian metric on X. Then we take e > 0 such that the closed ball B(y, e) c X centered at a point y E X is compact, and the transitiveness of the G-action gives us that this is so at any point x E X. We now assume that M is a complete (X, G)-manifold. It is obvious that M is covered by the space X. Then from what has preceded we have that M satisfies (3), which completes the proof of the theorem.

§6. The eight 3-dimensional geometries It follows from the previous section and a theorem of Singer [1], that any studying of a complete (X, G)-manifold M is equivalent to studying of a complete locally homogeneous Riemannian metric on M, by which we mean a metric such that, for

given x, y E M, there are neighborhoods U and V of x and y and an isometry (U, x) -* (V, y). Therefore, for classification of 3-dimensional locally homogeneous geometries, we only need to consider all pairs (X, G) where G acts transitively on a 3-manifold X with compact isotropy groups Gx. Here we identify two geometries

(X, G) and (X', G') if there is a diffeomorphism X -* X' which conjugates the

26

1. Geometric Structures

actions of the (isomorphic) groups G and G. Also it is natural to restrict our attention to a few (X, G)-geometries satisfying the following conditions: 1. The space X is simply connected (in the opposite case, one can consider a structure in the (X, G)-manifold M defined by the universal covering space X).

2. The group G is unimodular, i.e. there exists a measure on G invariant under multiplication on the right or on the left. Otherwise, X would possess a G-invariant vector field which expands volume, so there could be no (X, G)manifolds which are compact or which even have finite volume. So we will say that our (X, G)-geometry admits a compact quotient. 3. G is a maximal group of homeomorphisms of the space X with compact stabilizers. In the opposite case, if G were contained in a larger group G', then any (X, G)-manifold would be a (X, G')-manifold, so our (X, G) -geometry would be redundant. This approach has been used by W. Thurston for the following classification of all 3-dimensional geometries:

Theorem 1.16. Any maximal, simply connected, 3-dimensional geometry which admits a compact quotient is equivalent to one of the eight geometries (X, Isom X) where X is one of R3, H3, S3, If][2 X R, S2 X R, SL2 IR, Nil or Sol. Before we describe these eight geometries, we note that the most interesting, most complicated, and most frequently encountered among them is the hyperbolic one. The other seven are exceptional geometries, as Euclidean and spherical geometries in the two-dimensional case. In particular, for six of them (with the exception of the Solgeometry), all closed 3-manifolds carrying such geometries are Seifert fibered spaces. Moreover, any Seifert fibered 3-space admits such a geometric structure, see Scott [6] and Chapter 6. When we describe these eight canonical 3-dimensional geometries represented

by homogeneous spaces X, we shall indicate a simply connected space X and its isometry group G = Isom X having compact isotropy subgroups G, X E X, that are isomorphic either to 0(3) or 0(2) or the trivial group O(1). The first three geometries have Gx - 0(3), so their sectional curvatures are constant, i.e., either positive or zero or negative (see §4, Chapter 1).

G-1. Spherical geometry. One can think of the 3-sphere S3 either as the unit sphere S3 (0, 1) in Euclidean 4-space ]R4, or as the group of unit quaternions or as the unit sphere f Z = (z 1, z2) E C2 : I z I = 11, using whichever point of view seems convenient.

So, the first Euclidean approach shows that Isom S3 = 0(4), and the classification of spherical 3-manifolds is equivalent to classifying the finite subgroups of SO(4) which act freely on S3 (because such manifolds must be orientable, as any orientation reversing homeomorphism of S3 has a fixed point). These groups were classified by Goursat [1] and H. Hopf [1]. For a very careful and accessible description of all 3-dimensional spherical forms, we refer a reader to Postnikov [3]. Among these

1.6. The eight 3-dimensional geometries

27

manifolds one can find RIP3, the Poincare homology sphere, prism manifolds and lens spaces L(p, q); all of them are Seifert fibered spaces, see Scott [6].

G-2. Euclidean (= flat) geometry. As any isometry g E Isom R3 can be expressed as g(x) = U(x) + b, U E 0(3) and x, b E R3, the map g H U defines a surjective homomorphism Isom 1R3 -+ 0(3) with the group of translations of R3 as the kernel. So we have the (split) exact sequence:

0--> R3-) IsomR3-*O(3)--> 1. In order to classify closed flat 3-manifolds (equivalently, torsion free discrete subgroups I' C Isom R3 having a free Abelian subgroup of rank 3), we can apply the celebrated Bieberbach Theorem 2.6 and obtain the exact sequence

1->ZxZxZ-aF-+H-a1

(1.40)

where H C 03 is a finite subgroup. There are only ten non-homeomorphic closed flat 3-manifolds six of which being orientable. Each of them is finitely covered by the 3-torus T3 = S1 x S' x St and admits a Seifert fibered space structure. Existence of a natural Seifert fibration on R3/ I' is related to a somewhat surprising fact that, for any torsion free discrete group F c Isom R3 satisfying (1.40), there exist a direction on 1R3 left invariant by r (it is not true for groups with torsion). More exactly:

Theorem 1.17. Let F C Isom R3 be a non-cyclic discrete torsion free group. Then F leaves invariant some family of parallel lines in R3 and the flat manifold II83/ F is Seifert fibered by circles which are the images of these lines.

In order to prove this fact, one has to consider the action of r on T = Z x Z x Z (see (1.40)) by conjugation, which induces an action of the finite group H on T. Then, using the classification of finite subgroups of SO(3) (see Wolf [1]), one can prove that some cyclic subgroup e C Z x Z x Z is left invariant by this finite action. Since G is a group of translations, it determines a direction in R3 which must be 1'invariant. Further, as r contains translations in this direction, namely the elements of C, it follows that the parallel lines in this direction descend to give a natural Seifert fibration on JR3/ F, see Scott [6] for details. In Chapter 6, §3, we will give a criterion when a flat n-manifold is a circle bundle, see Theorem 6.40 and Corollary 6.41. G-3. Hyperbolic geometry, (IHI3, Isom 1H13). This is the most rich and complicated 3geometry whose several models have been defined in §4. Discussion on this geometry will be continued in the rest of the book.

1. Geometric Structures

28

For the next four geometries, we have the isotropy subgroups Gx = 0(2).

G-4. (S2 x R) -geometry. This is the simplest of the eight geometries. Its isometry group is the product Isom S2 x Isom R. There are only seven (S2 x R)-manifolds without boundary(includingS2x118itself)withfourclosedonesamongthem. Between these four closed (S2 x R)-manifolds, only two are orientable: the Hopf manifold S2 X S1 and 1RP3#R1F3. The latter one is the only one closed 3-manifold which possesses a geometric structure and is a non-trivial connected sum. G-5. (H2 x R)-geometry. Its isometry group is Isom H2 x Isom R. Unlike the previous product geometry, there are clearly infinitely many 3-manifolds with (H2 x R)geometry because they include any product of a hyperbolic surface with R or S1. Furthermore, a compact 3-manifold M has some (H2 x 1l8)-structure if and only if it is finitely covered by the product F x S1 where F is a hyperbolic surface, see Scott [6]. That is why we call such (H2 x 1[8)-manifolds virtually trivial Seifert manifolds, see Chapter 6 for more details.

G-6. SL2-geometry. The space SL2R denotes the universal covering for the 3dimensional Lie group SL2 JR of all 2 x 2 real matrices with determinant 1. The group SL2 ll$ is itself a Lie group and so admits a metric invariant under left (or right)

multiplication. However, we will describe the metric on the space SL2 R (which is homeomorphic to an open solid torus B2 x S1) in a more useful way. Namely, we consider a natural Riemannian metric on the unit tangent bundle Tl H2 of the hyperbolic plane H2. Since for any isometry g E Isom H2 its differential Dg is an isometry of TH2, we have a description of isometrics g of T1H2 which act in each fiber by their derivatives (rotations of S1). Now there is a natural identification of T1ll2 with PSL2 JR, the orientation preserving isometry group of H2, so we have a Riemannian metric on PSL2 R. Finally, PSL2R is double covered by SL2R and so its universal covering is SL2 JR whose induced metric is the one which we are interested in. Therefore, the space SL2R is naturally identified with a line bundle over H2, whose isometry group, Isom SL2, is the product of 1[8 and the universal covering of the group Isom H2, and preserves this bundle structure. Here the H2-isometries act by their derivatives and JR acts as a simultaneous rotation of all the vectors that fixes all points in H2, and we have the exact sequence:

0 -+ 118 -* Isomo SL2 -* PSL2 R -* 1,

(1.41)

where Isomo SL2 is the identity component of the group Isom SL2 (which has only two components, both orientation preserving, see Scott [4]).

1.6. The eight 3-dimensional geometries

29

Figure 10. (2,3,7)-tessellation of 1H12 *)

All closed SL2-manifolds naturally inherit structures of Seifert bundles (with non-

zero Euler numbers). An example of a 3-manifold with that SL2-structure is the space of unit tangent vectors to any hyperbolic surface. Another example is a manifold obtained by factorizing Tl H2 by the action of a group of orientation-preserving automorphisms of some regular tessellation in 14112, e.g., a (2,3,7)-tessellation, see Figure 10.

G-7. Nil-geometry (Heisenberg group 3e3). The space 7C3 is the 3-dimensional (nilpotent) Lie group which consists of all 3 x 3 real upper triangular matrices of the 1 x z form 0 1 y under multiplication. This group is called the Heisenberg group 0 0 1 *) Figure 10 is reprinted with permission from: D. B. A. Epstein et al., Word Processing in Groups, Copyright 01992 by Jones and Bartlett Publishers, Sudbury, MA, USA, www.jbpub.com.

1. Geometric Structures

30

and is a central group extension of the form:

0 -a R -->. je3

R2 --> 0,

(1.42)

where the center IR consists of those elements of N3 for which x = y = 0. As a Lie group, Jf 3 admits a Riemannian metric invariant under left multiplication (see Marenich [1] for its description and equivariant compactification by St) and becomes a line bundle over a 2-cell -- R2, see for details Goldman [5]. To describe this line bundle, one can consider R3 with a contact structure given by the following (nonintegrable) 2-plane field. At points of the z-axis, {(x, y, z) E R3 : x = y = 01, planes r of the field are orthogonal to the axis. Along any ray emanating from the z-axis and orthogonal to it, a plane r at a point on the ray always contains the ray itself, and its slope in the direction orthogonal to the ray increases linearly (namely, it equals one half the distance from the z-axis). In other words, such plane r (x, y, z) is spanned by vectors et - (y/2)e3 and e2 + (x/2)e3 where el, e2, e3 are coordinate vectors in IR3. The isometry group Isom 3e3 acts on JR3 as a group of affine transformations preserving that contact structure (the field jr)). Namely, any of its elements is a lift of a Euclidean isometry of R2 to an affine map of R3 preserving the field {r} and distances along lines parallel to the z-axis. In fact, there is a whole 1-parameter family of such lifts. Note that, as in Isom SL2, all isometries of 3e3 preserve orientation; in particular, an orientation reversing isometry of JR2 lifts only to maps which reverse the z-direction. The group Isom R3 has exactly two components (3f3 x U(1) is the identity component), and there is an exact sequence:

0 -- R -a Isom je3 -+ Isom R2 --* 1.

(1.43)

A natural example of a Nil3 -manifold is obtained as the quotient space of the Heisen-

berg group 3f 3 by the subgroup To C Je3 consisting of all matrices with integer entries. This manifold, X3/ Fo, is a non-trivial orientable circle bundle over the torus T2 (with Euler class 1). In coordinate form, this group Fo is generated by two affine maps in 1R3,

(x,y,z)

(x+1,y,z+y/2)

and

(x, y, z) H(x, y + 1, z - x/2), (1.44)

whose commutator is the vertical translation by the unit vector e3. Any other closed Nil3 -manifold is finitely covered by the manifold 3e3/ To. For another examples and properties of (not necessary compact) Heisenberg manifolds R3/ T' where r C 3C3 x U(1), see Apanasov-Xie [1].

G-8. Sol-geometry. This last geometry has the least symmetry of all the eight geometries as its isotropy subgroups have trivial identity components. As before, the space X is a Lie group So13 defined as a split extension of 1R2 by R. So one has the exact sequence 0 -+ JR2 -_+ So13 -* JR -+ 0,

(1.45)

1.7. Four-dimensional geometries

31

where R-action on R2 is linear, with determinant one and distinct real eigenvalues. Namely, for a given t 0, one has (x, y) H (etx, e-ty). We can identify the group So13 with R3 so that the plane {z = 0} corresponds to the normal subgroup 1R2 in (1.45). Then the multiplication in So13 is given by

V V' = (x, Y, z) (x', Y', z') _ (x + e-Zx', y + ezy', z + z'),

(1.46)

where (0, 0, 0) clearly is the identity and a left invariant metric on 1[83 is given by the formula ds2 = e2zdx2 + e-2zdy2 + dz2 . (1.47)

Here the fact that {z = 01 corresponds to the normal subgroup 1R2 is represented by the following property of the multiplication: (x, y, 0) (x', y', z) = (x', y', z') (ez'x a-z'Y, 0).

The full isometry group G = Isom(So13) has eight components, while the identity

component is the group So13 acting on itself by left multiplication. The isotropy subgroups Gx are isomorphic to the dihedral group D(4) of order eight and, for the origin 0 = (0, 0, 0), the stabilizer Go consists of eight linear transformations in 1183,

(x, Y, z) i-- (±x, ±Y, z)

,

(x, Y, Z) H (±Y, ±x, -z) ,

(1.48)

which really are isometries and isomorphisms of So13. Due to (1.46), one can see that isometric to R2 planes {z = constant} form a Ginvariant foliation on 1[83 identified with So13, and therefore, all closed 3-dimensional Sol-manifolds are non-trivial torus bundles over Sl with orientation preserving hy-

perbolic gluing maps with distinct real eigenvalues et and a-t (which are algebraic integers).

§7. Four-dimensional geometries Although, due to Freedman-Quinn [1], the techniques of high-dimensional manifold topology have been successfully extended to the topology of 4-manifolds, the results of Donaldson [1] show that one must seek quite a different pattern in studying smooth 4-manifolds. Here low-dimensional techniques, especially geometric approaches may be more appropriate. Since the most successful and coherent account of the relations between 3-dimensional topology and geometry is given by Thurston's geometrization approach (see Chapter 6), this motivates the study of geometrical (locally homogeneous) structures in dimension four. As in dimension three, we consider 4-geometries as pairs (X, G) of spaces X and their isometry groups G which act transitively on X with compact isotropy subgroups

1. Geometric Structures

32

Table 1

Isotropy

Riemannian manifold

(identity component)

S4, R4, H4

SO(4)

U(2)

CP2, CH2

SO(2) x SO(2)

S2 X S2, S2 X R2, S2 X H2, R2 X H2, H2 X H2

SO(3)

S3xR,H3xR

SO(2)

Ni13 xR, SL2 R x R, Solo

(S')

F4

{id}

Ni14, Sol,4n n (in particular, So13 xR), Soli

G. Also we use the same three natural conditions for such geometries as in Section 3, i.e., we only consider maximal geometries (X, G) with simply connected complete Riemannian manifold X having the isometry group G = Isom X such that G acts transitively on X and contains a discrete subgroup F with X/ r of finite volume. There are 19 maximal 4-dimensional geometries, one of which is in fact an infinite family of closely related geometries. The list of them (due to Filipkiewicz [1]) arranged by the identity components of isotropy subgroups is presented in the Table 1, where

(S'),n,,, denotes the image of S1 in U(2) C SO(4) by z H (zm, zn) From Table 1, we see that besides easy cases of the product geometries, we also have the irreducible Riemannian symmetric spaces: the 4-sphere S4, the hyperbolic 4-space ff1I4, the complex projective space CIP2 and the complex hyperbolic space CH2 = (SU(2, 1)/S(U(2) x U(1))), which is different from H4 (see detailed description in Chen-Greenberg [1], Goldman [5] and Apanasov-Xie [1]). Among remaining geometries we have only nilpotent and solvable geometries. Namely, there are the R3 x y",', R nilpotent Lie group Nil4 = R3 x# R and the solvable Lie groups Solm where the action of the quotient is given respectively by 0

1

0

,e(t) = exp t 0 0

1

0

0'

,

ym,n(t) = exp It

'0

a

0

0 0

b 0

')1 0 c

(1.49)

Here a > b > c are real, a + b + c = 0, and eQ, eb, eC are the roots of the polynomial (A3 - m),2 + nA - 1) with positive integers m and n. We can identify So14n m with Sol3 xR where S013 is the 3-dimensional Sol-geometry. In general, solvable geometries n and Solm, n, are isomorphic if and only Sol,4n

1.8. Geometry of orbifolds

33

if the corresponding matrices (1.49) are proportional. It gives an infinite series of solvable geometries.

Note that the conditions on the numbers a, b and c exclude the case of equal roots of the polynomial, i.e., the case m2n2 + 18mn = 4(m3 + n3) + 27. In this degenerate case, we have a semidirect product Solo = 1[83 a y JR with the action y (t) (x, y, z) = (etx, ety, e-2tz). This geometry has isotropy SO(2) because of isometric rotating the first two coordinates. A further solvable geometry So14 is conveniently represented as a matrix group

0

b a

c a

0

0

1

1

:a,a,b,cER,a>0

.

(1.50)

Finally, we have the last geometry 1l' with isometry group R2 x SL2 R where SL2 R acts naturally on J2. This space is diffeomorphic to JR2 x H2 and has a non-product metric with alternating curvature. The isometry group of this space, Isom 1F4 - R2 x SL2118, acts on 1182 x 12 as follows. Let (x, y) E JR2 and z E 12 = {z E C : Imz > 0}. Then JR2 acts on R2 x 11112 as (u,

z),

and SL2 J acts on JR2 x H2 as a c

).xYz)=(ax + by, ex + dy, d b

az+b cz + b

The combination of these actions gives R2 x SL2-action on 1182 X H2 whose stabilizer

of the point (0, 0, i) is SO(2) C SL2. For a detailed description of all these 4-geometries, we refer a reader to the original Thesis of Filipkiewicz [1] (see also Wang [1]) and, for preferred complex structures carried by them, to Wall [2, 3]. The results of the latter work appeals to the extensive classification theory available for corresponding complex surfaces, due for the most part to Enriques [1] (in the algebraic case) and to Kodaira [1] (in general); see also Friedman-Morgan [1]. Another descriptions of some 4-manifolds with those geometries can be found in Apanasov [50], Apanasov-Xie [1], Goldman [5], Hillman [1, 2] and Ue [1, 2].

§8. Geometry of orbifolds As we have seen, locally homogeneous Riemannian manifolds M (i.e., (X, G)-manifolds) can be represented as quotient spaces X/ F by a discrete free action on X of the fundamental group iri (M). It is time now to enlarge our vocabulary, so that we

1. Geometric Structures

34

could work with the quotient spaces of not necessarily free group action. Such spaces, introduced by I. Satake [1] and W. Thurston [1], tend to be simpler than manifolds and often give easy, graphic examples of phenomena involving 3-manifolds. Shortly, a smooth n-dimensional orbifold O is a Hausdorff space X = Xo with a countable basis and some additional structure which is locally modeled on a quotient space of a finite group action on a smooth n-manifold. More precisely, an n-orbifold O is covered by an atlas { (Eli, Gi, cpi, Ui) } i,=-, of folding charts each of which consists of a smooth connected n-manifold Ui, a finite group Gi acting on Ui smoothly and effectively, an open subset Ui of (9 and a folding map cci : Ui -> Ui which induces a natural homeomorphism Bpi / Gi : Ui / Gi --* Ui. As for a manifold, these charts must satisfy a certain compatibility condition. Namely, whenever Ui C Uj there is to be an injective homomorphism fij : Gi -+ d j and an embedding cpi j : Ui y Uj equivariant with respect to fij (i.e., for g E Gi and X E Ui, cpi j (g (.x)) = fij (g)gpi j (x)), such that the diagram below is commutative: Vii

Uj

Vii

I

1 (0rj lfij )

(1.51)

Uj l.fij Gi

I Ui C Uj

(p'

j

Uj/Gj.

In other words, for every x E Ui and y r= Uj such that Vi / Gi (x) = V j / G j (y) is in Ui fl Uj, there is a diffeomorphism ipxy between neighborhoods of x and y such that cpxy(x) = y and cOjioxy = qqj.

We note that the embeddings Vi j are defined modulo composition with elements of the group Gj, and correspondingly, the homomorphisms fij are defined modulo conjugation by elements of Gj. In general, it is not true that (Pik = Ojk o cpij when Ui C Uj C Uk; however, there exists an element g E Gk such that g o fik = (pjk 0 Wij

and

g o fik(gi) . g-t = fjk o fij (gi) ,

gi E Gi .

(1.52)

It is clear that the orbifold structure should not be dependent on the covering {Ui }: two coverings give the same orbifold structure if their union satisfies the definition. Also, since we operate in the smooth category, we have to assume that the action of the group Gi in the tangent space at its fixed point is defined by some subgroup of GL(n, R). Due to the finiteness of Gi, this action should be determined by the action of a subgroup of the orthogonal group O(n). Similarly to (X, G)-structures on manifolds, we can define locally homogeneous geometries on n-orbifolds by using in the definition of orbifolds all the mappings and

1.8. Geometry of orbifolds

35

group actions related to the (X, G)-category. In that sense, we can speak about Euclidean, spherical, hyperbolic, conformal orbifolds, or more generally, about orbifolds modeled on (X, G)-geometries. Considering these (X, G)-geometries, we will restrict ourselves to geometries satisfying the identity condition (IC) in §2, in particular, real analytic geometries. Decreasing the size of neighborhoods U, of a point x in an orbifold (9, i.e., the size of folding charts (0i, Gi, oi , Ui) in the orbifold's definition, we see that the finite groups Gi stabilize to a group Gx which we call the local group at a point x E (9. Now we can define the singular set Eo of an orbifold (9 as the set of those points x E Xo that have non-trivial local groups G. So an orbifold 0 is in fact a manifold

ifEo=0.

Since intersections of the singular set Ed with folding charts are closed there, the singular set Eo is closed in the orbifold and nowhere dense. The latter follows from the fact that a non-trivial homeomorphism of a manifold that leaves an open set fixed cannot have a finite order. In order to illustrate these concepts, we consider some examples of orbifolds. First of all, a manifold M with boundary can be given an orbifold structure in which its boundary becomes a "mirror". Namely, in this structure, any point on the boundary has a neighborhood U C M modeled on R'1 /Z2, where Z2 acts by reflection with respect to a hyperplane in 1W', see Newman [1]. Example 1.18 (A rectangular pillow). Let G be the group of Euclidean isometries of 1[82 generated by reflections in the sides of a rectangle R C JR2 (which is a model of a billiard table). This group G is isomorphic to the product D,,, x D,,,, of two infinite dihedral groups Dw = Z2 * Z2, and the quotient space 1R2/G is R.

Let us consider the subgroup Go C G of index 2 consisting of all elements in G preserving the orientation. Its quotient (9 = R2/Go is obtained by identifying the sides of two adjacent rectangles R and R' by the reflection in their common side. Topologically, the orbifold (9 is a sphere. However, it has four singular points corresponding to points in ]R2 having non-trivial isotropy groups (pi - Z2). This sphere has a Riemannian metric having zero curvature in the complement to these four points pi and the curvature Kp; = it, concentrated at each of the corner pi of our "pillow". The neighborhood of each point pi is a cone, whose vertex angle (on the

surface) is it = 2ir - Kpi. Example 1.19 (Borromean Rings). We consider three families of straight lines in R3 of the form:

{x EIR3:x=(t,n,m+1/2)}, {xE1R3:x=(m+1/2,t,n)}, {xER3:x=(n,m-1-1/2,t)},

(1.53)

where n and m are integers and t is a real parameter. These lines correspond to a family of planes parallel to the coordinate planes, which cuts off unit cubes with half-integer vertices on the lattice in R3 as shown in Figure 11.

36

1. Geometric Structures

Figure 11

Figure 12

Let G C Isom R3 be the group generated by 180° rotations around these straight lines. This group acts discontinuously in 1R3, and one can consider a unit cub Q

1.8. Geometry of orbifolds

37

whose vertices have half-integer coordinates, as a fundamental polyhedron for G, U{g(Q) : g E G} = JR3 (see §4, Chapter 2). Furthermore, as generators of the group G, we can consider the rotations around those lines (1.53) that intersect sides of the cube Q. Identifying the G-equivalent points on the cube sides, we get the quotient space 0 =1[83 / G. In that identification, each cube side folds up twice as an open book. Here we have to examine the images of the rotation axes, which form the singular set

E0, see Figure 12. The quotient space O = R3 / G is homeomorphic to the sphere S3. Its singular set EO consists of three linked circles in the form of the Borromean rings, see Figure 13. In the complement S3\Eo, we have a Euclidean structure, i.e., a Riemannian metric having curvature 0, while at points Eo we have cone-type singularities with angles r along the link circles.

Figure 13. The Borromean rings

As in all our examples, a basic type of orbifolds appears to consist of the quotient space U/ F by a group F acting on a manifold U smoothly, effectively and properly discontinuously (see Chapter 2). This type of an orbifold is said to be good. The notion of being good can be defined more formally in terms of orbifold coverings. Definition. An orbifold & is said to be an orbifold covering of another orbifold O (or a covering orbifold) if the following conditions hold:

i) there exists a continuous projection p: Xo -* X0 between the underlying spaces;

ii) every point x E Xp has a neighborhood U = U/Gx such that each component Vj of p-1(U) is homeomorphic to U/Gd, where Gj C Gx is a subgroup; iii) the homeomorphism in (ii) respects the projection.

In other words, O is an orbifold covering O if the orbifold O admits an atlas of folding charts { (Vi, Gi, (pi, U)}1 such that, for each component V of p-1(Ui ), there exists a folding chart 1G: Vi -+ V in the maximal atlas of the orbifold O so that

(pi = p1.

38

1. Geometric Structures

Now a connected orbifold (9 is called good if there exists an orbifold covering p: 0 --> O such that 0 is a manifold. Otherwise O is called bad. As an example of a bad orbifold, one can take a 2-orbifold O with the underlying space X0 equal S2 and Eo equal a point, whose neighborhood is modeled on R2/7L where Z, acts by rotations. Using the notion of orbifold covering, one can define the universal covering orbifold (or the universal branched cover), (9, of any given orbifold O as follows.

Let Xo be the underlying space of 0, X E X0 a base point for O, and O with projection p: O -* (9 a connected covering orbifold for O with a base point i, p(i) = x. Then the orbifold O is called universal if, for any other covering orbifold

O' with projection p': 0' --* O and a base point x', p'(x') = x, there is a lifting q: 0 -+ O' of the projection p to a covering map of the orbifold 0', i.e., the following diagram is commutative: O

P

N /P

O

It is clear that any good orbifold O is covered by a simply connected manifold M. So one can directly show that this manifold M is the universal covering orbifold 0 by proving that every covering orbifold O' -* O is isomorphic to the quotient MI G, where G' is some subgroup of the group G = G(M, O) of the deck transformations of M over O. In fact, this is the general situation for geometric orbifolds, i.e. orbifolds admitting (X, G)-structures (which need not to be Riemannian ones), because of the following Thurston's uniformization theorem for such orbifolds. Theorem 1.20. Let a (X, G) -geometry satisfy the identity condition (IC). Then every (X, G)-orbifold O is good, i.e., O is covered by a manifold.

Now it is natural to define the fundamental group 7r rb(O) of an orbifold O as the deck transformation group G(O, O) for the universal covering orbifold O of the orbifold O. Generally, this group.7ri rb (O) differs from the fundamental group nrl (X0)

of the underlying space X0 of an orbifold O. Namely, .ni rb (O) = Trl (Xo) only in the case when (9 is a manifold. However, applying Armstrong Theorem 2.18, one can immediately see a precise relationship between these fundamental groups in a quite general case:

Theorem 1.21. Let (9 be a good orbifold with its underlying space X0 and let N be the normal subgroup of the fundamental group rrirb (0) generated by those deck transformations g E G(O, (9) _ 7rl rb(O) that have fixed points. Then

nl(Xo) = nirb(O)/N.

Notes

39

Finally, we should make a few remarks about geometric orbifolds. Due to Thurston's uniformization Theorem 1.20, for any (X, G)-orbifold (9, one can easily define the developing map d: (9 X and the holonomy homomorphism d*: nirb(0) -+ G Furthermore, if the orbifold 0 is either closed or complete as a metric space, then its developing map is a covering, i.e., (9 is a complete (X, G)-orbifold. In particular, for an orbifold (9 with a geometric structure modeled on a simply connected space X, the universal covering orbifold O is the space X itself, 7rlrb(0) acts on X as a discrete subgroup r of the group G, and we have that (9 = X/ F. As we have seen, the orbifold concept provides us with a tool for studying discrete group actions on manifolds, which generalize free group actions. For such actions, most of the concepts defined for manifolds are transferred to orbifolds. It allows us define Euler characteristic X (0), orientability of an orbifold, suborbifolds and Haken orbifolds, see Thurston [1] and Scott [6].

Notes For a general treatment of Riemannian geometry, see do Carmo [1]; for the hyperbolic case, see Thurston [1, 9] and Benedetti-Petronio [1] and, for properties of the Buse-

mann function (and horospheres in negatively curved spaces), see Eberlein-O'Neil [1]; for conformal geometry and Mobius transformations, see Ahlfors [7], Apanasov [36] and Kulkarni [4]. The last two treatments contain different proofs of Theorem 1.6 which is originally due to Liouville (1850) and has been considerably extended since then (by diminishing the degree of required smoothness), see Gehring [ 1], Reshetnyak [1, 3], Heinonen-Koskela[l] and Capogna [1]. This is related to the theory of quasiconformal mappings in space, see Vaisala [1] and Reshetnyak [2], while the compact-

ness principle is originally due to Belinskii [1, 2]. Theorem 1.5 is due to GehringMartin [1]; it reformulates the compactness principle for quasiconformal mappings in terms of convergence groups, see Chapter 2. For the linear representation of the Mobius group we follow Apanasov [3]; see

also Kobayashi-Nomizu [1] and Greenberg [1]. The last work contains a proof of Theorem 1.12 with a classification of Lie subgroups of the Mobius group Mob(n). Theorem 1.13 is based on works by Selberg [1] and Malcev [1], and on the linear representation of Mob(n); for a simpler proof of Selberg's Lemma, see Cassels [2]. In the presentation of geometric structures we basically follow Thurston [1] and Scott [6]. Example 1.10 is due to Gieseking [1]. The classification of 4-geotnetries is due to Filipkiewicz [1] (see also Wang [1]). The notion of orbifolds was introduced by I. Satake [1] and W. Thurston [1]. For 3-dimensional orbifolds see also Bonahon-Siebenmann [1 ], Dunbar [1] and Scott [6]. The unifonmization Theorem 1.20 for geometric orbifolds was originally formulated in Thurston [1]; for its proof, see Matsumoto-Montesinos [1].

Chapter 2

Discontinuous Groups of Homeomorphisms

§1. Convergence Groups Here we shall consider first simple properties of discrete groups of homeomorphisms.

The structure of such general groups is not as well understood as that of groups of conformal (Mobius) homeomorphism, a class of groups which has been studied intensively for many years. But there exists a class of groups of homeomorphisms acting on a domain in the n-dimensional sphere S' , a class which lies between the two classes

mentioned above. This class includes in particular groups of homeomorphisms of uniformly bounded dilatations, i.e. of K-quasiconformal homeomorphisms. Current research of several authors discovered many beautiful properties of such groups and their deep relations to the low-dimensional geometry and topology. Presenting main concepts of this theory, we will simultaneously clearing up what happens in the special case of Mobius groups. We will see that some proofs and features of Mobius groups depend not on their conformal nature but on dynamics of discrete convergence groups.

1. Compactness principle. Let ) C S" = W n be a domain and G a subgroup of the group of topological automorphisms of D. For K > 1, the group G is called K-quasiconformal if each element g E G is a K-quasi-conformal homeomorphism of 0. As we saw in Chapter 1, families of quasi-conformal mappings have the compactness property formulated in Theorem 1.5. This property plays a crucial role in what follows and motivates the following notion introduced by F. Gehring and G. Martin in [1].

Definition. A group G C Homeo(S') of self-homeomorphisms of S" is said to be a convergence group if each infinite subset of G contains a sequence {gi } of distinct elements such that one of the following is true. (i) There exists a self-homeomorphism g of S" such that

lim gi = g and

i-oo a uniformly in Sn.

lim g = g-1 1

i-*oo

2.1. Convergence Groups

41

(ii) There exist points xo, yo in S" such that

lim gi (y) = yo

and

V-->00

lira gi 1(x) = xo

i-->o0

(2.2)

uniformly in compact subsets of S"\{xo} and S"\{yo}, respectively. It follows immediately from Theorem 1.5:

Corollary 2.1. Arbitrary K-quasi-conformal group G acting in S' is actually a convergence group. In particular, every Mdbius group is a convergence group. El It can be easily seen that, for a convergence group G, its conjugation f Gf by a self-homeomorphism f of S" is a convergence group. Hence, there exist convergence groups which are neither Mobius nor quasi-conformal groups. It is naturally to ask:

When is a convergence group G C Homeo(S') topologically conjugate to a Mobius group F C Mob(n)? For the first time, the similar question in quasi-conformal category was posed by F. W. Gehring and B. Palka [1]. We will discuss that in Section 4. Now we restrict our attention to the case of discrete groups.

Definition. A group G of self-homeomorphisms of a domain £ C S" is said to be a discrete group if it does not contain any infinite sequence of distinct elements converging to an element of G uniformly in compact subsets of D. This is equivalent to say that G is a discrete subset of the group Homeo()) in the compact-open topology.

In the case of discrete subgroups G C Mob(n) of the Mobius group, we have discrete subsets G of the topological group Mob(n). Here the topology of Mob(n) can be introduced in two equivalent ways. First, this topology is the topology induced by the topology of the Lorentz group O(n + 1, 1) due to the linear representation of Mob(n) (see Chapter 1, §4). The second approach to the topology of Mob(n) is obtained from the following. Due to (1.10), each element g E M6b(B"+1) can be represented in a canonical form:

g=Ug,; ga(a)=0foraEBn+1; UEO(n+1). Therefore, we can introduce the product topology on Mob(n), i.e., the topology of Since Mob(n) = M6b(Bn+1), this topology is equivalent to the previous one.

O(n + 1) x

B"+1.

Definition. Let a group G of self-horneomorphisms of D act on an open subset X C D. One says that it acts discontinuously (= properly discontinuously) on X if, for each compact subset K C X, there exists only a finite number of elements g E G such that g(K) n K 0 0. A group G C Homeo(S") is called a discontinuous group in S" (or Kleinian one in the case of G C Mob(n)) if there exists a non-empty, open set X C S' on which G acts discontinuously.

2. Discontinuous Groups of Homeomorphisms

42

It is clear that the discontinuity of a group G involves its discreteness (the absence of elements close to the identity). The converse is not true. A classical example of this is provided by the Picard group, which consists of all transformations w E Mob(2),

w(z) = (az + b)l (cz + d), having integer complex coefficients. It is clear that this group is discrete, but it does not act discontinuously in any open subset of the extended complex plane C = C U {oo}: the images of any point in the plane are dense in U.

For a group G C Homeo(,), Z) C Sn, let 9(G) C £ be the set of all x E £ for which there exists a neighborhood U C £ such that g(U) fl u = 0 for all but finitely many g E G. This set 52 (G) is called the discontinuity set of G in £, and its complement A(G) = £\S2 (G) is called the limit set of G. It is easy to check that 0 (G) and A (G) are open and closed G-invariant sets, respectively. An immediate consequence of definitions given above is the following fact.

If G C Homeo(Sn) is a discrete convergence group, then for each infinite sequence of distinct elements in G there exist a subsequence {gi} C G and points xo, yo E Sn such that the condition (2.2) holds. Moreover, the points xo and yo are limit points of G and, for each G-invariant closed set E C Sn which contains at least two points, xo, yo c A(G) C E E.

(2.3)

For a proof of the latter inclusion in (2.3), suppose that G does not act discontin-

uously in Sn\E, i.e., for a compact F C Sn\E and a sequence {gi} C G, we have gi (F) fl F 0. The convergence property of G gives us a subsequence {gi1 } and points xo, yo E E such that gi, (x) -+ yo uniformly in F. Then, for a small enough neighborhood V of yo, v fl F = 0, and for large j, this contradicts to our assumption because gig (F) C V. As a corollary of the above fact (2.3) we see that the limit set A (G) of a discrete

group G C Mob(n) coincides with the limit set of its extension in the (n + 1)dimensional sphere Sn+1 = Rn+1 U {oo} (a discrete subgroup of MSb(n + 1)). Hence,

the group G C Mob(n) C Mob(n + 1) acts discontinuously in balls (half-spaces) Rn+ 1 U Rn+1

Sn+1\Sn.

Another consequence of easy arguments from definitions of a convergence group G and its limit set A (G) shows that each point of A (G) is the limit of an orbit gi (x) where {gi } C G and X E 0 (G).

Lemma 2.2. Given a discrete convergence group G C Homeo(Sn) and a point yl E A(G), there exist a point x1 E A(G) and a sequence {gi } of distinct elements in G such that lim gi(y) = y1 and lim g71 W = xl 1 +oo

1-->oo

uniformly in compact subsets of Sn\{x1} and Sn\{y1}, respectively.

Proof. As yl E A(G), we can choose a sequence {hi} C G such that

hi(VV)flVi#0

(2.4)

2.1. Convergence Groups

43

where Vi, i = 1 , 2, ... , are spherical ri-neighborhoods of yi with limi, w ri = 0. Due to the convergence property of G, we can assume that lim hi (x) = yo and Jim hi 1(x) = xo uniformly in compacts of complements of some points xo, yo E S', respectively.

If xo = yj then the conclusion of the lemma is true for xl = yo and gi = hi t Otherwise, for xo ; yl, we can find (due to (2.4)) for each number i a point zi E Vi with hi (zi) E Vi. We have limi,,, hi (zi) = yo because, for large io, the set {zi i > io} U {yl} is a compact subset of S"\{xo}. Thus, for the spherical distance :

function p,

p(Yi, yo) < Jim (p(yt, hi(zi)) + p(hi(zi), yo)) = 0. 1-400

Now we have yo = yi and may take xi = xo and gi = hi. Then, due to the convergence gl (x) -3 xl, we have that 1

g1-1(x)Egi1(U)nU#0 for any x E U\{yi } and large i. It shows that the group G does not act discontinuously in any neighborhood U of xi, that is xl E A(G).

Lemma 2.2 shows that limit points of a discrete convergence group G are limit points of any G-orbit G(x) = {g(x) : g E G} of a point x E S2 (G). Moreover, if A (x, G) is the set of all y E SI for which there exists a sequence {gi} C G with limi,. gi (x) = y then A(x, G) C A(G) for all x E S", with equality whenever x is not fixed by all g E G (i.e., G(x) x). The latter fact is derived from the following. For any x E S' with G (x) x and any limit point yi E A (G), we have (in the setting of Lemma 2.2) that there exists g E G such that either x # xi or g(x) # xi. Hence the point y1 is the limit either of gi (x) or of gi (g (x)), so we have that yj E A(x, G). Theorem 2.3. There exists a dichotomy that the limit set A(G) of a discrete convergence group G C Homeo(S') either contains fewer than three points or is perfect.

Proof Suppose card A(G) > 3, and let yl, x1 E A(G), {gi} C G be as in Lemma 2.2 and x2 E A(G)\{xl, y1}. If gi (x2) yj for infinitely many i, then {gi (x2)} is an infinite subset of the limit set A(G) which has yj as a limit point. Otherwise we may assume that gi (x2) = Y1 for all i .

In the latter case, we have an alternative: either yj # xl or yj = xl. If yj ; xl then lim gi (yi) = Y1 due to Lemma 2.2 where gi (yl) # yi because gi 1(Yi) = x2 ; Yt . If yi = xl, then there exists a third limit point x3 E A(G) \{x2, xt = yj } with images gi (x3) converging to yl. Again, gi (x3) # yj because gi 1(yi) = x2 X3. Thus, in any case, arbitrary limit point yj E A(G) is a limit point of an infinite subset of A(G), and so A(G) is perfect.

44

2. Discontinuous Groups of Homeomorphisms

2. Elementary discrete convergence groups. Discrete convergence groups G C Homeo(S") corresponding to the exceptional case of the Theorem 2.3 dichotomy, i.e.,

those that have at most two limit points in the limit set A(G), card A(G) < 2, are called elementary groups. Otherwise we say that G is non-elementary. We shall describe all elementary groups by the following statement which becomes a criterion in the case of Mobius groups.

Theorem 2.4. If a discrete convergence group G C Homeo(S") is virtually Abelian, then G is elementary. Moreover, if an elementary group G is a Mobius one, then G is virtually Abelian, i.e., G has an Abelian subgroup of finite index. For the proof we need to classify all cyclic discrete convergence groups.

Definition. Let g be a self-homeomorphism of Sn generating a discrete convergence group (g). We say that g is: O elliptic if it is of finite order, ord(g) = inf{m > 0 : gm = id) < oo; (ii) parabolic if ord(g) = oo and g has a unique fixed point; (iii) loxodromic if ord(g) = oo and g has two distinct fixed points. We remark that this definition is exhaustive, i.e., each g E Homeo(S'1) generating a discrete convergence group is either elliptic or parabolic or loxodromic homeomor-

phism. This fact will be derived from the following result of Gehring and Martin [1, §5] which is classical in the case of Mobius groups.

Theorem 2.5. Let G C Homeo(S') be a discrete convergence group, n > 1. Then the limit set A(G) has the following properties. (i) A(G) = 0 if and only if G is a finite group of elliptic elements.

(ii) A(G) consists of a unique point, xo, if and only if G is an infinite group consisting of parabolic and elliptic elements g which fix the point xo. (iii) A(G) consists of exactly two points, xo and yo, if and only if G is an infinite group which contains only loxodromic elements fixing xo and yo, and elliptic elements which either fix or interchange xo and yo. In addition, G must contain at least one loxodromic element and at most finitely many elliptic elements which fix xo and yo or are of odd order. Proof of Theorem 2.4. In order to prove the first claim, it is enough to prove that each Abelian group G is elementary. Suppose the contrary, i.e., that A(G) contains at least three points, yl, y2, y3. Lemma 2.2 shows that there are points xl, x2, x3 E A(G) and sequences {gl,i }, {92,1}, {$3,i } of distinct elements of G such that for k = 1, 2, 3,

i m gk,i (x) = Yk

iaoo

uniformly in compact sets in S"\{xk}. By symmetry and relabelling, we may assume that one of the following two cases occurs: Yl

X2,

Y2 0 xi ;

or

Yl

X2,

Y2 54 X3,

Y3

(2.5)

xl .

(2.6)

45

2.1. Convergence Groups

In the first case of (2.5), choose disjoint neighborhoods U1 and U2 of yl and y2, respectively, so that

U1CSn\{x2},

U2 CS'\{x.1}.

For a fixed x E S' \{x1, x2}, there exists an integer i2 such that g2, i (Ul U {x}) C U2 for i > i2 because U1 U {x} is compactly contained in S"\{x2}. Similarly, there exists an integer i1 such that gl,i(U2U{x}) C U1 fori > i1. Hence, fori > io = max(il, i2), we have g1,i o g2,i (x) E U1 , g2,1o gl,i (x) E U2 .

This contradicts to the fact that G is Abelian and U1, U2 are disjoint. Suppose next that the case of (2.6) holds and choose pairwise disjoint neighborhoods U1, U2, U3 of the points Y1, y2, y3, respectively, so that

U1 C Sn\{x2},

U2 C Sn\{x3},

U3 C S"\{xl}.

If X E Sn\{x1, x2, x3}, then as above there exists an integer io such that for i > io, g2,i (Ul U {x}) C U2 ,

g3,i (U2 U {x} C U3 ,

g1,i (U3 U {x}) C U1 .

Therefore, we have g1,i o g3,i 0 92,i (x) E U1 ,

g2,i O 91,i 0 93,1 (X) E U2

for i > io. This again contradicts to the fact that G is Abelian and completes the proof of the first claim. Due to Theorem 2.5, the second claim splits into cases of one and two point limit sets because the case of a finite group is clear, and we shall give just a sketch of the proof.

In the case of A(G) = {xo, yo}, all but finitely number of elements g E G are loxodromic with fixed points xo and yo, which can be labeled so that

lim g` (x) = yo

i-+00

and

Jim g-' (x) = xo i-oo

uniformly on compacts in Sn\{xo} and Sn\{yo}, respectively. Here yo is called the attractive and xo the repulsive fixed point for g. Moreover, for any other loxodromic element f E G, there exist non-zero integers i and k such that f i = gk (see Corollary 2.11). Therefore, if G is a Mobius group, its loxodromic elements are translations in the hyperbolic space H"+l along axis y = (xo, yo) with a minimum of translation distance dG > 0,

dG = inf{d(x, g(x)) : x E y, g E G is loxodromic}. Hence we can find an Abelian subgroup Go C G of finite index. The last case of one point limit set is exactly the case of the classical Bieberbach theorem:

46

2. Discontinuous Groups of Homeomorphisms

Theorem 2.6. Let G be a discrete group of Euclidean isometries. Then: (i) If Rn/ G is compact, the normal subgroup G* of translations is afreeAbelian group of rank n having finite index µ(G*, G) in the group G. (ii) If 118n / G is not compact, there exists a finite index normal subgroup G* C G that is a free Abelian group of rank k, 0 < k < n - 1. Then up to conjugation of the group G by an isometry of ', the groups G and G* have the following structure. We write Rn = IRk x Rn-k and let g: x H U x +a (U E O(n), a E IRn)

be any element from G; then U(Rk) = Rk, U(In-k) = Rn-k and a E IRk x {(0, ... , 0)}. If g is an element in the group G*, then UInk = id, i.e., guRk is a translation.

We will prove Bieberbach Theorem (so, finish the proof of Theorem 2.4) in Chapter 3, simultaneously with Margulis's Lemma (Theorem 3.1).

We remark that the second assertion of Theorem 2.4 is not true in general case of (non-Mobius) convergence groups. Namely, the example below provides a construction of a large class of discrete convergence groups which are elementary but have no Abelian subgroups of finite index. Actually, it shows that we may find, as such elementary discrete convergence group G C Homeo Sn, a group isomorphic to the fundamental group of agiven n-dimensional negatively curved manifold M because the universal covering M is homeomorphic to Rn.

Example 2.7. Let r be a discrete convergence group having an invariant domain B, which is homeomorphic to RI (in particular, one could take an arbitrary discrete subgroup of Isom IEIIn as the group For a homeomorphism f : B --* Rn, the conjugacy group G = If y f -1 : y E P } = f r f -1 acts discontinuously in ]R' because r acts discontinuously in B (since A(P) C 8B). Putting g(oo) = 00 for each element g E G, we obtain an extension of the G-action in llln to discrete convergence group action on Rn U {oo} = Sn because r is a convergence group and the boundary 8B is 17-invariant. Denoting this action again by G, we have an elementary discrete convergence group G C Homeo(Sn) (with one point limit set, A(G) = fool) which generally is not virtually Abelian; in particular, F may be isomorphic to the fundamental group of a closed hyperbolic n-manifold (for n = 2, of a surface of genus more than one).

Corollary 2.8. Each element g of a discrete convergence group G is either elliptic or parabolic or loxodromic. Moreover, g and gk are always elements of the same type for each integer k # 0.

Proof Using Theorem 2.4, we see that, for a given g E G, the cyclic group (g) is Abelian, and hence elementary. Therefore g is elliptic, parabolic or loxodromic due to Theorem 2.5. It is clear that g is of finite order if and only if gk is. Next, if ord(g) = 00 then Theorem 2.5 implies that each element of (g) other than the identity is parabolic or loxodromic depending on whether A ((g)) contains one or two points, respectively.

47

2.1. Convergence Groups

As a corollary of the above classification of elements of a discrete convergence group and a classical Newman [1] result which says that a periodic homeomorphism of the sphere S" pointwise fixing an open set is the identity, we have the following uniqueness result which resembles analyticity and is important for the geometrization of manifolds and orbifolds (see the identity condition (IC) in Chapter 1, §5).

Corollary 2.9. Let G C Homeo(S") be a discrete convergence group and f, g E G.

If f = g on an open set, then f - g. 3. The limit set of a convergence group. From the point of view of dynamics of a self-homeomorphism g of the sphere S" generating a discrete convergence group (g), one could rewrite the above classification of such homeomorphisms in the following equivalent form: (i) g is elliptic if it has a finite order; (ii) g is parabolic with a fixed point xo E S" if

lim g` (x) = xo

i--+00

and

lim g-` (x) = xo

iioo

(2.7)

uniformly on compacts in S"\{xo}; (iii) g is loxodromic with attractive and repulsive fixed points xo and yo if

lim g` (x) = yo

and

Jim g-` (x) = xo

i->oo

(2.8)

uniformly on compacts in S"\{xo} and S"\{yo}, respectively. It is obvious that the latter case is equivalent to the condition that there exists a

topological ball B C S" containing the attractive fixed point, yo E B, having the repulsive fixed point in its complement, xo E S"\B, and such that g'(-B) C B for some i > 1. As in the classical Mobius case, the following result shows that loxodromic and parabolic fixed points of elements of a discrete convergence group G have antagonistic behavior.

Lemma 2.10. Let G be a discrete convergence group and f, g E G have a fixed point in common. If g is loxodromic then f and g have two fixed points in common and there exists a non-zero integer k such that fgk = gkf. Proof. Suppose that xo and yo are respectively the repulsive and attractive fixed points

for g and f (xo) = xo. Choose disjoint round balls U and V centered at xo and yo, respectively, so that g(U) fl v = 0, and let E = S"\(U U V). Then there exists an integer io such that

.fg-` (E) C U\{xo},

gkfg (E)\U ; 0

(2.9)

for i > io and sufficiently large k. Let k(i) denote the smallest k for which (2.9) holds

and set hi = gk(i) fg-t Then hi (xo) = xo ,

hi (E) f1 E

0,

2. Discontinuous Groups of Homeomorphisms

48

for i > io, while

lim hi (yo) = yo

i-*00

since limi,w k(i) = oo. Hence for large i, the following closed sets

El ={xo},

E3 = E

E2={hk(yO):k>i}U{yo},

are pairwise disjoint. This implies that at most finitely many elements g E G satisfy the following conditions:

g(Ej)nEj:A0

for

j=1,2,3.

Therefore there exist il, i2 with iI < i2 for which hil = hit. In other words, if i = i2 - iI and k = k(i2) - k(i1) then gk

= fgi f-I

and

gkm = fgim

0

f-1

for all integers m. From this it follows that f (yo) = yo. We want to show that k = i. If k < i, we can choose m > 0 such that

g(k-i)" (f(U)) C U and g(i-k)m(f(V)) C V . Then, for a topological ball Uo = gim(U), yo E S'\Uo, we have:

.f(UO) =.f(gim(U)) = gkm(f(U)) C gkm(g(i-k)m(U)) = UO.

This shows that f is loxodromic with the attractive fixed point xo. Similarly Vo = g-" (V) does not contain xo in its closure, while .f

CVO)

=

f

(gim CV-))

= g-km (f CV)) C g-km (g(k--i)m (V))

= VO

Thus xo cannot be an attractive fixed point for f and we conclude that k >_ i. Reversing the roles for f and f -t in the above arguments shows that k < i and hence 0 that k = i. Thus fgk = gk f Corollary 2.11. Loxodromic elements f and g of a discrete convergence group G

satisfy the alternative: either fix(f) n fix(g) = 0 or fix(f) = fix(g). In the latter case, there exist non-zero integers i and k such that f i = gk.

Proof. Lemma 2.10 shows that fix(f) and fix(g) are either disjoint or coincide. For the proof of the existence of the integers i and k in the second claim, assume that xo and yo are, respectively, the repulsive and attractive fixed points for f as well as g and let U, V, E be as before (replace f by f -I if necessary). Then

f(E) c U\{xo}

for i > io,

49

2.1. Convergence Groups

and, as in the proof of Lemma 2.10, for each such i, we can find an integer k(i) such that hi (E) n E 0, hi = gk(i) a f -`

As above, considering pairwise disjoint closed sets El = {xo}, E2 = {yo} and

E3 = E, we obtain that hi, = hi, for some it < i2. It shows that f' = gk for

i =i2-it andk=k(i2)-k(ii).

As we saw in Theorem 2.3, either a discrete convergence group G C Homeo(SI) is elementary, or its limit set A(G) is perfect. Moreover, for each x E S' which is not fixed by all g E G (i.e., if x ¢ fix(G)), the limit set A(G) is actually the set A(x, G) of all y E S" for which there exists a sequence {gi} C G of distinct elements with limi gi (x) = y. We now apply Lemma 2.10 to show that it is the case of nonelementary groups (when the limit set A (G) is the closure of the set of fixed points of loxodromic elements).

Theorem 2.12. If a discrete convergence group G C Homeo(Sn) is non-elementary, then fix(G) = {x E S" : g(x) = x for all g E G} is empty, and A(x, G) = A(G) for each point x E S". Proof. As we saw before, it is enough to show that fix(G) = 0. Applying Theorem 2.5,

we see that G contains at least one loxodromic element f. We shall find a second element g E G such that fix(f) n fix(g) = 0. Applying Lemma 2.2 to a point yl E A(G)\ fix(f), we find a point xl E A(G) and a sequence {gi} C G which converges to yl uniformly on compacts in S"\{xl}. The proof will be derived from the fact that one of the following fails for some i :

fix(f) n fix(gi) # 0 fix(f) n gi (fix(f) o.

(2.10)

(2.11)

Namely, for such i, we may take g = gi or g = gi fgi 1, respectively. So, if both (2.10) and (2.11) hold for all i, then xl E fix(f) because otherwise gi would converge to yl uniformly in fix(f) and fix(f) n gi (fix(f )) = 0 for large i. Hence x1 ; yl and all but a finite number of the elements gi are loxodromic. The latter claim follows from the fact that, for a round ball B centered at yi and disjoint xl, gi (B) C B for all sufficiently large i. Therefore, fix(f) = fix(gi) for large i due to (2.10) and Corollary 2.11, and we have:

fix(f)\{xl} = Jim gi(fix(f)\{xl}) = yt . a->oo

This contradiction to yl ¢ fix(f) completes the proof.

Corollary 2.13. Let G C Homeo(Sn) be a discrete non-elementary convergence group. Then: (i) G contains infinitely many loxodromic elements gi no two of which have a common fixed point, and A (G) is the closure of the set of loxodromic fixed points.

2. Discontinuous Groups of Homeomorphisms

50

(ii) If G contains a parabolic element then A(G) is the closure of the set of parabolic fixed points.

(iii) If G contains an elliptic element g with fix(g) # 0 then A(G) lies in the closure of the set of elliptic fixed points. (iv) A(G) is the smallest closed non-empty G-invariant set in S".

Proofs of all claims follow from the fact that A(G) = A(x, G) for each x E S" (Theorem 2.12).

Next we establish a significant extension of the first assertion of Corollary 2.13 which generalizes the so-called "double density" property of discrete Mobius groups. Namely, we say that loxodromic fixed points (i.e., fixed points of loxodromic elements) are dense in A (G) x A (G) if, for any disjoint open neighborhoods V1 and V2 of arbitrary limit points x1 and x2, there exists a loxodromic element g r= G with one fixed point in VI and the other in V2. Theorem 2.14. Loxodromic fixed points of a non-elementary discrete convergence group G C Homeo(S") are dense in A(G) x A(G). Proof. We may clearly assume that V1 and V2 from the definition of the double density of loxodromic fixed points are round balls which both meet A (G). Corollary 2.13(i)

shows that, for i = 1, 2, there exists a loxodromic element gi E G with fixed points xi, yi and yi E Vi. By replacing gi by gi 1, if necessary, we may assume that the point yi E Vi is the attractive fixed point for gi. Due to V1 fl V2 = 0 and Corollary 2.11, we have x1 x2. Similarly, considering a loxodromic element go E G with a fixed point distinct from points xi and yi, i = 1, 2 (Corollary 2.13), we see that its repulsive and attractive fixed points, xo and yo respectively, are distinct from the points xl and x2. Let Uo, V0, U1, U2 be pairwise disjoint neighborhoods of xo, yo, XI, x2, respectively.

Then

V2 C S"\{Y1},

U1 C Sn\{xo},

Vo C Sn\{x2},

and, for large i,

gl i(V2) C U1 ,

go(Ui) C Vo,

92t(V1) C U2,

g0t(U2) C UO,

g2(Vo) C V2 .

Similarly,

gl(UO) C Vi,

for large i. Hence, considering hi = g2 o go o gj we can choose i such that

hi(V2)CV2,

hi '(-VI) C VI.

Therefore hi is a loxodromic element which has fixed points in disjoint balls Vi and V2.

2.1. Convergence Groups

51

4. Conjugation problem. As we observed before, the conjugation of a convergence group (in particular, a Mobius group) by either quasiconformal or topological automorphism is a natural way to construct new convergence groups. Is it the only method for doing that? In other words, Is every K-quasiconformal (or convergence) group a quasiconforinal (or topological) conjugate of a conformal (or quasiconformal) group? This question is a very deep problem related to many other problems in geometry and topology. A good example of such a relation has been provided by recent (independent) solutions of this problem for convergence group actions on the circle S1 by Casson-Jungreis [1] and Gabai [1, 21:

Theorem 2.15. G C Homeo(Sl) is a discrete convergence group if and only if G is conjugate in Homeo(S1) to a discrete subgroup of Mob(1). This result has a lot of important consequences as the following: (1) a new proof for the Nielsen Realization Problem (see Theorem 4.39); (2) an affirmative solution of the Seifert Fibered Space Conjecture (see Theorem 6.7); (3) the torus theorem for 3-manifolds (see Theorem 6.8). In the quasiconformal case, the answer to this conjugation problem is affirmative in dimension one (where quasiconformal maps are more commonly called quasisymmetric maps) and in dimension two, see Hinkannen [1], Sullivan [1] and Tukia [1], and generally negative in higher dimensions. Namely, in any dimension n > 3, Tukia [2] constructed (non-discrete) quasiconformal groups non-isomorphic (as topological groups) to Mobius groups; thus they are not conjugated to Mobius groups. Martin [2] modified Tukia's method in order to apply it to discrete quasiconformal groups. Freedman-Scora [1, 2] gave a different approach to constructions of "non-standard" (non-conjugated to Mobius groups) quasiconformal groups in Sn, n > 3. In the general case of convergence groups, the answer to the above problem is largely unknown, although there are good positive results showing that some additional hypotheses are clearly necessary, cf. Martin [5]. We ask the question with regard to conformal groups since we are primarily interested in lower dimension phenomena or cases where a natural geometric candidate for a possible conjugacy is a conformal group, for instance as in the case of topological analogies of Schottky groups. As a simple example, the group of complex biholomorphisms of the unit ball in the complex space Cn (i.e., the group of isometries of the complex hyperbolic space CH'), n > 2, acts as a convergence group on the boundary sphere S2n-1 but, by Margulis [7, 12] super-rigidity theorem, this group or any of its discrete co-compact subgroups cannot even be isomorphic to conformal groups as the enveloping algebras of their Lie algebras are different. Actually, these groups cannot even be topologically conjugate to quasiconformal groups (in the standard conformal structure on the sphere). In higher dimensions, it is known that the fundamental group of a strictly negatively curved manifold acts as a convergence group on the sphere at infinity of the

52

2. Discontinuous Groups of Homeomorphisms

universal covering space, see Martin-Scora [1]. Thus reasonable affirmative solutions to the problems of conjugation and extension to the (n + 1)-ball would imply results concerning the homotopy type of negatively curved compact manifolds. This is especially interesting in the three-dimensional case where it is thought that every negatively curved compact 3-manifold is hyperbolic, see also Chapter 6, §3. The answer to this question would essentially be provided by a solution of the conjugation problem for convergence groups acting on the two sphere and having this sphere as the limit set: Are they topologically conjugate to Kleinian groups? For partial results along this line see Theorem 7.8, Martin-Scora [1] and MartinTukia [1]. There is a relationship here between these questions and those of M. Gromov

concerning his so-called hyperbolic groups, see Gromov [5] and Chapter 4, §6.5. M. Freedman [3] has shown that, in higher dimensions, the question of the existence of an extension of the group action on the n-sphere to the action on the (n+ 1)-ball for even a rather restrictive class of convergence groups is equivalent to the surgery problem. In particular, the 4-dimensional surgery problem, at present unsolved, is equivalent to the following question concerning a topological characterization of certain Schottky groups acting on the 3-sphere:

Problem. Given a convergence group G acting on the 3-sphere with the compact quotient, isomorphic to a free group on a finite number of generators and having a Cantor set as the limit set, does the G-action extend to a convergence group action on the 4-dimensional ball? Finally we remark that the question about a topological conjugacy of every element of a discrete convergence group to a Mobius transformation is unclear, either. Let us examine this question for all three classes of elements.

Elliptic elements. In dimension two, each periodic self-homeomorphism f of S2 is topologically conjugate to an orthogonal transformation due to a theorem of BrouwerKer6kj6rt6-Eilenberg (see Eilenberg [1]).

The situation is different in higher dimensions. Montgomery-Zippin [1] constructed a topological involution f of S3 whose fixed set fix(f) is a wild knot. Such f cannot of course be conjugate to an orthogonal transformation. More exotic examples can be found among counter-examples to the generalized Smith conjecture. In particular, for n > 4 there exist smooth, and hence quasiconformal, periodic orientationpreserving homeomorphisms of S" whose fixed points sets are smoothly knotted codimension two spheres, Giffen [1]. Hence there is no hope of topological conjugacy. However, the affirmative solution of the Smith conjecture shows that periodic diffeomorphisms of the 3-sphere with a nonempty fixed point set are conjugate to orthogonal transformations. Nothing is known if the fixed point set is empty. Parabolic elements. Such self-homeomorphisms g of S' are characterized by (2.7). If the fixed point xp is oo, the (2.7) says that

lim g` (x) = oo and

t-*oo

Jim

i-oo

g-i

(x) = oo

(2.12)

2.1. Convergence Groups

53

uniformly in compact sets in W1. This condition (2.12) is known as Sperner's condition

and homeomorphisms of S' with this property are called quasitranslations. Results due to Sperner and Kerekjarto (Kerekjdrto [ 1]) show that each quasitranslation in 1182 is

a topological translation, i.e. it is topologically conjugate to f (x) = x + el. However it is not true in higher dimensions where there are counter-examples of Kinoshita [1] (for n = 3) and Husch [1] (for all n > 3). Here is an outline of a construction of a counter-example in dimension four, see Martin [5]. Let W3 be a Whitehead 3-manifold, i.e., W3 is a contractible 3-manifold not homeomorphic to R3, see Hempel [1]. It is known that W3 x R is homeomorphic to R4. Therefore, W3 x S1 and 1[83 x S' have the same universal covering space R4. The covering action extends to a quasitranslation of S4 and these two actions are topologically distinct as their quotients are not homeomorphic.

Loxodromic elements. Such a self-homeomorphism g of the n-sphere Sn = Rn U {oo} with xo = 0 and yo = oo as its repulsive and attractive fixed points is characterized by the following condition:

lim g` (x) = cc and

i->oo

Jim g-' (x) = 0 i-oo

(2.13)

uniformly in 118n\{0} and R', respectively. The orientation preserving homeomorphisms g which satisfy (2.13) are called topological dilations. Results due to Kdrekjarto [2], Homma-Kinoshita [1] and Husch [2] and recent proof of the Annulus Theorem in dimension four (see Freedman-Quinn [1]) imply that topological dilations or their inverses are always topologically conjugate to the dilation x r 2x. Here is an outline of the proof of this fact. We may assume that there is a topological sphere S c S" = Rn of codimension one separating the origin 0 E Rn and oo and such that g(S) fl S = 0. The convergence property (2.13) implies that every sphere S separating 0 from oc moves off itself as it is compactly supported away from the limit set, A((g)) _ {0, oo}. In low dimensions, one can use a finite cut and paste process to modify such an S to find one that moves off itself. In higher dimensions, one must use more sophisticated methods. When n > 3, S'\{0, oo} is simply connected. The action of (g) in Sn\{0, oo} is effective and must cover a compact manifold with the fundamental group Z. This manifold must fiber over the circle and, using some homotopy theory, one can show that there is a sphere transverse to the fibration which can be lifted. This is the sphere we are looking for.

Now set S,n = g' (S). The convergence property (2.13) implies that (S,,,) is a disjoint collection of spheres with 0 and oo as their only accumulation points. The Annulus Theorem implies that the region between SY1 and S,,+l is homeomorphic to an annulus. Therefore we can use this annular structure to construct the desired conjugacy of g and {x -+ 2x}. The existence of a topological conjugacy implies of course that (S' \{xo, YO})/(g) ^

Sn-t X S1

.

(2.14)

54

2. Discontinuous Groups of Homeomorphisms

It then follows from Sullivan's [1] proof of the Hauptvermutung for quasiconformal manifolds in dimensions n 4 that, for quasiconformal g, this topological conjugacy can be made quasiconformal.

§2. Group action on the discontinuity set 1. The quotient space. Let G C Homeo(S") be a discrete convergence group acting discontinuously somewhere in the sphere S', i.e., G is a discontinuous group. Then the discontinuity set n (G) C S' is the maximal open subset in S' with discontinuous action of G (Lemma 2.13(iv)), and we can consider the space of the orbits G(x), x E 0 (G), i.e., the quotient space 0 (G)/ G. Let 7i : 0 (G) -* 0 (G) / G be the natural projection. Proposition 2.16. Let G C Homeo(S") be a discontinuous convergence group. Then the quotient space S2 (G)/ G is an n-dimensional orbifold. Moreover, if the group

G acts freely on 0 (G), then Tr: (G) -+ 0 (G)/ G is a covering and ) (G)/ G is a manifold.

Proof. As it is obvious that the structural conditions for an orbifold are met, it is sufficient to prove separability for the topological space 0 (G)/ G. We consider two

distinct orbits G(xl) and G(x2), G(xl) fl G(x2) _ 0. Due to the discontinuity of the group G, any compact neighborhood of a point x1 E S2 (G) contains only a finite number of images g(x2) of any point x2 E S2 (G). Hence, there exists a neighborhood Ul C 0 (G) of the point x1 such that U1 fl G(x2) = 0. Furthermore, then x2 fl G(U1) = 0 and, as above, there exists a neighborhood U2 C 0 (G) of the point X2 such that G(U1) fl G(U2) = 0. This proves that the quotient space 0 (G)/G is a Hausdorff space.

Similarly, if G acts on 0 (G) without fixed points, for any point x E S2 (G), one can find a neighborhood U that does not intersect the images g(x), g E G\{id}. In that neighborhood the projection it is a homeomorphism, so it is a covering map and 13 S2(G)/G is a manifold. Remark 2.17. If G C Homeo(S") is not a convergence group then the quotient space 0 (G) / G need not be Hausdorff. An easy example (Thurston [ 1]) is a cyclic group (g) generated by a self-homeomorphism g of 1k2, g(xl, x2) = (2x1, X2/2), see Figure 14. Here (g) acts discontinuously in 1R2\{0}, and S = (1[82\{01)/(g) is a surface with irl (S) = Z ® Z. However, S is not a Hausdorff space since points on coordinate axes as (1,0) and (0,1) do not have disjoint neighborhoods.

Since each periodic self-homeomorphism of 2-sphere S2 is topologically conjugate to an orthogonal transformation, the quotient space 0 (G)/ G for discontinuous convergence group G C Homeo(S2) is always a 2-manifold. Similarly, for a Kleinian group G C Mob+(3), the quotient space 0 (G)/ G is an orientable 3-manifold because

2.2. Group action on the discontinuity set

55

the quotient space B3/ r of a round 3-ball B3 by a finite group r C 0(3) consisting of rotations is homeomorphic to B3. However, in higher dimensions, even the quotient space 0 (G) / G of a Kleinian group G C Mob(n) need not to be a manifold but only a pseudo-manifold because of finite order elements in G. Actually, the quotient space W41 T of a closed round 4-ball by a finite cyclic group r c 0(4), r = Zk, which acts freely on the boundary 3-sphere )B4, is an example of such a pseudo-manifold. The quotient space W41 r is homeomorphic to a cone over its boundary which is a lens space S3/ r and hence is not a topological ball.

9 CSC c

v

Figure 14

In the case of the simply connected discontinuity set n (G) of a discontinuous convergence group G, one can understand topological properties of the quotient space S2 (G)/G due to the following Armstrong j1] theorem:

Theorem 2.18. Let X be a path connected, simply connected, locally compact metric space and G C Homeo(X) a discontinuous group. If H C G is the normal subgroup generated by those elements which have fixed points in X then nl (X/G) - G/H.

From the geometrical point of view, for G C Mob(n), the natural projection Tr: 0 (G) --; 0 (G)/G defines the structure of a conformal orbifold on the orbifold S2 (G)/G. This orbifold branches over the projections of subdomains Sk fl 0 (G) of k-dimensional spheres Sk, 0 < k < n - 2, with non-trivial stabilizers

GSk={gEG:g(x)=x forallxESk}, which act as structural groups for the orbifold 0 (G) / G.

2. Discontinuous Groups of Homeomorphisms

56

2. Basic Classes of Kleinian Groups. Fuchsian groups. A Kleinian group G C Mob(n) is called Fuchsian on the n-sphere Sn = 3 if it leaves invariant a certain round n-ball (half-space) B" C Rn. The limit set A(G) of a Fuchsian group G lies on the (invariant) boundary sphere Sn-1 = 8B". If it coincides with the boundary sphere, G is called a Fuchsian group of the first kind. If, on the other hand Sn-1 \A (G) is not empty, G is called a Fuchsian group of the second kind. In latter case, G acts on the boundary sphere Sn-1 as a Kleinian subgroup of Mob (n - 1), and either it is elementary or its limit set A(G) is a perfect set, nowhere dense in the sphere Sn-1. A Fuchsian group of the first kind has two invariant components, int Sn-1 = B" and ext Sn-1 = Rn\Bn. There is one invariant component for a group of the second kind, the discontinuity set SZ (G) itself.

Figure 16

Figure 15

Quasi-Fuchsian groups. These groups represent a direct generalization of Fuchsian groups. Namely, a quasi-Fuchsian group is a group G whose discontinuity set 0 (G) is the union of two G-invariant topological balls 0o and S2i, with the limit set A (G) as their common boundary topological (n - 1)-sphere. We note that, for n > 3, it is insufficient to require this property only for the closure of no alone because there exist groups G whose topological (n - 1) -spheres A(G) do not split up 118n into two contractible components, see Chapter 5, §5.

An important subclass of quasi-Fuchsian groups consists of conjugations f I' f -1 C Mob(n) of Fuchsian groups on the n-sphere by its quasiconformal selfhomeomorphisms f compatible with 17-action, see Figure 16. We will study such groups in Chapter 7 in connection with deformation spaces. This approach provides first examples of non-trivial quasi-Fuchsian groups in S", n > 3 (i.e., ones distinct from Fuchsian groups on S"). This is the type of a quasi-Fuchsian group Go in the following example.

Example 2.19. Let Q = {x E ]R3

lxi 1 < 1) be the unit coordinate cube. Circumscribe from its vertices spheres Si, 1 < i < 8, having radii -v1r3_/3. As the spheres Si, :

2.2. Group action on the discontinuity set

57

9 < i < 14, we take six spheres that do not intersect each other and whose centers coincide with the centers of the faces of Q and have radii (1/2 /6). All intersection dihedral angles of these spheres Si, 1 < i < 14, equal to rr/3, and the balls bounded by these spheres cover the boundary of the cube Q. We consider a group Go C Mob (3) generated by reflections x i with respect to the spheres S; , i = 1, ... , 14. Due to Corollary 2.39, this group is a Kleinian one. Its fundamental set splits up into a bounded component F0 C Q and an unbounded one, F1 C 1R3 \ Q, and can be taken in the form 14

F(Go)nextSS=FoUF1. i=1

It shows that the discontinuity set cZ (Go) splits up into two (invariant) components S2o and SZ1, S2o U S21 = S2(Go). In fact, supposing the contrary, i.e. that Q(Go) is connected, we obtain arcwise connectedness of S2 (Go) because it is open. Let

y c cZ (Go) be a curve joining the origin 0 E F0 and oo E Fl. One can assume that y intersects polyhedra g(Fo) and g(F1), g E Go, avoiding their edges, that each intersection consists of not more than one component,and that the number of intersected polyhedra is minimal (due to the compactness of y c 0 (Go), this number is always finite). Then the intersected polyhedra contain some polyhedra gl (Fo) and g2 (F1), gZ E Go, having common faces, which is clearly impossible. As Fo U F1 = F(Go) is a fundamental domain for Go, we apply Lemma 2.22 to come to the conclusion that 0 (Go) splits up into two invariant components: 2o = Go(Fo) = U{g(Fo) : g E Go) C Q,

01 = Go(F1).

(2.15)

Applying a result of Brown [1] that the union of an increasing sequence of open topological balls is homoeomorphic to a ball, we see that 0o and 721 both are homeomorphic to a ball and, moreover, G is a quasi-Fuchsian group. In fact, methods of Chapter 7 enable us to show that Go is obtained by a non-trivial quasi-conformal deformation from a certain co-compact Fuchsian group in 193, cf. Figure 15 and 16.

It is not difficult to construct infinitely generated quasi-Fuchsian groups G C Mob(n) distinct from quasiconformal conjugations of Fuchsian ones, see Tetenov [1]. However, a similar question for finitely generated groups is still open.

Schottky groups. By definition, a Schottky group on the n-sphere S" is a Kleinian group G C Mob(n) with generators gl,... , g associated with 2m disjoint closed topological balls Bt, Bi, ..., Bm, B;n C R" and such that gi (ext B1) = int i = 1, . . . , m. It is readily seen that G is free and that all its elements are loxodromic transformations. Following Myrberg [1], one can show that any non-elementary

58

2. Discontinuous Groups of Homeomorphisms

Kleinian group G C Mob(n) contains non-cyclic Schottky subgroups.

Figure 17. Classical and non-classical generators of a Schottky group

A subclass of Schottky groups is the class of classical Schottky groups, for which the balls in the definition are round balls, see Figure 17. Dimension arguments (Marden [3]) show that this subclass is proper. For constructive examples of non-classical Schottky groups in two and four dimensional spheres, see Yamamoto [2] (cf. Figure 17) and Gusevskii [7, 8], respectively.

Web groups. A Kleinian group G C Mob(n) is called a web group if stabilizers Gl of all components 0j of S2 (G) are quasi-Fuchsian groups. Often, one requires also that the group G is finitely generated. The limit set A (G) of a web group G contains points not lying on the boundary of any component S21 C 0 (G). Example 2.20. Here we present one of possible constructions of finitely generated web groups in dimensions n > 3. §5 in Chapter 6 contains another constructions of web groups based on different ideas. Let r be a free Abelian group of rank 3 generated by Euclidean translations by the coordinate vectors el, e2, and e3 in R3, and Go C M6b(3) be the quasi-Fuchsian group from Example 2.19 associated with the unit coordinate cube Q C 1[83. Using notations from Example 2.19, we shall show that the group G = (Go, F) is a finitely generated web group in S3. Namely, the group G = (Go, F') is Kleinian, since there are no G-equivalent points within the polyhedron Fo. The stabilizer Gw C G of oo coincides with r, while each element y E r translates the isometric sphere of any element g E G\r to the isometric sphere of ygy-t E G\r, see Figure 18. Then Lemma 2.31 and the fact that the sides of Q (a fundamental polyhedron for r) are pairwise F-equivalent imply that F0 is a fundamental domain for G. The discontinuity set of the group G,

S2 (G) = U g(Fo) gEG

2.2. Group action on the discontinuity set

59

splits up into a countable set of non-invariant components S2i, i = 0, 1, 2, ..., the first component S20 being defined by (2.15), while the others being images of S2o

under yi E G\Go, S2i = y2),

i > 1. This shows that stabilizers GQ; of all components j are t h e subgroups Gi = yiGoyl 1 C G, i = 1, 2, ... , Go = GE20. It is clear that all these stabilizers GQj are quasi-Fuchsian groups. Hence G = (Go, F) is a finitely generated web group.

Figure 18. A web group

Degenerate groups. This name is given to non-elementary finitely generated Kleinian groups in IR', n > 2, whose discontinuity sets are (connected) contractible domains, i.e., have trivial homotopy groups. So far, there have been no constructive examples given of such groups (even in dimension two). In the planar case (n = 2), Bers [3] and Maskit [2] have given a proof of the existence of such groups, which is based on the properties of Teichmiiller spaces and dimension arguments. Of course, extending the action of a degenerate group G C M6b(2) from the plane into the space r, n > 3, we obtain a degenerate group there. However, it is not clear whether there are specifically spatial degenerate groups, such as, for example, ones for which the dimension of the limit set is n - 1. Example 2.21. Groups with analogous properties, but infinitely generated, are readily constructed in any dimension. For this purpose, it is sufficient to take a smoothly embedded (n - 1)-disk D C R'2 and consider a countable set E of non-intersecting spheres ai touching the disk D from one side in such a way that the points of contact are dense in D and the family E' of spheres obtained by reflecting spheres in E with

60

2. Discontinuous Groups of Homeomorphisms

respect to corresponding tangent planes to D consists of disjoint spheres ul . Then the group G C Mob(n) generated by infinitely many parabolic transformations gi with the isometric spheres of E E (the spheres on E E' are isometric for gi 1) has the limit set A(G) containing the closed disc D and looking like a (n - 1)-dimensional tree, see Figure 19, and its discontinuity set S2 (G) is contractible. These properties follow from Lemma 2.22.

Figure 19. The limit tree of a degenerate group

Function groups. These are Kleinian groups in R' , each of which has an invariant component of its discontinuity set. In particular, Fuchsian, quasi-Fuchsian, Schottky and degenerate groups belong to this class (see also examples in §2, Chapter 5). A particular place in the class of function groups is taken by finitely generated ones, each having an invariant contractible component. We call these B-groups. Spatial B-groups will be considered in Chapter 6, §5.

3. Connected components of the discontinuity set. The discontinuity set S2 (G) itself splits up into connected components S2i, the set of which can be either finite or countable. Shortly, they also are called components for the group G. The stabilizers

Gni = {g E G : g(52i) = 52i}

(2.16)

of these components are subgroups in G and are discontinuous convergence groups themselves. If the stabilizer Go; of a component SZi coincides with the group G, the component 52i is called invariant. If 521, n2,. .. is a complete set of components not equivalent to each other, the orbifold 52 (G)/G is represented in the form:

0(G)/G = 01/GO, U 02/Gc2 U U.

.

(2.17)

Therefore, the following questions are of certain interest: How many components S2i of the discontinuity set 0 (G) and how many equivalence classes of these components may have a given discontinuous group; how many of these components maybe G-invariant, and what is the homotopy type of these components?

Answers to these questions have long been known for planar Kleinian groups. Namely, the number of connected components for a Kleinian group G C Mob(2) may be either one, or two, or oo. Not more than two of them may be invariant. If there

are two invariant components, the other components of the group G are so-called atoms, i.e., simply connected domains having trivial stabilizers (Accola [1]). All

2.2. Group action on the discontinuity set

61

finitely generated Kleinian groups G C M6b(2) having two invariant components are quasi-Fuchsian groups (Ahlfors [1] and Accola [1]), hence they do not have any other components of the discontinuity set. The number of equivalence classes of connected components for a finitely generated Kleinian group G C Mob(2), i.e., the number of connected components of 0 (G) / G, is finite due to the Ahlfors finiteness theorem, see Chapter 4. The situation alters completely as the dimension increases. Firstly, for any dimension n > 3, there exist finitely generated Kleinian groups G C Mob(n) whose discontinuity sets each consists of two invariant components S2o and 521, one of which is contractible and the other is not even simply connected (in fact, has non-trivial homotopy groups ,rk(521), 1 < k < n - 2), see Theorem 7.49. Secondly, for any n > 3, there exist analogous groups G C Mob(n) with Q (G) Qo U 521, having two invariant components with non-trivial homotopy groups nk ( i ),

i = 0, 1, k = 1, ... , n - 2, see Theorem 7.49. Thirdly, for n > 3, there exist finitely generated Kleinian groups G C Mob(n), each having two invariant components S2o, S21 C 0 (G) and, in addition, each having an infinite number of non-invariant components: 00

O(G)\(QoU01)US2; J=2

Here the non-invariant (contractible) components 52j, j > 2, are not atoms, i.e., they have non-trivial stabilizers Gj C G. These three types of finitely generated (and even geometrically finite) Kleinian groups G C Mob(n), n > 3, containing two invariant components each are clearly not quasi-Fuchsian (see above, for n = 2). The first two classes will be constructed in Chapter 7 as the boundary groups for deformation spaces of some Fuchsian groups I' C Mob(n). An example of the third type will be constructed in Example 4.60. As Theorem 4.49 clearly shows, a finiteness condition on a discontinuous group G is necessary for any study of the above questions. Given a discontinuous convergence group G C Homeo(Sn ), its invariant connected components of the discontinuity set, 521 c 0 (G), can be recognized by the following easy criterion.

Lemma 2.22. Let G C Homeo(Sn) be a discontinuous convergence group with a component 520 of the discontinuity set 52 (G) containing a certain point xo. If here, for each generator g of the group G, the points g(xo) and xo can be joined by a path lying in S2 (G), then the component no is G-invariant. The next characterization of invariant components is topological:

Lemma 2.23. Let 0o be a connected component of the discontinuity set 52 (G) C Sn of a torsion free discrete convergence group G C Homeo(Bn+1) acting in the (n + 1)-ball, Bn+1, and Go = {g E G : g(52o) = S2o} its stabilizer in G. Then

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2. Discontinuous Groups of Homeomorphisms

the component S2o is G-invariant if and only if the homomorphism of the fundamental groups,

i*: n1(SO) -* n1(M(G)) = G

(2.18)

,

induced by the inclusion

is So = p0/Go C M(G) = (Bn+l U S2(G))/G,

(2.19)

is actually an epimorphism.

Proof Suppose 0o be a G-invariant component. For each non-trivial loop a E nl (M(G)), letus fix anelement g r= G\{id} representing a. For this g and for a point x E S2o, a path [x, g(x)] C S2o connecting points x and g(x) x is projected by the natural projection n : 00 -a S2o/ G to a non-null-homotopic loop in S2o/ G. Hence i* is an epimorphism. On the other hand, if the homomorphism i * in (2.18) is non-epimorphic, then there

exists g E G = n1 (M (G)) such that g (x) ¢ 0o for some x E 00 (otherwise, the loop n([x, g (x)]) would be a nontrivial preimage of g in nl (S2o/ Go)). Thus the component 0o cannot be G-invariant, which completes the proof. Lemma 2.24. Let S2o be a component of the discontinuity set S2 (G) C Sn = 3Bn+i of a torsion free discrete convergence group G C Then 0o is simply connected if and only if the homomorphism i,, in (2.18) is a monomorphism. Homeo(Bn+1).

Corollary 2.25. A connected component 0o C 0 (G) C S' for a torsion free discrete convergence group G C Homeo(Bf+1) is contractible and G-invariant if and only if the inclusion i in (2.19) induces a homotopy equivalence of the boundary component So = lo/Go to the manifold

M(G) = (Bn+1 U S2(G))/G.

(2.20)

Proofs of Lemma 2.24 and Corollary 2.25 are similar to the proof of previous Lemma 2.23. Also we note that all these facts are still true for groups G with torsion if one formulates them in the orbifold category. 0

V. Fundamental domains A fundamental domain for a discontinuous convergence (Kleinian) group G C Homeo(S') is an open subset F c 0 (G) of the discontinuity set not containing G-equivalent points and such that S2 (G) C G (F). One needs to add some of bound-

ary points to F to get a fundamental set of the group G, i.e., a set containing one point from each orbit G(x), x E S2 (G). The orbifold S2 (G)/ G is obtained from F by identifying G-equivalent points on the boundary 8 F.

2.3. Fundamental domains

63

The concept of a fundamental domain plays a special role in the theory of discontinuous transformation groups. It is employed as a combinatorial-geometrical method for studying the structure of a Kleinian group as an abstract group (i.e., to derive its generators and defining relations) and thus to establish a relation between the theory of Kleinian groups and combinatorial group theory, see Lyndon-Schupp [1], Magnus-Karrass-Solitar [1], and Cannon [1]. On the other hand, it enables one to prove discontinuity and to find a fundamental domain for a transformation group having given generators.

1. Fundamental polyhedra. Here we consider an important class of closed fundamental domains having some geometric characteristics, fundamental polyhedra for discrete MObius groups. Since Mob (n -1) acts naturally in the round ball Bn bounded by the (n - 1) -sphere Six-1 as the isometry group IsomlHI of the hyperbolic n-space IHIn (Poincare model), we can restrict our attention to the hyperbolic case. First of all, let us define a hyperbolic polyhedron P C lH[n as a closed subset of lH[n with the following properties:

(i) P = cl1n (intHn P). (ii) The boundary a P = alln P belongs to the union of hyperplanes in IHI' which is locally finite. If the set of such hyperplanes is finite, P is called finitely-sided.

(iii) A side (or a (n - 1)-side) of P is a set S C aP such that S lies in some hyperplane H C IH[n, intH S is a connected component of intH(H fl 8P), and S = clH(intH S).

Since a side S of a polyhedron P C if is a (n - 1)-dimensional polyhedron (in Hn-1 - H C IH[n), we can define inductively an i-dimensional side of a polyhedron P as a side of an (i -I-1)-dimensional polyhedron which itself is an (i + 1)-side of the polyhedron P. If our polyhedron P (or int P) is not connected, we define its n-side as the closure of a connected component of intnn P.

Definition. We call a polyhedron P C IHI' a fundamental polyhedron for a given discrete group G C Isom Hn if P satisfies the following conditions: (F- 1) int P is a fundamental domain for the group G action in IEIIn. (F-2) The sides of P are pairwise identified by elements of G.

(F-3) The tessellation of IH[n by polyhedra g(P), g E G, is locally finite, which means that each point x E H' has a neighborhood that intersects only a finite set of distinct polyhedra g(P), g E G. We can formulate the latter condition (F-3) for fundamental domains F as well. In this case we say that this fundamental domain F has local finiteness. At first sight, this condition is redundant, but in fact it is not a consequence of (F-1) and (F-2) even for polyhedra. Due to Beardon [3], a necessary and sufficient condition for a fundamental domain F C H2 for G E Isom IH[2 to have local finiteness is the existence of a homeomorphism between the surfaces IH[2 / G and (cln2 F)/ G. Another such sufficient condition on a fundamental domain F C 113[2 having the accessible

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2. Discontinuous Groups of Homeomorphisms

boundary 8F is that, for any x E 8F, its multiplicity n(x),

n(x) = card{g E G: g(x) E 8F}

(2.21)

is at least two. This implies that, for a convex fundamental polygon P C IFII2, the condition (F-3) does follow from (F-1) and (F-2). As an example of a fundamental domain without (F-3)-property, one can take the polygon P = PI U P2 C R+ in Figure 20 (see also Figure 50), which is a fundamental domain for a cyclic group G = (g) C Isom IFl[2, g(x) = 2x.

Figure 20

The situation with local finiteness of a fundamental domain F C Hn becomes more complicated in dimension n > 3. We shall show that neither the existence of a homeomorphism (clEn F)/ G -* H/ G, nor side pairing on the boundary 8F of a fundamental polyhedron F, nor its convexity are sufficient conditions for having the (F-3)-property of local finiteness.

... } be a regular decomposition of the whole solid angle in IR" (centered at the origin) into polyhedral angles oi such that any two angles Lemma 2.26. Let E = {ol, o2,

of and oj have a common k-dimensional side, 0 < k < n - 1, and any point of the space belongs either to the interior or a k-side of some o1. Then there exist a discrete group GE C IsomlFII" and its fundamental domain (polyhedron) PE C Un with pairwise GE-equivalent sides, such that all vertices of PE are GE-equivalent, and PE has polyhedral angles at these vertices equal to the angles of of the regular decomposition E. Proof. Let E be a given regular decomposition of the unit ball Bn (0, 1) = IHI" centered at the origin, see Figure 21. Then we consider disjoint hyperplanes S2, S2, S3, S2, .. .

in IH[" (disjoint (n - 1)-spheres in IR") such that Si c int of and Si C int of, i > 2, and define the discrete group G C Isom IHIIn as a free group generated by hyperbolic

2.3. Fundamental domains

65

translations gi, gi (Si') = Si, i > 2. Obviously, the connected component, P, of the complement IHFZ\ Ui (Si U Sj') containing the origin is a fundamental domain for GE.

Let denote Pi = P fl ai, i = 1, 2, .... Then the fundamental polyhedron we are looking for has the following form, see Figure 21:

PE = U[P1 U gi(F )) .

(2.22)

i>2

Figure 21

Theorem 2.27. For any n > 3, there are discrete groups G C Isom IHI" and their convex fundamental polyhedra P C ]HP` with pairwise G-equivalent sides such that IHIn/G and PIG are homeomorphic but the tessellation {g(P) : g E G} of Hn is not locally finite.

Proof. To construct the group G C Isom 1H[" and P C lHln with claimed properties, it is enough to construct an infinite regular decomposition E _ {0.1, 62, ... } in Lemma 2.26 such that infinitely many angles Qi have a common edge (geodesic ray in IHIn). Equivalently it is enough to find a regular cell decomposition of the sphere at infinity aHn = Sn-1 having a vertex common to infinitely many cells. To do so, let us assume for simplicity that n = 3 and consider a finite triangulation

of S2 whose triangles are bounded by arcs of great circles on S2. Then we obtain a desired infinite triangulation E in Lemma 2.26 after replacing a triangle ABC in

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2, Discontinuous Groups of Homeomorphisms

that finite triangulation of S2 by infinite set of triangles shown in Figure 22. Here the vertex B is common to an infinite set of triangles. This finishes the proof.

Figure 22

Despite such pathological examples, there are sufficient (exact) conditions for local finiteness.

Theorem 2.28. Let G C Isom H", n > 2, be a discrete group and P C H" a convex polyhedron satisfying conditions (F-1) and (F-2) and having finite multiplicities,

n(x) < oo, for all boundary pints x E B P. Then P is a fundamental polyhedron for G. Before proving this fact, we remark that any finite sided polyhedron can always be decomposed into finite convex polyhedra. It implies:

Corollary 2.29. Let P C H" be a finite sided polyhedron satisfying (F-1) and (F-2) for a discrete group G C Isom H. Then P is a fundamental polyhedron for G. Proof of Theorem 2.28. We assume for simplicity that n = 3 (though the arguments do not depend on dimension n), In order to prove the claim, i.e. the local finiteness of the tessellation G(P) in H3, we suppose the contrary. So let x E H3 be such that any of its neighborhood intersects infinitely many distinct polyhedra g1 (P), gi E G, and points xi E gi (P) tend to x. For a fixed pointy E int P, let yy be curves connecting the points gj (y) and x; inside of gi (P). Since the distances d(x, gi (y)) tend to oo, each sphere S2 (x, r) of an arbitrary large radius r > 0 intersects infinitely many curves yi, and there are limit points for such intersections. Topological arguments imply that

2.3. Fundamental domains

67

there exists a limit curve y that passes through x and intersects all S2(x, r), r > 0. Obviously, neighborhoods of any point on y intersect infinitely many gi (P), gi E G. Since the curve y cannot intersect either int gi (P) or interiors of sides of gi (P), it consists only of vertices of polyhedra g(P), g E G or of segments of their edges. If w is such an edge, we can find some sequence of polyhedra gi,n (P), m = 0, 1, 2, ... which contain w and are pairwise adjacent along (n -1)-sides. However, for all z E w, the multiplicity n(z) < oo, so this sequence cannot be infinite, and a neighborhood

of int w belongs to the union of finite polyhedra g(P), g E G. It shows that points x E y cannot be interior points of any edge of g(P), g E G. On the other hand, y cannot consist entirely of vertices of g(P), g E G. This contradicts to the fact that int P is a fundamental domain for G and completes the proof. Later on, we will also consider polyhedra in non-hyperbolic geometries but in the same way as in H'. In particular, in the conformal geometry (S", Mob(n)), (n - 1)dimensional round spheres play the same role as hyperplanes in H. In the next three sections, we shall describe some canonical ways for constructions of different types of fundamental polyhedra.

2. Ford and isometric polyhedra. Theorem 2.30. Let G C Mob(n) be a Kleinian group whose discontinuity set 0 (G) contains oo and G. = {id}. Then the set

P(G) _ {x E Kn :

sup

Ig'(x)I < 1 j= next I (g)

gEG\{id}

(2.23)

gEG\(id}

is a fundamental domain for the group G.

Remark. This fundamental domain P (G) in (2.23) is called the isometric fundamental polyhedron. Making this name compatible with the polyhedron definition in the above section, one has to take the closure of P (G). Also we note that, due to (2.23), this isometric polyhedron is the intersection of (n + 1)-dimensional isometric polyhedron for action of G in the upper half-space R+1 with its boundary, R". In this half-space, isometric spheres I (g) for g E G are hyperbolic hyperplanes, so the latter (n + 1)-dimensional isometric polyhedron in the upper half-space Rn+ 1 is in fact hyperbolically convex polyhedron (compare Theorem 2.33).

Proof. Since each element g E G\{id} maps ext I (g) into int I(g-1), it is clear that P (G) does not contain G-equivalent points. In order to prove that 2 (G) is tessellated by g(P(G)), g E G, we shall firstly show that, for any k > 0, the radii rg of isometric spheres I (g), g E G, have the property

card{gEG:rg>k}
(2.24)

The conditions of the theorem imply that there exists a sufficiently large ball B containing the orbit G(oo)\{oo} whose points are the centers of all isometric spheres

2. Discontinuous Groups of Horneomorphisms

68

1(g). Extending G-action in Sn+1 = R"+1 U too), we may assume that G (or, more precisely, yGy-1 for y E Mob(n + 1)) acts in a certain (n + l)-ball. It means that isometric spheres I (g) are uniquely determined by their centers. So, for two elements g1 and $2 E G with arbitrarily close centers of their isometric spheres and radii greater than k, the image g 1 g2 1(oo) would not lie in the ball B. This proves (2.24) finiteness. In turn it follows from this that there exists a sufficiently large number R such that all isometric spheres I (g), g E G\{id} lie in a ball having the radius R.

Now, let us assume that P (G) is not a fundamental domain. It means that there exists a point xo E SZ (G) and its neighborhood Bo = B" (xo, po) such that Bo fl G(P(G)) = 0. It is clear that then there exists go E G such that xo E int I (go). It follows from our assumption that Bo does not contain the center qo = go 1(oo) of the isometric sphere I (go) with a radius ro > 0. It implies that po < ro < R. The image B1 = go(Bo) is a ball having a radius p1. Using the distortion property (1.27), we see that P1 ?

2 ar

PO

ro -po2

> K po,

(2.25)

where the constant K = (1 - po/R2)-1 > 1, and the equality attains if xo E I(go). We replace Bo by the ball B1 whose center x1 lies inside of the isometric sphere I ($1) of a certain g1 E G. Let rl be the radius of I (S1) and p2 > 0 the radius of the ball B2 = g1 (B1). Due to pt > p, we have by analogy with (2.25) the following inequalities:

ri pt

P2

pt

> K pt > K rt - pt2 > 1 - pt2 /R2 2

2

po

(2.26)

Continuing this process, we obtain a ball, namely the image g (Bo) of the ball Bo = B(xo, po), which has an arbitrarily large radius pm > K'po (because (2.26)) and, by assumption, does not intersect G(P(G)). This contradiction proves the theorem. 0

Note that the conditions G,,, _ {id} and 00 E 7 (G) do not restrict the generality of Theorem 2.30, since one always can find h E Mob(n) such that these conditions will be satisfied for the conjugate group hGh-1. However, this sometimes leads to an unnecessary complication. We give below a direct construction for a fundamental (Ford) polyhedron for a Kleinian group G C MSb(n) having a non-trivial stabilizer G,, of the infinity. Lemma 2.31. Let P,, be a certain fundamental polyhedron (with pairwise equivalent

sides) for the infinity stabilizer G,,. _ {g E G : g(oo) = oo} in a Kleinian group G C Mob(n). Then P,. fl {x E supgEG\G. Ig'(x)I < 1} is a fundamental domain for the group G.

The proof is elementary (compare Ford [1, §35]) and relies on the fact that el-

ements h E G. translate isometric spheres I(g) = {x E R" : Ig'(x)I = 11 of elements g E G\GOO to isometric spheres I (hgh-1) of their conjugates hgh-1.

0

2.3. Fundamental domains

69

3. Dirichlet and isometric polyhedra. A disadvantage of the using of the isometric fundamental polyhedron (2.24) is a special role played by the infinity as the center of P (G). This deficiency can be eliminated. In this section, we shall give a construction of fundamental polyhedra for a Kleinian group G c Mob(n) centered at any points x E 0 (G), Gx = {id}. Before doing this, we introduce a generalized Jacobian of a Mobius transformation g, similar to det Dg(x). Namely, for a given y E ][8n, we consider a transformation y E Mob(n) that transfers y to the infinity and, for a transformation g E Mob(n), define:

ay(g, x) = det[Dygy-1(y(x))]

.

(2.27)

We shall show that this definition is independent of a choice of the transformation y c Mob (n), y (y) = oo. Let yl E Mob (n) be another such transformation, yl (y) _

0o and yl ; y. Then y yl 1 is a similarity transformation: det Dy yi 1(x) equals to some constant r 54 0 and det Dyl y-1(x) = 1/r. Hence

detDYgY-1(Y(x))=detDyyj 1ytgyj 1Yiy-1(Yl(x))

(2.28)

= det Dyt gYl 1(Yl (x)).

The definition of ay (g, x) implies directly that: J1. x) = det Dg(x);

J2. ay(g, x) = ay(y)(ygy-1, y(x)) for ally E Mob(n); D. ay (yg, x) = ay (y, g(x)) ay (g, x) for all g, y E Mob(n). This function also has two important properties: A. ay(g, x) = (Ix - yl/lg(x) - yl)2n det Dg(x);

J5. ay(g, x) = ax (g-1, y), except the case x = y = g(y). To prove the property J4, we consider the inversion with respect to the sphere Sn-1 (y, 1) as the mapping y in property J2. Then y-1 = y, and the Jacobian matrix takes the form Dy(x)=(E-2Qx-y)llx-y12, (2.29)

where E = (Sjj) is the unit matrix and the matrix (E - 2Qx_y) is orthogonal (corresponds to a reflection), with the determinant -1. One can therefore write the following chain of equations, which demonstrates J4:

ay(g, x) = aoo(Ygy-1, y(x) = detDYgY-1(Y(x))

= (Ix - y1 /1g(x) - YI)2n det[(E - 2Qg(x)-y) . Dg(x) (E - 2Qx-y)]

= (Ix - yl/Ig(x) - y1)2n det Dg(x). We now prove the property J5. Let x = y. Then for any transformation y E Mob(n), y (y) = oo, we have ay (g, y) = detDYgY-1(00)

2. Discontinuous Groups of Homeomorphisms

70

(2.30)

and

ay(g-t, y) =

detDyg-ty-t(oo).

These two Jacobians (2.30) become zero (which proves J5) if the mapping ygy-t is not a similarity, i.e., if g(y) ; y. Now let x ; y. From J2 we see that it is sufficient to prove J5 for x = 0 and y = oo. Then the left-hand side of J5 is det Dg(0), while the right-hand side takes the form det[Dxg-11(0)], where 1(x) = x/Ix12. Using the matrix representation derived in Chapter 1 §4 and (2.29), one can show by direct calculations that this Jacobian is precisely det Dg(0). This completes the proof of J5.

We now fix a point y E Rn and, for a transformation g E Mi b(n), g(y) # y, define an (n - 1) -dimensional sphere Iy (g) (a generalized isometric sphere for g): Iy(g) = {x E I[8n : Iay(g,x)I = 1}.

(2.31)

When y = oo, this sphere coincides with the isometric sphere I (g) of g,

I.- (g) = I (g) .

(2.32)

It is easy to show that the transformation g maps Iy (g) to Iy (g-1), therefore, their radii are mutually inverse. In analogy with Theorem 2.30, one can prove:

Theorem 2.32. Let G C Mob(n) be a Kleinian group and y E Q (G) have the trivial stabilizer, Gy = {id}. Then

Py(G) _ {x ERn :

sup

Iay(g,x)1 < 1} = n extIy(g)

gEG\{id}

(2.33)

gEG\{id}

is a fundamental polyhedron for the group G.

We call this polyhedron Py(G) a generalized isometric polyhedron for G centered at y. Due to (2.32), it coincides with the isometric fundamental polyhedron (2.23), if y = oo. It has a clear metric description if G is a hyperbolic isometry group or, in other words, if G has an invariant round ball or a half-space.

Theorem 2.33. Let G C Mob(B') be a discrete group either in the unit ball B" _ B'(0, 1) or in the upper half-space IIB+ (equipped with the Poincare hyperbolic metric d(*, *) converting it into the hyperbolic n-space IHn) and a point y E Hn have the trivial stabilizer Gy = {id}. Then, in IHn, the fundamental polyhedron Py(G) coincides with the fundamental Dirichlet polyhedron centered at the point y:

IEII" fl Py(G) = Dy(G) = n {x E 1HIn : d(x, y) < d(g(x), y)} . gEG\{id}

(2.34)

2.3. Fundamental domains

71

Proof. For concreteness, let us assume that G acts in the unit ball B" (0, 1). To prove the theorem, we note that, for g E G\{id}, the points y and g-1(y) are symmetric with respect to the generalized isometric sphere Iy (g) which is orthogonal to the boundary sphere Si-1(0, 1). So, the intersection Iy(g) fl B" constitutes the set of points hyperbolically equidistant from y and g-1 (y):

Iy (g) fl B" _ {x E B" : d(x, y) = d(x, g-1(y))} .

(2.35)

As the polyhedron Py (G) sides lie on generalized isometric spheres (2.31), this proves (2.34).

We note that, in the above situation, Py (G) C R" remains unaltered if its central point y is replaced by the point y* = y/Iyj2 which is symmetric to y with respect to Sn-1 (0, 1). It implies that, for the isometric fundamental polyhedron P (G) = P" (G) of a discrete group G C Mob(B"),

B" fl P(G) = Do(G) = n {x E B" : d(x, 0) < d(g(x), 0)).

(2.36)

gEG\{id}

Among boundary points of fundamental polyhedra Py (G) or Dy (G) we distinguish those that are fixed by parabolic or elliptic elements g E G. These may form cycles, each consisting of a set of G-equivalent k-dimensional sides, 0 < k < n -1. However, we can avoid such situation due to a choice of the center y:

Lemma 2.34. The set of points y E S2 (G), for which the isometric polyhedra Py (G)

of a Kleinian group G C Mob(n) have equivalent parabolic or elliptic points at their boundaries, lies in the union of an at most countable set of (n - 1) -dimensional spheres. Proof. Let y E 0 (G) be such a point that 8 Py (G) has G-equivalent (distinct) parabolic

or elliptic points p and g(p), where g E G\Gp. In the elliptic case, denote by Sp the maximal pointwise fixed side (of dimension < n - 1) of Py (G) such that p E Sp

and g(Sp) c 8Py(G). In the parabolic case, we put Sp = {p} where p is the parabolic fixed point. It is obvious that, for any point q E Sp, we have Jay (g, q) I = 1. Applying J5-property of the generalized Jacobian, we conclude that, for all q E Sp,

!aq(g-1, y) I = 1. The set of points y E r that satisfy this equation for q E Sp is a sphere of dimension m, m < n - 1:

S'" (P) = n Iq(g-1). gESS

Thus, the set of points y E S2 (G) for which Py (G) has G-equivalent elliptic or parabolic points at its boundary is contained in the union of spheres S'", m < n - 1, which has the same cardinality as a set whose elements are either parabolic fixed points or k-dimensional spheres, pointwise fixed by some elements g E G. Since the group G is countable, this completes the proof.

72

2. Discontinuous Groups of Homeomorphisms

4. Pseudo-Dirichlet polyhedra. In dimensions n < 3, isometric and Dirichlet polyhedra Py (G) and Dy (G) for discrete groups G C Mob(B") have some stability with respect to the choice of their central point y in some open subset of Bn, see Macbeath [3] and Jtrgensen-Marden [2]). In particular, this stability implies the following alternative (see Theorem 4.4): for a given group G, either all of such polyhedra have finitely many sides or all of them are infinite sided. However, in dimensions n > 4, this alternative has no place. It will be shown in §2, Chapter 4. Here we shall introduce a new kind of fundamental polyhedra for discrete groups G C Isom 113' in the hyperbolic n-space Hn. We call them pseudo-Dirichlet polyhedra Dy(G) centered at points y E 1HIn. Actually, such a pseudo-Dirichlet polyhedron

Dy (G) may be different from the Dirichlet polyhedron Dy (G) only in dimensions n > 4. In addition, as it will be shown in Chapter 4, §2, pseudo-Dirichlet polyhedra possess some kind of stability of the number of sides with respect to a choice of central points y E Hn. Due to its definition (2.36), each Dirichlet polyhedron Dy (G) for a given discrete

group G C Isom 113[n is the intersection of half-spaces Sy (g) C H' bounded by Dirichlet hyperplanes Sy (g),

Sy(g) = {x E lIP : d(x, y) = d(x, g(y))},

g E G\{id},

(2.37)

where y E Sy (g). Preserving the main idea of that definition of a Dirichlet polyhedron as the intersection of half-spaces, we will adjust the definition (2.37) for the boundary planes Sy (g) of these half-spaces according to types of elements g E G. For simplicity, we assume that the group G C Isom 1H[1 is torsion free.

Let us start with some observations on parabolic elements of a discrete group G C Isom lHln. Let p E aHn = Sn-1 be a parabolic fixed point of G and Gp C G its stabilizer. The stabilizer Gp acts on (n - 1)-dimensional horosphere Sp C 111171 centered at p as a discrete group r of Euclidean isometries of ][8n-1 = Sp (see §3.5, Chapter 1). Hence, up to conjugation of Gp, each element y E Gp may be represented in the form

Y(x)=U(x)+a,

U E O(n - 1)

,

a E I[8n-1.

(2.38)

The element y E GP is called an irrational parabolic element (=irrational Euclidean twist translation) if the orthogonal transformation U in (2.38) has infinite order. In this case, we say that the parabolic fixed point p and the parabolic subgroup Gp C G have irrational types. Applying Bieberbach theorem (Theorem 2.6) to r = Gp, we have the Bieberbach decomposition of the horosphere Sp, Sp = Rn-1 = Rk X Rn-k-t in accordance with the action of the stabilizer Gp on Sp as the group r of Euclidean isometries. Namely,

the action of any element T E F on Rn-1 = Sp is of the form T (x) = U(x) + a, a E Rk x {O) C Rn-1, where the orthogonal transformation U E O(n - 1) can be

73

2.3. Fundamental domains

represented in the form:

U=[ 0 01, U2EO(n-k-1),

U1EO(k).

(2.39)

U2

Also, in the horosphere Sp, we see that there exists a k-dimensional Gp-invariant horocycle LP corresponding to the minimal F-invariant subspace IRk c Rn-1 such that GP acts on LP as a Euclidean crystallographic group, FINk, with the

compact quotient L p / G p = IRk / r. Hence, the orthogonal transformation U1 in (2.39) has a finite order. Let Yo E LP be the orthogonal projection of a given point y E IfIIF with trivial

stabilizer Gy = {id}. This point yo can be taken as the center of a compact kdimensional Dirichlet polyhedron for the action of GP on the horocycle LP:

Dk(Gp)=LpflDyo(Gp)=Lpf1Dy(Gp).

(2.40)

The Gp-orbit of this Dirichlet polyhedron Dk(Gp) defines a co-compact tiling on the horocycle LP, which is specified by the following (k - 1)-dimensional subhorocycles:

LP fl Sy(y) = Lp fl Syo(y) ,

y E Gp,

(2.41)

which are Dirichlet (k -1)-planes (2.37) with respect to the isometric r - Gp-action on LP with the induced Euclidean metric. At this moment we are ready to define our pseudo-Dirichlet planes Hy (g) bounding

half-spaces Hy (g), y E Hy (g), as follows (compare Figure 23).

Figure 23. Dirichlet and pseudo-Dirichlet planes

2. Discontinuous Groups of Homeomorphisms

74

Definition. Let p be an irrational parabolic fixed point of G C IsomlHF. For any element g from the stabilizer-subgroup Gp of the point p, a pseudo-Dirichlet hyperplane Hy (g) C EI" is the hyperplane orthogonal to minimal Gp-invariant horocycles LP, that intersects LP along the subhorocycle

Sy(g)nLp=Syo(g)f1Lp, in other words, Hy (G) = Syo (g). If an element g E G does not belong to any parabolic subgroup of irrational type, then we define its pseudo-Dirichlet plane Hy (g) as the Dirichlet plane Sy (g).

Now we shall define pseudo-Dirichlet polyhedra Dy (G) by means of some reconstruction of Dirichlet polyhedra Dy (G). For this, we take a subset YD C EH" consisting of points y with trivial stabilizers Gy = {id} and satisfying the following condition: any Dirichlet polyhedron Dy (G) centered at a point y e YD is a simple polyhedron, which means that no two of its boundary parabolic fixed points are G-equivalent. Due to Lemma 2.34, YD can be described as the complement in II" to the union of an at most countable set of (n -1)-

planes. Therefore, for y E YD and for any g E G\{id}, Dirichlet planes S,(g) and Sy(g-1) do not contain distinct parabolic fixed points p and g(p) $ p. Similarly, the pseudo-Dirichlet planes Hy(g) = Syo (g) for elements g from irrational parabolic subgroups Gp C G contain parabolic points distinct from p only in the case when the corresponding centers y E YD lie in an at most countable set of planes which are orthogonal to minimal Gp-invariant horocycles Lp C H. Therefore, after removing from the set YD this at most countable set of planes, we have a set Y(G) C II" such that: (i) for any y e Y(G), neither Dirichlet planes Sy(g) and Sy(g-1) nor pseudoDirichlet planes Hy (g) and Hy (g-1) contain distinct parabolic points p and

g(p)0 p; (ii) Y(G) is the complement in H' to the union of an at most countable set of hyperbolic planes (having zero measure). Definition. Apseudo-Dirichlet polyhedron Dy (G) for a discrete group G C Isom }Ell"

centered at a point y E Y(G) C ]HI" is said to be the polyhedron obtained from the Dirichlet polyhedron Dy (G) after replacing all Dirichlet planes Sy (g), g E G, by corresponding pseudo-Dirichlet planes Hy (g). In other words, pseudo-Dirichlet polyhedra Dy (G) are obtained from Dirichlet ones, Dy (G), as the result of the following reconstruction. We enumerate conjugacy classes of irrational parabolic subgroups in the group G

by their fixed points p1 = oo, P2.... lying on the boundary of a simple Dirichlet polyhedron Dy(G) centered at y E Y(G), and denote D°(G) = Dy(G). Now we have an algorithmical process of our polyhedra reconstruction:

D°(G) -> Dy (G) _. ... i D , (G) -a ... -a Dy (G)

(2.42)

2.3. Fundamental domains

75

where

D`y(G) = n{Hy (g) : g E G1 \{id}} n Ri, Ri = int (U{g(Dy 1(G)) : g E Gp; }>

i > 1.

Here the polyhedra Dy (G) are fundamental polyhedra for the group G with pairwise G-equivalent sides. It can be easily seen from the following two facts. Firstly, if it is true for polyhedra DY'-1 (G) (i.e., in the (i - 1) step), then U{g(Ri) g E G\Gpi } U Ri is a tessellation of the hyperbolic space ]HI" with G\Gpi-equivalent sides.

Secondly, the polyhedron

Dy*(GA) = n{Hy (g) : g E Gp;\{id}} is a fundamental polyhedron for the parabolic subgroup GP, because it is equal to the Dirichlet polyhedron Dy1(G A) centered at the orthogonal projection, yi, of the point y to a minimal G pi -invariant horocycle L1. Another essential remark is that all obtained in this way fundamental polyhedra Dy (G) are simple in the sense that each of them has no distinct G-equivalent parabolic fixed points on its boundary. It follows immediately from the definition of the set Y (G) of center points y.

As a result of this reconstruction process we finally obtain, for a given center y E Y(G), a simple fundamental polyhedron which is called the pseudo-Dirichlet polyhedron Dy (G) centered at y. It can be done in some non-degenerate cases, in particular for groups G with a finite set of conjugacy classes of irrational parabolic subgroups, or for groups G all whose irrational parabolic fixed points are in fact cusp points. We will see this in Chapter 4, §3 (Lemma 4.11). We note that, during this reconstruction process, we have replaced all Dirichlet planes whose closures contain irrational parabolic points by pseudo-Dirichlet planes. This fact allows us to formulate the following conjecture.

Conjecture 2.35. At leastfora geometricallyfinite group G C IsomR', the following (hyperbolically convex) polyhedron

n{Hy (g):gEG\{1}},

yEY(G)CHn,

is likely to give an equivalent definition of the pseudo-Dirichlet polyhedron Dy**(G). So, pseudo-Dirichlet polyhedra Dy (G) are convex.

Remark 2.36. In a discrete group G C Isom IH
2. Discontinuous Groups of Homeomorphisms

76

5. Polyhedral representations of groups. Let G be either a discrete group of isometries of a simply connected space X of constant sectional curvature or a Kleinian group in n-sphere S' with an invariant simply connected component X of the discontinuity

set 0 (G). In both cases, let P C X be a fundamental polyhedron for G-action in X

_

and

gp = {g E G :

dim(P fl g(P)) = n - 1}

(2.43)

be the set of side-pairing elements of the group G that identify (n - 1)-sides of P.

Theorem 2.37. For a given fundamental polyhedron P C X of a discrete group G acting in X, G is generated by elements of the set p. Moreover, defining relations for the group G = (g.p) can be taken as the following edge relations:

g1g2 = id forge, g2 E 9p , or

glg2..... gk = id

(2.44)

where

gi E gp , gi gi+1 i4id , g1 g2

gm =Aid for m < k

and

dim{P fl g1(P) fl g1g2(P) n ... n gl ..... gk-1(P)} = n - 2.

Proof. Let us fix a point xo E P. Then for any element g E G\{id} there is a path a c X connecting points xo and x = g(xo), because of connectedness and Ginvariantness of X. Discontinuity of the action of Go = (9.p) C G in X implies that there exists a closed domain ) C X such that a c int I and £ is covered by a finite number of images y (P), Y E Go. In particular, one can find yo E Go such that yo(P) covers a neighborhood of the point x = g(xo). It then follows that g = yo E Go. This proves the first assertion of the theorem that G = (gp). The second assertion is obtained by considering all the cycles consisting of Gequivalent (n - 2)-sides of P. Here, it is crucial to use simply connectedness of X and that a discrete group does not contain infinite-order elliptic elements. For details, see Gerstenhaber [1] and Macbeath [1]. The next theorem establishes a reverse relationship.

Theorem 2.38. Let P be a convex polyhedron in Euclidean, spherical, or hyperbolic n-space X, and let to each (n - 1)-side Ai of P there be assigned an isometry gi of X such that: 1) for each Ai there exists another side Aj such that gi gj = id; 2) for each (n - 2) -side E of P there exists a sequence A1, A1, ... , Ak of its (n - 1)-sides such that 81g2 . . gk = id, -

Pflg1(P)ng1g2(P)n...ng1g2.....gk-1(P))=E, and the polyhedra in (2.45) have pairwise disjoint interiors.

(2.45)

2.4. Convex polyhedra and reflection groups

77

Then the group G C Isom X, generated by the transformations {gj }, is discrete. Moreover, if P is bounded in X, P is a fundamental polyhedron for G. This theorem is usually known as the Poincare theorem on fundamental polyhedron. It follows directly from a more general Aleksandrov's [ 1] result on filling of a space by polyhedra, see also Abels [1]. In the case of an unbounded polyhedron P C X = H",

these conditions are insufficient to fill the whole space H" by g(P), g E G, and P might be not a fundamental polyhedron, either. Chapter 4 §5 deals with this topic as well as gives a proof of a generalization of Aleksandrov theorem to the non-compact (unbounded) case.

Changing the space X in Theorem 2.38 to a simply connected domain X C S" and Isom X to conformal automorphisms Mob(X) and using a generalization of Aleksandrov's theorem from §5, Chapter 4 on a conformal filling of X, one can generalize Theorem 2.38 for the case of (non-convex) compact polyhedra P C X. Here P will be a fundamental polyhedron for the group G = (gi, i = 1, 2, . . ). A crucial condition for such a generalization is simply connectedness of X. .

The following fact immediately follows from Theorem 2.38:

Corollary 2.39. A group G acting in a constant curvature space X and generated by reflections in the (n - 1) -sides of a convex polyhedron P C X is discrete if and only if..

1) magnitudes of all dihedral angles of P are integer parts of 27r, and 2) if, for a dihedral angle at an (n - 2)-side E of P, its magnitude is an odd

part of 2rr, then P has a symmetry plane passing through E.

Applications of these results are not restricted only on Kleinian groups G C Mob(n) that are groups of spherical, Euclidean, or hyperbolic isometries (i.e., elementary or Fuchsian groups). One can use them for arbitrary Kleinian groups because their canonical extensions to the half-space i+ t are discrete isometry groups in the Poincare model of the hyperbolic space ]HI"+1

§4. Convex polyhedra and reflection groups 1. Coxeter diagrams and Gram matrices. Let X" be an n-dimensional Euclidean, spherical or hyperbolic space, P C X" be a convex polyhedron bounded by hyperplanes Hi, i E I, and G C Isom X" be a discrete group generated by reflections gi with respect to the planes Hi. Corollary 2.39 gives necessary and sufficient conditions for the group G to be a discrete group with P as a fundamental polyhedron. These conditions state that P must be a Coxeter polyhedron, i.e., for any i, j E 1, i # j, the hyperplanes Hi and Hj either do not intersect or else form a dihedral angle containing P, with the magnitude equal to it/nib where nib E Z, nib > 2. Here the relations

g? = id,

(gigi)n" = id

78

2. Discontinuous Groups of Homeomorphisms

are the defining relations for G. We call such groups G reflection groups. Weyl [1], Cartan [1, 2] and Coxeter [1, 2] have studied discrete reflection groups in Euclidean space and thus in the sphere. In particular, a complete classification has been obtained there in terms of symmetric Gram matrices. In the hyperbolic space 1H[", n > 3 (the case n = 2 was described by Poincard [1]), such a description of cofinite reflection groups, i.e., groups generated by reflections in sides of a finite-volume convex hyperbolic polyhedron P, has been given by Vinberg [1, 3, 4] (we subsequently call such groups crystallographic reflection groups). His results are formulated in terms of Gram matrices, and the proofs are based on the Perron-Frobenius theorem on matrices containing non-negative elements. Now we formulate an existence theorem. Let V, be the linear model for the hyperbolic n-space, described in Chapter 1, §3. Recall that points in this model are rays in Minkowski space M"+1 lying inside the light cone L+, i.e., in {x E M"+1 : [x, x] _ -xp + xi -I+ xn < 0, xo > 0}.

Let P C V, be a convex polyhedron bounded by hyperplanes Hi. We denote by ei E 109["+1 the vectors perpendicular to the i-th hyperplanes Hi, in the metric of the Minkowski space defined by the bilinear form [x, y], see §3, Chapter 1. We assume that vectors ei are directed away from P and normalized by [ei, ei ] = 1. There are no linear relationships between these vectors ei containing non-negative coefficients. For any two hyperplanes Hi and Hj containing faces of P and forming a dihedral angle with the magnitude aij, we put

aij = [ei, ej]

-cosaij

and, if these planes do not intersect, then

aij=[e1,ej] The obtained symmetric matrix A = (aij) is called the Gram matrix for a convex hyperbolic polyhedron P. It has the following properties:

1) the rank ofAisn+1; 2) aii = 1, aij < 0 for i

j;

3) A is indecomposable, i.e., it is not a direct sum of two matrices. Theorem 2.40 (Vinberg [1]). A symmetric matrix A = (aij) is the Gram matrix for a certain hyperbolic polyhedron if and only if it satisfies the conditions 1)-3) and the following two additional conditions: 4) A has at least one principal submatrix that is either a positive matrix having order n (C+-matrix) and rank n - 1 or is a non-negative matrix having order n + 1 (CO-matrix) and rank n - 1 (such submatrices are called nodal ones); 5) for any nodal submatrix BI and any its C+-submatrix B having rank n -1, there exists a nodal submatrix B2 containing B and different from B1. Symmetrical matrices satisfying conditions 1)-5) are called C--matrices.

2.4. Convex polyhedra and reflection groups

79

This theorem in particular implies a complete description of all discrete groups generated by reflections in faces of a hyperbolic simplex. For that purpose we only need to enumerate the corresponding C--matrices whose non-diagonal elements have

the form ai j = - cos r/mi j or aij = -1. In that description, it is convenient to use the so-called Coxeter diagrams, which are constructed as follows. We take vertices in a number equal to the matrix order. Vertices vi and vj are joined by (mi j - 2)-fold line (or by a simple line having index m1), where ai j = - cos nl mi j . If aij < -1, the vertices are joined by a dashed line, with statement of the number -ai j . Then for a description of groups generated by reflections in faces of a hyperbolic simplex, one merely needs to sketch any connected diagram having the following properties: 1) the number of vertices is n + 1; 2) the diagram is not contained in the diagram tables of Coxeter C+ and COmatrices for Euclidean reflection groups (see Coxeter-Moser [1]); and 3) any subdiagram composed of n vertices either is the union of some diagrams from the Coxeter C+-matrix tables or is contained in the Coxeter C0-matrix table.

5

4

n = 4

5

5

5

0-50

0<

W Figure 24. Cocompact Lanner groups

Each subdiagram of the first type in condition 3) is obtained by deleting a vertex of the diagram corresponding to a face of the simplex P opposite to its ordinary vertex. A subdiagram of the second type corresponds (in the same way) to a vertex of P at infinity.

2. Discontinuous Groups of Homeomorphisms

80

Such reflection groups for which the simplex P is bounded were first described by Lanner [1], from a combinatorial point of view; they exist only for n < 4. There are nine such groups for n = 3, and only five for n = 4. Figure 24 shows their diagrams. Here there are two the most striking facts about the obtained classification of hyperbolic Coxeter groups, that is, the reflection groups whose fundamental polyhedra are unbounded (finite volume) hyperbolic simplices:

(1) They exist only in dimensions n < 9; (2) There are only finitely many such groups in each dimension n, 3 < n < 9. The final result is summarized in Figure 25.

n=3

4M

40--0 4 04-0 40-4

4 0--0

4

0---a-O6 0

6 5

6

0"'o"

81

2.4. Convex polyhedra and reflection groups

n5

4

4

4

4

o-o-b-o--o

4

n = 6

4

n=7

0---0--0

o-o.-._p =-o-o

0j

Lo-

n= 9

4

0

Figure 25

0-o

o--o c*-o

0

0

2. Discontinuous Groups of Homeomorphisms

82

2. Andreev classification of hyperbolic reflection groups. Andreev [1, 2] has derived a complete classification of crystallographic reflection groups in three-dimensional hyperbolic space, see Kaplinskaya [1] for the case of simplicial prisms. This classification amounts to an exhaustive description of the Coxeter polyhedra P in ll1[3, whose most important property from the geometrical viewpoint is that their dihedral angles are integer parts of it and they are acute-angled: none of the dihedral angles in the polyhedron exceeds 7r/2. Andreev's work on characterization of convex hyperbolic polyhedra was later putin a broader context (related to the hyperbolization theorem for 3-manifolds, see Morgan [1] and Chapter 6, §3) by W. P. Thurston [1, 2], who also noted some applications to conformal geometry via the theory of circle packings. Due to the fact that an acute-angled spherical polyhedron not containing opposite points is a simplex (see Cartan [2] and Coxeter [1]), polyhedral angles at vertices of a bounded acute-angled polyhedron in H' or W are all simplicial. A convex polyhedron having that property is called a simple polyhedron, and its combinatorial type is called a simple combinatorial type.

Theorem 2.41 (Andreev [1]). Given a simple combinatorial type of a polyhedron P which is different from a simplex and a triangular prism and given its dihedral angles not exceeding n/2, the hyperbolic 3-space 1H13 contains a bounded acuteangled polyhedron P of the prescribed type if and only if the following conditions are satisfied:

1) f three sides of P meet at one vertex, the sum of the dihedral angles between them is greater than n; 2) if three sides of P are pairwise adjacent but do not meet at one vertex, the sum of the dihedral angles between them is less than n;

3) if four sides of P are adjacent on a circle (as the side faces of four-sided prism), the sum of the dihedral angles between them is less than 2n. An unbounded acute-angled polyhedron having finite volume in IH[3 may have four

sides meeting at some vertex in the sphere at infinity 8H3. The following criterion applies for such polyhedra.

Theorem 2.42 (Andreev[2]). Given a combinatorial type a polyhedron P different from simplex and triangular prism types and given its dihedral angles not exceeding 7r/2, the hyperbolic 3-space IH[3 contains a finite volume unbounded convex polyhedron

of the prescribed type if and only if the following conditions are satisfied: 1) if three sides meet at some vertex of P, the sum of the dihedral angles between them is not less than n;

2) if four sides meet at some vertex of P, all the dihedral angles between them

are n/2; 3) f three sides are pairwise adjacent but do not meet at one vertex, the sum of the dihedral angles between them is less than 7r; 4) f a side Si C a P is adjacent to two sides S2 and S3 C a P, while S2 and

2.4. Convex polyhedra and reflection groups

83

S3 are not adjacent but have a common vertex p V Si, then the sum of the dihedral angles formed by Si with S2 and S3 is less than ir; and 5) if four sides are adjacent on a circle but do not meet at a single vertex, the sum of the dihedral angles between them is less than 2rr.

In general case of the hyperbolic n-space H", n > 4, such a classification is impossible because a simple convex polyhedron in W, n > 4, may have more dihedral

angles than there are degrees of its freedom. Recall that m-sided n-dimensional polyhedron depends on mn - n(n + 1)/2 parameters. For n = 3, this number of parameters and the number of edges in P coincide, and this is the reason for the compatibility in the linear inequalities represented by the Andreev conditions. As I. Rivin pointed out, the condition on acute angles of polyhedra in Andreev's theorems may be omitted. Namely, he obtained a new characterization of the shapes of convex hyperbolic polyhedra (with arbitrary dihedral angles < 1r) in terms of generalized Gauss map by using of great ideas of A. D. Aleksandrov [1] work on convex polyhedra. Rivin's characterization of hyperbolic convex polyhedra can be used to derive the Andreev's theorems presented above. We refer the reader to RivinHodgson [1] for a detailed treatment of this work; see also Hodgson [1]). To explain Rivin's main result, let us firstly define the Gauss map G associated with a compact convex Euclidean polyhedron P E 1W3. It is a set-valued function defined on P and whose values are subsets of the unit sphere S2. This map G assigns to each point p E 8 P the set of outward unit normals to support planes to P at p. Thus a whole face s C P is mapped under G to a single point G(s) - the outward normal

to s. An edge e c P is mapped to a geodesic segment G(e) C S2, whose length is easily seen to be the exterior dihedral angle at e. A vertex v E P is mapped by G to a spherical polygon G(v), whose sides are the images under G of edges incident to v, and whose angles are easily seen to be the angles complimentary to the planar angles of the faces incident to v. Thus, G(el) and G(e2) meet at the angle (7r - a) whenever et and e2 meet at an angle a. The polygon G(v) is also called the spherical polar of the link of a vertex v c- P. Here, by the link of a vertex v E P, we mean the intersection P and a small sphere centered at v, resealed so that the radius is 1, see Figure 26.

rr - a

Figure 26. Gaussian image of a vertex with dihedral angles A, B, C

2. Discontinuous Groups of Homeomorphisms

84

Gathering the above observations together, we see that G(P) is combinatorially Poincare dual to P, while metrically it is the unit sphere S2. A similar construction can be applied to a convex hyperbolic polyhedron P C H3. Namely, associate to each vertex v E P a spherical polygon G(v) spherically polar to the link of v E P (as in Figure 26). Glue the resulting polygons together into a closed surface by using the rule that two sides of G(vi) and G(v2) are identified isometrically whenever vl and v2 share an edge. The resulting metric space G(P) (with the path metric (1.1)) is clearly homeomorphic to S2, still being Poincare dual to P, as the complex. Metrically, however, it is no longer the round sphere. To see this, consider G (s) which is the single common point of the spherical polygons G (v, ),

where vi are the vertices of the face s. The angle along the edge G(v,) incident to G(s) is the exterior angle of s at vi, and so by the Gauss-Bonnet Theorem, the sum of these angles is 27r --Area(s) # 27r. Thus G (s) is a cone-like singularity, or a cone point, with the cone angle greater than 27r. This analogue of the Gauss map turns out to have rather remarkable properties, which are studied in detail in Rivin [1] and Rivin-Hodgson [1]. The main result is a precise characterization of those surface that can arise as G(P) for a compact convex polyhedron P C lHl3:

Theorem 2.43. A path-metric space (S, g) homeomorphic to S2 can arise as the Gaussian image G(P) of a compact hyperbolic convex polyhedron P C 1H13 if and only if the following conditions hold:

(1) The metric g is piecewise spherical, with constant curvature 1 away from a finite collection of cone points ci. (2) The cone angles at ci are greater than 27r. (3) The lengths of closed geodesics in (S, g) are all strictly greater than 2,r.

Furthermore, the metric on G(P) determines the hyperbolic polyhedron P uniquely (up to isometry).

Remark 2.44. In condition (3) above, the word "closed geodesic" can be interpreted as either "local" geodesic or as a closed "global" geodesic (i.e. the image of a circle under an isometric mapping). The theorem holds with either interpretation of "geodesic". In a space with piecewise spherical metric, any geodesic looks like spherical geodesic (arc of a great circle) away from the cone points, and subtends an angle > 7r on either side if it passes through a cone point.

The necessity of conditions (1) and (2) is immediately apparent from the above discussion of the Gauss map G. The necessity of (3) follows from a hyperbolic version of Fenchel's theorem ("the total geodesic curvature of a closed curve in H3 is greater than 27r"). The proof of uniqueness closely follows the argument used by Cauchy [1] in the proof of his celebrated Rigidity Theorem for convex polyhedra in R 3, see also Aleksandrov [1]. The proof of sufficiency of conditions (1)-(3) is based upon a "continuity" method, as used to great effect by Aleksandrov [1] in his work on convex polyhedra.

2.4. Convex polyhedra and reflection groups

85

Similar characterization of ideal hyperbolic convex polyhedra P C ]H[3 (with arbitrary dihedral angles, compare Andreev's Theorem 2.42) is obtained in Rivin[l], see also Hodgson-Rivin-Smith [1]:

Theorem 2.45. Given a combinatorial type of a convex Euclidean polyhedron P with

weights w (e) assigned to the edges e c P, let P* be the planar dual (or Poincare dual) of P, where the edge e* dual toe is assigned the dual weight w* (e*) = it - w (e). Then P can be realized as a hyperbolic convex polyhedron P C IH[3 with all vertices on the sphere at infinity and with dihedral angle w(e) at every edge e if and only if the following conditions hold:

(1) 0 < w*(e*) < it for all edges e. (2) The sum of dual weights of edges ei , e2, . equal to 2ir.

.

.,

e* bounding a face in P* is

(3) The sum of dual weights of edges e*, e2, ... , e* forming a circuit that does not bound a face in P* is strictly greater than 27r. Furthermore, this hyperbolic realization of the polyhedron P is unique up to isometries of IHI3.

Finally, we remark that one of the geometric and combinatorial consequences of that characterization of hyperbolic convex polyhedra is a combinatorial characterization of convex Euclidean polyhedra in I[83 all of whose vertices lie on the unit sphere. To see this direct corollary, it is enough to notify that hyperbolic convex polyhedra (in the projective Klein model of the hyperbolic space 1HI3 in the unit ball B3, see Chapter 1, §3) with all vertices on the sphere at infinity correspond precisely to convex Euclidean polyhedra inscribed in the sphere S2 = aB3. Therefore, a polyhedron is of inscribable type exactly when it admits an edge-weighting that satisfies the condition in Theorem 2.45 (see Hodgson-Rivin-Smith [1]). That resolves a problem posed by Jacob Steiner in 1832. 3. Co-finite reflection groups in low dimensions. The maximal dimension of the hyperbolic space lHI for which bounded Coxeter polyhedra (or, equivalently, co-compact crystallographic reflection groups) are known is eight. Makarov [4] has geometrically derived some examples of such polyhedra in 1H[4 and 1H15, while Bugaenko [1, 2] has described some such polyhedra in 1H[6, IHI7 and lH8 by their Coxeter diagrams.

Makarov's examples also illustrate his general geometric methods of constructing crystallographic groups in the hyperbolic spaces, see Makarov [7]. For n = 6, 7, Bugaenko has constructed his groups as linear transformation groups over the ring of integers in the field Q(k), that preserve quadratic forms (see Chapter 3, §8):

+l x2+xi 2 -}-...+xn;

2

0

n=6,7.

Figure 27 shows the Coxeter diagrams for these polyhedra as well as for a compact

86

2. Discontinuous Groups of Homeomorphisms

Coxeter polyhedron in 1H18.

Figure 27

The maximal dimension of the hyperbolic space IF for which crystallographic reflection groups are known is 21, see Borcherds [1]. Geometrical constructions of these examples for n < 10 are contained in papers by Makarov [1-7] and, for n < 19, Coxeter diagrams for such groups can be found in papers of Vinberg [1, 2, 4, 6] and Vinberg-Kaplinskaya [1], while for n = 18 see Meyer [1]. Crystallographic reflection groups in H' and groups commensurable with them constitute a very particular case of co-compact and co-finite discrete groups G C IsomIHF'. For example, the arithmetic discrete groups in IHI2n+1 related to noncommutative algebraic extensions over the field Q are never commensurable with crystallographic reflection groups, see Vinberg [1]. Also, for any n, there are infinitely many incommensurable discrete arithmetic co-compact groups in IH"; for example, see the construction in Chapter 3, §8. However, surprising results due to Nikulin [3, 4] (in the arithmetic case) and to Vinberg [9, 10], Prokhorov [1] and Khovanskii [1] (general case), show that there are no discrete crystallographic reflection groups at all for sufficiently large dimensions n (see Chapter 3, §9.4).

§5. Discrete group action on the limit set The goal of this section is to show how the conformal geometry of a discrete group action in the n-sphere S" or in the hyperbolic space H'+1 determines dynamics of

the action of a given discrete group G C Mob(n) = Isom ]H[1+1 on its limit set A(G) S; S" = 8H,+1 This continues the study (in §1) of such dynamics of a convergence group G on its limit set and, more generally, on the whole sphere at infinity and on the set of hyperbolic geodesics, that is on S" X S".

The action of a group G C Mob(n) on S" x S" or (what is the same) on the set of hyperbolic geodesics in IHIn+1 is called ergodic if, for any measurable function cp in S' X S", its invariance with respect to the G-action (which means that

2.5. Discrete group action on the limit set

87

(p(x, y) = co(g(x), g(y)) for all x, y E S' and g E G) implies that (p is constant almost everywhere. Given a discrete group G C Isom IEI"+1, its action on the sphere Sn = 3 fl2+1 splits (up to a zero-measure set) into dissipative and conservative parts. The dissipative part

of the G-action is the union of pairwise disjoint measurable sets, which are freely permuted by elements of G. This is quite analogous to the G-action in 1n+1 or in the discontinuity set 0 (G) C S" (see §2). So, we can split the dissipative part of the G-action on Sn into the discontinuity set S2 (G) of G (if any) whose quotient by G gives a conformal orbifold S2 (G)/G, and the dissipative part (if any) of the G-action on the limit set A (G). The second, conservative (or recurrent) part X of the G-action on Sn, is a subset of the limit set A(G) and may be characterized as follows: given any subset X C .7C of positive measure Mn (x) > 0, there exists a sequence of distinct

elements g' E G such that x fl gk(X) has non-zero measure for all i = 1, 2, ... . Now, in terms of any measure of finite mass on S" (for instance the solid angle measure as viewed from a point inside the hyperbolic space IEIn+1), this partition into dissipative and conservative parts of the G-action on Sn divides the points of S" into

those for which the sum (over the group) of area distortions is finite and those for which it is infinite. We consider these two cases separately. 1.

Infinite solid angle of G-orbit and ergodicity on S" x S". We assume that

the group G C Isom lHIn+1 acts in the Poincare ball model in B"+1 = B"+1 (0, 1) whose sphere at infinity has the spherical metric dx/(1 + Ix I2). Since any hyperbolic isometry g is a composition of a hyperbolic translation (along a geodesic in B1+1 joining antipodal points) and a rotation of the ball B"+1 (see Chapter 1, §3), the linear distortion dg(x) of the spherical metric by g varies in an interval [1/A, A] with

A=,1g>1. We split the set of all discrete groups G C Isom IHIn+I into two classes according to convergence or divergence of the series FgEG A9 An. This condition is known as the

finiteness condition for the solid angle of the G-orbit. Its meaning is clear from the following.

For a fixed x E B"+1 = H'+' and a point y E Bn+t, y

x, let 9(x, y) be

the reciprocal of the hyperbolic area of the sphere passing through y with the center x. Clearly, 0(x, y) is the (density of) solid angle of one point as viewed from the other.

Lemma 2.46. Given a discrete group G C Isom Iln+t the series >gEG )'9 n is divergent if and only if the total solid angle of an orbit, >gEG 0 (x, g(y)), viewed from any point x E H"+1 \G(y) is infinite. Proof. Assuming the upper half-space model of II1+1, we consider a hyperbolic

translation g as a homothety g: x r* x/X with a very small linear derivative 1/.k. Then, for points x and yo lying on the (vertical) x"-axes above the origin, 0 (x, yo) and

9(x, g(yp)) = O(x, yo/X) are approximate in the ratio 1/A". Therefore, in general situation with a fixed x E III1+1 \G(y) and elements g E G with large Ag, the order of

88

2. Discontinuous Groups of Homeomorphisms

the term O(x, g(y)) is in a bounded ratio to 1/X9'. So, our series converge or diverge o simultaneously. Another point of view on this condition uses Green's functions. Namely, let gx (y) be the Green's function of the hyperbolic space with the pole x E lHIn+1 That is, gx (y) is a function that is positive and harmonic in ]H[1+1, symmetric about x and tends to

zero at infinity. Then the integration along a geodesic (y, oo) C H'+1 gives us the exact formula:

Y

gx(Y) =

J 00

O(x, z)dz

Since U(x, y) exponentially tends to zero as the distance d(x, y) increases to oo, 8(x, y) for points the integral is approximately the upper limit, and hence gx(y) with large d(x, y). Therefore, for a torsion-free group G, the convergence of the series EgEG 1/A1 is equivalent to the existence of abounded Green's function gx (5y), X y E M = IH[n+1 / G gx(y) _

gx(h(Y))

(2.46)

hEG

on the hyperbolic manifold M.

This observation is the main idea of the proof of the following theorem on the dynamics of G in Sn x Sn.

Theorem 2.47. Given a discrete group G C Isom ffi[n+1, its action on the set of hyperbolic geodesics (or on S' x Sn) is ergodic if EgoG A8 n = oo. Proof For a point p E 1171+1, let gp be the Poisson measure on Sn = alffn+1 The measure A p (X) of a set X C Sn is defined as the solid angle of X viewed from the point p. It is clear that the measure µp is the spherical average of the measures µq where q ranges over a sphere centered at p, because both these measures, p , and the average one, are invariant with respect to the full rotation group about p. Using the Poisson measure, we can extend any bounded measurable function p on the sphere Sn to a bounded harmonic function h on H1+1. Namely, we define such extension by using the Poisson formula:

h(P) _ f oodµp °

µp)

(2.47)

This extension h is non-constant if (p is non-constant almost everywhere on Sn. To show this, take x, y E Sn as density points of sets where p has values in disjoint intervals. Then, for points p E 11n+1 close to x, µp sees mostly the values of cp lying in the associated interval, and hence h(p) nearly lies in this interval. Similarly, for points q E ffi[n+1 close to y, h(q) nearly lies in the second (disjoint) interval, so

h(P) 0 h(q). We denote by P the averaging operator on functions and measures on H'+1 For a function f , the function P f value P f (x) is the average of f (y) over a hyperbolic -

2.5. Discrete group action on the limit set

89

ball B (x, ri) of a radius i > 0 centered at x. For a measure µ, the measure p P is such that, for the Dirac mass Sx at x E IHtn+l, Sx P is the uniform measure on B(x, 17) of total mass one. Then, in (2.47) notations, (f, AP) = (Pf, µ) when both sides make sense, and the operator P clearly commutes with hyperbolic isometries. Also, due to the above observation that µp is the average of pq over S' (x, r7) the extension h in (2.47) is P-harmonic, that is Ph = h. Now we need the following two lemmas on sequences of measures.

Lemma 2.48. There exists a constant C > 0 such that, for sufficiently large numbers N, the density of the measure _N t Sx P` at y E lpln+l is at least C9 (x, y). Proof. Let r) > 0 be a number fixed for our averaging operator P. Then the sequence of measures Sx P', i = 0, 1, 2.... begins at the point x and spreads out symmetrically by steps of length at most a7. This sequence converges to oo in the sense that, for large enough i E N, almost all the mass is eventually outside any fixed sphere S centered at x E pJn+l. This fact follows from the observation that any ball B(z, 77) centered at a point z E S has more than half of its volume outside S. Thus, for any spherical shell S,i = B (x , R) \ B (x , R - )7) of width 77 centered at x, all the mass of the measure S, P' passes through it for i E oo. Since the density function of the measure Y°_1 811P, depends only on the distance d(x, y) and decreases as this distance increases to oo, the desired inequality for it follows from the fact that this density puts at least mass 1 in each shell S., Y E S,7, whose volume is proportional to 1/9(x, y). To formulate the second, key lemma, we define the space T as the quotient (ln+1 x

Sn)/G by the diagonal action of G. Then, using the averaging operator P in each H'+'-level, we can provide the space T with an averaging operator P on measures and functions. Also, using the family of Poisson measures µp on factors {p} x Sn and the natural measure on 1p1n+l, we define a G-invariant volume element dm' on IHIn+t x Sn, and hence a natural smoothdm = dm'/G on the space T. In this notation, let gp be the image measure resting on a sphere of T. So we have the following key lemma: Lemma 2.49. For N increasing to oo, the density of the measure Y_Nt i pPi relative to dm converges to 0o at almost all points of T.

Proof. We take a G-orbit (y, s), (g(y), g(s)), ... in IH[n+l x Sn, compute the density of the measure VN = FN 1µp (P x i d)t in Hn+1 X Sn relative to dm' and add these densities up along the G-orbit. Let d v be the natural hyperbolic volume element and g, (y) the density of the Green's measure E°_16, Pi relative to dv. In terms of the (non-G-invariant) product measure dv x up on 1H[n+1 x Sn, the limit measure dimN.c,,, VN has density gp (h) at each point (y, s) E IHI1+1 x S1. Rewriting this density in terms of the G-invariant measure dm', one introduces the Radon-Nikodim factor (d,sp/dlzy)(s) at each point

90

2. Discontinuous Groups of Homeomorphisms

(y, s), and the desired density takes the form

E gp(h(y))d1h(y) (h (s))

.

(2.48)

hEG

Since g, (y) = gy (x) and h-1 transforms (p, h(s), h(y)) into (h-1 (p), s, y), we can rewrite the density in (2.48) as follows:

gy(h(p))dAh(p) (s). dtty heG

(2.49)

To finish the proof, we shall show that this density is infinite at almost all s E S'. To do that we will use the fact that EhEG gy (h (p)) = oo, which follows from Lemmas 2.46 and 2.48.

Let B = B(p, r) C 1H["+1 be a sufficiently small ball, so that its G-orbit G(B) is the union of disjoint balls g(B), g E G. Also, let irB be a G-invariant function whose value rrB (y) at y E IHI1+1 is the probability that a random walk starting at y with P as the transition operator hits the orbit G(B). We claim that nB (y) = 1, that is almost all paths starting at y hit G (B). Namely, since TCB (y) < 1 and PzrB < JrB, the G-invariant function rrB - PJrB would be greater than s > 0 on the G-orbit G(B') of some smaller ball B' C B, if 'rB is not identically 1. Using (2.47) and the property

(f, AP) = (Pf, s) of the averaging operator P, we write the identity:

(rtB - PnB , Sy + ByP + ... + 8yPN) = (7rB, Sy) - (7rB, SyPN+t)

(2.50)

Since its left-hand side is at least s EheG gy (h (p)) as N tends to oo, and the righthand side is uniformly bounded with respect to N, we have that 7rB = PJrB. This shows that 7rB (y) = 1.

We shall apply this fact to show that, for almost all so E S', almost all paths starting at y E 1H["+1 and ending at so must also hit the orbit G(B). Indeed, for a fixed xo, the function awx (so) considered as a function of x, is P-harmonic due to the 0 above property of a measure FGp to be the average of µq, q E S(p, r). The boundary values of this function are zero at S"\{so} and +oo at so. So, using a standard limiting procedure (see Kemeny-Knapp-Snell [1]), one sees that gx (y) dA (so) is the corresponding Green's density for the random walk constrained so that its limit at S" is so. Then, due to nB (y) = 1, Fubini's theorem shows that, for almost all so E S", almost all paths (y, so) hit G(B). Finally, let us suppose that, for some so E S" and K > 0, gy(h(p)) hEG

dµh(p) (so)

< K.

dAy

Then this inequality holds for all points p in a small ball B C Hn+1, and hence the constrained Green's measure of the orbit G(B) is finite. This contradicts the fact

2.5. Discrete group action on the limit set

91

that almost all conditional paths starting from any point hit G(B), and thus hit this orbit G(B) infinitely often by the Markov property. This completes the proof of Lemma 2.49.

Now, continuing the proof of Theorem 2.47, we suppose that q. (x, y) is a Ginvariant non-constant characteristic function on the set S" x Sn of hyperbolic geo= oo. Due to Fubini theorem, we can suppose that (after desics, and >JgeG X9'

interchanging x and y if necessary), for a set of points y of positive measure, q0(x, y) is non-constant a.e. function of x. Extending each such function cp (x, y) on Sx x {y} to a P-harmonic function (see (2.47)),

h(p, y) =

f

P(x, y)dl-tp(x)

on the level lHIn+1 x (y), we obtain a function h(x, y) on the space T because of the G-invariantness of cp(x, y). Since this function is P-harmonic on each level, Ph = h. On the other hand, h is non-constant on a set of levels of positive measure because (p(x, y) has such a property. Let h+ and h - be the positive and negative parts of the function h - 2 . Then, adding the inequality P I h - -11 > I P (h - 1) I = I h - 11 to the equality F (h - 1) = h 1,

2we have that Ph+ > h+. Moreover, using the arguments in the proof of Lemma 2.49 2

2

2

2

that 7rB (y) = 1 (see (2.50)) and Lemma 2.49 itself, we obtain that Ph+ = h+ almost everywhere. The latter fact implies that the subsets of levels where h+ is zero cannot be connected by action of P with their complements. This contradiction completes the proof of Theorem 2.47.

2. Conical limit points and ergodicity on S" x S". Applying Lemma 2.2 to the Mobius action of a discrete group G C Isom HI in the Poincare ball-model of lHln, we see that limit points z E A(G) are approximated by a G-orbit, G(x), of an arbitrary fixed point x E Sn. Generally, this approximation is not uniform, but one can distinguish a class of limit points having a good approximation. Definition. A limit point z E A(G) c BIHI" is called a conical (approximation) limit point if, for some geodesic ray 8 C IHIn tending to z, and for some point x E IHI", there exists a sequence {gj } C G of distinct elements such that the orbit {gj (x)} approximates z in a bounded neighborhood of the ray P. It is clear that this definition does not depend on a choice of f and x.

Theorem 2.50. Let a discrete group G C Isom IHIn act in the unit ball B, B = Bn (0, 1), as a Mobius group, and z be its limit point. Then the following statements are equivalent:

_

(i) z is a conical limit point. (ii) For some (and hence every) point x E S2 (G) C JRn, there exist a constant K > 0 and a sequence {gi } C G of distinct elements with isometric spheres

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2. Discontinuous Groups of Homeomorphisms

I (gi) having radii ri such that

Igm(x) - zI < Kr? . (iii) For some (and hence every) point x E llln, x # z, there exist a constant K > 0 and a sequence {gi } C G of distinct elements, such that Igi(x) - g1(z)l 2 K.

(iv) There exists a sequence {gi } C G of distinct elements such that distances Igi (x) - gi (z) I are uniformly bounded from zero on compacts in l1.' \{z}. (v) There exist a sequence {gi } C G of distinct elements and two distinct points z', z" E a B, z' o z", such that lim gi (z) = z' and lim gi (x) = z" where the latter limit is uniform on compacts in R \{z}. (vi) For a given hyperbolic geodesic ,B C B ending at z, there exists a compact C C B and a sequence {gi } C G of distinct elements with gi (1B) fl C # 0.

Proof Due to (1.27), for any transformation g E Mob(n), g(oo) 0 oo, and any points x, y V too, g-1(oo)}, we have

r2lx

- yl

Ig(x) - g(y)I = Ix - g-1(oo)1 ly - g-1(00)1

(2.51)

where r is the radius of the isometric sphere I (g). This shows that, for any given points x, y r= SZ (G), there are positive constants K1 and K2 such that

K1r2 < I g(x) - g(y)I < K2r2

(2.52)

for all but finite number of g E G. Therefore, if (ii) holds for some x E 0 (G) then it holds for any other y E 2 (G). Now, assuming (ii) and taking in (2.51) y = z and g = gi 1, we see that for any

fixed point x, x # z, there exist a constant K > 0 and a sequence {gi } C G (of inverse elements of the sequence in (ii)) such that I gi (x) - gi (z) I >- K, i.e., (iii) holds.

Another application of (2.51) shows that the bound in (iii) for some x, x ; z, implies a similar uniform bound for all points x of a compact in IRn\{z}, therefore (iv) holds. Choosing a subsequence of {gi } and applying Corollary 2.1, we immediately have an equivalence of statements (iv) and (v). Now, for the sequence {gi} C G in (iv) and for a hyperbolic geodesic 48 C B joining points x, z E aB, the statement (iv) on a uniform positive lower bound for I gi (x) - gi (z) I implies that geodesics gi (0) intersect some compact Euclidean ball Bn(0, r), 0 < r < 1. It shows that (vi) holds for C = Bn(0, r) C B.

Increasing the compact C in (vi) to Co = {x E B : dhyp(0, x) < do} for some do > 0, we obtain for any x E Co that dhyp (x , gi (48 )) < 2do because gi (48) f l C # 0. In

2.5. Discrete group action on the limit set

93

other words, the sequence gl 1(x) approximates z in 2do-neighborhood of the geodesic so z is a conical limit point. Finally, if an orbit ai = gi 1 (0) approximates z in a bounded K-neighborhood of the geodesic f ending at z and passing through 0, we can write gi = Ui g,', where Ui E O(n) and isometries gai have the canonical form (1.29). In particular, radii ri of the isometric spheres I (gi) of these transformations gi are such that r? = (1- jai 12)/ l at I2 where I ai I tends to 1 and dhyp (ai, a) < K. Triangle inequality and (1.24) then show that I z - ai I < K'- (1- I ai 12) / I ai 12 = K'r? for some constant K' > 0. This completes the proof of (ii) and of the whole theorem. o

In the case when G acts in the upper half-space W+, and oo E 0 (G), one can easily apply the statement (vi) of Theorem 2.50 to describe conical limit points z E A (G) c aR+. Namely, taking the geodesic P to be the vertical ray (z, oo), we have that z is a conical limit point if and only if there exist a constant K > 0 and a sequence {g1 } C G such that

Ig(oo) - g(z)I > K.

(2.53)

Secondly, we note that all loxodromic fixed points z E A(G) of a discrete group G C Isom H' are conical limit points. This follows from (vi) in Theorem 2.50 where the sequence {gi } consists of powers of a loxodromic element g E G, g(z) = z, and is the axis of g. Also, conical limit points cannot be parabolic fixed points. It is obvious for cusp points because there exists a precisely Gp-invariant horoball centered at a cusp point

p, where the group G action is the Gp-action, and hence there is no G-orbit approximating p along a geodesic ending at p, see Figure 28. For a noncusp parabolic point, one derives this fact by considering Margulis domain at that point, see SusskindSwamp [ 1 ]. This dichotomy between parabolical and conical limit points holds even in more general situation of noncompact symmetric rank one spaces, see Apanasov-Xie [1, 3].

C Figure 28

Lemma 2.51. Let P C IHI'2 be a fundamental polyhedron of a discrete group G C Isom En. Then its boundary at infinity does not contain approximation points.

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2. Discontinuous Groups of Homeomorphisms

Proof Let us use the Poincare ball-model of }IP and consider a geodesic ray g connecting a boundary limit point z ET fl A (G) = P* and an inner point xo E int A, Due to the local finiteness condition (F-3) on P (see §3), Euclidean diameters of any infinite sequence gi (P) must tend to 0. In particular, we have this property for any sequence {gi } having lim gi (z) = z', so lim gi (fi) exists and equals z'. It contradicts 0 to (v) in Theorem 2.50 and proves that z cannot be a conical limit point.

Now we shall describe the ergodicity of a G-action on S" x S" in terms of G-action on the set of conical limit points.

Lemma 2.52. Let G C Mob(n) be a discrete group and Y C S" a G-invariant set. Then this set Y has full measure, m" (S"\Y) = 0, if its density point is a conical limit point of G. Proof. Let XY be the characteristic function of Y and {gi) C G a sequence of distinct elements such that, for p E IH["+t, {gi(p)} conically approximates a density point so E Y. For the Poisson measure gp, we have

mn(Y) = AP(Y) = f XYdgp =

J

giXYdgp =

J

Xvdggi(P)

Then the right-hand side tends to the full measure because gi (p) approximate the density point so in a cone.

Corollary 2.53. If the set Y C A (G) of conical limit points of a discrete group G C Mob(n) has a positive measure then Y has the full measure.

Lemma 2.54. The action of a discrete group G C MSb(n) on S" X S" is dissipative if the set Y C A(G) of conical limit points has zero measure.

Proof For p E H"+1 and each point q E G(p), we define a set Dq C Sn X Sn consisting of pairs of points on S" such that the connecting hyperbolic geodesic is

closer to q than to any other point of the orbit G(p). Obviously, D. fl Dqf = 0 for q # q'. At the same time, for almost all pairs (s, s') E S" X Sn, we can find such a closed point q E G(p) by swelling up a geodesic tubular neighborhood of the geodesic (s, s')H until it first meets the orbit. The existence of such a "first time point" q E G(p) follows from mn(Y) = 0. On the other hand, the set of pairs (s, s') not having unique "first time points" in the tubular neighborhood of (s, s')H is the countable union of submanifolds in S" x S" of lower dimensions. Hence, up to a zero measure set in S" X Sn, such first point q E G(p) is unique, and the family of sets Q Dq, q E G (p), provides the desired partition of S" x S", where G acts freely.

Corollary 2.55. The set Y C A (G) of conical limit points of a discrete group G C Mob(n) has the full measure on S" if G has infinite solid angle, EgEG A9 n = oo. Proof. It immediately follows from Theorem 2.47 that G acts ergodically on S" X S,. So Lemma 2.54 completes the proof. 0 The above facts imply:

2.5. Discrete group action on the limit set

95

Corollary 2.56. The set of all conical limit points of a discrete group G C Mob(n) has either zero or full measure. In the first case, G-action on S" x S" is dissipative. In the second case, it is ergodic. Lemma 2.57. Let G C Mob(n) be a discrete group whose set Y of conical limit points has a positive measure. Then EgEG Aan = co. Proof. Let a (s) be the minimal angle of a cone in H1+1 with vertex s E Sn that contains

a G-orbit {gi (p)} approximating s. This is a positive measurable function on Sn. Furthermore, since mn(Sn\Y) = 0 due to Corollary 2.53, one has that a(s) > a > 0 on some subset Y0 C Y of positive measure. The angle a > 0 determines the radius of a ball B = B(p, r) whose orbit approximates points s e Yo in cones of the angle a. So if the total solid angle of the orbit of B were finite, we could cast out finitely many balls gi (B) so the remaining solid angle would be arbitrarily small. This shows that the orbit viewed from p with arbitrarily small solid angle approximates the set Yo of a positive measure, in cones with angle a > 0. This contradiction completes the proof.

Theorem 2.58. For a discrete group G C Mob(n), the following assertions are equivalent: 1.

EgEG)`sn=oo.

2. The set of conical limit points of G has the full measure on Sn. 3. G-action on Sn X Sn is ergodic.

Proof. It follows from Lemma 2.57 ((2)x(l)), Theorem 2.47 ((l) .(3)) and Lemma 2.54 ((3)=(2)).

3. Horospherical limit points and recurrence. In order to study dynamics of a discrete group G C MSb(n) on S', we want to define new classes of limit points for G, in addition to the classes of parabolic fixed points and conical limit points. A points s E A (G) is called horospherical limit point if, for some (any) point

y E H'+', the G-orbit G(y) intersects any horoball centered at s. We denote the subset of horospherical limit points by Ah (G) C A (G). Obviously, Ah (G) contains all conical limit points of G. On the other hand, from the construction of Kleinian groups G C Mob(n), n > 3, in the proof of Theorem 3.14, one can easily see that there are (infinitely generated) discrete groups G C Mob(n) whose parabolic fixed points simultaneously are horospherical limit points of G, see also Waterman[3]. We however remark that even such "bad" parabolic fixed points cannot be conical limit points of the group G. This observation is due to A. Starkov's [2] dynamical arguments, see also Apanasov-Xie [3] and Tukia [19]. Ifs E Sn is not a horospherical limit point then, for a fixed orbit G(y), y E lHIn+1 we can increase the size of horoballs B, centered at s (that is, swell up horospheres

S, = aBs) until some of them hits G(y), B,, fl G(y) 0 0. In the case when there is a unique first hit q, {q} = B, fl G(y), we call this point q the "closest orbit point to s" and denote the set of points s E Sn which have a closest orbit point by Ocl(G). It

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2. Discontinuous Groups of Homeomorphisms

is clear that, up to countably many lower dimensional submanifolds of the sphere Sn (i.e. up to a set of zero measure), the set O(i (G) contains the discontinuity set SZ (G). Throwing out the mentioned above countably many lower dimensional submanifolds of S" which are bases of horospheres with more than one point of the reference

orbit G(y), Y E i}fi+1, and taking the complement in the remaining set to the set Ah(G) of horospherical limit points, we obtain the set AG C A(G) of so-called "Garnett points." A Garnett limit point z E AG has a critical horosphere SZ so that the closed horoball bounded by SS has no points from the orbit G(y), but every larger horoball Bz based at z contains infinitely many orbit points.

Lemma 2.59. The set AG of Garnett points of a discrete group G C Mob(n) has zero n-measure. Proof Assuming m1z (AG) > 0, we can apply the Luzin theorem (see Natanson [11) to find a subset X C AG of positive measure where the radius of the critical horosphere SZ is a continuous function (we assume that H'1+1 =- DK 1). Let zo be a Lebesgue density point of the set X and So its critical horosphere of radius ro > 0. Then there are infinitely many points qi E G (y) outside So, which converge to zo but enter every larger horosphere S... Since points of the set X close to zo are bases for horospheres whose radii are approximately ro (due to continuity) and which contain no points from the orbit G (y), each point qi has an associated "forbidden region" Fi C S" which does not contain points from X. One calculates easily that mn (F,) is a definite proportion

of the measure mn(Bi), m,z(F1) > K mn(Bi), K > 0, where Bi C iR' = S'\{oc} is the smallest ball centered at zo containing the vertical projection of qi E R'+1 This contradicts the fact that zo is a density point of X. -

Theorem 2.60. Let G C MSb(n) be a discrete group and G(y) the G-orbit of an arbitrary pointy E Hn+1 Then the union Ah (G) U Oci (G) of the horospherical limit points and of the points in the sphere Sn which have a closest point form a set of the full n-measure. Furthermore, this dichotomy is precisely the partition of the group G action on Sn into its conservative (= Ah (G)) and dissipative (= Oc1(G)) parts.

Proof. The first part of the theorem on mn (S"\ (Ah(G) U Oci(G))) = 0 directly follows from the claim in Lemma 2.59 that mn (A G) = 0. As to the second claim, we associate to any point q E G (y) a set Aq consisting of those points z E Sn for which q is closest in the above sense. It is clear that the sets Aq are freely permuted by elements of the group G. Hence, the union U{Aq : q E G(y)} is a dissipative set. On the other hand, the set Ah (G) of horospherical limit points may be characterized as the set of limit points z E A(G) such that supgEG Ig'(z)I = oo. This implies that

there is no dissipative part in Ah(G) because, for almost all dissipative orbits G(x), the sums of Jacobians are finite:

E det Dg(x) < oo. gEG

2.5. Discrete group action on the limit set

97

The convergence of this series follows from Lemma 5.15 (the union of volumes is bounded). We see from the above proof that the notion of Dirichlet fundamental polyhedra Dq (G), q e G (y), has been extended for the group G action on the set 0,1 (G) of points with a closest orbit point. It implies:

Corollary 2.61. The dissipative part of the action of a discrete hyperbolic group G C Isom Hn on the sphere alHln is precisely the set UgEG g(alHln fl Dq (G)).

Corollary 2.62. The action of a discrete group G E Mob(n) on the n-sphere Sn is conservative if a fundamental Dirichlet polyhedron Dq (G), q E jn+l, intersects Sn along a zero measure set. We may reformulate these facts in terms of the growth of hyperbolic volume of the quotient space M = IH[n+l / G. Namely, in the hyperbolic orbifold M, we consider

a subset M(r), r > 0, of those points whose distance from a fixed point xo E M is at most r. If G is trivial, M is the whole hyperbolic space 1H[1+1, and M(r) is a hyperbolic ball D(r) of radius r > 0. Theorem 2.63. For a discrete group G C Isom H+ 1, the following assertions are equivalent: (1) The group G action on the sphere Sn = aH'+1 is conservative;

_

(2) The horospherical limit set A12 (G) has full measure on Sn; (3) For a convex fundamental polyhedron P C ]Htn+1 of G, mn, (P fl Sn) = 0;

=0 (4) ]lmr volM(r) vol D(r)

Proof. We have already obtained the equivalence of the first three assertions. Now we shall show how the last assertion follows from (3). The converse is also simple and we will check this in Chapter 7, §2.4, see (7.29)-(7.30).

Let co(r) be the solid angle of the intersection S'(q, r) fl P, where S"(q, r) is a sphere centered at a fixed point q E P with a radius r > 0. Due to convexity of P, the angle w(r) decreases as r increases to infinity. It follows from (3) that limr.oo co (r) = 0. Denoting a (r) the hyperbolic n-measure of the sphere Sn (q, r), we obtain: 0 a(r)w(r)dr volM(r) (2.54) vol D(r) 0 a(r)dr On the other hand, for any s > 0, there is a large enough ro > 0 such that co (r) < e for r > ro. This and (2.54) implies the bound:

fp° a(r)w(r)dr vol M(r) < vol D(r) f0 a(r)dr

+

s

a(r)dr f0 a(r)dr fro

(2.55)

The first term of this bound tends to zero as r increases to infinity. The second one

is less than the chosen arbitrarily small s > 0. This completes the proof that the left-hand side in (2.55) is arbitrarily small as r -+ oo.

2. Discontinuous Groups of Homeomorphisms

98

4. Conformal dynamical systems and Sullivan's microscope. Generally speaking, a conformal dynamical system consists of a countable collection of partially defined conformal transformations in some Riemannian manifold X, closed under composition whenever this makes sense. Usually, the manifold X is either R" or S" = R'2 U {oo}. Although this definition does not formally include (uncountable) flows, those who are interested in ergodicity can look at flows by taking many transversals. Along this way, the flow defines a countable dynamical system. As examples of conformal dynamical systems, we have the following:

(1) Actions of discrete groups G C Mob(n) on the n-sphere S". _ (2) Iterations of complex analytic maps on the extended complex plane C. (3) Foliations of a Riemannian manifold with conformal holonomy.

To study topologically transitive closed sets X C S" invariant with respect to a given dynamical system, that is sets with a dense full orbit, we consider here conformal dynamical systems each of which contains an expanding cover of the closed invariant

set X C S". Such an expanding cover of X is characterized by the property that, for any point x E X, there exists a conformal transformation g in the dynamical system such that its linear distortion Ig'(x)I is more than one, see (1.5). In other words, we require that X be covered by interiors of isometric spheres 1(g) of elements of a given dynamical system. The main observation of D. Sullivan [7] about such dynamical system is that a conformal measure m relative to a topologically transitive expanding cover is ergodic and hence is unique given its exponent and the total mass. Important applications of this fact are for the Julia set of analytic map iterations and for the limit set A (G) of a discrete group G C Mob(n). The most interesting for us is the later case which we consider in details.

The existence of an expanding conformal cover for the limit set A(G) C S" of a discrete group G C Mob(n) implies that there is a finite cover of A(G) by open sets U1, ... , Uk, together with elements gi, ..., gk E G, such that the maps gj: (A(G) fl U1) --* A(G) are expanding. Lemma 2.64. Let G C Mob (n) be a convex co-compact group, that is a geometrically

finite group with no parabolic elements. Then its limit set A(G) has an expanding conformal cover.

Proof We notice first that, due to Theorem 2.50, the limit set A (G) has an expanding conformal cover if it consists of conical limit points. Then the finiteness Theorem 4.16 completes the proof. Our aim here is to show that the existence of an expanding conformal cover for the limit set A (G) C S" forces A (G) to be quasi-self-similar, that is its each small piece can be expanded to a standard size and then mapped into A (G) by a K-quasi-isometry. More precisely, following D. Sullivan [7], we call a set A C S" quasi-self similar if there are uniform constants K and ro such that, for any x E A and a ball B(x, r)

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2.5. Discrete group action on the limit set

centered at x with radius r, 0 < r < ro, there exists a K-quasi-isometric bijection f, A n B(x, r) +A, (2.56) r which distorts distances in the interval between 1/K and K. In other words, the distortion of an unlimited "microscoping" of the set A can be uniformly bounded.

f

Theorem 2.65. A set A C S' having an expanding conformal cover is quasi-selfsimilar.

Proof. We may assume that oo 0 A, so A is a closed compact subset in r with a finite expanding cover. Let ro > 0 be the Lebesgue number of this cover, that is any subset A n B (x, r) of diameter less than ro lies inside of one of finitely many open balls U1, ... , Uk of the expanding cover of A. We can iterate by using a given conformal transformation g until the diameter of the image ball, B(xm, rm), is no longer smaller than ro. Using the chain rule, we see that the total distortion of the iteration of g equals the product of distortions of the transformation g. Going backwards as in Figure 29, we see that each step has a compression, and the total compression can be obtained by using a geometric series:

log(ratio) _

I log(Ig'(xi)1) - log(Ig'(Yi)1)I i

Figure 29

This and the mean value theorem implies

log(ratio) = = Ko (g', g"") E

Ixi - Yi 15 Ko(g', g")

Ixi - YiI

?i Ix - yI < K(g, A),

where the constant Ko(g', g") does not depend on the choice of a point x E A and a radius r, and the constant K(g, A) is universal, too. This completes the proof.

Corollary 2.66. Let G C Mob(n) be a convex co-compact group. Then its limit set A(G) is quasi-self-similar. Proof. It directly follows from Theorem 2.65 and Lemma 2.64.

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Notes The treatment of convergence groups in § 1 mainly follows Gehring-Martin [1]. Theorem 2.6 is due to L. Bieberbach; for its proof, see Chapter 3, § 1. The double density of loxodromic fixed points is originally due to Gotchalk-Hedlund [1]; its present proof by Gehring-Martin [ 1] closely follows Greenberg's [1] proof for Mobius groups. The example of a non-Hausdorff quotient (Remark 2.17) is due to Thurston [1]. Lemma 2.23 generalizes a result of Accola [1] for planar Kleinian groups. Examples 2.19 and 2.20 were constructed in Apanasov-Tetenov [1] and Apanasov [11]. The latter one is the

first example of a web group on the 3-sphere (see Krushkal'-Apanasov-Gusevskii [1, 3]; for the original definition of web groups, see Abikoff [1]). Example 2.21 generalizes Accola [1]. We retain the name B-groups because the first such groups (distinct from quasi-Fuchsian ones) were obtained as boundary groups of Teichmiiller spaces of Fuchsian groups, see Bers [3] and Maskit [2].

The method of fundamental polyhedra goes back to studies by Poincare [1, 2] on Fuchsian and Kleinian groups on the plane. the construction and properties of generalized isometric polyhedra in §3.3 are due to Apanasov [8]. The concept and properties of pseudo-Dirichlet polyhedra are due to Apanasov [31, 34]. It plays a significant role in geometrical finiteness (see Chapter 4). The list of diagrams of reflection groups with unbounded simplices as fundamental polyhedra is taken from Chein [1]. It was produced on the base of a computer program by N. Spiridon; see also Koszul [1]. Conical limit points have been introduced by Hedlund [1] and to great success used by Beardon-Maskit [1], where their equivalent definitions (i)-(iv) and (vi) (in Theorem 2.50) were obtained. Results in §5 on dynamics of discrete hyperbolic group action on the limit set and on the set S" x S" of hyperbolic geodesics are due to D. Sullivan [1], where the reader can find a history of these problems; in particular,

in dimension n = 2, Corollary 2.56 was originally proven by E. Hopf [1]. The "microscoping" machinery in Section 4 of § 5, which is basic for D. Sullivan's results on conformal dynamical systems, was developed in Sullivan [7].

Chapter 3

Basics of Hyperbolic Groups and Manifolds

§1. Margulis's Lemma and splittings of hyperbolic manifolds In the study of hyperbolic groups and manifolds, an essential tool is the Margulis Lemma which describes (roughly speaking) "uniformly thin" parts of hyperbolic manifolds as those submanifolds that have virtually Abelian fundamental groups. More precisely, for a given Riemannian manifold/orbifold M and s > 0, we can define the s-splitting M = M(o,,) U M[,,,,,) of M into c-thin and s-thick parts. Here the s-thin part M(o,,] consists of those points x c M through that there pass homotopically non-trivial (piecewise differentiable) loops y c M with length not exceeding E. By definition, the s-thick part M[,,,,,) is the complement in M to the interior M(o,,) of the s-thin part. Of course, for a compact manifold M, we have Mto,EJ = 0 whenever s is small enough. To examine this splitting, it is natural to consider the universal covering M which, in the case of a complete hyperbolic manifold M, is the hyperbolic space H, and to consider subgroups GE (x), x E H1, of the fundamental group nl (M) = G C Isom H"

generated by all elements g E G for which the hyperbolic distance d(g(x), x) < s. For a fixed x, we define G' (x) c G to be the subgroup of G,.(x) generated by the elements whose derivatives also differ by s from the identity. Then we can formulate Margulis's Lemma describing the s-splitting of a hyperbolic manifold/orbifold M as follows:

Theorem 3.1 (Margulis's Lemma). For all n there exist s = s(n) > 0 and v = v(n) E N such that, for any discrete group G C IsomH" and x E H", the subgroup GE (x) c G is virtually Abelian and its subgroup G' (x) is Abelian of index at most v.

Before we go on to prove this statement, we remark that it can be generalized to the case of Riemannian manifolds X having bounded curvature K, -1 < K < 0. In this generalization, the property of GE (x) to be a virtually Abelian subgroup of a hyperbolic group G must be replaced by virtually nilpotent, see Ballman-Gromov-

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3. Basics of Hyperbolic Groups and Manifolds

Shroeder [1]. The proof of this fact mainly follows the same scheme presented here for the hyperbolic case. As the first step of the proof, we shall identify nilpotent subgroups of isometrics of H' (or 1[81, S'1) as being generated by elements with small "rotational parts". Here a group G is said to be nilpotent if it has a finite series of commutator subgroups

G1 = [G, G] = ([g, h] : g, h E G),

G11+1=[G,Gm], m>1.

(3.1)

A group G is called virtually nilpotent (or Abelian, etc.) if it has a nilpotent (respectively, Abelian, etc.) subgroup of finite index. Now assuming that d (x, y) is the corresponding distance function in either or Hn, let us define

Sn,

Rn

U(S') = {g E Isom S1 : d(x, g(x)) < it/2 for all x E Sn} ,

U(RI) = {g E SimRn : rot(g) E U(Sn-1)},

(3.2)

U(Hn) = {g E IsomHn : rot(g) E U(Sn-1)}, where rot(g) is the rotational part of g defined as follows.

If g is a similarity in R1, g(x) = r V (x) + b, r > 0, b E 1W2, V E 0(n), it acts isometrically on Sn-1 identified with the set S(Rn) of parallel classes of straight rays in Rn, and we have a homomorphism, rot: Sim Rn -a Isom Sn-1, rot(g) = V. In the hyperbolic case, we consider the Poinca d conformal model of Hn in the unit ball B' C R. Any g E Isom En with a fixed point y E Bn induces (after scaling) an isometry of the unit tangent space Ty (Rn) at the point y, hence, an isometry of Sn-t For any other fixed point z E Bn of g, one can check that the induced isometrics in Ty (Rn) and T 1(Rn) are conjugate. The corresponding conjugacy class in Isom Sn-1 is called the rotational part, rot(g), of the isometry g.

Theorem 3.2. Let X be either Sn, R'2 or Hn and U (X) defined as in (3.2). Then, for any nilpotent group G C Isom X, its subgroup generated by G fl U(X) is Abelian.

Furthermore, there exists A = µ(n) such that the index of the Abelian subgroup (G fl U(X)) in G is at most µ.

Proof First, assuming X = Sn(Q, 1) C Rn+1, we can extend the action of g E Isom S" on Rn+1 to its action on 0+1 which preserves the standard Hermitian form (,) on C1+1 Since, for z = x + iy E Cn+1 with x, y E Rn+t, one has Re(z, g(z)) =

(x, g(x)) + (y, g(y)), it follows that g E U(Sn) if and only if Re(z, g(z)) > 0 for each z E (C1+1 \{0}. This fact allows us to prove that a given g E Isom Sn commutes with some y E U(Sn) if g commutes with the commutator [g, y].

Namely, it follows from [g, [g, y]] = 1 that g commutes with y-1gy, so that they have common eigenspaces and are simultaneously diagonalizable. If V is an eigenspace of g different from y (V), which is an eigenspace of y -1 gy, then there exists some other eigenspaceV'ofy-1gy(orthogonal toy(V))with some v E VnV', v 0 0. Since y (v) E y (V) is orthogonal to V, we have (v, y (v)) = 0. On the other

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103

hand, Re(v, y(v)) > 0 because y E U(S'). This contradiction shows that V = y(V) for any eigenspace V of g. Therefore, g and y are simultaneously diagonalizable, and hence commute.

Now suppose that, for given a, b E G fl U(S'), we have a non-trivial element d = [cl, d1], d1 = [c2, ... [c, , cn+1] . ]], such that each ci is either a orb and the "length" of d, the number n, is maximal. Then [a, d] = [b, d] = 1, and we can apply

the above fact for g = di and y = cl. So we have that dl commutes with cl and d = 1. This contradiction shows that any elements a, b E G fl U(S'1) commute, and the group (G fl U(S')) is Abelian.

Finally, we obtain a bound it = µ(n) for the index [G : (G fl Um))] of this Abelian subgroup from the compactness of the group Isom S'. Namely, for an open symmetric neighborhood V (1) C Isom Sn of the identity, there is an upper bound µ(n) of the number of disjoint g(V(1)), g E Isom Sn, embedded in Isom Sn. This completes the proof of the theorem for X = S'. In Euclidean case, X = W2, we can use induction on the dimension because, for a plane R' c Rn, U(1 ') is the intersection of U(1W) with the stabilizer of this plane in Sim Rn. In this case, what we are going to prove is a generalization of the statement of the theorem for nilpotent groups G C Sim Rn:

(*) (G fl u (R,)) is an Abelian subgroup in G C Sim lR' of index at most

N = g(n) To prove this let us define, for any g E Sim 1W', a maximal plane tg C Rn (maybe a single fixed point) where g acts either trivially or by translation:

rg = {x E Rn : d(x, g(x)) is minimal} .

Since, for a non-translation g E Sim Rn, tg is a proper subspace in Rn, we can use induction to observe that, for an Abelian group G C Sim Rn, the plane r (G) = n{rg : g E G} is a non-empty G-invariant plane, and G acts on r(G) by translations. This planer (G) is naturally foliated by the set of minimal G-invariant planes (the affine spans of some G-orbits). This defines a canonical Bieberbach decomposition R' = V1 ® V2 ® V3 where the subspace VI corresponds to leaves of the foliation in r (G), V2 is determined by the orthogonal complement to V1 in r (G) and V3 is the orthogonal complement to r(G) in Rn. We call this decomposition trivial if dim Vi = n for some i. In this situation we have: (i) G is a pure translation group if dim V1 = n; (ii) G is trivial if dim V2 = n; (iii) G has a unique fixed point in 1W' if dim V3 = n. Now, assuming that G C SimRn is nilpotent and G = (G fl U(Rn)), we shall show that G is Abelian. To do that, we consider the canonical decomposition 1W' = V1 ® V2 ® V3 with respect to the center Z(G) of G, which is also respected by G. So G C Sim R'n 1 x Sim 1l m2 x Sim R13 , and its projections on Vi are nilpotent, i = 1, 2, 3. Applying induction on the dimension, we see that it is enough to show

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3. Basics of Hyperbolic Groups and Manifolds

that, for trivial decompositions (i.e., for the case when mi = n for some i), such G-projections are trivial. According to (i)-(iii), we have the following:

(1) Z(G) is a translation group with no non-empty proper invariant plane in R" if ml = n. So, for any g E G, the (Z(G)-invariant) plane rg must be equal to 1R', and g is a translation in R. The fact that translations commute shows that G is Abelian. (2) Z(G) is trivial if m2 = n. Then nilpotency of G implies that Gm+1 = {1}

forsome m,andhence Z(G) has a unique fixed point in lR' if m3 = n. Then G C R+ x Isom Ss-1 where G acts on Sr-1 by rotational parts, and rot G is nilpotent and generated by elements of U(S" ). So, rot G is Abelian, and hence G is Abelian. This completes the proof of (*) because the index of (G fl U (]R")) in G has abound dependent only on n (as in the spherical case). Finally, we shall consider a hyperbolic nilpotent group G acting in the unit ball B = B'(0, 1) C lR' as a Mobius group. First, we observe that G has a common fixed point, i.e., fix(G) = {x E B : g(x) = x, g E G} ,-E 0. Indeed, let o- = fix(Z(G)). As we have seen before, in Chapter 2, the set of points x E B fixed by an Abelian group r C Isom H' consists of either one or two points in 8 B, or is the closure of a hyperbolic subspace in H" (maybe, a point). So or , 0, Go = {g E G : g(x) = x, x E o}

is a normal subgroup of G, and the nilpotent group G/Go acts effectively on a. Furthermore, if o- degenerates to a point, the group G fixes this point, and or = fix(G). If a contains more then one point, we apply induction on dimension and may finally assume that a consists of precisely two points, x, y E 8 B. Assuming that GI Go is non-trivial and considering some g E G, g(x) = y, we see that Z(G) must pointwise

fix the hyperbolic geodesic (x, y) C B. This contradiction with the definition of a = fix(Z(G)) shows that G/ Go = {id) and o- = fix G. Now, in order to show that (G fl U(IHI' )) is Abelian, we consider a fixed point x E B of this nilpotent subgroup of G. If X E B, we have reduced the problem to the spherical case. If X E 8 B, we have reduced it to the case of Euclidean similarities. So, in both cases, we have already seen that this subgroup is Abelian, and its index in G is bounded by some p. = µ(n). This completes the proof of Theorem 3.2. Now, to finish the proof of Margulis's Lemma, we shall show how nilpotent groups arise from discrete isometry groups.

In any Lie group H, the commutator mapping [, ]: H x H -+ H has a zero derivative at the identity (e, e). Therefore, for any h, h' E H sufficiently close to the identity, we have I [h, h'] I < C Ih I I h' j for some constant C and a smooth norm

I * I on H. Thus, one can find a bounded symmetric neighborhood O(H) C H of the identity e such that whenever h, h' E O (H), we have [h, h'] C O (H) and [h, h'] I < I h 1 /2. Let M > 0 be the size of O (H) and m > 0 a lower bound of the norms I g I of non-trivial elements of a discrete subgroup G C H. Then, for any integer k, k > log2(M/m), any k-fold commutator of elements of G fl 0 (H) is trivial.

3.1. Margulis's Lemma and splittings of hyperbolic manifolds

105

Repeatedly applying the identity [ab, c] = [a, [b, c]] [b, c] [a, c], we see that any k-fold commutator in the group (G fl 0 (H)) is trivial. So we have

Lemma 3.3. Let G be a discrete subgroup of a Lie group H and O(H) C H the bounded symmetric neighborhood of the identity defined above. Then (G fl 0 (H)) is nilpotent. Now, dealing with our three geometries in turn, we start with the Jordan Lemma:

Lemma 3.4. Discrete subgroups G C Isom Sn are uniformly virtually Abelian.

To prove this fact, we consider 0 (H) C Isom Sn in Lemma 3.3 so small that it is contained in U(Sn) (see (3.2)), and denote it by Uo(Sn). We may also assume that Uo(Sn) is conjugacy invariant. Then, due to Lemma 3.3 and Theorem 3.2, (G fl Uo(Sn)) is nilpotent and even Abelian, with index [G, (G fl Uo(Sn))] at most the uniform constant v = v(n).

For the Euclidean geometry, we pick up some s > 0, put the neighborhood 0 (H) in Lemma 3.3 equal

O(H) = {g E IsomJl

: d(0, g(0)) < s and rot(g) E Uo(Sn-1}

(3.3)

and define

Uo(R') = {g E IsomRn : rot(g) E Uo(Sn-1}. Now, for a discrete group G C Isom Rn and r > 0, we define Gr = ({g E G fl Uo(R") : d(0, g(0)) < rs})

(3.4)

and denote gr (x) = rx. Then we see that the groups

gr'Grgr=({gEgr1Ggrf) Uo(ll ):d(0,g(0))<s}),

r>0,

belong to 0 (H) in (3.3), and due to Lemma 3.3 and Theorem 3.2, are nilpotent and even Abelian. It shows that (G fl Uo(IRn)) = Ur Gr is Abelian. It is also easily seen that index [G : (G fl U0(IIBn))] is uniformly bounded by the constant v = v(n), and we have the following form of the Bieberbach Theorem (compare Theorem 2.6): Theorem 3.5. Discrete subgroups G C Isom Rn are uniformly virtually Abelian.

Finally, in the basic hyperbolic case, we fix a point x E Hn and define a norm * IIX on Isom ]Hln by the formula:

(3.5) IIgilX =max{d(x,g(x)), IIDg-EIIo(TXEn)}, g EIsom]HIIn, where the latter term measures a distance between the rotational part of g and the identity transformation E. One can easily check that Ilh-1ghllX = II gllh(X), which

means that (3.5) defines a conjugacy invariant. So, for the group H = Isom IH[n and a given x E IH[n, we can define the neighborhood 0 (H) in the form: IE,(x) = {g E Isom1Hl" : IIgIIX < s1},

(3.6)

where 61 > 0 is so that 1,1 (x) C U(IH[n) in (3.2). Now, for a discrete group G C Isom fin, one can rewrite: G' 1 (x) = (G fl I., (x)), and apply Lemma 3.3 and Theorem 3.2 to deduct the following:

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106

Lemma 3.6. Let G C Isom H' be a discrete group and x E 1HI' a fixed point. Then the subgroup G'E1(x) = (G n lel (x)) is Abelian.

We willfinish the proof of Margulis's by finding a small s = s(n) = 1/N, such that the subgroup GE (x) is virtually Abelian and, moreover,

[Ge(n)(x) : (GE(n)(x) n le, (x))] _ [Ge(n)(x) : G,(,)(x) n GE1(x)] < N.

(3.7)

The constant s = s (n) is called the Margulis constant. As the index constant v (n), we take the maximum of the Bieberbach constant in Theorem 3.5 and the number N,

e(n) = 1/N. Also, in (3.6), we have chosen s1 = s(n). Note that for0 < s < s(n), we have the index [G,(x) : GE(x)nG'E(n)(x)] < v(n). So, by intersecting all subgroups g(GE(x) n GE(n)(x))g-1, g E Gs (x), we see that GE(x) contains a normal Abelian subgroup of bounded index, where the bound is independent of G. To prove the statement (3.7) and finish the proof of Theorem 3.1, we shall provide

the following arguments of Thurston [1] about subsets YT of the group G, where Yr = {g E G : g is expressible as a word in elements of Y of length at most r}: Lemma 3.7. Let W be a neighborhood of the identity e E H in a (locally compact) topological group H, and let WW, i E N, be a sequence of symmetric neighborhoods of e such that W1 is compact and (Wi )' C W1 for each i. Then there exists some N E N

such that, for any discrete group G C H,

[(G n WN) : (W n (G n WN))] < A.

(3.8)

To deduct the final statement (3.7) from this Lemma 3.7, it is enough to take

W =IE1(x)andWW _{g EIsomIEln :d(x,g(x))
V-1 V C W, there is an upper bound, say (N - 1), of the number of disjoint right translates V- g of V with g E W1.

For a discrete group G C H, let p be a maximal number of disjoint translates V

ai, i = 1, ... , p, where ai E W1 n (G n WN) and p < N - 1. Denoting

GN = (W n (G n WN)), we claim that {GN ai : i = 1, ... , p} includes a complete set of cosets for GN in (G n WN), so that (3.8) holds. To show this, consider GN h with a given h E (G n WN). Taking gi E G n WN,

suppose that h = jI_1 gi and E > N. Then we see that sets V V. hj, j = 1, ... , N, cannot all be disjoint if hj _ fl( gi E (WN)N C_ W1. Therefore, we can represent h = ally, where a$ E W1 and 0. Sowehave afa-1 E V-1 V C W, a,Ba-1 E GN. Hence GN h = GN (aPa-1)ay = GN h', where h' = ay has a reduced length. Applying induction, we see that there exists h" E W1 such that

GN h = Gn h" and, for some a1, V h" n v ai # 0. Thus h"ai 1 E W, and GN ai = GN h" = GN h, which completes both proofs of Lemma 3.7 and Theorem 3.1.

3.2. Injectivity radius of hyperbolic manifolds

107

§2. Injectivity radius of hyperbolic manifolds Given a point x in a hyperbolic n-manifold M, we define the injectivity radius of M at x as the supremum injm(x) of all positive numbers r such that the r-ball in M, BM (x, r) = {y E M : d(y, x) < r}, centered at x is isometric to the metric r-ball B(r) C IH[n. It is easy to observe that typically a hyperbolic manifold M of infinite volume has no upper bound on its injectivity radius. In particular, it is the case if such a manifold M has a geometrically finite non-cusp end, see Chapters 4 and 5. To see that, one may find a hyperbolic half-space H+ in the universal cover M = H, that does not intersect the convex hull C(A(G)) of the fundamental group

ni (M) = G C Isom if, see § 6. However, due to a corollary of the Canary [3] filling theorem (see Chapter 5, § 4), which generalizes a deep result of W. Thurston [1, Th. 9.5.13], such an upper bound for the injectivity radius does exist for the convex core C (M) of a topologically tame hyperbolic 3-manifold M, that is, for a hyperbolic 3-manifold M (with finitely generated fundamental group G) homeomorphic to the interior of a compact 3-manifold:

Theorem 3.8. Let M be a topologically tame hyperbolic 3-manifold. Then there exists an upper bound K such that injM (x) < K for all points x E C (M) in the convex core of M.

Clearly, if a hyperbolic manifold M has a parabolic end E, the injectivity radius injM (x) tends to zero as the point x goes to the infinity of E. Moreover, the work of C. McMullen [3] suggests that typically, even in the absence of parabolic ends, there is no lower bound on the injectivity radius of a geometrically infinite, topologically tame hyperbolic 3-manifold (see also a concrete construction by Bonahon-Otal [1]). Nevertheless, the purpose of this section is to find a positive uniform lower bound for maxXEM injm(x) which depends only on dimension of M.

Theorem 3.9. There exists a universal constant r = r(n) > 0 dependent only on dimension n, n > 2, and such that any complete hyperbolic n-manifold M allows an isometric embedding of the r-ball B(r) C H. Before proving this theorem, we shall study convergence of sequences of hyperbolic isometries gi E Isom in .

Lemma 3.10. Given a sequence {gi } of hyperbolic isometries gi converging to the identity, there exist its subsequence {gig } and another sequence {y, I C Isom IHI' of conjugations such that g = limj,w yjgig yj-t is a parabolic transformation.

Proof We shall consider three cases in accordance with whether each isometry gi has

(i) one fixed point pi E aHn, or (ii) at least one fixed point pi E 1H[n, or (iii) two different fixed points pi, qi E 81H[n.

3. Basics of Hyperbolic Groups and Manifolds

108

In the first case, we define isometries yj E Isom H' such that yjgiJ y[ 1 act in the half-space I[8+ = IH[" as Euclidean motions, x r-a Ui, (x) + aj, where Uij E O(n - 1),

aj E I[8n-1 C of+, Uij (aa) = aj, Iaa I = 1. Furthermore, the subsequence (gig} is chosen so that the limits limm.,,,, aJ = a E Sn-1 (0, 1) and lime,, U,, = id exist. In the second case (ii), we consider the Poincare ball model of HI in Bn (0, 1) and take wi E Isom IH(n such that wi (pi) = 0. Then, due to compactness of the orthogonal group 0(n), there exist w E 0(n) and a subsequence {gig } such that the orthogonal transformations h1 = wwii gi, wig 1w-1 converge to the identity.

Let rj = let - h1(e1)l and aj = 1-Ti el E B"(0, 1). Then we define conjugating isometries yj as yj = vjwwi, where vj E Mob(B") are hyperbolic translations such that vj(-aj) = 0 and vj(oo) = aj/(1 - rj). These conditions on yj make sure that the sequence (vjhjvJ -- 1 = yi gig yi- 11 converges to a parabolic

transformation g E Mob(B'), g(el) = el, whose isometric spheres I (g) and I (g) are unit spheres centered at (1, q2, ..., q") and (1, -q2, ..., -q"), F-7=2 q? = 1, and are tangent at the fixed point el = (1, 0, ... , 0). In the last case (iii), we can assume that the fixed points pi and qi converge to some points p and q in a1HI" = IR" such that

wi (pi) = oc ,

1

so that there exists a sequence of wi E Isom nn

wi (qi) = 0,

w = lim wi E Isom IHI" .

(3.9)

We use (3.9) to get representations wi gi wi 1(x) = ri Ui (x) + ai ,

(3.10)

where the orthogonal transformations Ui converge to the identity, the positive numbers ri converge to one, and the lengths of the vectors ai E R"\{0} tend to zero. To finish our proof, we consider a subsequence (gi; } such that the corresponding unit vectors ai, /l ail I converge to some a E Sn-1 (0, 1). Then the desired conjugations are performed by transformations yj = wig /lai; I E Mbb(R+).

Lemma 3.11. Given a point xp E ffi[" together with a sequence {gi } C Isom IH[" such that lim gi (xo) = xo, there exists a sequence {mi } of natural numbers such that

limii00 97' = id. Proof. Let us consider the hyperboloid model of Hn and assume that xo = eo = (1, 0, ... , 0). We can represent each gi E Isom 1H[" by a product gi = Ui Vi of Ui E 0(n + 1) and a matrix Vi where Vi corresponds to a hyperbolic rotation and limVi(eo) = eo and lim. 11 V1 II = 1.

Given a vanishing sequence {e1 } of positive numbers, there exists a sequence {Kj} of positive numbers such that, for any orthogonal transformation U, one can find an integer m, I m I < Kj, for which 11 U' - E 11 < sp Now, let ij E N be a number such that

JIIViIIK'-1I<ej,

forall

i>ij.

(3.11)

3.2. Injectivity radius of hyperbolic manifolds

109

Then, for a number i, ij < i < ij+l, we choose mi E Z so that

ImiI <-Kj,

IIUm` -Ell <ej.

(3.12)

Conditions (3.11) and (3.12) directly imply that limi, gm` = id. Proof of Theorem 3.9. Let us suppose the contrary. Then, for any integer m, there exists a discrete torsion-free group G. C Isom ffffn whose any Dirichlet fundamental polyhedron does not contain hyperbolic 1/m-ball. In other words, in the Poincare ball model of the hyperbolic n-space H, for a given integer m and any isometry w E Isom Hn, the centered at the origin Dirichlet polyhedron for the conjugation WG,nw-1 of our group Gm C IsomlH[n does not contain the hyperbolic 1/m-ball B(0, 1/m) centered at the origin:

B(0, 1/m) ¢ D0(WGmw-1), m E N.

(3.13)

This shows that the groups Gm, M E N, contain such elements gm E Gm that d(0, gm (0)) < 2/m. So we have lim n, gm (0) = 0 and, due to Lemma 3.11, we can assume that limm- gm = id. Now, without changing our notations, we transfer all group actions from B' (0, 1) to the half-space hyperbolic model in 1R such that the origin in Bn(0, 1) corresponds to

en = (0, ... , 0, 1) E R. Let hyperbolic isometries Wm E M6b(II8+) = MSb(n - 1) be such that their isometric spheres I (wm) and I (w, 1) pass through en and the centers of I (wm) are fixed by the corresponding transformations gm E Gm. Then wmgmwm1(x) = rmUm(x) +am

(3.14)

where Um is an orthogonal transformation, rm > 0 and am E R n-1 x {0} c R.

Since lim gm = id, we easily get that lim Um = id, lim rm = 1, lim am = 0. Applying Lemma 3.10, we get a sequence {yj } C Mob(n -1) and an integer sequence {m3 } such that the limit transformation, -1 g = lim (yj wmj gm; wmi yj

1

i+00

)

(3.15)

is parabolic. Using the initial assumption (3.13) with w = yj wmj , we find again in the groups

-1 -t GIi = Yi W mj Gmi wmi Yj

(3.16)

some elements vj that converge to the identity and do not have common fixed points with the transformations h j = yj wmi gmi wmj yi- 1 Thus we get a sequence of subgroups (v j , h j) C G. whose generators have no common fixed points, are not elliptic, and converge correspondingly to the identity and the parabolic transformation g in

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3. Basics of Hyperbolic Groups and Manifolds

(3.15). This already contradicts somehow to the discreteness of these groups because h j are close to a translation by a unit vector and isometric spheres I (v j) have arbitrarily large radii (compare Theorem 3.15). However, we can show non-discreteness of groups (v j , h j) with j large enough by using more clear arguments. Note that such arguments simultaneously provide a proof of more general fact formulated in Corollary 3.12. Let S j E Mob(R+) be elements of a sequence converging to a Mobius transformation 3 and such that Sj (en) = en and

Sjvj8J-- 1(x) =rjUj(x)+aj,

(3.17)

where orthogonal transformations Uj converge to the identity, positive numbers rj tend to 1, and translations vectors aj E RI-1 x {0} have zero limit. Such sequence {Sj } exists because of lim vj = id. Assuming that numbers j are sufficiently large, we drop subscripts and denote the conjugate groups Sj (vj, hj)Sj - 1 again by (v, h).

Since v and h have no common fixed points, we see from (3.17) that h is not a similarity. Hence we can apply Corollary 1.8 to get the following representation:

h(x) = pVI(x -q)+q',

(3.18)

where q and q' are the centers of the isometric spheres I (h) and I (h-1) with radii 'fp-, 1(x) = x/x12 and V E O(n). The center q in (3.18) cannot be fixed by our generator v. To prove that, we suppose

the contrary, which means that (a - q) = rU(q). Using this, we can write

h1(x) - hvh-1(x) = r-1VUV-1(x - q) +q'

(3.19)

where r, V, U and q' are from (3.17) and (3.18). Therefore, the element hl of the group (v, h) is a similarity and, due to Lemma 2.10, fixes the second fixed point of v, q. This fact and (3.19) imply that q = q', which means that the transformation h must be elliptic. Impossibility of that completes our proof of the fact that v(q) q. Now we consider a sequence {hi) C (v, h) with elements

hl = hvh-1, ..., hi+1 = hivhi 1, ....

(3.20)

Since v and h are arbitrarily close to the identity and a parabolic transformation, respectively, there exists a constant K > 0 independent of h and v (i.e., K is the same for all sufficiently large subscripts j) such that

d (e, h(en)) < K and d(x, v(x)) < K for all x from the closed hyperbolic ball B (en, K). In particular, we have that d(vh-1(en), h-1(en)) < K, which implies the following bound:

d(h1(en), en) = d(hvh-1(en), en) < K .

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111

Induction on i extends this bound to all elements hi of our sequence in (3.20):

d(hi(e,,),en) < K,

i > 1.

(3.21)

In order to show that our sequence consists of distinct transformations, hi # hj j, we can use Lemma 1.9 to calculate canonical representations of hi in the form (see Corollary 1.8): hi (x) = pi Vi x (x - qi) + qf where pi are squares of radii of the isometric spheres I (hi). In particular, for i = 1, we have: for i

piVtz(x - q1) + qi = hi (x) = hvh-1(x) = px[rpV UV-1 x(x - q') + V(v(q) - q)] + q',

(3.22)

where v(q) - q ; 0, as has been shown. This fact and Lemma 1.9 allow us to find

pi = rp2/iv(q) - q12. In this formula, p > po > 0 because of (3.21), and r is arbitrary close to 1. Also the distance I v (q) - q J 0 0 is arbitrarily small because v is arbitrarily close to the identity and JqI is uniformly bounded from above (q is close to (lim h)-1(oc), the center of the isometric sphere for the limit parabolic transformation). That is why we have p1 > 2p. Using similar arguments and inducting on i > 1 we see that pi+1 > 2pi, which means that isometric spheres I (hi) have strictly increasing sizes. Hence hi # hj for i # j. This fact and the boundedness (3.21) of the orbit hi (en) show that, for j large enough, the groups (vj, hj) are non-discrete. This completes the proof of the theorem.

Simultaneously, we have proven the following fact on algebraic limits of discrete Mobius groups Gi where the limit group is generated by transformations obtained as limits of generators of Gi. Corollary 3.12. Let {G,,, } be a sequence of discrete Mobius groups Gm = (gi,m,

,

gk,m) C Mob(n), n > 2, 2 < k < oo, such that any two generators gi,m and gj,m, 1 < i < j < k, have no common fixed points and are not elliptic. Then none of the generators gi = limmioo gi,J of the algebraic limit G = ($1, ... , gk) is the identity.

§3. Thin cusp submanifolds According to Margulis's Lemma, we have to study discrete Abelian subgroups I' C Isom Hn in order to understand thin submanifolds of hyperbolic n-manifolds. For n > 3, we have the following possibilities: (i) r = Z is cyclic, with either loxodromic or parabolic generator; Zk, k <_ n - 1, is a free Abelian group of rank k and is conjugate to a (ii) I' discrete subgroup r0 C Isom Rn-1; the subgroup 1,0 is generated by (screw)

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3. Basics of Hyperbolic Groups and Manifolds

translations whose translation vectors span a subspace J1 and rotational parts pointwise preserve this k-subspace.

(iii) P - Z x Z,n is a group of hyperbolic translations and rotations around some geodesic in 1FF;

(iv) P is a product of a free Abelian group of rank k, k < n - 1, and a finite Abelian group conjugate to a subgroup of O(n - k - 1) C O(n). (v) P is a finite group conjugate to a finite Abelian subgroup at O(n); if n = 3,

P-Z2xZ2. Obviously, the groups r in the last three cases cannot serve as subgroups G' (x) C G C Isom 1H[n of the fundamental group G = nl (M) of ahyperbolic n-manifold M provided sufficiently small a > 0 (at most the reciprocal of the order of corresponding finite

Abelian subgroups in r). Moreover, taking a > 0 less than the minimal translation length of loxodromic generators in G, we see that sufficiently thin submanifolds in M correspond to parabolic subgroups r c G C Isom 1111". These subgroups are conjugate to discrete subgroups of Now let us consider a torsion-free discrete group r C Isom R` 1 which also acts in the half-space 1R+ bounded by R'-1 as a hyperbolic isometry group (Poincard half-space model of 11In). For any point x E R'-1 and a > 0, there exists a number to = to (x) > 0 such that PE (x) = P for any point x on the geodesic ray a (x , a), Isom1FF-1

a(X, a) _ {(x, xn) E Ilg+ = IEBn-1 X 118.1.: xn > te(x)}

.

It shows, one more time, that the statement of Bieberbach theorem (compare Theorems 2.6 and 3.5) on virtually Abelian discrete subgroups of Isom 1[8n is a particular case of Margulis's Lemma (Theorem 3.1). On the other hand, it is useful to make bounds for te(r). Let us consider three different cases.

(A) F C Isom R' 1 consists of pure translations. Conjugating such group P by a dilation, we can make the minimal translation length equal to 1. Then, using elementary trigonometry on the hyperbolic plane 1EI2 (see Beardon [4], §7.2), for

g(x) =x+a, lal = 1, we have that d(x, g(x)) = 2 Arc sinh(Ix - g(x) I/2 xn g(x)n) = 2 Arc sinh(1/2xn) . (3.23) Denoting by £ - the minimum of lengths of translation vectors (in r), we see from (3.23) that, for a discrete group F C Isom R' of Euclidean translations, the s-thin part of 18TH/ P has the form:

(Hn/ P)(p e] = {X E IIS+ : xn > tc(F)}/ r ,

te(P) = £r/2 sinh(a/2) .

(3.24)

(B) F C Isom RI-1 does not have irrational screw translations. In other words, for any g E P\{id}, there exists an integer m so that g'n is a translation. Moreover, due to Bieberbach Theorem 2.6, there exists a number mr such that, for any g E P, its m-power g' with Im I < mr is a pure translation. Hence, if £r is the minimum of

3.3. Thin cusp submanifolds

113

lengths of translation vectors in IF, (3.24) easily provides an upper bound, t6 (F), for

numbers te(x), x E R'_1)

it/2 sinh(s/2) < te(x) < Qr mr/2 sinh(s/2).

(3.25)

These inequalities show that in the case (B), the s-thin part (IH[.n/ F) (0,,] is the union of the I,-quotient of the horoball (3.24) and the manifold

{x = (x, x,) E R' : te( ) < xn < te(r)}/ r , which is compact. Compactness of the latter manifold is obvious if k = n - 1. If k < n - 1, it follows from the following two facts. Firstly, the translation vectors of elements in IF form a basis in Rk (where F acts co-compactly). Secondly, for points x E Rn-1 with sufficiently large Euclidean distance de(x,Rk) to the I'-invariant subspace Rk, de (x, Rk)2 > 212. (m12, - 1) /4, we have tE (x) = to (r).

(C) r C Isom R"-1 has irrational screw translations. In this case, let us consider a decomposition of I1 ! 1 (similar to the Bieberbach decomposition in the proof of Theorem 3.2), Rn-1 = I8k x Rn-k-1 where Rk is the minimal r-invariant subspace (which is generated by vectors of (screw) translations in r). Then any g E IF acts on Rn-1 (up to a choice of the origin in Rn-k-1) as follows:

g(x) = g(xl, x2) = (Ui(xl) + v, U2(x2)) , where v E Rk is a translation vector, and U1, U2 are orthogonal transformations in Rk and R,-k-1, respectively. This and (3.23) show that, for unit translation vectors v E Rk and U1 = id and for x = (x1, x2, xn) E l[8+7

d(x, g(x)) < 2 Arc sinh( 1 +41x312/2xn) . This inequality implies that, if x = (x 1, x2) E of translation vectors in F, then

Rn-1

(3.26)

andLr is the minimum of lengths

Lr/2 sinh(s/2) < tE CO < fr 1 + 41x3/.tr 12/2 sinh(e/2).

(3.27)

Summarizing all three cases, let us denote a parabolic s-neighborhood of the parabolic cusp point oo (which is the fixed point of a parabolic group I') by

V00 (r, s) = U a'(x, s) = Ix = (x, xn) E R+ : xn > tE(x)}

(3.28)

xERn-I

This domain is the universal covering of an s-thin neighborhood of the cusp end of IH[n/ r, i.e., it serves as a replacement of a horoball (3.24) centered at a parabolic fixed point having a pure translation stabilizer. As it follows from (3.24), (3.25) and (3.27), we have the following inclusions: X E R : x" > Lr1 + 4Ix3/Lr IZ 2 sinh(e/2) C

/ r c V00(r, s)/ r = (H"/ r)(o,el

{xER:xn

>

2 sinh(s/2) }

/ IF .

(3.29)

114

3. Basics of Hyperbolic Groups and Manifolds

Thin ends of oriented 3-manifolds. Since all elements of torsion-free discrete subgroups r C Isom+ 192 are pure translations, one can give a complete easy description of s-thin parts of oriented hyperbolic 3-manifolds M = ]EII3 / G. For sufficiently small s > 0 which, generally speaking, depends of M, they appear as either quotient spaces of horoballs centered at parabolic fixed points p E 8IHl3 by the action of their stabilizers Gp C G (cyclic or free Abelian group of rank two) as in (3.24), or quotient spaces of hyperbolic cylinders Dr(f) = {x E IH[3 : d(x, 2) < r} with geodesic axes £ C 1H13 and radii r > 0 by the action of hyperbolic (screw) translations along £.

In fact, s > 0 having the above property, can be taken as a universal so for all discrete groups G C Isom+ H3: Theorem 3.13. There exists a constant so > 0 such that each component of M(o,,], s < so, of any complete oriented hyperbolic 3-manifold M is either: (1) the quotient of a horoball modulo the action of groups Z or Z x Z, or (2) the quotient of a cylinder Dr (t), r > 0, along a geodesic .£ C IH13 modulo a loxodromic 7G-action.

Proof. Before we go on to prove this statement, let us remark that a given hyperbolic structure on the interior of a compact 3-manifold M of finite topological type cannot

possess arbitrarily short closed geodesics. So one can consider sufficiently small s = s (M) such that s-thin components of M(o,,] consist only of quotients of horoballs

(3.24). However, there exists no universal constant a having this property for all hyperbolic structures (even with the same topological type of M), i.e. a constant that strengthens Margulis constants from Theorem 3.1 and 3.13. This follows from a construction of Bonahon-Otal [1] (see also Thurston [5]) which shows how to construct, for any compact 3-manifold whose interior has a hyperbolic structure of infinite volume, a new hyperbolic structure with arbitrarily short geodesics. We shall prove Theorem 3.13 by direct arguments. For that, let us consider a point x E M(o,EI where s = 83 is the Margulis constant and x E H3 a lift of X. Then there exists some non-trivial g E Ge (i) where rrl (M) = G C Isom H3. If g is loxodromic, we denote its axis by i. All rotations around £, hyperbolic translations along £ and uniform contractions of ]Hl3 along geodesics, orthogonal to t, commute with g. This shows that Mto,ei contains a cylinder Dr (f) where r = d (t , z),

since d(y, g(y)) < s for ally E Dr(t). If g is parabolic with a fixed point p E 81E1l then M(o,E) contains a horoball centered

at p and bounded by a horosphere passing through x. Therefore, M(o,,] is the union of horoballs and solid cylinders Dr (f). Whenever two of these are not disjoint, they correspond to two deck transformations gi, $2 E G8 (x"). Theorem 3.1 implies that gi and $2 must commute, so the corresponding horoballs or solid cylinders must be concentric. The precise invariance of these horoballs and solid cylinders with respect to their stabilizers completes the proof. 0

Thin ends of n-manifolds, n > 4. In dimension 4 already, the properties of s-thin parabolic ends of hyperbolic manifolds change drastically in both topological and

3.3. Thin cusp submanifolds

115

geometrical senses. A topological difference comes from a discovery by D. Ivansic [1, 3] that there are finite volume ends of such manifolds that are non-embeddable into compact 4-manifolds as complements of their submanifolds, see Chapter 6, §3. To see a geometric difference, we observe that in the proof of Theorem 3.13, we have made a substantial use of the fact that any discrete group G C Isom IEIl3 has precisely Gp-invariant horoballs Bp C 1H13 centered at its parabolic points p E 81EII3, i.e., such horoballs Bp that:

y(Bp) = Bp for all y E Gp

and g(Bp) fl Bp = 0 for all g E G\Gp.

(3.30)

The next theorem shows that this fails in dimension more than three.

Theorem 3.14. For any n > 4, there exist oriented hyperbolic n-manifolds M such that the universal coverings of any of their s-thin parabolic ends (with arbitrarily small s) do not contain any horoball. Proof. We shall give a direct construction of a discrete group G C Isom IH[4 acting in IlB+, having oo as a parabolic fixed point and such that any horoball {x E R+ : X4 > t} t E R+, is not precisely Gam-invariant. The extension of this group actions to Eon, n > 4, provide similar group actions in high dimensions. The group G C Isom 1Ell4 that we are going to construct is in fact a Kleinian group on 3-sphere ]R3, G = (Go, h), whose parabolic generator, h(x) = U(x) + el, has an orthogonal transformation U E 0(3) of infinite order as its rotational part, U (e t) = et, and whose other generators (infinitely many hyperbolic translations gi E MOb(I18+), i E N) generate a free (Schottky) subgroup Go and have the isometric spheres

I(gi) = {x E R4 : Ig(x)I = 1} = S3(zi,me),

I(g- t) = S3(-zi,me),

(3.31)

of arbitrarily large radii mi. So large isometric spheres I(gi) and I(gi t) intersect (for sufficiently large i >_ ir) any horoball Ba(r) = {x E ](8+ : X4 > r} centered at the parabolic fixed point oo. Hence gi (Bw(r)) fl Ba(r) # 0 for i > it , and Ba(r) cannot be precisely invariant for the parabolic stabilizer G. = (h) of oo. In order to find such generators gi, let us fix some increasing sequence of integers mi and consider a minimal angle ti between one of the vectors e2 and -e2 and vectors from the orbit {Uk (e2), 0 < k < 2mi },

ti = min

O
{I arccos(e2, Uk(e2))I , I arccos(-e2, Uk(e2))I } .

(3.32)

Then we set 2i = mi / sin(ti /2) and zi = Lie2 so that, due to lim ti = 0, we have for all i > 1 (up to a subsequence):

1/2 < ti - mi < 2i + mi < Li+1 - mi+t Now, considering balls

Bo = B(0, 1/2) ,

Bi = B(zi, mi),

B' = B(-zi, mi),

(3.33)

116

3. Basics of Hyperbolic Groups and Manifolds

bounded by disjoint spheres So, Si and Sill we define hyperbolic translations gi so that

I (gi) = Si and I (gi 1) = Sl . Due to (3.33), for any element g of our free subgroup Go, there exists a number io such that g (n (ext Si fl ext S')) C Bio U B a .

(3.34)

To complete the proof, we only need to show that group G = (Go, h) is discrete. We will prove even more, namely, that G is a Kleinian group in Il83 = a1E[4, i.e. it has non-empty discontinuity set SZ (G). This will immediately follow from the fact that

g*(Bo) fl Bo = 0 for all

g* E G\{id} .

(3.35)

To prove (3.35), note that for each non-zero integer m and all i,

Bo fl hm (Bo) = 0 .

(B1 U B1) fl hm (Bi U B,) = 0 ,

(3.36)

For Bp, this is true because h is a translation by a unit vector. For i > 1 and any integer in, I m I > 2mi, balls BI and B, of diameters 2mi do not intersect their images under translation h' by the vector met of length at least 2mi. For any other non-zero integer m, I m I < 2mi, (3.36) follows from the definition (3.32) of the numbers ti. In fact, the choice of the numbers ti and £i ensures that the sectors Vi = {tx : x E B1, t > 0) and Vi '= {tx : x E BZ, t > 0) do not intersect the setU{Um(Vi) : m E Z\(O), ImI < Mil. For all numbers i and m # 0 and the set D = U{hm (Ui Bi U BZ U Bo) : m E Z\{0}}, we immediately get from (3.36) that hm (Bi U BI) C D

,

hm (Bo) C D

,

(Bi U B;) fl D = 0

.

(3.37)

At the same time, (3.34) and (3.36) imply that, for any g E Go\{id}, there exists an integer io such that g(Bo U D) C Bio U B o.

(3.38)

Now, let us write any element g* E G\{id} as a word of the form

g* =

91hn1

, ... , hnkgk+l ,

(3.39)

where elements gj E Go are non-trivial if j gb 1 or j # k = 1 and n.j ; 0 for all

j

Then (3.37), (3.38) and induction on the length of words (3.39) finish the proof of (3.35) and complete our construction. .

3.4. Precisely invariant horoballs

117

§4. Precisely invariant horoballs As we saw in the previous section, for a given discrete group G C Isom H" with parabolic fixed points p E aH1, it is important to know its action in small horoballs Bp C H1 centered at p. It is obvious that such G-action in a domain D is the action of its stabilizer subgroup

H C G if D is precisely invariant with respect to this stabilizer H, i.e.

y(D)=D for y E H and g(D) fl D= o for g E G\H .

(3.40)

Conjugating G CMob(I[8+) so that its parabolic fixed point p becomes oo, we have that all horoballs Bp centered at p = oc have the form of half-spaces parametrized by a > 0, Ha = {x E 1I : x, > a}. To find precisely Gm-invariant horoballs Ha (in contrast to Theorem 3.14), one can apply the following:

Theorem 3.15. If a discrete group G C Mob(n), n > 2, with a parabolic stabilizer G,,,, C G either contains a Euclidean translation or its limit set A(G)\{oo) lies a finite distance apart a minimal G,, -invariant subspace E C IRn, then there exists ao > 0 such that the horoballs Ha C ][8+ tare precisely Gam-invariant for all a > ao. Moreover, if G contains a Euclidean translation by a vector a E R1, then ao < Ial. Proof. Let initially G,,o contain a Euclidean translation, h (x) = x +a, by a vector a E R1. Considering the group AGA-1, A(x) = x/lad, we have a Euclidean translation in AGA-1 by a unit-length vector a / l a I where la I is the ratio of radii of isometric spheres

1(g), g E G\GOO, and I (AgA-1). Therefore, in order to prove the statement, it is enough to prove that radii of isometric spheres 1(g) do not exceed one if G contains a translation by a unit-length vector a E IR'. Let us suppose the contrary, which means that there exists g E G\GOO such that its isometric sphere I (g) has radius greater than one. Then, conjugating G by aEuclidean translation, we can assume that I (g-1) = Sn-1(0, r), r > 1, and define the following

sequence of elements gi E G\G.:

go = g,

gi+l = gthgi

1

.

(3.41)

Due to Corollary 1.8, g1 (x) = r? U,: Z (x - q1) + q where Ui are orthogonal transformations, 1(x) = x/fix 12, and r1 > 0 are radii of isometric spheres I (gi) and

I (g-1) centered at ql and qc, respectively. Having r = ro, U = Uo, q = qo, and q' = qo = 0 and applying Lemma 1.9, we see that (3.42) gl(x) = ghg-1(x) = r2UI[r2U-1x(x) +a} = r2U(r2Ra,Z"(U-1(x) + r2a) + a) = r4URaU-lz(x + r2U(a)) + r2U(a).

where Ra is the reflection with respect to the subspace P (a, 0) orthogonal to a. Therefore,

q1 = -r2U(a)

,

ql =r 2U(a), rl = r2

,

U1 = URaU-1 .

(3.43)

3. Basics of Hyperbolic Groups and Manifolds

118

Similarly to (3.42), we get for i > 2 that 1

gi(x) = gi-Ihgi-1(x) =

r12

Ui.1'(x - qi)

+qi/

=r41U1RaUi i (x-qi-1+r21Ui-1(a))+q;_1+r? i-2

i-1

j=0

j=0

iUi_1(a)=...

MIX - E r2i+i Uj(a) + r2` U1_1 (a)] + E r2i+1 Uj (a) .

=

(3.44)

Then formulae analogous to (3.43) follow: qi

=

i-2

r2i+'Uj(a) -r2,Ui-1(a),

=r2, ri

,

j=00

Ua qi=Ei() j i-1 r2i+1

U1 =

I

JJ( U(

2

R

1 UR R a=[, a)! a] 2`-' a (3.45)

L j=0

12

J

Here Ui are reflections with respect to subspaces P(U1-1(a), 0) orthogonal to Ui-1(a). Hence, due to compactness of O(n), we can assume that lim Ui = Ra. Applying (3.45), we shall obtain a contradiction with discreteness of G by showing that the orbit {gi (en+l)} is bounded in IH[n+1 _ R+ 1. To do that, we consider the isometry (P: Hn+1 .. I[8+ 1 between the hyperboloid and half-space models of Hn+t

ip(eo) = en+1 (see (1.31)), that conjugates Mob(n) and O+(n + 1, 1)/{±11. Then Lorentz transformations T. = cp-1gmcp transform the vector eo E Hn+1 into the first column-vectors of their matrices (qp, n+1

T. eo = (tom0, t10, ... , n+1,0) ,

(tpo)2

= 1. - E(i0)2 i=1

(3.46)

Taking into account the bounds I :0 I < I too I, 1 < i < n + 1, that follow from (3.46), we see that the uniform boundedness of the orbit {gm(em)) follows from a uniform bound for the term too of the Lorentz matrices Tm. To obtain such a bound, we shall apply formulas (1.35) and (3.45) to study how the terms too behave when m tends to oo. Namely, denoting (*, *) the dot product in ]Rn and ti an asymptotic equality up to quantities of orders not exceeding the order increase of rm_1 = r2"`, r > 1, we get the following asymptotic equivalences describing the behavior of too as m -+ oo: (1 + Iq2)(1 + I qm I2) 2r2m(gm, Um(gr'n)) ti 2rm1gm1 Jq,

r4y

- 2r3. =

r2m+2

'- 2r.4 - 2r.3 =

-

2r3-2m 2r2m+2

-

(3.47) 2r3.2m

This asymptotic and (1.35) show that too are uniformly bounded and the orbit {gm (en+1) } is contradictionally bounded, too:

d(gm(en+1), en+1) < C.

(3.48)

3.4. Precisely invariant horoballs

119

This completes the proof that radii of isometric spheres I (g), g E G\GOO, are at most IaI if G,, has a translation h, h(x) = x + a, by a vector a E R n.

Now let G,, contain only irrational screw translations h(x) = V (x) +a, V E O(n), a E lR' \{0}, V (a) = a. We shall show that there exists a constant depending only on G, which bounds radii of isometric spheres I(g), g E G\Gw. To do that, let us again assume the contrary. Due to Theorem 2.12, the condition on the limit set is equivalent to the existence of a certain Euclidean t-neighborhood Ut (E) of the minimal Gam-invariant subspace E C Ilk" containing the G-orbit of oc whose points are centers of isometric spheres for g E G\GOO. Due to this observation we may assume that, for a small s > 0, the norm II V 11 of the orthogonal transformation

V is less than s, and there exists g E G\GOO, g(x) = r2UZ(x - q) + q', such that its isometric spheres have a radius r > 1 + s and their centers q and q' are at Euclidean distances less than s/2 from the subspace E C R' spanned by translation vectors of elements g E Gam. In fact, to show this deduction, one can replace the screw translation h by its sufficiently large power, h1c, and conjugate G by the dilation With such an adjustment, we define a sequence {gi } C G similar to (3.41): go = g,

gi+1 = gi hgr 1 for i > 0. Then calculations analogous to (3.42)-(3.45) show that, for the elements gi (x) = r? Ui x (x - qi) + q' of this sequence, we have the following: ri r2;_? 11lail

,

Ui =Ui-1RaiVUi-1

q; =qi-1+r 1Ui-1 x(ai), qj =qi-1 -r2

1Ui-lV-1X(ai),

(3.49)

where ai = V (qi _ 1) - qi _ 1 + a is orthogonal to the reflection subspace P (ai, 0) of the reflection Rat .

Then, since our assumptions imply ro = r > 1 + s > Iai I > 1, we see from (3.49) that the radii ri > r(312)1 tend to oo as i increases and, for sufficiently large i, the orthogonal transformations Ui are close to Ra V because the corresponding i-th reflection is defined by a vector close to qi. These two facts and (3.49) imply the following asymptotic equivalences:

(1 + Iqi 12)(1 + Iqi I2) " r4 - 2(riri-1)2 ,

2r?(Ui(q ),qi) -- 2rt IgillgiI

2r4-2(riri-i)2

(3.50)

where, as in (3.47), -- denotes an asymptotic equivalence up to quantities of orders not exceeding the order of increase of r4. The latter term tends to infinity for large i because r4 > r(3/2)`. Applying, similarly to (3.48), the asymptotics (3.50) and formulas (1.35) for terms of matrices Ti = co-1gico, we obtain the uniform boundedness of the orbit {gi (e,,+1) }. This contradicts to discreteness of G because elements of our sequence {gi } are pairwise distinct. The latter follows from the fact that radii ri of the isometric spheres I (gi) are all different (and even arbitrary large). This contradiction

3. Basics of Hyperbolic Groups and Manifolds

120

shows again that, in any case, there exists a uniform bound Ro for radii of isometric

spheres 1(g), g E G\G,,,,. It implies that for any a > Ro and g E G\G,,, we have g (Ha) fl H. C int I (g-1) fl Ha = 0, which means that Ha is a precisely GO-invariant horoball. This completes the proof. Let us finally remark that the above proof in fact proves a stronger result which, in the case of existence of Euclidean translations, can be formulated as follows:

Corollary 3.16. Let G C Mob(n), n > 1, contain a Euclidean translation h by a vector a E R' and G\ G,,,, contain an element g with the radius of its isometric sphere I (g) greater than jai. Then h is the limit of the sequence {gi } C G, gi+1 = gi hgi 1

go=g. Proof Let us transform the sequence {gi } into the commutator sequence {gi [gi, h] = gi h-1 E G. Then we see that

lim gi = h 4' lim g"i = id .

(3.51)

Also, we see that in order to prove (3.51), one can change the transformation g E G\Gw by any transformation f E Mob(n) of the form f = high2 where hl and h2 are arbitrary Euclidean translations in I R" . Namely, for new sequences {fi } and {f },

1i+1=fihfi 1, 1i+1 = [1i, h] =

.fo=f=h1gh2, 1ih--1

,

and

(3.52)

11 = [f, h],

we have fi = hi gi hi 1 and fi = hi gi hi 1. Therefore,

lim gi = h q lim fi = id

.

(3.53)

Next observation, like one in the proof of Margulis's Lemma, is that the commutator sequence in (3.52) converges to the identity if the initial element f is close enough to the identity. This follows from the following two facts:

1. For any two elements c and d of a Lie group H suffisiently close to the identity, the sequence of commutators, c1 = [c, d], ... , ci+1 = [ci, d], . , converges to the identity; and 2. The translation h canbe replacedby ShS-1, S (x) = x/K, which is sufficiently close to the identity, if we replace f in (3.52) by S f 3-1. Finally, we apply (3.49) in order to get a new element g E G\G,,, with sufficiently large isometric sphere I (g). Then, due to Corollary 1.8, there are a Euclidean motion w, two Euclidean translations h 1 and h2 and a sufficiently close to the identity M6bius transformation v such that g = h1 vwh2, i.e., h11gh21 = vw. Replacing g by vw, we have the above situation with the initial element close enough to the identity. So, the corresponding commutator sequence converges to the identity, and (3.53) completes the proof.

3.5. Group action on the set of horoballs

121

§5. Group action on the set of horoballs Let us consider an isometry between the hyperboloid model, H' C Mn+1, of the hyperbolic space HI and the Poincare ball model in B'(0, 1) as the radial projection from the point -eo = (-1, 0, ... , 0) in the Minkowski space Mn+1 (see Chapter 1, §4), Hn -) Bn(0, 1) C {x E Mn+1 : xo = 0}. For this isometry, points in the sphere at infinity, aHn, which are determined by rays from the origin in the light cone L+, correspond to points in Sn-1(0, 1) = aJH(n. Furthermore, a point v E L+ on such a ray corresponds to a horosphere {w E Hn : [w, v] _ -11 which bounds a horoball {w E Hn : 0 > [w, v] -11 in Hn, see Figure 30. ]' xo

Figure 30

The center of the horosphere is the ray going through v. As the point v E L+ moves away from the origin along the ray, the horoball contracts towards its center at infinity. This bijection between the points of the cone L+ and the set of horoballs becomes a homeomorphism if we use the so-called Vietoris topology on the space of closed horoballs in the closed unit ball Bn (0, 1), which employs the Hausdorff metric on closed subsets in Ilgn = Rn U {00}. Namely, for two closed subsets X, Y C Rn and the spherical metric s(x, y) on Rn (see (1.10)), we define the Hausdorff distance,

_

p(X, Y) = sup{s(x, Y), s(X, y) : x E X, y E Y).

_

(3.54)

122

3. Basics of Hyperbolic Groups and Manifolds

Therefore, in order to study the action of discrete groups G C Isom H' on the set of parabolic horoballs in IEI[n, we can consider corresponding discrete subgroups in O+ (n, 1) and their actions on the light cone L+. However, in contrast with discontinuous G-action on EP, the action of discrete subgroups of 0+ (n, 1) on L+ is usually non-discontinuous.

For example, if G C SO+(n, 1) is a discrete co-compact subgroup (a uniform lattice in 0+ (n, 1)), then due to Ellis-Perrizo [1], the stabilizer N of a point v E L+ acts strictly ergodically on SO+(n, 1)/G; hence the G-orbit of each point v E L+ is dense on L+. Similarly, for co-finite discrete groups G C O+(n, 1), their G-orbits G(v) = {g(v) : g E G} are dense in L+ for almost every point v E L+. Despite such density of G-orbits of almost all points v E L+, G-orbits of points on parabolic fixed rays in L+ are discrete in good cases:

Theorem 3.17. Let G C O+(n, 1), n > 2, be a discrete group and let v E L+ be a point with parabolic stabilizer G and such that the corresponding horoball in Ifi[n is precisely G (v) is discrete, closed in L+ and does not accumulate at the origin.

Proof. From the definition of a precisely horoball, it is clear that its G-orbit is discrete and closed in the space of all hyperbolic horoballs. Moreover, normalizing the G-action in the Poincare half-space model of 1Eln (i.e., considering a conjugate group AGA-1, A(x) = rx for an appropriate r > 0), we can assume that the maximal precisely horoball in R+ is {x E Il8+ : x, > 1). For the Poincare ball model of Hn in Bn(0, 1), it corresponds to the situation when the Euclidean diameters of the horoballs are bounded by one. This implies that the origin in the Minkowski space Mn+1 D L+ is not a limit point of the orbit G(v) of the point v E L+. Due to the above homeomorphism between the set of parabolic horoballs in If1[n and the set of points v E L+ lying on fixed parabolic rays, this completes the proof.

We immediately have:

Corollary 3.18. Let G C O+(n, 1), n > 2, be a discrete group with parabolic stabilizer G of a point v E L+, which either satisfies the conditions in Theorem 3.15

or has finite co-volume vol(O+(n, 1)/G) < oo. Then the orbit G(v) is discrete, closed in L+ and does not accumulate at the origin.

Therefore, for a given discrete group G in Corollary 3.18 and each its parabolic fixed point p E alln (i.e., a fixed ray a C L+), we can consider a point v E a and define radii of horoballs corresponding to its images g(v) E L+, g E G, equal to the radius of the initial horoball corresponding to v. This implies that G preserves radii of horoballs. To define the radius of the initial horoball, we can assume that it is the maximal precisely Gp-invariant horoball Bp that corresponds to some vo E or C L+. Then we define its radius as PG (vo) = 1. For smaller horoballs Bp C Bp centered at the same point p E 8H" and corresponding to v E a, II vo II < II v II, their radii pG (v) are monotonically decreasing to 0 as the Euclidean length of vectors v increases to oo.

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Also, this radius-function PG(v) is 0+(n, 1)-equivariant in the sense that, for any

y E o+(n,1), PG (v) = Pycy-1(Y (v)) .

(3.55)

However, in contrast with dimension n = 3 where all discrete groups G C ISOm H3 act discretely on the set of parabolic horoballs (see Corollary 3.18), there are discrete groups G C Isom1H[n, n > 4, whose orbits G(Bp) of parabolic horoballs (or corresponding orbits G(v), V E L+) are not discrete. Such (infinitely generated) groups G can be obtained by a little adjustment of our construction in the proof of Theorem 3.14. Namely, the main observation is as the following. For numbers i large enough, the

images xi (Hl) of the horoball H1 = {x E R+ : x4 > 1} under reflections Xi with respect to the spheres S3 (Z,, mi) from the construction in Theorem 3.14 are horoballs arbitrarily close to Hl and centered at the points zi. An inductive adjustment changes each of generators gi with isometric spheres I (gi) = S3 (Z,, m,) to a new generator, gi, so that the horosphere gi (Hl) is still centered at zi and passes through e4 = (0, 0, 0, 1). Then it becomes clear that horobals {gj (H1)} converge to H1 in the Hausdorff metric (3.54).

It is still unknown whether or not finitely generated discrete groups G E Isom IH[n, n > 4, may have such non-discrete orbits of horoballs.

§6. Convex hull constructions Here we shall present two constructions of convex hulls in the hyperbolic space IHI' and in the Minkowski space M1+1, which are related to discrete group actions and play important roles in the study of hyperbolic manifolds. As usually, a set C in a Riemannian manifold X is called convex if, for any two points xl, x2 E C, the geodesic segment [x1, x2] also belongs to C; C is called strictly convex if the open interval (x 1, x2) belongs to the interior of C. The convex hull C (A) of a set A C X is defined as the smallest convex subset in X containing the set A.

Hyperbolic convex hulls. For a given discrete group G C Isom Hn, we define the convex hull CG for G as the hyperbolic convex hull of the limit set A(G) C MIEn. In other words,

CG = C(A(G)) = n{Y c fflln u akl[n : Y fl IH[n is convex}.

(3.56)

Due to the canonical (isometric) action of Mob(n - 1) in IH[n, we also can speak about convex hulls CG for discrete Mobius subgroups G, CG C Nn-. For example, for any discrete Mobius group G C Mob(n - 1) preserving a round hypersphere Sn-2 in Sn-1 = 81H[n (in particular, for Fuchsian groups G on Sn-'1), its convex hull CG belongs to the closure of the hyperbolic (n - 1)-plane H C IH[n having aH = Sn-2. This is a degenerate case because intH,= CG = 0. For any other group G C Mob(n -1)

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having no G-invariant round (n - 2)-sphere in Sr-1, one can see directly from (3.56) that intun CG # 0. It also follows from the fact, that CG contains n-simplices whose vertices xp, xl, ... , x1z e A(G) do not belong to any round (n - 2)-sphere. Another trivial example is the convex hull CG for a discrete group G C Isom 1111" whose limit set A(G) is the whole sphere Sii-1 = 81 l'n at infinity. It is clear that CG =1H[". 81H[" and A(G) does not belong to any round In the generic case, when A(G) (n-2)-sphere, wehaveintnn CG ; 0, clan (intHn CG) = CG\A(G), andtheboundary BJnCG is a pleated hypersurface in IHP'. This pleated surface 8C(G) is bent along (n - 2)-dimensional hyperbolic convex polyhedra and has complete disjoint geodesics as one-dimensional skeleton that intersect only at aRP. In terms of Riemannian geometry, the exterior curvature of this boundary surface is 0. The interior curvature (in the sense of Aleksandrov's spaces, see Aleksandrov-Berestovskii-Nikolaev [1] and Nikolaev [1-5]) of this boundary surface depends on dimension n. For n = 3, it is the sectional curvature of the surrounding space 1111, which is -1. For n > 4, the boundary surface 8 p CG may have additional cone singularieties as it shown in Section 5 of §6.8. The geometric properties of this pleated surface also depend on dimension n, and we can describe them more easily by using projective Klein model of 1111" in B" (0, 1) where hyperbolic planes are intersections of Euclidean planes with Bn (0, 1) (see Chapter 1, §3), and CG is the Euclidean convex hull of A(G) C Sn(0,1). Since any discrete group G C Isom EHn preserves its convex hull CG and acts discontinuously on CG \A (G) by isometries, we can define a convex hyperbolic orbifold with boundary, (3.57) OG = [CG\A(G)]/G,

which is a hyperbolic manifold with boundary if G is torsion free. There exists a natural retraction p,

p: O(G) = [EP\A(G)]/G -- OG = [CG\A(G)]/G,

(3.58)

induced by the following G-equivariant retraction p: IHI'i -+ CG. The retraction p is the identity on CG. We define p5(x) as the closest point to x in CG if x E H\ CG. On the other hand, if x E 81H1n\A (G) = Q (G), we define p(x) as the first intersection point of CG and the family of increasing horospheres centered at the point x, see Figure 31. This definition is correct because CG fl H" is convex and hyperbolic metric balls and horoballs are strictly convex in IH[".

Theorem 3.19. Let G C Isom lIP be a discrete non-elementary group. Then the orbifold OG = [CG\A(G)]/ G is the minimal convex retract of the hyperbolic orbifold H'/ G (and O (G) ), whose inclusion induces isomorphism of the fundamental groups.

Proof. Convexity of the orbifold (9G and the existence of (identity) isomorphism norb(OG) = G -- G = 7.irb(0(G)) follow from the definitions. So we only need to prove that any convex suborbifold X C O (G) with the isomorphism irirb (X) - G = 7riorb (O (G)), induced by inclusion contains our suborbifold OG.

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125

Since X is convex (with hyperbolic interior x fl int (9(G)), we see that its development, d, maps its universal covering space X C (9 (G) = H' U S2 (G) to some set D C IEIIW which is G-invariant (since d,k(Jrirb(X)) = G) and has convex intersection with IHIn. Due to (2.3), the limit set A(G) is a subset of D fl aHl. Hence CG C_ D and (9G C X.

x Figure 31

Example 3.20 (Mickey Mouse). Let r C Isom 1H13 be a discrete group acting at infinity aIHI3 = K2 as a Fuchsian group uniformizing a closed Riemann surface S2 = B2(0, 1)/ I' of genus two. Then 3-manifold M(F) = (1B13 U 0 (T))/ r is the product S2 x [-1, 1] whose minimal convex retract Mr, is the middle convex surface S2 x {0} = H/ F, where the 2-plane H C IIi13 has the circle at infinity equal aB2(0, 1).

Let us consider a closed geodesic a C S2 (see Figure 32) which is covered by the hyperbolic axis & of a cyclic hyperbolic subgroup (y) C F. Here y identifies the sides a and a' of the regular isometric fundamental polygon P(F) C B2(0, 1), lying correspondingly on isometric circles I (y) and I (y-1), see Figure 33. These sides a and a' (homotopically) correspond to a simple closed geodesic 8 C S2 intersecting a. Increasing radii of circles I (y) and I (y -1), we can move them in such a way that their angles of intersection with other circles bounding P (F) remain the same

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(= .7r/4) but angles of intersection with S1 (0, 1) become less than n/2, see Figure 33.

Figure 32

Figure 33

Identifying the corresponding sides of the obtained (P (F) -deformed) polygon by hyperbolic translations g (not preserving the disk B2(0, 1) any more), we generate a new discrete group G C Isom H13 conjugate to F by a quasi-conformal homeomorphism f, G = f Ff -1 (see details in Chapter 7, §3). This new group G is not Fuchsian. Its convex hull CG has non-empty interior bounded by two pleated cells Ho, HI C H3 with the common boundary quasicircle A(G) ti S1. One of those boundary

cells, Ho = p(f (B2(0,1))), can be visualized as follows.

Figure 34

Figure 35

The generator g of G that corresponds to the generator y of r, does not preserve the plane H C H3. Moreover, the hyperbolic geodesic $ = H fl g-1 (H) is the lift

3.6. Convex hull constructions

127

of the geodesic P on the convex surface S2 x {0}. Points at infinity of the union of two open half-spaces H+ and g-1 (H+) C 13[3 bounded by H and g-1 (H) belong to the discontinuity set Q (G) C 8113. Considering the union of all such half-spaces in the orbit G (H+), we obtain some domain in IHl3 = ]18+ (bounded in Euclidean sense) whose boundary in 113 is our pleated 2-cell Ho. This 2-cell is almost everywhere totally geodesic, except (bending) geodesics G(,B). One can see the orthogonal projection of this cell on the boundary plane R2 of IIB+ in Figure 34. The corresponding boundary

pleated surface of the minimal convex retract MG C H3/ G S2 x [-1,1 ], Ho/ G, is obtained from S2 = H/ F by its bending along the geodesic ,B by some angle, see Figure 35. The other pleated surface, H1 / G, is more complicated, see §8, Chapter 6. Minkowski convex hulls. Let G C Isom H' be a discrete co-finite group with at least one parabolic fixed point, i.e., IH[" / G is non-compact and has finite volume. Due to the finiteness Theorem 4.16, every parabolic fixed point p E 811II" of G has precisely Gp-invariant horoballs (in fact, p is a cusp point and represents one of finitely many G-orbits of such cusps), and we can apply results of §5 on the G-action on the light cone L+ in the Minkowski space Namely, suppose that p1, ... , Pk E all" represent all (distinct) G-orbits of parabolic fixed points with stabilizers Gi = Gp; C G. Then, due to Corollary 3.18, we can consider disjoint G-orbits G(Bi ), i = 1, ... , k, of disjoint horoballs Bi C 131" (of radii less than one), which correspond to discrete orbits G(vi), i = 1, . . . , k of some points v1, ... , vk E L+ on non-equivalent parabolic fixed rays in L+. Now we define M"+1.

C(G) = C (G; v1, ... , vk) = C (U{g(vi) : g E G, i = 1, ..., k}) C M"+1 (3.59)

to be the closed convex hull of the discrete subset Ui G(vi) C L+ in the Minkowski space, which does not accumulate to the origin.

Lemma 3.21. C(G) fl L+ is the set of points of the form av, where a > 1 and v E G(vi) for some i = 1, ... , k. Proof. Let us consider x and ax E L+, a > 1, which are not of the stated form. If E is the horizontal (n - 1)-dimensional plane contained in {z E M"+1 : zo = axo} and tangent to L+ at the point ax, we can slightly rotate the n-dimensional plane tangent

to L+ at ax around E so that its image, P, intersects L+ in a long thin ellipsoid nearly equal to the ray [0, ax] and containing ax. Since there are only finitely many points from the orbit G(v1, ... , vk) whose 0-th coordinates are less than axo, we can consider this rotation around E as small as the point x lies on the opposite side of P from the orbit G(v1, ... , vk). It shows that x does not belong to C(G). Now let us show that av E C(G) for any a > 1 and for any given point v from

G(v1, ... , vk). Due to Corollary 2.16, G-orbit of the parabolic fixed ray r C L+ passing through v is dense in L+. So we can choose a sequence of such rays rj = gj (r) C L+ converging to r, gj E G. Since the orbit G(v) is discrete, the points gj (v) tend to oo. It shows that, for sufficiently large j, the segments [v, gj (v)] C C(G)

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128

contain points arbitrarily close to the point av, a > 1. Hence av E C(G) because C(G) is closed.

Lemma 3.22. The dimension of C(G) C Mn+l is (n + 1). Furthermore, each ray from the origin lying inside L+ meets 8C(G) exactly once.

Proof. Due to Lemma 3.21 and the density on L+ of the set of parabolic fixed rays g(v), g E G, v E G(vi, ... , vk), the first claim is obvious. Now let us consider a ray r C int L+ emanating from the origin, and let z be a point of the projective Klein model of 1H[n in B" (0, 1), z = r fl {x E Mn+1 : xo = 11. Due to the density of the G-orbit of parabolic fixed points on 8B" (0, 1), the point z lies in the interior of the hyperbolic convex hull in B" (0, 1) of finitely many parabolic fixed points corresponding to W 1 ,.. . , w," E G(v1, ... , vk). It shows that the ray r meets the convex hull C(wi, ... , w,) C C(G). If zo is the first intersection point of r and C(G) as we proceed from the origin, we see (as in the proof of Lemma 3.21) that any point of r above zo lies in the interior of C (G). This completes the proof.

Lemma 3.23. The boundary 8C of C(G) in Mn+1 is the union of C(G) fl L+ and a countable, locally finite in int L+ set of codimension-one faces F1, i E N. Each Fi is the convex hull of a finite number of points in G(v1, ... , vk), and the intersection of its affine hull, Ai, with the light cone L+ is ellipsoidal with respect to the Euclidean structure on Mn+1

Proof. Let us firstly prove that, for any point zo E 8 C\L+, a support plane P C M"+1 for C(G) at zo meets L+ in an ellipsoid. Since the ray from the origin going through zo meets int C(G) (Lemma 3.22), the support plane P cannot contain the origin, and we can find some w E M"+1 such that

P = {x E Mn+1 : [x, w] _ -1} ,

C(G) C {x E M"+1 : [x, w] < -1} .

(3.60)

The vector w in (3.60) cannot lie outside L+. Otherwise, if [w, w] > 0, we choose xo E int L+ so that [xo, w] = 0 and [xo, x0] = -[w, w]. Then x0 + w E L+, [x0 + w, w] = [w, w] > 0, and there exists a neighborhood U of xo + w such that

[x, w] > [w, w]/2 for any x E U. Due to density of parabolic fixed rays in L+ (Corollary 2.13), we can find some y E G(v1, ... , vk) and small enough s > 0 such

that sy E U. Then [y, w] = [sy, w]/s > [w, w]/2s > 0. This contradiction to (3.60) and the fact that y E C(G) show that [w, w] < 0.

If [w, w] = 0 then w E L+ because [zo, w] = -1 for our point zo E intL+. Furthermore, w cannot lie on a parabolic fixed ray in L+, i.e., w ; )g(vi) for any X > 0, g E G and i = 1, ... , k, because [g (vi), w] < -1. It implies that w corresponds to an approximation point of G, wo E aH" (see Theorem 4.9), and hence

there exists an orbit gj (w), gj E G, accumulating to the origin. For this sequence and for any v e G(v1, ... , vk), [gj-- 1 (v), w] _ [v, gj (w)] tends to 0 in contradiction with (3.60) and the fact that gJ -- 1 (v) E C(G). This finally proves that [w, w] < 0,

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129

w E int L+, and p n L+ is an ellipsoid. Conjugating in O+(n, 1) so that the plane P becomes horizontal, {x E M[n+1 : xo = const}, we see that this intersection becomes an (n - 1)-sphere, so P n L+ is spherical in the Euclidean structure of the affine hull A of P.

Now we can find a support plane P at zo E aC\L+ which contains n affinely independent points from the orbit G(vl, ... , vk). Namely, if some affine subspace A of P containing P n G(v1, ... , vk) has dimension at most (n - 1), we can rotate P about A in order to meet more points from G(vi, ... , vk) and make the dimension of the affine hull of P n G(vi, ... , vk) maximal, i.e., equal to n. To finish the proof, we only need to show that the set of faces Fi C Pi is locally finite inside L+. Otherwise, we can choose a sequence {Fi } of distinct faces of

C(G) meeting a compact Co C int L+ so that intersection points xi E Fi n Co converge to some x E Co and the affine hulls Ai of Fi converge to a limit A, X E A, which is a support plane for C(G) intersecting L+ in an ellipsoid. It shows that this intersection contains only finitely many points from the orbit G (v i , ... , vk) and hence (Ui Ai) n G(vi, ... , vk) is finite. Therefore the initial set (Fi} is finite.

§7. Tessellations of manifolds by ideal hyperbolic polyhedra For a given non-compact, finite volume hyperbolic n-manifold M = Hn/G, n > 2, we can apply the above C(G)-construction of Minkowski convex hulls to obtain a canonical (finite) decomposition of M into ideal hyperbolic polyhedra with disjoint interiors.

Namely, the decomposition of aC(G)\L+ into n-dimensional boundary faces Fi (see Lemma 3.23) is G-invariant due to the G-equivariantness of C(G)-construction. Since this cell structure is locally finite, we can use Lemma 3.22 to project it homeomorphically to a locally finite tessellation of the projective Klein model of 1FF in B"(0, 1) C {x E M[n+t : xo = 1}. This tessellation is G-invariant, consists of ideal hyperbolic n-polyhedra (of finite volume) and depends only on our choice of

points vl, ..., vk E L+ representing disjoint orbits of precisely Gi-invariant parabolic horoballs in 1EI1.

Replacing the initial points vl, ... , vk on parabolic fixed rays in L+ (i.e., disjoint horoballs in 1In) by points alv1, ... , akvk with ai > 0, we obtain a new G-invariant tessellation of 1E1 which is the same if al = = ak because of the radial projection property. Therefore, we have a (k - 1)-parameter family of G-invariant tessellations of fn by ideal hyperbolic polyhedra such that the interior of a polyhedron of any dimension isometrically injects into M =1EIn/G. In some cases, such ideal polyhedra decomposition can be subdivided into an ideal simplex decomposition of a given noncompact hyperbolic n-manifold M of finite volume. To do that, one has to ensure a G-compatible simplex decomposition of the boundary of the ideal polyhedra. It is trivial in dimension n = 2. Also, due to results

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of M. Wada and H. Yoshida [1] and Wada-Yamashita-Yoshida [1], it can be done for a 3-manifold M if one of the following conditions holds:

(1) M allows a decomposition into at most two ideal convex hyperbolic polyhedra, or (2) M is obtained by side identification of n ideal convex hyperbolic polyhedra

P1, ... , P, in such a way that each side of the first (n - 1) polyhedra Pi,

i = 1, . . . , n - 1, is pasted with a side of the last polyhedron Pn, and the possibly remaining sides of the last polyhedron P, are glued in pairs. In general situation (even in dimension n = 3), it is not yet known whether a given noncompact hyperbolic n-manifold M of finite volume can be decomposed into ideal hyperbolic simplices.

§8. Hyperbolic arithmetics Here we shall apply "arithmetic methods" to the study of closed and finite volume hyperbolic n-manifolds. The fundamental groups G of such manifolds are lattices G c Isom IEII' , and their isometric actions in the hyperbolic space W' have compact (or finite volume) fundamental polyhedra in IEII. For such groups, there is a general construction of arithmetic lattices. It arisen, for n = 2, in works by Fricke [1] and Bianchi [1] (see also Fricke-Klein [1]) and, for n > 3, in Venkov [1]. Later great developments were due to the works Mostow-Tamagawa [1] and Borel [1, 2]. Moreover, due to a celebrated superrigidity theorem of Margulis [7, 12], in the case of a semi-simple Lie group 9, of the JR-rank greater than one (the dimension of a maximal

torus is greater than one, so 9, ; IsomW), every lattice in 9. is arithmetic, that is all corresponding (finite volume) manifolds are arithmetic. Similar arithmeticity of lattices is true for all non-compact symmetric rank one spaces different from IHI" and CH'. This is due to superrigidity theorems by Corlette [2] and Gromov-Schoen [1], see also Apanasov-Kim [1, 2].

The arithmetic construction of finite volume hyperbolic manifolds is based on consideration of automorphism groups of indefinite quadratic forms. To show clearly the key ideas of this arithmetic construction, let us present its simple version. Namely, we consider a quadratic form,

fa(x) _ -,Idxp + x1 +... + xn ,

(3.61)

in W+1 where d is a natural number, free from squares and greater than one. Then, as in Chapter 1, §3, we obtain a hyperboloid model of the hyperbolic space IHE" by taking the

upper half of the hyperboloid {x E R'+1 : fd(x) = -1, xo > 01. Its isometry group is the index two subgroup O'(fd, R) in the group O(fd, IR) c consisting of those linear transformations of space lRi+1 that preserve both the quadratic form fd and the upper half of the hyperboloid.

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131

Now, for any n > 2, we define a group G = G(n, d) as the group O'(fd, Ad) over the ring Ad of integers in the quadratic field Q(om). In other words, taking for simplicity d 1 (mod 4),

G(n, d) =O'(fd, Ad) = {M = (aij) E GLn+1 (R) : aij = nij +mjj'ld, .fd(Mx) = fd(x); nij, jnij E Z). (3.62) Theorem 3.24. For all n > 2 and for all natural numbers d distinct from squares and from one, G = G (n, d) is a discrete co-compact isometry group in the hyperbolic space IHIn.

Proof. First, we shall give a proof of co-compactness for an analogous group G = O(f, Z) C GL4(R) which preserves the quadratic form

f(x)=-7xp+xi+x2+x3 and consists of integer matrices.

The discreteness here obviously follows from the discreteness of GL4(Z) in GL4(IR). Also, the action of Off, ]I8) - IsomlHl3 on H3 is transitive. So, to prove compactness of the orbifold 1H13 / G, we only need to prove compactness of the factor

0(f, IR)/ O(f, Z). This space can be identified with the O(f, R)-orbit of the integer lattice Z4 C ]R4, L0 = 0(f, R) (Z4) C L, where by L we denote the set of all lattices in Ra

In general, lattices in RI are characterized (up to a compact set of possibilities) by two invariants. First, by the volume V of the fundamental domain, V = I det M 1, where matrix M is composed of the basis vectors and, second, by the minimal distance, S, between the origin and points in the lattice. We introduce in L a topology when two lattices are close if their basis vectors are close. This topology is equivalent to the quotient topology GL,, (1R) / GL,, (Z) = L. We shall prove the compactness of the 0(f, III)-orbit L0 by applying the following Hermite-Mahler compactness criterion, see Mahler [1]: Theorem 3.25. A subset of lattices L0 has a compact closure in the whole space L if infLo S > 0 and supLo V < Do-

Now we shall show that infLo S > 0 on the set L0 which is the Off, R)-orbit of the integer lattice Z4 C 1R4. Indeed, after reducing form f modulo 8, we see that the restriction of the form f to 7Z4\{0} C 1R4 takes only non-zero integer values. Let

U = {x E R4 : f (x) < 1/2} be a neighborhood of the origin. As f (0) = 0 and f is continuous, U does not contain points from Z4\{0}. So, no point from any lattice from the Off, H)-orbit L0 of Z4 is contained in U, because the form f is preserved by the group 0(f, H). This implies that infLo S is positive. Next, we have supLo V < 1. In fact, V (Z4) = 1, while all elements M of the group Off, R) leave form f invariant, so det M = 1.

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Finally, let us show that Lo is closed in L. In fact, if a sequence {Lj } C La converges to a lattice L,, E L, then the restriction of the form f (continuous and integer on points of lattices L j) to a point in L,k takes integer values. Let { V; } and (U1)

be correspondingly bases for L* and Lj with sufficiently large number j. Choosing subsequences, if necessary, one can assume that Uj is close to Vi, f (Ul) = f (Vi) and f (UL ± Uk) = f (VV ± Vk), which implies f (Ui, Uk) = f (V1, Vk) for 1 < i, k < 4. Therefore, a linear transformation that transforms Lj into L. leaves the form f invariant, i.e., it is an element of the group Off, R). It shows that the lattice L,k lies in the set La. These properties of the O(f, R) -orbit L0 together with Theorem 3.25 complete the proof of compactness of 1H[3/G for our special quadratic form. In the general case of G = O(fd, Ad), it is possible to modify the above proof in the following standard fashion, where one only has to prove separately the discreteness

of the group G (because the ring Ad of integers in the field Q(/) is not discrete in R any more). We note firstly that the image of the embedding

(m+n-,Id-) H (m+n-va,m-nv/d-),

m,n EZ,

of the ring Ad is a discrete subring in the product R x R. This embedding induces an embedding of the group G = O'(fd, Ad) into the group GLn+1(R x IR) = GLn+I (TR) x GLn+1(][8)

,

(3.63)

and forms there a discrete subgroup. As the group G = G(n, d) specified in (3.62) leaves the form fd in (3.61) invariant, its image in the second cofactor GLn+l (R) in (3.63) leaves the form

+xi +...+x2 invariant, i.e., this image is a subgroup of a compact orthogonal group. This immediately implies discreteness of the image of G in the first cofactor GLn+l (R) in (3.63), which coincides with the group G itself. This completes the proof of Theorem 3.24.

0 The above construction is a particular case of a general construction of real arithmetic lattices due to A. Borel and Harish-Chandra [1]. To describe this generalization, let us consider a Lie group g with finitely many connected components and whose connected component of the unity 9° is the direct product of non-compact simple Lie groups without center. Furthermore, let G be an irreducible algebraic group, defined over a field K of algebraic numbers. We shall assume that:

i) the field K is totally real, ii) the group G1 is locally isomorphic to 9, iii) for any non-identity morphism o : K -> IR, the group (G°)R is compact, where G° is an algebraic group, defined over K°, which obtained by applying o" to all of the coefficients of the equations defining G.

3.9. Arithmetic groups generated by reflections

133

Given a faithful matrix representation p of G, let us consider the group GII = {g E GIR : p(g)jj E 1l}

where 11 is the ring of integer elements of 1K. It is well known that the group GII does not depend (up to commensurability) on the representation p. Here two groups are called commensurable if they have a common subgroup of finite indexes in both groups.

Definition. Let a class A(G, K) consist of those subgroups of 9' that either have the form cp(G5) where p is a homomorphism of (GIR)D on 9° or are commensurable with the latter. Then a subgroup r C 9' is called an arithmetic subgroup if it belongs to the union:

U A(G, K) .

(3.64)

G,K

By A. Borel and Harish-Chandra [1] (compare Theorem 3.24), any arithmetic subgroup F C 9' is discrete with finite volume quotient 9/ F, i.e., r is a lattice. We obtain the definition of quasi-arithmetic subgroups in 9' if, in the above definition of arithmetic subgroups, we change the class .4(G, K) to a class QA(G, K)

of those discrete subgroups F C 9, that have finite co-volume, vol(9/ F) < 00, and whose finite index subgroups are contained in (p(FK), where cP is an appropriate homomorphism of (GR)e on ge.

In the next section, we shall describe all (real) arithmetic and quasi-arithmetic subgroups of isometries of the hyperbolic space H" generated by reflections. In that case, the group 9' = Isom lHII' consists of two connected components and go is a non-compact simple Lie group, without center.

§9. Arithmetic groups generated by reflections 1. Vinberg's criterion for real-arithmeticity. Here we shall present a criterion for (real) arithmeticity and quasi-arithmeticity of discrete groups of isometries of the hyperbolic space HI generated by reflections, which has been obtained by E. B. Vinberg [1].

First we fix some notations. If G is an algebraic group over a field IK, then by Ant G and Ad G we denote the groups of its automorphisms and inner automorphisms, respectively. If L is a field containing the field IK then any morphism a of the field L in C, identical on K, will induce an isomorphism g i-+ g° of GL onto G°t,. We shall

assume that g" = (gt)°. Given a quadratic form f with coefficients in K, let O(f) (correspondingly, SO(f )) be the group of its automorphisms (unirnodular automorphisms), as an algebraic group

3. Basics of Hyperbolic Groups and Manifolds

134

over 1K. So the hyperbolic isometry group 9 = Isom J" can be identified with the + xn . On the other group (Ad WON = Ad O(A)R, where fo = -xo + xi + hand, 9, = 0(1, n) can be embedded as a subgroup of index 2 in O(fo)a, see Chapter 1, §3.

Let I, be a discrete co-finite subgroup of O(1, n). Then the question of whether or not the group r is quasi-arithmetic reduces to the question of the existence of a totally real field K and of a basis {ui } in the Minkowski space M'+1 satisfying the following conditions: (a) All elements in I, can be represented in the basis {ui } as matrices with entries in 1K.

(b) The scalar products bit = (u,, uj) he in K. (c) The matrix (biz) is definite for any non-identity morphism a: K -a R. This fact follows from the following Lemma, see Vinberg [1, Lemma 7]:

Lemma 3.26. Let r be a Zariski-dense (over 1) subgroup of 0(1, n) containing reflections. Suppose, furthermore, that G is an irreducible algebraic group defined over the real number field K, and that o is an isomorphism of Ga onto 0(1, n)0. If r fl cp(Gx) is a subgroup of finite index in r, then 1) G = Ad SO (f ), where f is a quadratic form with coefficients in K, and cp can be (uniquely) extended to an isomorphism of the group (Ad O(f ))a on O(1, n). 2) If r is generated by reflections, then r C cp (Ad 0(f)K)For our purposes, it is convenient to reformulate the above conditions (a)-(c) in such a way that no linear independence is required for the vectors {ui }. This can be done as follows:

(a)' All elements in I' leave invariant the K-submodule EK of the space M"+t generated by the vectors {ui }. (b)' The scalar products bi.l = (ui, uj) lie in 1K. (c)' for any non-identity morphism a : K -+ IR, the matrix (b 1) is semidefinite.

We remark that the conditions (a)' and (b)' imply that dima EK = n + 1. The following result of Vinberg [1] presents two main necessary and sufficient conditions for a discrete co-finite group P C Isom 1HIn generated by reflections to

be real arithmetic. First, the Gram matrix A = A' = (ai j) for the group r must have non-negative entries for all those morphisms in C that are morphisms of the field generated by the numbers 2ai j and are not identical on a subfield generated by numbers of the form ai, i2 . a1213 . ai,,, j j. Second, the numbers 2ai j must be completely real algebraic integers.

Theorem 3.27. Let F C Isom 1HI" be a discrete group generated by reflections with a fundamental polyhedron P C 1HI' of finite volume and let A = be the Gram matrix of P. Denote by K the field generated by the numbers aid, and by K the field

3.9. Arithmetic groups generated by reflections

135

generated by the numbers aili2 . ai2i3 .

. aim-lim . aimi, .

(3.65)

Then the group F is a quasi-arithmetic group if and only if K is a totally real field of algebraic numbers and for any morphism a: K -+ R, not identical on 1K, the matrix

A° = (a) is nonnegative. Moreover, the group P is arithmetic if and only if the algebraic numbers 2aij are integers.

Remark 3.28. If the fundamental polyhedron P of r is non-compact, the conditions in Theorem 3.27 can be greatly simplified. In this case, P is quasi-arithmetic if and only if the numbers (3.65) are rational. For the proof, let us consider the principal CO-submatrix B of A, which corresponds

to some vertex of P at infinity. Its rows are linearly dependent but the rows of A which pass through it are linearly independent. This property is preserved under any morphism a: K --+ R. On the other hand it cannot hold for a nonnegative matrix. Hence for a group r to be quasi-arithmetic, it is necessary that K = Q.

Remark 3.29. It is easy to verify that the principal C+-submatrices of A remain positive under any morphism cr : ]K --> R. Therefore if we denote by D the determinant of any principal submatrix of order n +1 adjacent to a principal C+-submatrix of order

n, the condition A° > 0 will be equivalent to the condition D° > 0. Since D E K and D < 0, the quasi-arithmeticity condition can be satisfied only if D is a primitive element of the field K. Hence the quasi-arithmeticity criterion can be reformulated as follows: The number D must be a primitive element of the field K, and all its conjugate numbers must be positive.

Moreover, if all numbers aid > -1, i ; j, i.e., no dotted lines occur in the Coxeter diagram of the group F, then the numbers 2aij will all be integers. Hence, in this case, the quasi-arithmeticity of F implies its arithmeticity. In general, this is not true.

2. Makarov's groups. The first examples of non-arithmetic discrete groups with fundamental domains of finite volumes in a symmetric space different from the hyperbolic plane IEi[2 were constructed by V. S. Makarov [2] (a Selberg conjecture has been disproved by this). Namely, for sufficiently large integers m, Makarov has constructed some nonarithmetic groups r,, C Isom IEi[3 generated by reflections in sides of polyhedra P,,, C IEl[3 of the combinatorial type of a triangular prism and represented by the diagram in Figure 36 where the integer m > 5 and the number C = (cos 21r/m)-1/2 cos it/m. Due to Theorem 3.27, the groups P,,, are not quasi-arithmetic for any m ; 6. Similarly, in a previous paper of Makarov [1] (where the question of whether or not a group is arithmetic was not posed), he constructed another series of groups P,' C Isom H3 generated by reflections with compact quotients l3 / Fm . The diagram of one series of such groups (their bounded fundamental polyhedra) is shown in Figure 37,

3. Basics of Hyperbolic Groups and Manifolds

136

where the number C = (3 cos(2n/m) - 1)1/2 (4 cos(2n/m) - 2) -1/2 and the integer

m>7. O-- C-

m

Figure 36

v-m

O- CFigure 37

It was pointed out in Vinberg [1] (as an application of Theorem 3.27) that these

groups rm are arithmetic for m = 7, 8, 9, 10 and 14. They are quasi-arithmetic, but not arithmetic for in = 12, 18, 24 and 30. Finally, they are not quasi-arithmetic for all other integer m. Another examples of non-arithmetic crystallographic groups generated by reflections in hyperbolic spaces Hn, n < 10, were found by Rusmanov [1].

3. Complex-arithmeticity for reflection groups. Due to the A. Borel and HarishChandra [1] method, it is possible to consider not only real-arithmetic but complexarithmetic hyperbolic lattices as well. In fact, there are more arithmetically definable subgroups than those considered in Theorem 3.27. The reason appears to simply follow from the fact that C contains more algebraic number fields than R. The relationship between real- and complex-arithmetic groups is given by the following theorem of A. Reid, see Maclachlan-Reid [1]:

Theorem 3.30. The set of complex-arithmetic subgroups of Isom+ H3 that contain Fuchsian subgroups (lattices in Isom+ H2) coincides with the set of real-arithmetic subgroups of Isom+ 1H13

However, in the case of groups generated by reflections, real-arithmeticity is no more special than complex arithmeticity. This follows from the above Theorem 3.30 and the following result of Hilden-Lozano-Montesinos [4]: Proposition 3.31. Let G C Isom IH13 be a crystallographic reflection group. Then G contains a Fuchsian subgroup. Proof The reflection planes corresponding to elements of the group G decompose H3 in finite volume polyhedra. Each of them is a fundamental polyhedron for the group G. Take a plane H containing a two-dimensional side of one such polyhedron. Then the stabilizer GH C G of H contains a Fuchsian group F. In fact, H is tessellated by copies of 2-sides of a polyhedron. Taking a copy for each 2-side of the polyhedron, we obtain a union of finitely many 2-sides in H. This union is a fundamental polygon for r. This completes the proof.

Corollary 3.32. Every complex-arithmetic subgroup G C Isom I113 which is commensurable with a group generated by reflections is real-arithmetic.

3.9. Arithmetic groups generated by reflections

137

4. Finiteness theorems for reflection groups. First we introduce some notations and definitions. Let 1K be a totally real algebraic number field, with A its ring of integers. The quadratic fonn n

f (x) = E aijxixj

(aij = aji E A)

i, j=0

with signature (n, 1) is called admissible if the following quadratic form is positivedefined for any non-identical embedding a': K -+ R n

fa(x)

a ?lxixj . i, j=1

Mostow and Tamagawa [1] have shown (the case K = Q was considered for the first time in Venkov [1]; see also the proof of Theorem 3.24) that the group O'(f, A) of integer linear transformations that preserve the form f and each connected component of the hyperboloid {x E 118"+1 : f(x) = -1} is a discrete co-finite subgroup in

Isom H. Moreover the group O'(f, A) is co-compact if, additionally, K Q, or if K = Q but f does not represent zero in Q, see Venkov [1]. By Or (f, A) we denote the subgroup of O'(f, A), generated by all reflections in 0'(f, A). If this subgroup has a finite index, form f is called reflective. A crystallographic reflection group G C Isom TES" is arithmetic if and only if it is a

finite index subgroup of the group O'(f, A) where f is an admissible quadratic form (automatically, it is reflective) over totally real field TIC (= the field of the group G). This follows from the definition of arithmetic subgroups in a semi-simple Lie group 9 and from Corollary 3.32. Numerous finiteness theorems have been proved recently for the crystallographic and arithmetic subgroups of Isom HI, mainly by Nikulin and Vinberg. The first result of this type was a theorem of Nikulin [1, 2] which states that, for a fixed dimension of 11", there exist only finitely many hyperbolic reflective lattices over Z having rank > 5. Later, Nikulin extended that theorem to any hyperbolic lattice with any base field and then proved that in general, for any fixed n and any given degree of the base field K, the number of reflective forms is finite, up to proportionality and integer equivalence. A consequence of this result is the finiteness (up to commensurability) of the number of arithmetic discrete reflection groups for given dimension n and degree of the base field 1K. Moreover, there are no hyperbolic lattices S over 7G of rank S > 20 for which the automorphism group of S is generated (up to a finite index) by 2-reflections. The following finiteness theorem of Nikulin [4] is a generalization of this result.

Theorem 3.33. If the degree N of a base field 1K is sufficiently large (N > No, where No is some calculated constant), then there are no arithmetic groups generated by reflections in the hyperbolic space 1E" of dimension n > 10; respectively, there are

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3. Basics of Hyperbolic Groups and Manifolds

no hyperbolic lattices S, rank S > 11, whose automorphism groups are generated by reflections (modulo finite index).

In the proof of this result, some results on combinatorics of convex polyhedra having simplicial sides (Stanley [1] and Danilov [1]) play an important role (along with some interesting arithmetical properties of Gram matrix elements which are close to the properties of Pissot-Vijayaraghavan numbers, see, for example Cassels [1, Ch. 8]). These results (by virtue of the duality) can be reformulated for simple polyhedra. This allows one to obtain an upper bound for the average number of 1-dimensional sides

in a k-dimensional side of a simple polyhedron P for l < k < 21, see Nikulin [4]: Theorem 3.34. Let P be a convex closed polyhedron in the hyperbolic space 1H[", n >

2k - 1, k > 2, with simplicial angles. Then the average number of vertices of its k-dimensional sides does not exceed 2Cn/ (C[nl2] + Cn-[n/2]) .

This bound implies that, for a given k, the average complexity for k-dimensional sides of a simple polyhedron P C Hn tends to that for the k-dimensional cube as the dimension n increases. This observation for three-dimensional sides plays a key part in proving the above Nikulin's finiteness theorems. Let us now consider the following particular case of Nikulin's bounds. Theorem 3.35. The average number of vertices (sides) of a two-dimensional side for a simple convex polyhedron P C Hn is less than

( 4(n - 1)/(n - 2) for even n , t 4n/(n - 1) for odd n. In other words, this means that a simple polyhedron in the high dimensional hyperbolic space HI should have a large number (in a certain sense) of quadrangular and triangular two-dimensional sides. Vinberg [9, 10] used this fact to obtain the following finiteness theorems (cf. Theorem 3.33) which, with Nikulin's results, are promising a classification of all reflective forms.

Theorem 3.36. There are not reflective forms in the hyperbolic space Hn having dimension n > 30. In other words, in H, n > 30, there are neither co-compact nor arithmetic reflection groups.

Theorem 3.37. For n > 22, there are no arithmetic co-compact reflection groups in the hyperbolic space )E]In having the base fields different from Q(%), Q(v) and Q(cos 2rr/7).

Theorem 3.38. For n > 14, there are no arithmetic co-compact reflection groups in Hn having base fields different from

Q6/2), Q(J), Q(/), Q(4), Q(J, -/3-), Q(8, -13), and Q(cos 2 r/m) , where m = 7, 9, 11, 15, 16 or 20.

3.9. Arithmetic groups generated by reflections

139

Using a generalization by Khovanskii [1] of Nikulin's estimates of the side complexity for simple polyhedra (Theorems 3.33 and 3.34) in the case of almost simple polyhedra (including finite volume polyhedra), Prokhorov [1] extended Vinberg's finiteness theorem to the general case of crystallographic groups with unbounded fundamental polyhedra as follows:

Theorem 3.39. There are no crystallographic reflection groups in the hyperbolic space lA(n of dimension n > 995.

5. Applications to algebraic geometry. Research on crystallographic and arithmetic groups is important and interesting not only in itself but also for algebraic geometry. We describe probably the first and certainly the most interesting example of such an application. This is related to type K3 surfaces, which are non-singular simply connected projective algebraic surfaces over C whose canonical class is zero. Examples are:

1) a two-sheet cover over the projective plane CIP2 branched over a non-singular curve of degree six; 2) an intersection of three quadrics in CIP5; 3) an intersection of a quadric and a cubic in CIP4; and 4) a quartic in CIP3, the so-called Kummer surface which can be represented in homogeneous coordinates as follows: K3 = {[Zo, Z1, Z2, Z3] E CIP3

Z :

0 +Zi +Z2 +Z3 = 0} .

(3.66)

For each K3 surface X, there is an intersection form in the lattice of its algebraic cycles, see Chapter 6, §2.3. This is an even integer quadratic form having signature (1, n) where n + 1 < 20 is the dimension of the algebraic cycle lattice. Let fx be the opposite form having signature (n, 1). In 1971, Piatetski-Shapiro and Shafarevich [1] proved the Torelli's global theorem for K3 surfaces X and, as a corollary, obtained a description (in terms of form fx) of the automorphism group of a surface X within a finite central extension. In particular, Aut X is finite if and only if the group O'(fX, Z) contains a finite-index subgroup generated by reflections associated with integer vectors e for which fx (e) = 2, see also Nikulin [1, 2]. Such reflections are called 2-reflections, and an integer quadratic form of the signature (n, 1) whose automorphism group contains a finite-index subgroup generated by 2-reflections is called a 2-reflective form. Algebraic surfaces were classified in classical studies of the Italian algebraicgeometry school early in this century. For that classification, see, e.g., Shafarevich [ 1]. However, those studies left open the problem of description of birational automorphism groups for two classes of algebraic surfaces: K3-type surfaces and Enriques surfaces, the quotients of K3 surfaces by involutions without fixed points. Such a description should include, in particular, a solution of the finiteness problem.

140

3. Basics of Hyperbolic Groups and Manifolds

To solve this problem, Nikulin [1, 2] described all 2-reflective even integer quadratic forms having (n, 1) -signature for n >_ 4. The number of them is finite and they are all related in the above fashion to K3-type surfaces. The maximal possible n for them is 18. The even forms cover all possibilities, since an odd form is 2-reflective if and only if the canonically related even form is 2-reflective. This description was completed by the proofs of finiteness for 2-reflective forms in the cases n = 3 (Vinberg) and n = 2 (Nikulin), see Theorems 1.3 and 2.5 in Nikulin [6]. These results with the finiteness theorems for arithmetic groups complete the description of K3-type surfaces X having finite groups Aut X. Finally, Nikulin's [7] description of Enriques surfaces X having finite groups Aut X completes the description of algebraic surfaces having finite birational automorphism groups.

§10. Non-arithmetic groups of Gromov and Piatetski-Shapiro In contrast with finiteness theorems for arithmetic lattices in the previous section, here we shall describe "the interbreeding construction" of M. Gromov and PiatetskiShapiro [1] of non-arithmetic lattices G C IsomW (both co-compact and co-finite) which works in all dimensions n = 2, 3, ... . These non-arithmetic groups G C Isom ]IF will be obtained by interbreeding two arithmetic torsion free subgroups G 1 and G2 of Isom IHi" (compare § 8) as follows. Let us consider arithmetic torsion free groups G1, G2 C Isom IHl' such that: a) The hyperbolic (finite volume) manifolds Mi = ]EII'1 / Gi , i = 1, 2, contain

connected submanifolds Mt C Mi of dimension n with totally geodesic boundaries Si = 8M+ C Mi. That is, the universal covering of Si is a hyperplane in the universal covering W of Mi (in particular, Si is an (n - I)dimensional hyperbolic manifold). b) The hyperbolic (n - 1)-manifolds S1 and S2 are isometric. Gluing together the submanifolds Mi and M2 according to an isometry between S1 = 8M1 and S2 = 8M2 , we produce "the hybrid manifold" M which carries a natural metric of constant negative curvature coming from those on Mi and M2+. This metric is complete if the lifting of the glum map has no loxodromic fixed points in the boundaries of the universal coverings S1 and S2 (- Hr-1). Then the universal covering of M equals W1 (compare Theorems 7.5 and 7.8), and G is the fundamental

group of M acting isometrically in lIP, M = IEP/G, volM < oo. Note that G is a co-compact group if G1 and G2 are co-compact (i.e., M is compact if Mi, i = 1, 2 are compact). The main idea of Gromov and Piatetski-Shapiro [1], which allows to construct a non-arithmetic group G C Isom lIP by the interbreeding of two arithmetic subgroups Gi and G2, is the idea of taking the non-commensurable arithmetic groups G1 and G2. Two groups G1 and G2 are called commensurable if G1 fl G2 has finite index in G1 as well as in G2. Equivalently, for torsion free groups, there exists a hyperbolic

3.10. Non-arithmetic groups of Gromov and Piatetski-Shapiro

141

manifold admitting locally isometric finite covering maps onto M1 = WIG1 and onto M2 = H/ G2. In the case of non-commensurable groups G1 and G2, the obtained group G cannot be arithmetic. Indeed, arithmeticity of G would imply commensurability between G and G 1 as well as between G and G2 and hence, commensurability between G 1 and G2.

As a source of arithmetic groups G1, G2 C Isom IHI" which we need for a realization of the above idea, we can apply the arithmetic construction of §8 or, more generally, the construction of arithmetic groups O'(f, A) in the previous section. Namely, for

the ring A of integers in a totally real field K C R of algebraic numbers and for a non-singular admissible (see §9.4) quadratic form f of the signature (n, 1), the group 0'(f, A) C GLn+1(118) is the group of f -orthogonal matrices with entries from A. Actually, for our applications, we shall use the fields Q and Q(J). In the latter case, arithmetic groups O'(f, A) are co-compact but, in the case K _ Q, groups O'(f, A) may be either co-compact or non-cocompact but with co-finite volume (the group O'(f, A) is co-compact if and only if f has non-trivial zero in Q, see Venkov [1] and Borel [2]). It is possible to explain the action of O'(f, A) on the hyperbolic space H' in the following way. Given an admissible form f of the signature (n, 1), consider the hyperboloid

S=Sf={x ER"+1 : f(x)=-1}. This hyperboloid S has two connected components isometric to IH' in the metric induced by the pseudo-Euclidean metric f on R'+1. Thus S/ {+ 1, -11 = H' and O(n, 1) = O'(f) acts isometrically on 1H n. We can obtain subgroups of O'(f, A) acting free on lIP in the usual way. Namely, take a prime ideal p in the ring A of integers of the field K and define the congruence subgroup Gp (F) C O'(f, A) by

Gp(.f) _ {g E O'(f, A) : g = id (mod p)} .

(3.67)

It is well known that, for sufficiently large j p j, the group G p (f) has no torsion and its action on 1HIn is free, see Borel [2] and Raghunathan [1]. There is a way to describe totally geodesic submanifolds in arithmetic hyperbolic

manifolds. Namely, take a (k + 1)-dimensional linear subspace Ro C Rn+1 which meets the pseudo-sphere S = Sf C IE8n+1. Then the intersection So = S fl Ro is a totally geodesic submanifold in S of dimension k (see Chapter 1, §3). For a subgroup G c O'(f, A), denote by Go C G the stabilizer subgroup of Ro. If the subspace R0 is K-rational and Go has finite index in G, then Go is arithmetic. That is, the image of Go in the full isometry group O'(f) = O(k, 1) gives a proper immersion So/Go -+ SIG. Lemma 3.40. Let to E O(n, 1) be the orthogonal reflection of R"+1 with respect to R0. If to normalizes the group G, then the canonical map So/Go --> S/G is a proper embedding, provided G has no torsion.

3. Basics of Hyperbolic Groups and Manifolds

142

Proof Suppose two distinct points x and x' from So go to the same point in S/G, i.e., g(x) = x' for some g E G. Since x and x' are fixed by 10, the commutator h = g-11ogIo-1 fixes x. Since 10 normalizes G, this h is contained in G and, as G has no torsion and acts freely on So, we get h = id.

Since So = fix(Io), the equality h = [g, lo] = id implies that g E Go. This completes the proof.

Corollary 3.41. If the group generated by G and IOGIo 1 is discrete and torsion free, then the map So/Go -> S/G is an embedding.

For the proof, it is enough to notice that, for every subgroup G' C G, the map So/ Go -+ SIG' is an embedding provided So/ Go -* SIG is an embedding.

Interbreeding hyperbolic manifolds. Now we can realize the above idea of constructing non-arithmetic (co-compact or co-finite) groups G C Isom H, n > 2. Let fo be a quadratic form in variables xo, ... , xn_1 over K C R of the type (n - 1,1) and let

fi = fo -f- al xn

for ai > 0 in K, i = 1, 2.

(3.68)

Then the reflection Io in the hyperplane Ro = {xn = 0} C Rn+1,

lo: (xo, ... , xn-1, xn) H (x0, ... , xn-1, -xn), lies in O'(fi, A) and the previous discussion applies to the congruence subgroups G p (fi) C O' (fi , A) with sufficiently large I p I (i.e., torsion free subgroups). Therefore, the hyperbolic manifold

M(fo, p) = Hn-1/Gp(fo) (where we identify HI-1 with So/(+1, -1)) embeds isometrically into both manifolds M(fi , p) = lH[n/Gp (fi ), i = 1, 2. Note, that for odd p, all manifolds M(fi , p), j = 0, 1, 2, are orientable. In fact,

if -10 1 (mod p), then Gp(fi) C SO(n, 1), i = 1, 2, and Gp(fo) C SO(n - 1,1). The totally geodesic hypersurface M(fo, p) does not necessary bound in M(fi, p). (In fact, for a large Ip1, it does not bound). However, there exists an obvious double covering M(fi, p) of M(fi, p), such that the lift of M(fo, p) to M(fi, p) consists of two dijs oint copies of M(fo, p) which do bound some connected submanifold MM C M(fi, p), i = 1, 2. (The boundary of Mj+ is the union of two copies of M(.fo, Now, if n+ 1 is even and a1 /a2 is not a square in the field K then the forms f1 and f2 are not similar over K and the groups O' (f1, A) and 0'(f2, A) are not commensurable. This follows from the following fact, see 2.6 in Gromov-Piatetski-Shapiro [1]:

3.10. Non-arithmetic groups of Gromov and Piatetski-Shapiro

143

Commensurability Condition. Let fl and f2 be two forms over K of type (n, 1) for n > 2, such that the corresponding arithmetic groups O'(fl, A) and 0'(f2, A) are commensurable in the following sense. There exists an isometry a of the hyperbolic n-space H1 = Sf,/{+1, -11 onto H2 = Sf2/{+1, -1} which sends some subgroup

of finite index G1 C O'(fl, A)/{+1, -1} (acting on H1) into O'(f2, A)/{+l, -11 (acting on H2). Then the forms ft and f2 are similar over IK, i.e., there exists a linear K-isomorphism II8n+1 _ Jn+1 sending fl to ).f2 for some .l E K.

Corollary 3.42. Let f1 and f2 be diagonal,

fl

n+1

aix2

n+1

and

i=1

f2 = E bix? , i=1

with ai and bi in K.

Then, if (n + 1) is even, the ratio of the discriminants, nn+; ai / iln±1 bi, lies in (K,)2 Proof. A linear transformation over K with determinant D multiplies discriminants by D2 and similarity f Xf multiplies the discriminant of f by xn+1

Example 3.43. a) Let IK = Q and

fl f2

-xo+xi+....+xn-1+xn -xo+xi +...+xn-1+2xn

Then, for (n + 1) even, the groups O'(fl, A) and O'(f2, A) are not commensurable as 2 is not a square in Q. Also note that these groups are not co-compact as fi (x) = 0

forx=(1,1,0,...,0)andi=1,2. b) Let K = Q(,/2-) and

fl = -

x1

- ... + xn-1 + X.

f2 = - v Gx p -+x2 --... -l- xn-1

+ 3xn .

Here again the corresponding groups are not comensurable for (n + 1) even, but now these groups are co-compact (K # Q). Therefore, for (n + 1) even, we have as result that the manifold M obtained by

gluing Mi to M2 along the totally geodesic boundary is non-arithmetic complete hyperbolic n-manifold, i.e., its fundamental group ri (M) = G C Isom Hn is not arithmetic.

Finally, if (n + 1) is odd, we consider a K-rational hyperplane R' C R'+1 normal to Ro. For example, let fo = E n-1 i=O bix?, where b1 > 0, and take R'={x1=0}CllSn+1.

144

3. Basics of Hyperbolic Groups and Manifolds

Then the corresponding hypersurfaces MM C M(f p), i = 1, 2, are normal to M(fo, p). Therefore, their "halfs" Mi n Mi and M 2 n M2 can be glued together and give a totally geodesic hypersurface M' C M. If M is arithmetic, then so is M', see 1.4 in Gromov-Piatetski-Shapiro [1]. But, for n - 1 = dim M' > 2, the manifold M' is non-arithmetic due to the previous arguments ((n - 1) is even), and thus the non-arithmeticity of the manifold M (i.e., of its fundamental group rrl (M) = G C Isom IHI") is established for all n > 3. The case of n = 2 is well known.

§11. Fibonacci manifolds If a group G is only given by a finite presentation, it is usually very difficult or even impossible to decide the simplest questions concerning the structure of G, especially the question whether G is the fundamental group of a manifold. A well-known example are the Fibonacci groups:

F(2, m) = (x1, x2,

, xm; Xixi+l = xi+2, i mod m) ,

(3.69)

which are (as we shall show) the link between certain objects in 3-dimensional topology, in 3-dimensional hyperbolic geometry, in the theory of discontinous transformation groups in rank one Lie groups, and in the knot theory.

The first problem which arises is whether F(2, m) is trivial. This can easily be settled by computing the commutator quotient of F (2, m), which turns out to be always

finite and non-trivial for m # 1, 2. The next question is considerably more difficult. It asks whether F(2, m) is infinite. This has been the concern of most of the existing literature. It is now known that

F(2, m) is finite if and only if m = 1, 2, 3, 4, 5, 7.

(3.70)

The case m = 9, which had been opened for some time, was solved recently by M. F. Newman [1]. R. C. Lyndon [1] used small cancellation theory to prove that F(2, m) is infinite for all m > 11, see also corrections by Chalk-Johnson [1]. However the geometric study of the Fibonacci groups initiated by H. Helling, A. C. Kim and J. L. Mennicke [1, 2] makes the subject much more easier and answers many questions. To see that, we shall construct a nice 3-dimensional complex which has F(2, m) as its fundamental group. Starting from the presentation (3.69) for F(2, m), we take m triangles with the oriented edges labeled as in Figure 38. Then we patch these triangles according to the labelling of the edges to get a connected finite 2-complex with the fundamental group F(2, m). Namely, we have an oriented polyhedron bounded by 2m triangles. For each triangle, define a mate whose boundary consists of m pairs of triangles and such that the triangle inherits opposite

3.11. Fibonacci manifolds

145

orientations from either side. The resulting 3-complex is an orientable pseudomanifold whose fundamental group is F(2, m).

In the case when m is an even integer, m = 2n, the above construction presents a generalized icosahedron in Figure 39 which defines the above pseudomanifold denoted byMn,n1(Mn) = F (2, 2n). Our icosahedron has a central band of 2n triangular faces bounded above and below by n-pyramids, and the pseudomanifold Mn is obtained by identifying pairs of triangular faces as indicated in Figure 39. In particular, M3 and M5 are obtained by identifying faces in a cube and an icosahedron respectively. Q

P2n-2 F2n-2

L\5

F2

F2n

F2n-3

F2n-1

P2n-1

P1

F3

F1 xl

::,a

Figure 39

P3

F4

x3

X5

P5 ZZO

3. Basics of Hyperbolic Groups and Manifolds

146

Applying Theorem 5.11, we have that the obtained 3-complex (the pseudomanifold Mn) is a 3-manifold if and only if its Euler characteristic vanishes. Indeed, for all even m = 2n, we have zero Euler characteristic x (Mn) = 0, and the obtained 3-complex

Mn is always a 3-manifold called a Fibonacci manifold, with nl (Mn) = F(2, 2n). Moreover, due to Helling-Kim-Mennicke [1], these 3-manifolds Mn have complete hyperbolic structures for all n > 4, and M3 is an affine Riemannian 3-manifold. Following to Hilden-Lozano-Montesinos [5], we see that rotation of the polyhedron in Figure 39 by 2n/n about the axis PQ defines a cyclic action Cn of order n on the manifold Mn. The quotient M lCn of the action is homeomorphic to the 3-sphere S3. It may be obtained by taking a fundamental domain for the rotation of our polyhedron and making identifications, see Figure 40.

Q

X1

X1

Figure 40

Figure 41

The image of the axis P Q is the figure eight knot K, and the isotropy group of any point of Mn not on P Q is trivial. Figure 41 shows a Heegaard diagram for this quotient, where the grey line is the axis P Q. It lies below the diagram, inside the ball whose boundary is being identified along the disc pairs F1, F1, and F2, F2. Cancelling handles we obtain the sphere S3 and the figure eight knot K C S3, see (a), (b) and (c) in Figure 42. This completes the proof of the following theorem. Theorem 3.44. For any n > 2, the Fibonacci manifold Mn is the n -fold cyclic covering of the 3-sphere S3 branched over the figure eight knot K C S3. o

3.11. Fibonacci manifolds

147

Figure 42

Let (K, n) be the orbifold with the underlying space S3, with the singular set as the figure eight knot K, and whose isotropy group is cyclic of order n. Here we may consider n = oo which means that (K, oo) is the knot complement S3\K. Due to Thurston [1], such orbifolds (K, n) are hyperbolic for all n > 4. Therefore, this theorem in particular shows that the Fibonacci manifolds M, n > 4, allow hyperbolic structures which yield hyperbolic structures on the figure eight knot complement

S3\K, see also Howie [1]. On the other hand, since it was originally shown by different arguments in Helling-Kim-Mennicke [1] that these 3-manifolds Mn are hyperbolic for all n > 4, and M3 is Euclidean, Theorem 3.44 immediately implies:

Corollary 3.45. The orbifold (K, n) is Euclidean for n = 3, hyperbolic for n > 4, and spherical for n < 3. Actually, for n = 2, the Fibonacci manifold M2 is the lens space L(5, 2). Furthermore, due to Hilden-Lozano-Montesinos [5], we have:

Theorem 3.46. The group of the orbifold (K, n) (and the Fibonacci manifold Mn) is arithmetic exactly for n = 4, 5, 6, 8, 12, oo. As simple applications of the above results, one could now establish many important properties of the Fibonacci groups F (2, 2n) in (3.69). In particular, one has the finiteness (3.70). To see that, it is enough to take the list of possible finite fundamental groups of closed 3-manifolds, and prove (by computing commutator quotients) that F(2, 2n) is none of them for n > 3. We also obtain all properties satisfied by fundamental groups of 3-manifolds now for F (2, 2n). In particular, F(2, 2n) is a noetherian group, which means that every finitely generated subgroup is finitely presented, see Theorem 5.9. Since M3 is an affine Riemannian 3-manifold, the group F(2, 6) is a torsion free finite extension of Z3. Hyperbolicity of closed manifolds M,, for n > 4

148

3. Basics of Hyperbolic Groups and Manifolds

implies that the corresponding Fibonacci groups F(2, 2n) are torsion free, all their Abelian subgroups are cyclic, and all these groups F(2, 2n) have solvable word and conjugacy problems, see §6.5 in Chapter 4. Other interesting applications, due to Vesnin-Mednykh [1, 2], are related to hyperbolic volumes of Fibonacci manifolds Mn, n > 4. We start with an observation that the orbifold (K, n) has a rotational symmetry of order 2 whose set of fixed points is disjoint from the singular set of the orbifold. After factorizing by this symmetry we obtain the orbifold 62(2, n) whose underlying space is the 3-sphere S3 and the singular set is the link 62 (in the notations of Rolfsen [1]) of two components with branching indices 2 and n respectively, see Figure 43.

n

Figure 43. The link 62

In general case, let us denote by 62(m, n), m, n E N U {oo}, the orbifold whose underlying space is S3 and the singular set is the link 62 of two components with branching indices m and n respectively. Observe that the orbifold 62(m, n) can be obtained by the generalized Dehn surgery with parameters (m, 0) and (n, 0) on the two components of the link 62, see §1.5 in Chapter 5. Here the index oo indicates the removal of the corresponding component. In this case we deal with a noncompact orbifold.

So now we have the following diagram of coverings for the Fibonacci manifolds Mn and the orbifolds (K, n) and 62 (2, n):

Mn n (K, n) 2 > 62(2, n)

(3.71)

which implies the corresponding relation for hyperbolic volumes:

vol Mn = n . vol(K, n) = 2n vol 62 (2, n)

.

(3.72)

Now, let us consider noncompact manifolds related to the link 62. Let Tn, n > 2, be the link corresponding to the closed 3-string braid (Qto2 t)n. In this way we have a

3.11. Fibonacci manifolds

149

well-known family Tn of links in S3. In particular, T2 is the figure-eight knot, T3 is the Borromean rings in Figure 13, T4 is the Turk's head knot 818 and T5 is the knot 10123 in the notations of Rolfsen [1]. It was shown by Thurston [1] that the link complements

S3 \ T, n > 2, are hyperbolic and can be obtained as n-fold cyclic coverings of the non-compact orbifold 62 (n, oo). In particular, for their hyperbolic volumes we have: vol (S3 \ T,) = n vol (62 2(n, oo))

(3.73)

.

We can calculate hyperbolic volumes of the above compact hyperbolic orbifolds by means of the Lobachevsky function. To do that, we recall that a 3-dimensional ideal hyperbolic tetrahedron A with four ideal vertices is described (up to isometry) by a single complex parameter z with Imz > 0, see §5.2 in Chapter 4. In this case the dihedral angles of the tetrahedron A = Az equal arg z, arg z -:1 z , and arg l l zZ ; and each value occurs twice, for a pair of opposite edges of A. It is well known (see Milnor [6, 7] and Vinberg [12]) that the hyperbolic volume of an ideal tetrahedron Az is given by

vol(A) = A (arg z) + A (arg z z

1

\

I +A (arg

//J

1 1-z

,

(3.74)

where

A(x) =-J ln12sintIdt x 0

is the Lobachevsky function. It is an odd, ic-periodic function:

A(-x) = -A(x), A(x + n) = A(x). In terms of the Lobachevsky function, due to Vesnin-Mednykh [2], the hyperbolic volumes of the orbifolds (K, n) and 62(2, n) and the manifolds Mn are as follows. Lemma 3.47. The hyperbolic volume of the orbifold (K, n), n > 4, equals

vol(K, n) = 2(A(P + 8) + A(P - 6)), where 8 = n/n and ,8 = 1 arccos(cos(28) - 1/2). This and the diagram of coverings (3.71) imply that, for n > 4 and 8 = it/n and

P = arccos(cos(28) - 1/2), the hyperbolic volumes of the Fibonacci manifolds Mn, z the orbifolds (K, n) and 62(2, n) have the following values:

vol Mn = 2n (A( + 8) + A (P - 8)),

vol62(2,n)=A(8+8)+A(P -8),

(3.75) (3.76)

vol(K, 4) = A (3) ,

(3.77)

vol(K, 6) = 8 A (4) ,

(3.78)

2

vol(K , 10) = 2 (A

I

,r 10

+A ( 10

)

.

(3.79)

3. Basics of Hyperbolic Groups and Manifolds

150

The evaluation of hyperbolic volumes of the link complements S3 \ Tn is due to the following calculation by Vesnin-Mednykh [2]: Lemma 3.48. The orbifold 62(n, oo), n > 2, is hyperbolic and its hyperbolic volume equals

vol(62(n, oo)) = 4(A(a + y) + A(a - y)), where y =,r/2n and a = 1 arccos(cos(2y) - 1/2). This fact and (3.73) immediately give that, for n > 2 and the numbers y = 7r/2n arccos(cos(2y) - 1/2), the hyperbolic volume of the link complement 2 S3 \ T, equals

and a =

vol (S3 \ Tn) = 4n(A(a + y) + A(a - Y)).

(3.80)

The above calculations in Lemmas 3.47 and 3.48, (3.75) and (3.80) immediately prove that the volumes of compact Fibonacci manifolds M2, n > 2, coincide with those ones of our link complements S3 \ Tn, that is with the limit ordinals in the Thurston-Jorgensen description of hyperbolic volumes, cf. Theorems 6.35, 6.47 and Corollary 6.48:

Theorem 3.49. For n > 2, the following equality holds vol Men = vol (S3 \ Tn)

.

In particular, we have:

Corollary 3.50. The hyperbolic volume of the (closed) Fibonacci manifold M4 equals the hyperbolic volume of the figure-eight knot complement S3 \ K. Corollary 3.51. The hyperbolic volume of the (closed) Fibonacci manifold M6 equals the hyperbolic volume of the Borromean rings complement.

Many properties of hyperbolic manifolds are determined by arithmeticity or nonarithmeticity of their fundamental groups, see §§8-10. According to Theorem 3.26, the Fibonacci manifolds M are arithmetic for n = 4, 5, 6, 8, 12 and nonarithmetic for the other values of n. Due to Reid [2], the figure-eight knot T2 is the only arithmetic knot, hence the complements S3\T4 and S3\T5 to the Turk's head knot T4 = 818 and the knot T5 = 10123 are nonarithmetic. Furthermore, it is well known (due to Riley [3]) that the Borromean rings T3 (see Figure 13) is an arithmetic link as well. These facts and the description in Theorem 3.46 of arithmetic members among all Fibonacci manifolds show:

Corollary 3.52. Hyperbolic manifolds with the same volume can be both compact and noncompact as well as arithmetic and nonarithmetic: n M2n S3 \ Tn 2 3

4 5

arithmetic arithmetic arithmetic arithmetic arithmetic nonarithmetic nonarithmetic nonarithmetic

11

Notes

151

Notes Originally, the celebrated observation (Theorem 3.1), which has later been called "Margulis's Lemma", appeared in Kazhdan-Margulis [1], where it was used for aproof of A. Selberg's conjecture on unipotent elements. For different proofs of Margulis's Lemma (used in § 1), its relation to Bieberbach's Theorem (Theorems 2.6 and 3.5) and the more general situation of nilpotent groups, see Thurston [1], Ballman-GromovShroeder [1], Bowditch [1] and Vinberg [7]. The main Theorem 3.9 of § 2, providing a uniform bound for injectivity radii of hyperbolic manifolds (Apanasov [5]) is also related to a deep result on volumes by Kazhdan-Margulis [1]. For the case of surfaces (n = 2) it was proved in Harvey [1], Marden [2], Sturm-Shinnar [1]; see also Wielenberg [1] for another proof which allows an extension for hyperbolic 3-orbifolds. Waterman [1] provides numerical bounds for

these universal constants r(2) and r(3); see also recent bounds in Gehring-Martin [3, 5].

For a description of thin parts of hyperbolic 3-manifolds (Theorem 3.13) see Thurston [1]. For the contrasting result (Theorem 3.14) in dimensions n > 4, see Apanasov [19, 21]. Theorem 3.15 on the existence of precisely invariant parabolic horoballs was proved in Apanasov [3, 14]; in the case of Kleinian groups on the plane, it is well known as Shimizu's Lemma (Shimizu [1], see also Lehner [1]). Corollary 3.16 was suggested by G. Margulis (see Apanasov [36]). The approach in § 5, which links group actions on the set of horoballs and the actions

of these groups on the light cone in the Minkowski space, and related constructions (§§ 6,7) of Minkowski convex hulls and ideal polyhedra decomposition of hyperbolic manifolds are based on Epstein-Penner [1]. The construction of hyperbolic convex hulls (in the case of surfaces) is originally due to J. Nielsen [ 1]. For its extensive using in the case of hyperbolic manifolds (in particular, in Example 3.20), see Thurston [1]. The simplified version of the arithmetic construction in Theorem 3.24 follows the Sullivan Seminar [7]. For its further developments, see Mostow-Tamagawa [1], Borel [1, 2], and also compare § 9. In addition to § 9, the reader may find other arithmeticity criteria for hyperbolic groups as well as good surveys of arithmetic groups in Hilden-Lozano-Montesinos [6], Maclachlan-Reid [1], Neumann-Reid [1], Swinnerton-Dyer [1], and Takeuchi [1]. Besides original proofs of Theorem 3.39 due to Khovanskii [1] and Prokhorov [1], see also Nikulin [9]. There are improvements of finiteness results for reflective hyperbolic lattices. Regarding Theorem 3.36, there are no arithmetic crystallographic reflection groups in hyperbolic spaces H n, n > 22 (Esselmann [1]). Moreover, due to a recent result of Nikulin [9], in dimensions n > 43 there are no arithmetic hyperbolic lattices G over

Z which have subgroups F C G generated by reflections and such that G/ P is a parabolic group. The construction in § 10 of non-arithmetic groups in the hyperbolic space Hn is due to M. Gromov and I. Piatetski-Shapiro [1].

152

3. Basics of Hyperbolic Groups and Manifolds

The Fibonacci groups were introduced by J. Conway [1]; see references for their combinatorial study in Campbell [1]. Their geometric study has been initiated by H. Helling, A. C. Kim and J. L. Mennicke [1, 2] where they discovered that such groups F(2, 2n), n > 3, are the fundamental groups of either Euclidean or hyperbolic closed manifolds. The link of the Fibonacci groups to knot theory (Theorem 3.44), to Euclidean, spherical and hyperbolic orbifolds (Corollary 3.45) and arithmetic properties of those groups are due to Hilden-Lozano-Montesinos [5]. Evaluation of hyperbolic volumes of Fibonacci manifolds and discovering of hyperbolic manifolds with the same volume which can be both arithmetic and nonarithmentic or compact and noncompact is due to Vesnin-Mednykh [1, 2]. For further study, see HildenLozano-Montesinos [7-9], A. C. Kim [2, 3], Kim-Vesnin [2], Maclachlan-Reid [2], Mednykh-Vesnin [2] Vesnin-Mednykh [4] and Vesnin [3].

Chapter 4

Geometrical Finiteness

The subject of this chapter is a central concept in the study of geometric (in particular, hyperbolic) manifolds, the concept of geometrical finiteness. This notion describes simplest possible manifolds with geometric structures as those for which all the interesting geometry goes on in some compact set. The classical and intuitively simplest approach to describing such hyperbolic manifolds (and Riemann surfaces) is due to L. Ahlfors [2, 3] and L. Greenberg [2]. Geometrical finiteness is also related to other types of finiteness, in particular, to the group theoretic and the topological ones. Due to the basic finiteness Theorem 4.16, one can expect a simplest group theoretic condition on the fundamental group of a geometrically finite manifold, namely that this group be finitely generated and even finitely presented. Conversely, what does this group theoretic condition on the fundamental group tell us about the topology and geometry of our manifold? Our main aim here is to answer this question for geometries having conformal realizations.

§1. Classical finiteness for planar Kleinian groups Due to Theorem 1.13, one can study finitely generated discrete Mobius groups G C MSb(n) (which may have torsion) instead of corresponding manifolds whose fundamental groups are finite index subgroups of such Mobius groups.

Definition. We call a discrete group G C Mob(n) geometrically finite if, for its natural action in the hyperbolic (n + 1)-space En+1 (Poincare model), there exists a finite-sided fundamental polyhedron P (G) c 1111n+t First examples of geometrically finite groups are provided by Fuchsian groups on the plane or, in other words, discrete groups G C Isom 11112. A basic fact here is that the class of Fuchsian groups is very clear from the finiteness point of view. Namely, a discrete group G C Isom 11112 is finitely generated if and only if there exists a finitesided convex fundamental polygon P (G) C 1HI2. In the main case, when M =11112/ G

is an oriented hyperbolic surface with finitely generated nl (M) ti G, it turns out

4. Geometrical Finiteness

154

that M consists of a compact surface MM with boundary OMM = Uz Sl consisting of finitely many circles, together with a finite number of parabolic cusps and hyperbolic cylinders, see Figure 44. These ends of the surface M are quotients of horodisks and half-spaces in H2 by cyclic actions of their stabilizers. In other words, these ends are isometric to ends shown in Figures 45 and 46. 00

Figure 44

zyz+1 H2

00

Figure 45

Figure 46

This finiteness for surfaces has many geometric, topological, algebraic and metric consequences. For instance, any finitely generated discrete group G C Isom H2 has the following properties: 1) the surface M =11312/G has a finite topological type;

4.1. Classical finiteness for planar Kleinian groups

155

2) the covering 7r:11112

->. M is branched over at most a finite set of points; 3) the group G has at most finitely many conjugacy classes of maximal finite (elliptic) subgroups or, equivalently, a finite number of disjoint G-orbits of elliptic fixed points; 4) the group G has at most finitely many conjugacy classes of maximal parabolic subgroups or, equivalently, a finite number of disjoint G-orbits of parabolic fixed points; 5) the lengths of hyperbolic translations in G form a discrete subset in R; 6) the intersection of two finitely generated subgroups in G is again a finitely generated group; 7) the limit set A (G) C a1E1I2 consists only of two "good" classes of limit points, the conical points (see Theorem 2.50) and the so-called parabolic cusp points (see § 2 below). Now let us turn our attention to another, bigger class of discrete Mobius groups G C M6b(2) = Isom 1EII3 acting on the extended plane K2 = R2 U fool. By Poincar6

(see [2]), the study of such groups is equivalent to studying their action in the 3dimensional hyperbolic space 1H[3, and hence to studying hyperbolic 3-manifolds. In the case of 3-manifolds, the situation with finiteness is more complicated because there are finitely generated discrete groups G C M6b(2) that are not geometrically finite. A geometrically infinite nature of such groups (that do not have finite-sided fundamental

polyhedra in H3) has been made explicit by L. Greenberg [2]. Such geometrically infinite Kleinian groups G C Mob(2) were shown to exist by L. Bers [3] and B. Maskit [2]. Later, Jorgensen [3] and Jorgensen-Marden [1] have constructed some explicit examples of geometrically infinite groups whose limit sets coincide with 8H 3. As for the action of Kleinian groups on the plane, the situation is essentially rectified by the following fundamental Ahlfors finiteness theorem (Ahlfors [1]):

Theorem 4.1. Let G C Mob(2) be a finitely generated Kleinian group and 0 C 1182 its discontinuity set. Then the quotient St/G is a finite union of connected Riemann surfaces of finite topological type. That is to say, each of these surfaces is conformally equivalent to a compact surface with finitely many punctures. Furthermore, the natural covering n: 0 -k S2/ G is branched over an at most finite set ofpoints in these surfaces S2/ G.

Though Ahlfors's original proof was somewhat analytical, there are its geometric/topological interpretations, in particular by using the convex core of H13/G, see Kulkarni-Shalen [1] and Feighn-McCullough [1]. Another, more geometric formulation of the Ahlfors finiteness theorem is given by a Beardon-Jorgensen [1] result:

Theorem 4.2. A given finitely generated Kleinian group G C Mob (2), there exists a finite-sided fundamental polygon in K2 which need not to be connected and can be taken as the intersection of the closure in IEII3 of a Dirichlet polyhedron Dy (G) C 1H1 and the sphere 1182 = 81ElI3 at infinity.

This variant of Ahlfors's finiteness theorem shows that for a geometrically infinite,

156

4. Geometrical Finiteness

finitely generated Kleinian group G C Mob(2), the infinite set of sides of its fundamental polyhedron P(G) C 18[3 accumulates to points from P* = P(G) fl A(G) not lying at the boundary of the fundamental polygon T (-G) fl a]H[3 (because this boundary

allows only parabolic cusp points). Ahlfors and Sullivan [2] showed that this set P* has zero spherical measure. Also, there is a conjecture that this set P* is finite. It is related to a purely topological finiteness for 3-manifolds, well known as the compact core theorem (Theorem 5.9), which provides a compact submanifold MM C H3/G whose inclusion induces a homotopy equivalence. For such a compact submanifold MM whose boundary contains no 2-spheres, there is a bijective correspondence between the boundary components of 8MM and topological ends of H3/ G, so we deduce that ]EI3/G has only finitely many ends (see details in Chapter 5, §4). In particular, the toroidal components of 8 MM correspond to 7G ® 7G-parabolic ends of lH[3 / G, and we

have finiteness of conjugacy classes of Z ® Z-parabolic subgroups in G. Finiteness of conjugacy classes of cyclic parabolic subgroups in G can be obtained by applying a relative version of the compact core theorem (see McCullough [1] and FeighnMcCullough [1]) to the closure of the complement in II3/G to Z ® 7G-cusp ends. This finiteness of parabolic ends is due originally to D. Sullivan [5] and particularly confirms the conjecture on finiteness of P* = P fl A(G):

Theorem 4.3. Let G C Mob(2) be a discrete group having N generators. Then the number of conjugacy classes of maximal parabolic subgroups of G is finite and bounded by (5N - 4).

The original proof of this Sullivan's theorem is different from the topological approach indicated above and is based on a refinement of the original Ahlfors [1] construction in his proof of finiteness Theorem 4.1. In particular, it uses a construction of Borel series related to G-orbits of parabolic horoballs in TEl3 which are discrete in this case (see Corollary 3.18). Another important part of the original Sullivan's proof uses his ergodic theorem on G-invariant vector fields (see Theorems 7.15 and 7.17). Finally, let us consider the situation with elliptic elements in a finitely generated

discrete group G C Mob(2). Since any such group is virtually torsion free (Theorem 1.13), its elliptic elements belong to at most finitely many conjugacy classes in Mob(2), and there is a bound on their orders. Moreover, due to Feighn-Mess [1], the finiteness of conjugacy classes in G of finite subgroups of G holds for finitely generated discrete groups G C M6b(2). This question on finiteness of such classes was originally posed by Greenberg [4] and, as we will see in Chapter 5, §6, it has a negative answer in the case of other geometries (in particular, in 1EI4 and IH12 X IEEl2 geometries).

§2. Geometrical finiteness in higher dimensions As we will show in Chapter 5, §6, in our study of the finiteness problem for all finitely

generated Kleinian groups G C Mob(n), n > 3, we cannot obtain analogies of the

4.2. Geometrical finiteness in higher dimensions

157

compact core, Ahlfors and Sullivan finiteness theorems. All of them fail in those dimensions. So, in order to obtain a significant class of hyperbolic and conformal manifolds with visible geometric, topological and analytic properties, we shall consider geometrically finite discrete groups G C Mob(n). First of all, we shall discuss correctness of the definition of geometrical finiteness in higher dimensions.

1. Geometrical finiteness and Dirichlet polyhedra. In study of geometrically finite groups G C IsomlHln, n < 3, it is common to define such groups as the groups G having finite-sided (convex) Dirichlet polyhedra Dy (G) C 1HIn centered at points y e IE111. This definition correctly describes the class of geometrical finite groups in dimensions n < 3 because of the following Beardon-Maskit [1] equivalences.

Theorem 4.4. For discrete groups G C Isom 1H13, the following four definitions of geometrical finiteness are equivalent: (i) Some convex fundamental polyhedron P (G) C 1f1<3 has finitely many sides; (ii) All convex fundamental polyhedra P (G) C 11][3 are finite-sided; (iii) Some Dirichlet polyhedron Dy (G) C H3 has finitely many sides; (iv) All Dirichlet polyhedra Dy (G), y E ]El<3, are finite-sided.

A proof of this equivalence will be clearly understood from the future proof in a sequel of a corresponding result for pseudo-Dirichlet polyhedra in any dimensions (Theorem 4.9 and Corollary 4.12), because pseudo-Dirichlet polyhedra in dimension 3 coincide with the Dirichlet polyhedra. In dimensions n > 4, the definitions from Theorem 4.4 are not equivalent as we shall demonstrate below. That is why, in our definition of geometrical finiteness, we require the existence of some (maybe, non-convex) finite sided fundamental polyhedron

P(G) C W. As our construction below shows, the problem comes from parabolic subgroups in discrete groups G C IsomIHln.

Theorem 4.5. For a given dimension n > 4, there are geometrically finite groups G C Isom 1E1' all whose Dirichlet polyhedra are infinite-sided.

Proof We shall construct geometrically finite groups G as required in the claim as free products Z * Z of two cyclic parabolic groups G1, i = 1, 2, generated by irrational parabolic transformations gi E Isom 1E](n. Up to conjugation by hyperbolic isometries, such irrational parabolic transformations are Euclidean screw translations (in the Poincare half-space model 118+), x H UU (x) + at, with infinite order rotations U; preserving translation vectors ai. It is clear that we can restrict ourselves to dimension n = 4 and assume that one of the generators, say gi, fixes the infinity of Rn+. In other words, we assume that gl acts on aH4 =1R3 U {oo} as an irrational screw translation, gt (x) = Ul (x) +al, with an invariant straight line rl parallel to al. Lemma 4.6. Let al c ffil4 be the hyperbolic 2 -plane spanned by the invariant straight line rl C I[83 C aH4 and the fixed point pi = oo of an irrational parabolic transfor-

4. Geometrical Finiteness

158

mation g1 C Isom R3. Then the Dirichlet polyhedron Dy (GI) for the cyclic parabolic group G1 = (gl) C Isom H4 centered at a pointy E IH<4 is finite-sided if and only if

yEal. Proof First, it is clear that Dy (G1) is bounded by two parallel hyperplanes having p1 at infinity and orthogonal to al if the center y belongs to al. Now let us suppose that, for a center y E 1114\o-I, the corresponding Dirichlet polyhedron Dy (G1) is finite-sided, so that n

Dy(G1) = ns+ (g) y is the intersection of finitely many half-spaces bounded by Dirichlet planes Sy (gi ). We can identify the set of all straight rays in IRn with ]R X Sn-1 (0, 1), where the first coordinate fo E R' is the basepoint of a ray P = (flo, S1), and the second coordinate f l E Sn-I corresponds to the direction of the ray fi. Then, assuming the natural spherical metric on Sn-I, we see that Isom IRn acts isometrically on Sn-1 by rotational parts (see Chapter 3, § 1). Also, for a closed convex set P C Rn, we define

P1={f31ESn-1(0,1):f=(SO, P1)EP},

(4.2)

to be the (closed) set of directions of those rays that lie in P. The group G1 acts on the horosphere centered at oo and containing the center pointy by Euclidean isometries. So the intersection of Dy(G1) with this horosphere (= a horizontal plane identified with R3) is a closed convex Euclidean polyhedron, P C R3, the Dirichlet polyhedron for G I -action on the horosphere. Since the center y does not belong to the intersection of 61 and the horosphere (denote this intersection again by rl), we can consider the ray fi = (flo,;BI) in I[83 through y which meets ri orthogonally at its basepoint fio E tl. Moreover, if S+ C 1R3 is any Dirichlet half-space corresponding to half-spaces in (4.1), then the direction 61 of fi lies in int(S+)l C S2(0, 1) because y ¢ rl. Therefore, PI lies in the interior of P1 C

S2(0,1) which is a finite-sided spherical polygon (compare with the Gauss map in Chapter 2, §4.2). Now we see that GI acts non-discretely on S2(0, 1) by rotational (irrational) parts of g E G1, and hence g(P1) fl P1 has non-empty interior for some g E G1. It shows that g(P) fl P has non-empty interior. This contradiction shows that Dy(G1) in (4.1) is in fact infinite-sided. Continuing the proof of Theorem 4.5, we can find ahyperbolic half-space E+ C El"

bounded by a hyperplane E, which is disjoint from all its images under the cyclic parabolic group G 1 and from the 2-plane al C IE314. In fact, if H C lET4 is a hyperplane

orthogonal to al and passing through the fixed point pl = oo of G1, then we can take

4.2. Geometrical finiteness in higher dimensions

159

E+ equal to a hyperbolic half-space disjoint from Qt and lying in the domain D c 1HI4 bounded by H and gl (H) (see Figure 47). Now we define the second irrational parabolic cyclic group G2 as the conjugation of G1 by the reflection x E Isom 1H14 with respect to the hyperplane E C 1H14, see Figure 48. Then a2 = X (al) C R4 is the 2-dimensional hyperbolic plane from Lemma 4.6 where G2 acts without rotations, like a Euclidean translation group, and

o2n61 =0.

Figure 47

Figure 47

Finally, we define G = G, * G2 = G1 * x G 1 z C Isom 1EII4. From the construction

of G, we see that the polyhedron P(G) = D n 1(D) C 1HI4 is a convex fundamental polyhedron for G bounded by four sides. So G is a geometrically finite discrete group. On the other hand, applying Lemma 4.6, we immediately see that any of its Dirichlet

polyhedra D. ,(G) centered at an arbitrary point y E H4 must have infinitely many sides (in neighborhoods of either pt = oc or P2 = x(pl )) because its center y cannot lie simultaneously in both (disjoint) 2-planes al and a2, Q t n Q2 = 0. This completes the proof.

It is worth mentioning that the four-sided fundamental polyhedron P(G) C 1EII4 for the group G that we constructed above is in fact the pseudo-Dirichlet polyhedron Dz (G) for some z E I114, which has been defined in Chapter 2, §3. Namely, if the plane E has equal distances to planes H and gl (H), we can choose z to be the middle point of the segment I of a G1-invariant horocycle rl c E1 C H4 between H and g1(H). This point z projects orthogonally to the middle point of the segment 1(I) C I(rl) between planes 1(H) and $2x(H) where 1(al) = r2 is a G2-invariant horocycle. This observation will be crucial in our higher dimensional analogue of Theorem 4.4 (for pseudo-Dirichlet polyhedra). 2. Parabolic cusps. As we have already seen before, parabolic fixed points p E 8IEl' of a discrete group G C Isom HI may be of two types. The good one corresponds to

160

4. Geometrical Finiteness

the case of Theorem 3.15. An example of a bad one is given by the parabolic fixed point p of the group in Theorem 3.14, where we even have no horoball in JBI'1 centered

at p and precisely invariant for a parabolic subgroup fixing this point p. For a given parabolic point p E 81H1",i.e.,apoint fixed byaparabolictransformation

g e G, we have its stabilizer in G, Gp = {g E G : g(p) = p}. Since p cannot be fixed by a loxodromic element of the discrete group G (Lemma 2.10), its stabilizer Gp is a maximal parabolic subgroup in G and is conjugate by a hyperbolic isometry h, C Isom II8" preserving the upper h(p) = oo E 8R+, to a discrete group F = half-space IIB+. So we need to recall some facts about actions of such groups on RI-1 (identified with horospheres in R+ centered at oo). As in the proof of Theorem 3.2, we have the canonical decomposition, 1[8n-1 = V1 ® V2 ® V3, where F = acts on (4.3) Vi ® V2 =n [X E Rn--t : deuci(x, g(x)) is minimal} hGph-t

hGph--t

ger by translations, the subspace V1 corresponds to its (parallel) orbits that foliate V1® V2, and V3 is the orthogonal complement to the subspace (4.3) in R` 1. Furthermore, due to the Bieberbach Theorem 2.6, we have:

Lemma 4.7. Let r c Isom R" be a discrete group and r c lf8" a F-invariant subspace. Then r is the affine span of a IF-orbit, r(x), if and only if r/ r is compact. Moreover, any two such minimal F -invariant subspaces are parallel, and the perpendicular translation between them commutes with the r-action.

Similarly, for any minimal r-invariant horocycles r,, ap C lHI' (each of them is the intersection of a horosphere centered at the parabolic F-fixed point p = oo, and the hyperbolic plane a C Hn spanned by p and a minimal IF-invariant affine subspace r C li8n-1\p), we have that iP and rp are parallel Euclidean planes, and their r-quotients are compact. Definition. For a given parabolic group Gp C Isom Hn fixing a point p E 81HIn, or equivalently, for its conjugation r = hGph-1 fixing oo E aIIB+, we fix a minimal (Euclidean) Gp-invariant subspace r C aW \{p}, which is the boundary at infinity of the minimal Gp-invariant hyperbolic plane. Then we define a parabolic r-neighborhood Up (r), r > 0, to be an open subset Up(r) c 1Hl"\{p} such that h(Up(r)) = {x E IEB+ U BR n-1\fOO1 : deucl(x, r) > 1/r}

.

(4.4)

A parabolic fixed point p E 8 H[n of a discrete group G C Isom IIII" is called a parabolic

cusp point (=cusp point) of G if there exists a precisely Gp-invariant parabolic rneighborhood Up (r), Up (r) C ]HI" U 0 (G). We call such Up (r) a cusp neighborhood of p.

The simplest case is when the stabilizer Gp of a parabolic fixed point p has a free Abelian subgroup of rank (n - 1). Then hGph-1 contains Euclidean translations, the

4.2. Geometrical finiteness in higher dimensions

161

whole d1HIn\{p} is the minimal Gp-invariant subspace, and parabolic neighborhoods

U. (r) are horoballs in ffi centered at p. Due to Theorem 3.15, for r > 0 small enough, these horoballs Up (r) are precisely GP-invariant, and p is a cusp point. On the other hand, the parabolic fixed point o0 of the group G C Isom IHIz constructed in the proof of Theorem 3.14 is not a cusp point because some limit points of G are arbitrarily far from Gam-invariant line r.

Theorem 4.8. A parabolic fixed point p of a discrete group G C Isom Hn is a cusp point if and only if the quotient [A(g)\{p}]/GP is compact. Proof. As it follows from the above, we may assume that the maximal free Abelian

subgroup in GP has rank at most (n - 2). In other words, for p = 00 E BII n this means that the minimal Gp-invariant subspace r is a proper affine subspace of pn-1 = a1n\{p}. With this assumption, compactness of [A(G)\{oo}]/G(,, implies the condition in Theorem 3.15 that A(G)\{oo} lies a finite distance apart from the

subspace r, and we have that radii of all isometric spheres 1(g), g E G\G', are uniformly bounded. Therefore, all isometric spheres I (g) (which are centered at belong to the complement of some parabolic neigh9_1 (00) E A(G), g E borhood UO(r) with small enough r > ro > 0. Since g(ext I (g)) c int I (g-1) and Goo(Uo.(r)) = Uoo(r), we see that Uro(oo) is precisely G..-invariant, and p = 00 is a cusp point of G. The converse claim of the theorem is obvious because the Gam-quotient of the complement IEBn-1 \ Ur (oo) of a cusp neighborhood Ur (oo) C En U 92 (G) is always compact.

3. The Beardon-Maskit condition. Here we shall prove the main result of this section, which provides an equivalent definition of geometrical finiteness in terms of the limit set A(G). Originally, Beardon and Maskit [1] showed in dimension n = 3 that this condition on A (G) is equivalent to finite-sidedness of any convex fundamental polyhedron P (G) C IHi3 (Theorem 4.4). As we have already seen, this is not true in higher dimensions.

Theorem 4.9. A discrete group G C Isom H' is geometrically finite if and only if its limit set consists entirely of conical limit points and parabolic cusps. Before proving this claim, let us start with the following observation.

Lemma 4.10. Let P C HI be a finite-sided fundamental polyhedron of a discrete group G, P its closure in IHIn U 8W, and P* = P fl A(G). Then either P* is empty or consists of finitely many parabolic cusp points.

Proof. Due to Lemma 2.51, P* has no loxodromic fixed points. Hence, assuming that a point p E P* is 00 E dl[8+, we can apply the Bieberbach Theorem 2.6 to the stabilizer r = G p C G which is a group of Euclidean isometries in I[8+. So let I'* C P be a finite index free Abelian subgroup (maybe trivial), and for some yt, ... , )lm E r,

P=r*Ur*ylU...Ur*ym.

(4.5)

4. Geometrical Finiteness

162

Since the polyhedron P is finite-sided, there are at most finitely many points in P* which are G-equivalent to the point p = oo, say points gi (oo) E P* where go=id,g1,...,geEG.

Now suppose that oo E g(P) for some g E G and let ,6 be a geodesic ray in int g(P) ending at oo. Then, for some gi, 0 < i < .C, the ray gig-1(fl) tends to oc, and gig-1 E P = G. Moreover, due to (4.5), gig-1 belongs to one of the sets P*hj, hj E G, 0 < j < mt. This implies that F* is not trivial. Otherwise, a neighborhood of oc in ][8n+ would intersect only finite number of polyhedra g(P), and hence oo V A (G). If I'* has rank (n - 1), we have already shown that its fixed parabolic point oc

is a cusp point. So we can assume that I < k = rank r* < n - 2 and denote r a minimal I'-invariant (proper) k-plane in R". From the above, we have that any polyhedron g(P), g E G\G,., belongs to one of finitely many hyperbolic half-spaces SJ C IIB+1 0 < j < mt, which are bounded in the Euclidean sense, or to their images

y E F. It is obvious then that all these polyhedra g(P) lie in a bounded Euclidean neighborhood of the k-plane r, or equivalently, outside some

parabolic neighborhood U,,(r), r > 0. So Uo,)(r) is precisely Gam-invariant, and o p = oo is a cusp point. This completes the proof of the lemma. Proof of Theorem 4.9. Let us assume that G acts in the Poincare ball-model, 0 E int P C B" (0, 1), and A is a geodesic in Bn (0, 1) joining some point z E 8B and the origin. Then P intersects either finite or infinite number of images of the polyhedron P sides (when it goes to z). In the first (finite) case, there exists g E G such that g(z) E p,,, = P n Sii-1(0, 1) and, due to Lemma 4.10, either z belongs to S2 (G) or z is a cusp point. In the second case, z E A (G) and there exist a side S C a P with a sequence {yi } of points yi E S and a sequence {gi } C G of distinct elements such that gi (p) n s = {yi }. Picking a subsequence, we can assume that points yi converge (in Euclidean sense)

to a point y E S. If y E B'(0, 1), Theorem 2.50 (vi) shows that z is a conical limit point. Hence, due to Lemma 4.10, we can assume that the point y E Sr-1 (0, 1) is a cusp point.

Conjugating the group G, we pass to the Poincare half-space model of W where R"-1 our geodesic ;B joins the origin and z E = all8+\{oo}, and y = 00 E S. Let

P be a finite index free Abelian subgroup of the parabolic stabilizer G0 C G. If rank r = n - 1, the orthogonal projection of P onto Rn-1 is bounded, and hence n-th coordinates of points yi E S tend to oo. It implies that Euclidean distances between ends of geodesics gi (a), I gi (0) - gi (z) 1, tend to oo. Considering Euclidean translations

yi E IF so that yi gi (z) belong to some compact in R1- 1, we have a sequence yi gi (fi) of images of the geodesic fl which intersect a compact in R+. So z is a conical limit point. We proceed similarly in the case of rank r = k < n - 1. Here we consider a cusp neighborhood Um (r) c R+U S2 (G) which does not contain G-images of z. As above, we use the fact that lim yn = oo to get a sequence {yi } c F of Euclidean translations

4.2. Geometrical finiteness in higher dimensions

163

such that the images yigi(z) are uniformly bounded in Rn-1 but the other ends of geodesics yi gi (f ), yi gi (0), go to oo. So the geodesics y i gi (8) intersect a compact in

R', and z is a conical limit point. This completes the proof of the necessity of the conditions of the theorem.

To prove the sufficiency, let us suppose that A (G) consists of conical and cusp limit points and consider a Dirichlet polyhedron P = Dy(G) C IH[n. First of all, we shall prove that P* = aP n A (G) is a finite set of parabolic cusp points (where aP is the boundary in IElln).

Due to Lemma 4.10, we have to check only the case of infinite-sided polyhedra. Let P C Bn (0, 1) be an infinite-sided polyhedron and (Si } a sequence of its sides that accumulates at some point z E 3P. Due to local finiteness (F-3) of P, Euclidean diameters of hyperbolic planes Hi containing these sides Si tend to zero. Convexity of P implies that G-images of P he on either side of Hi, so z is a limit point and, due to Lemma 2.51, is in fact a cusp point. Moreover, Theorem 4.8 shows that [A(G) \ {z}]/ GZ is compact. Hence z cannot be approximated by other limit points in

8P, and P* is a finite set of cusps, say pi = z, ... , Pm. Now, having finitely many conjugacy classes of parabolic (cusp) subgroups in G, we can apply our reconstruction of the Dirichlet polyhedron P = Dy (G) (see (2.42) in Chapter 2, §3.4) to a finite-sided pseudo-Dirichlet polyhedron Dy (G). This pseudoDirichlet polyhedron D* (G) is centered at a point y E Y (G) C IHE" and has exactly m parabolic cusp points on the boundary. Let U1, ... , Um be cusp neighborhoods of these cusp points. Then the complement DY *(G)\ U Ui is compact in HI U Q (G), and hence it has finitely many sides. Therefore our proof can be completed by the following statement:

Lemma 4.11. Let G C Isom IEE" be a discrete group whose all parabolic fixed points

are cusp points. Then its pseudo-Dirichlet polyhedron D*(G) centered at a point y E Y(G) C HY has non-empty interior and, in any cusp neighborhood of each of its cusp points, is finite-sided.

Proof First of all, let us find a ball Bn (y, r) of a radius r > 0, contained in Dy* (G). In other words, this ball should be contained in all polyhedra of irrational parabolic subgroups Gi C G. From the contrary, we obtain parabolic cusp subgroups

G1 C G with fixed points po = oo, pi.... on the boundary aP such that lengths of (Euclidean) translations in Gi-invariant horocycles about pi tend to zero as i goes to oo. This means that subgroups Gi have parabolic elements gi whose isometric spheres I (gi) are arbitrary large. This is impossible because pi are cusp points (see Theorem 3.15). Now let p be a cusp point on 8Dy(G), LP a minimal Gp-invariant k-dimensional horocycle about p, and y* an orthogonal projection of y on Lp. Since G-action in a cusp neighborhood Up of p is the action of the stabilizer cusp subgroup Gp C G, it is enough to count the number of sides of the pseudo-Dirichlet polyhedron Dy**(Gp), which is obviously finite if p is rational parabolic fixed point. For an irrational cusp

164

4. Geometrical Finiteness

subgroup Gp, we know (see Chapter 2, §3.4) that:

Dy*(Gp) = [Lp n Dy*(Gp)] X

Rn-k.

(4.6)

This shows that sides of Dy*(Gp) are pairwise Gp-equivalent, and D*(Gp) is finitesided because LP n Dy* (G p) is compact. This completes proofs of both Lemma and Theorem 4.9.

An immediate consequence of Theorem 4.9 is the following generalization of Theorem 4.4:

Corollary 4.12. Let G C Isom IHIn, n > 2 be a discrete group. Then an alternative holds: either all pseudo-DirichletpolyhedraD**(G), y E Y (G) C 1H[n, are finite-sided O or all of them have countable sets of sides. Note that this statement in dimensions n < 3 is about Dirichlet polyhedra because in this case pseudo-Dirichiet polyhedra coincide with the Dirichlet ones.

4. The limit set measure. As an application of Theorem 4.9, we have: Theorem 4.13. A given geometrically finite group G C Isom 1Hln, its limit set A (G) either coincides with the sphere aIEdn or has zero spherical Lebesgue measure.

Proof. Assume that G acts in the Poincare ball-model of lflln. We shall strengthen Theorem 2.50(ii) by showing that there exists a uniform constant K > 0 such that, for any conical limit point z E A (G), there exists a sequence {gi) C G so that, for radii ri > 0 of isometric spheres I (gi),

Iz - gi(oo)I < Kr?.

(4.7)

In fact, as we have seen in the proof of necessity in Theorem 4.9, conical limit points z have appeared in connection with a sequence {yi } of points converging to one of finite

cusp points in the boundary of a fundamental polyhedron P for G. Corresponding constants, which characterize the conical approximation of z, have appeared as lower bounds of I g' (z) - g'(0) I for such sequences. So, our universal constant K > 0 corresponds to the minimum of those (finitely many) constants. Now let us assume that A(G) # Sn-1 (0, 1) and, for any number i, consider a covering of the set of conical limit points by balls of radii Iz - gi (oo) I. Increasing i to oo, we see from (4.7) and from the convergence of the infinite series Egc (see Lemma 5.15) that the (n - 1)-dimensional measure of A (G) is zero because the subset of cusp points is at most countable. It is possible to prove the stronger fact that the Hausdorff dimension of the limit set A(G) C Sn is less than n.

4.2. Geometrical finiteness in higher dimensions

165

Definition. To define the Hausdorff dimension and the Hausdorff measure, let us consider a set X from the Borel o--algebra in Rn. Then, for a fixed a > 0 and each e > 0, we put ha,, (X) = inf E de (Ei )a , where the lower bound is taken over all finite or countable coverings {Ei} of the set X by balls (or cubes) Ei whose Euclidean diameters de(Ei) do not exceed s. Then lime- o ha,e (X) = ha (X) is called the Hausdorff a-measure of X, and the number dimH X = sup{a : ha (X) # 0} is called the Hausdorff dimension of X.

Theorem 4.14. For a geometrically finite discrete group G C Isom Hn, n > 2, the Hausdorff dimension of A(G) is less than (n - 1). Proof. Before we go on to prove this statement, we remark that the geometrical finiteness condition is essential for it. This observation is due to a Sullivan [4] example of a (geometrically infinite) finitely generated Kleinian group G C Isom H3 which has zero Lebesgue 2-measure of the limit set A (G) C 1182 but whose Hausdorff dimension dimH A(G) is two. For simplicity, we shall demonstrate a proof of this theorem for the case of convex co-compact groups G C Isom H. These groups G are characterized by the property (see the next section) that their orbifolds 0 (G) = [Hn U 0 (G)]/ G are compact (such groups act co-compactly on the convex hulls CG\A(G)). Equivalently, these groups are geometrically finite and have no parabolic elements. Our proof is based on the idea of Tukia [3] to use some compactness argument

to prove the following property of the limit set A(G) C RI-1. Let Q be a (n - 1)dimensional cube intersecting A(G), and let K(Q, q), q E N, be the family of equal qn-1 subcubes of Q. Then there exists an integer q such that at least one subcube Q' E K(Q, q) does not intersect A(G). Let dEX be the Euclidean diameter of X C Rn and r = dE(Q fl A(G))/dE(Q). Since, for r < 1/2, we can easily take q = 4 as the desired number, we can assume that r > 1/2. LetsQ be the edge length of Q, zQ thecenterof Q, andzQ = (zQ, sQ) apoint in the upper half-space R , i.e., in Hn. Let us also consider points x, y E Q fl A(G) such that Ix - yj > dE (Q)/2, and a hyperbolic geodesic y joining x and y. Then, for the hyperbolic distance, d, we have:

d(y,zQ) < Co =log2+log(n- 1) + n _-1 . This shows that, for the convex hull CG of the limit set A(G), d(CG, zQ) < Co. In our case, CG/ G is compact, and therefore there exists a compact set K C Hn whose G-orbit, G (K), contains the point z Q associated with any cube Q C 1L8n-1 intersecting the limit set A(G). Now let Qo C Rn-1 be a fixed "standard" cube and A a similarity in 1[8+ such that

A(Q) = Qo. Then, for some h E IsomH", we can represent A in the form A = hg where g E G maps zQ into a point g(zQ) E K. Since h-1(Qo) E K, we see that the

4. Geometrical Finiteness

166

set of such hyperbolic isometries h is a compact subset in Isom ffl[n. Furthermore, we have A(Q n A(G)) = Qo n hg(A(G)) = Qo n h(A(G)) (4.8)

where sets h (A (G)) are nowhere dense in R`1 and h runs over a compact subset in Isom H. This implies the existence of a natural number q with the desired property. This property of q-subdivisions of Q provides us with the following inequality for

the family L(Q) = {Q' E K(Q, q) : Q' n A(G) 0 0}: dE(Q')' < CdE(Q) of

(4.9)

Q'E.C(Q)

where This inequality (4.9) remains true for slightly smaller a < n -1 and a slightly larger a=n-1andC=1-l/qn-1

constant C, 1 - 1/qn-1 < C < 1. Therefore, if we pass from £(Q) to the family £(Q') corresponding to Q' E £(Q), the inequality (4.9) and induction arguments will show that dimes A(G) < n - 1.

§3. Equivalent definitions of geometrical finiteness Continuing our discussion on geometrically finite groups G C Mob(n) described before in terms of fundamental polyhedra and the limit set A (G), we come to the context of hyperbolic (n+ l)-dimensional quotients ]H[n+t / G where several equivalent definitions of geometrical finiteness have been worked out. They are due principally to A. Marden [1] and W. Thurston [1] (who studied them for hyperbolic 3-manifolds) and based on properties of ends and convex cores of such quotients.

1. Standard cusp ends. Adjoining to M = W/G the quotient of the discontinuity set Q (G) C aI1F2 of a discrete group G C Isom H, we form the so-called Kleinian manifold/orbifold M(G) = [En U S2 (G)]/G (for detailed study, see Chapter 5). For geometrically finite groups G, we shall demand that these spaces M(G) have finitely many topological ends each, and that such ends can be identified with socalled standard cusp ends each of which is the unique end of M(I'), where I' C_ G is a maximal parabolic subgroup. Namely, such parabolic ends come out due to Margulis's Lemma (Theorem 3.1)

which claims that, for small enough s, 0 < s < s(n), s-thin components of M(G) correspond to parabolic fixed points of the group G. Assuming that such parabolic fixed point p is oo in the half-space model of ]l][n, we have that this s-thin component has the following form (see (3.30)):

Voo(G, s)/Goo = {('x, xn) E 1[8n-1 x 1R

: xn > te('x)}/Goo,

(4.10)

where Goo C G is the stabilizer of p = oo, and the function t8('x) is defined in (3.26)-(3.28).

4.3. Equivalent definitions of geometrical finiteness

167

However, instead of neighborhoods of s-thin ends of the form (4.10), we will use some different neighborhoods of parabolic ends of M(G). Here, by an end of a non-compact manifold M, we mean a family E Ui }i EI of subsets Ui C M such that: (1) each Ui C E is a connected non-empty open set having compact boundary aUi = (c1M Ui)\Ui; (2) if Ui, Uj E E, then there exists Uk E E such that Uk C Ui fl UJ; (3) niE1 Ui = 0; (4) if an open connected set U C M has a compact boundary 8 U and Ui C U for a certain Ui E E, then U E E. Any subset of M that contains a certain element from E is called a neighborhood of the end E. An end E is called isolated if it is not contained in any other end. Due to Husch and Price [1], if M is a 3-manifold with an isolated end E then, for any its neighborhood N, there also exists a neighborhood N of the end E such that:

(El) N C N; (E2) N is closed in M; (E3) N is a connected piecewise linear 3-manifold whose boundary is a connected surface; and (E4) N has exactly one end. The precise meaning of a standard cusp end is the following. Let a discrete group r C Isom IlB-1 act in the upper half-space lR as a hyperbolic isometry group. Then

M(I') = [R+ U IIBr-1]/ I'

(4.11)

is a Kleinian manifold/orbifold with a unique topological end er which we call a

standard cusp end. Assuming that rank r = k, 1 < k < n - 1, and the space Rn-1 = ]ISk x jpn-k-1 splits into r-invariant subspaces according to the Bieberbach

Theorem 2.6 (i.e., IRk is spanned by translation vectors of elements of P), one can describe bases of neighborhoods of these standard cusp ends. Namely, under these

assumptions, the parabolic cusp point p = oo of the group t has a cusp domain Up(s) C IR U I[8n-1 (see (4.4)):

Up(s)={xERn :xk+1+...+xn>

E,

xn>0},

s>0.

Then the standard cusp end er of the orbifold M(r) in (4.11) has a base of sneighborhoods consisting of suborbifolds Up(s)/ 17, s > 0. Note that each sneighborhood of a standard cusp end of rank k is finitely covered either by a punctured

solid torus Tn or by a punctured solid cylinder C,n,k, where k = n - 1 or k < n - 1, respectively.

Let G, G' C Isom IHIn be two given discrete groups with ends e and e' of their Kleinian orbifolds M(G) and M(G'). We call these ends equivalent (conformally or isometrically) if there are neighborhoods U(e) and U(e) which are conformally equivalent to each other (in the orbifold category). Note that these neighborhoods are orbifolds with boundary; their interiors however are isometric hyperbolic orbifolds.

168

4. Geometrical Finiteness

2. Convex core. For a given discrete group G C Mob(n), we have a canonical (minimal) hyperbolically convex subspace C(M) C M(G) where the inclusion induces homotopy equivalence. This convex subspace C(M), called the convex core of M(G), is the minimal convex retract of M(G) provided by Theorem 3.19, and has the form:

C(M) = MG = [C(A (G)) \A(G)] /G C M(G)

(4.12)

where the retraction p: M(G) -* C(M) is given by (3.58). For geometrically finite groups G, we shall have compact thick parts of these convex cores C(M). The s-thick part of M may be defined as in Chapter 3, § 1, that is as the set of points at which the injectivity radius is greater than or equal to s > 0. More precisely, using the subgroups GE (x) C G defined in Theorem 3.1, we set:

T, (G) = {x E H"+1 : G, (x) is infinite},

thin-.(M) = T,(G)/G C M(G).

(4.13)

Topologically, this is a disjoint union of s-thin connected components (Margulis cusps

and Margulis tubes). Then we write cusp, (M) for the union of all Margulis cusps (which correspond to conjugacy classes of maximal parabolic subgroups Gp C G) and call it the s-cuspidal part of the orbifold M(G). We write

thick, (M) = clM(M\thin, (M))

(4.14)

and call it the s-thick part of M(G). Another equivalent definition of geometrical finiteness demands that some uniform neighborhood U, (C (M)) of the convex core C(M) have finite volume and the orders of finite subgroups of the group G be bounded (if G has torsion). Due to Theorem 3.2 and the fact that any finite subgroup in G is conjugate to a subgroup of Isom S", we see that the last condition on the order of finite subgroups is equivalent to bounding of orders of all elliptic elements in G. The above condition that the orders of finite subgroups of G be bounded is redundant in some cases. Namely, this condition is implied by the finite volume condition on `U, (C (M)) if either dim M < 3, or M = )El[1+1 / G itself has finite volume. On the other hand, due to Theorem 1.13, we always have a bound on the orders of finite subgroups in G if G is finitely generated. Nevertheless, in any dimension n > 4, that uniform bound of the orders of finite subgroups is essential due to a construction by E. Hamilton [1] of discrete groups G C ]E3[" with the following properties:

Theorem 4.15. For n > 4, there exist discrete (geometrically infinite) groups G C Isom IH[" such that each of them has finite subgroups of arbitrarily large order, and the uniform neighborhood U 1 (C(M)) of the convex core C(M) C M(G) has finite volume.

The idea of the construction. We may assume n = 4. Then we define a sequence [Xi, r , gi }i EN of triples consisting of hyperbolic 2-planes Xl C 1,14, discrete groups

4.3. Equivalent definitions of geometrical finiteness

169

1'1 C Isom IH[4 which act effectively on Xi as subgroups of Isom 1H12, and elliptic elements gi E Isom 1E14 pointwise fixing Xi and having arbitrarily large orders as i goes to oo. Additionally we assume that among those 2-planes Xi the only adjacent ones intersect, Xi fl XJ = 0 for j (i ± 1) and j 0 i, and this intersection Xi n Xi+1 is orthogonal, along a point xi, see Figure 49. Then we define a desired group G as

G=(ri,gi:i (=- N).

Figure 49

In this group G, certain finitely generated subgroups have an amalgamated free product structure (see Chapter 5, §2), and there are finite subgroups of arbitrarily large order. The trick is to define our triples (in particular, the lengths of the hyperbolic segments Li = (xi _ 1, xi)) in such a way that for some K > 0, the uniform 1-neighborhood of the convex core C(M) of the orbifold M(G) is contained in the K-neighborhood of the union of the projection of long geodesic segments 2i C Xi. Furthermore, the groups Fi (which determine lengths of .ti) and elliptics gi need to be defined in such a way that the hyperbolic volume UK V01 (gi) is as small as °° i=1

length(ti)

<00.

Igi

On the base of that choice, U 1 (C(M)) has finite volume.

3. Basic finiteness theorem. The aim of this section is to prove the equivalence of various definitions of geometrical finiteness given above.

4. Geometrical Finiteness

170

Theorem4.16. Fora discrete group G C Mob(n) = Isom1EF+1, n > 2, the following statements are equivalent: GF1. The group G is geometrically finite: some fundamental polyhedron P c IPA +1 for G is finitely sided.

GF2. For some (any) s, 0 < s < s(n), the s-neighborhood UE (C(M)) of the convex core C(M) C_ M(G) has finite hyperbolic volume and there is a bound for the orders of finite subgroups of G. GF3. Forsome(any)s > 0, the s-thickpart thick8 C(M) of the convex core C(M) c M(G) is compact. GF4. The limit set A (G) consists entirely of conical limit points and parabolic cusps.

GF5. The Kleinian orbifold M(G) has finitely many ends, and each such end is conformally (isometrically) equivalent to a standard cusp end. Proof Since we have already proven the equivalence of GF1 and GF4 (Theorem 4.9), it is enough to prove the implications in the following diagram: GF1

GF2

GF5

> GF3

) GF4

Note also that, in all the definitions but GF2 (see Theorem 4.16), we may assume that

the group G is torsion-free. Indeed, in all these cases, G is finitely generated and, applying Selberg's Lemma (Theorem 1.13), we can pass to a finite index subgroup Go C G which has the same limit set A (Go) = A(G), and its convex core C (M0) has similar topology (up to finite covering) as C (M). The fact that G is finitely generated

if it satisfies GF3 (or GF5) follows from the existence of a canonical retraction of M(G) (or C(M)) onto a compact subset. This retraction is obtained by radiation from finitely many (standard) cusp ends.

M. Let P C IHIn+1 = R+ 1 be a finite-sided fundamental polyhedron for the group G. Due to Lemma 4.10, its limit vertices (that is, points of the set P* = P f1 A(G) if it is not empty) are parabolic cusp points. Assuming P* # 0, let us consider oc as one of these cusps. Then we have an alternative - either the GF1

stabilizer Gm C G of this cusp has a finite index Abelian rank n subgroup or the limit set A(G) lies in a set L C 1fin = 8Hn+1 which is Euclidean isometric to the set

{xER n:x2k+

n


t>0,

k=rank Go,)
In the first case of rank Gm = n, the convex hull C (G"') (z)) of the GOO-orbit of any finite limit point z E A(G)\(oo} contains a precisely GO,-invariant half-space

Ha = {x E

i+1

: x.+1 ? a}, a > 0. Hence Ha = CG fl Ha, which implies the

4.3. Equivalent definitions of geometrical finiteness

171

following inequalities:

[C(M) n (Ha/Goo)]

vol u(CG) n Ha n (Ci gi (P) i=1

(4.15)

< vol [Ha n (K x [0, oo))] < oo.

Here K C R" C 8H"+1 is a compact set obtained by the vertical projection of the finite union UT,=1 gi (P) of polyhedra such that the set {gi 1(oo) } , gi e G, is the set of all vertices of P which are G-equivalent to the cusp oo. Finiteness of this set follows from GF1 (Lemma 4.10). Compactness of the projection K is due to rank G. = n. The bound (4.15) shows that the hyperbolic volume of an E-neighborhood of the cusp end of the convex core C(M) corresponding to the parabolic cusp point oo is finite. We have a similar situation in the second case where the convex hull CG belongs to L x [0, oo). Namely, as the set L is bounded in linearly independent (n - k) directions, there exists a half-space Ha C I[8+ 1 a > 0, such that Ha n CG C Ha n (L x [0, oo)), and we have inequalities for vo121, (C(M) n (Ha/G,,.)) similar to (4.15): vol `L

[C(M) n (Ha/G.)]

< vol [`L(, (L x [0, oo)) n Ha n (K x [0, oo))] < oo.

(4.16)

Here K is a compact set obtained by intersecting L and the vertical projection of the finite union of polyhedra, UM 1 gi (P), gi E G, g-1 (00) E P. Compactness of K follows from the fact that vectors of translations in G,,o span the Goo-invariant subspace I[8k C IIS" determining the set L.

Therefore, (4.15) and (4.16) show that the sum of volumes of 8-neighborhoods of all (finitely many) ends of the convex core C(M) is finite. The compactness of the complement of these neighborhoods completes the proof that vol U. (C(M)) < 00. A bound for orders of elliptic elements of G (orders of finite subgroups of G) follows from finite-sidedness of the fundamental polyhedron P C lHl1+1 (a positive lower bound for its dihedral angles) because all defining relations (in particular, of the form gm = 1) can be taken as the edge relations (2.44) in Theorem 2.37.

GF2 = GF3. Suppose that Us (C(M)) has finite volume for some S > 0, and the orders of finite subgroups of G are bounded by some number m. Taking Margulis's

E-decomposition of M = 1+1/G into 8-thin and s-thick parts, 0 < e < E(n), we consider ri = min{S, I. Then, for any point x E thick6M, its 17-neighborhood ` L (x) C M has volume2at most V1 /m where V' is the volume of a hyperbolic j-ball B (z, rl) C 1H["+1 It shows that any packing of disjoint i-balls in M centered at

some points of c (M) n thick, (M) has at most m vol [UB (C (M) )] /v' elements. So, considering such a maximal packing, we see that (271)-balls with the same centers cover the E-thick part, c (M) n thick, M, of the convex core C (M). It follows that this s-thick part is compact.

4. Geometrical Finiteness

172

GF3 = GF4. Suppose that thick£C(M) is compact and let P C Jn+l be a fundamental polyhedron for the group G convex at its limit vertices, that is, for each z E P* = P n A(G), there exists a horoball BZ C 1n+1 centered at z and such that P n BZ contains all geodesic rays joining z with points x EP n B. It is obvious that all Dirichlet, isometric and pseudo-Dirichlet polyhedra (see Chapter 3, § 3) possess this type of convexity.

Lemma 4.17. If thickkC(M) is compact then, for any fundamental polyhedron P C Iffn+1 of the group G convex at its limit vertices, the set P* = P n A(G) is either empty or consists of parabolic fixed points.

Proof. As we have already observed, compactness of thickC(M) implies that the group G is finitely generated. So, passing to a finite index subgroup, we may assume that G is torsion-free. This finite index subgroup corresponds to a finite covering

of M whose s-thick part of the convex core is still compact. We can obtain the convex core C(M) by identifying G-equivalent sides of the polyhedron PH, PH = P n (CG\A(G)), where P is the closure of P in IEIIn+1 U aHn+1 If P* 0, we can assume that Hn+1 = R+ 1 oo E P* and PH contains the following geodesic ray o-,

a={x ER++1 :x=t-en+1, t> 1} CintPH. Let {yi} be the sequence of points yi E o- with the distance d(yi, yi+1) = 1, y' = en+l - exp(i), i > 0, and yi E Isomffl[n+1, yi(x) = x - exp(-i). We claim that there exists a sequence {gi } C G such that

Jim yi gi y;-1 = id

i-+co

(4.17)

.

To construct such a sequence, we follow the main idea of the proof of Theorem 3.9

(compare Kazhdan-Margulis [1]). Namely, for any s > 0, we apply compactness of thick6C(M) to find a natural number iE such that, for all i > iE, there are elements gi, E E G that translate points y t 1(en+l) = yi E o, into s-close points. These elements gi,s correspond to non-trivial s-small loops in C(M). We now take a small enough s > 0 and, for any i, obtain arbitrarily many those elements gi,g E G that translate the point yi = y,-1(en+l) into s-close points. Then, in the tangent space to IHIn+1, we consider the corresponding frame and, among its images, select s-near ones. This choice corresponds to selecting (among elements gi,s E G) of an element gi E G whose derivative at yi differs at most by s from the identity and d (y1, gi (yi)) < s. So we have a sequence {gi } C G with the property (4.17).

At the same time, we have d(en+i

,

Yigi+l Y;

1

(en+1)) = d

en+1 e

1

,

Yi+1gi+1Yi+1(en+1

e

))

,

which together with (4.17) implies that limi,w d (en+1, Yigi+1 Yl 1(en+1)) = 0 This property shows that, for i increasing to oo, the sequence { yi gi+i Yj 1 } converges to a non-degenerate transformation, and we can apply Corollary 3.12 to the

4.3. Equivalent definitions of geometrical finiteness

173

sequence { yi G y1 1 } of discrete torsion-free groups. It follows that, for large enough

i > io, elements Yi gi yi 1 and Yi gi+1 Yj 1 have a common fixed point. Obviously, for i > io, elements gi and gi+1 have a common fixed point.

Now we have that d (yi, gi (yi)) = d(en+1, Yigiyi-1 (en+1)) and, due to (4.17), limi_*oo d (yi , gi (yi )) = 0. Hence the geodesic rays gi (v) have a convergence point. Images of a geodesic in a discrete group having a common fixed point and consisting either of parabolic elements or of loxodromic ones converge either to that fixed point or to the common axis of the loxodromic elements. At the same time, fixed points

of loxodromic elements in the discrete group G cannot lie at the boundary of the fundamental polyhedron P (Lemma 2.51). So, for i > io, all elements gi are parabolic

transformations, and the point 00 EP * = P fl A(G) is their common fixed point. This completes the proof of the lemma. Corollary 4.18. Under the conditions of Lemma 4.17, there are at most finitely many distinct G-orbits of parabolic fixed points. Moreover, all such points are parabolic cusp points.

Proof Since each G-orbit of a parabolic fixed point is represented by a limit vertex p E P* = P fl A (G) of a fundamental Dirichlet polyhedron, it is enough to prove that P* is finite and its points (if any) are parabolic cusps. Let us again assume that P* ; 0 and no E P*. Due to Lemma 4.17, we have that the stabilizer Gm of this limit vertex of P is a parabolic group of rank k, 1 < k < n, acting in 118'+ 1 by Euclidean (screw) translations. Let E C Rn c aRn 1 be a G'0invariant k-space spanned by those translation vectors. If k < n and A (G) lies in some Euclidean t-neighborhood UL (E) C R' of E then, due to Theorem 4.8, no is a cusp point. Suppose now that k < n but there are limit points ym E A(G) arbitrarily far from E. Assuming E = li8k, we have: n

lim E (yi `)2 = oo .

m->oo

i=k+1

Applying elements from Gw, we may assume that these points ym lie in the projection of P onto Rn, that is in a fundamental polyhedron P (Go,) fl R'. Then we connect the

points ym to no by geodesics lying in the convex hull CG = C (A(G)) and see that each such geodesic intersects PH = P f1 CG. This fact contradicts the compactness of thick, C(M) due to the description of s-thin components obtained in (3.26)-(3.29). Hence no is in fact a cusp point, and P* consists only of cusps. Cutting off disjoint cusp neighborhoods Uz of all (cusp) points z E P* from the polyhedron PH, we must obtain a compact subset. It implies finiteness of the set P* and completes the proof of the corollary.

Now, continuing the proof of GF4, suppose z E A(G) and let or C CG be a geodesic ray tending to z. If this ray is projected by 7r: H'+1 -+ H'+1/ G = M to

4. Geometrical Finiteness

174

thin, M, that is or C T, (G) (see (4.13)), then or C T, (I,) where r C G is a maximal loxodromic or parabolic subgroup. As in the proof of Lemma 4.17, it follows that z is either a parabolic or loxodromic fixed point. In the first case, z is in fact a cusp point due to Corollary 4.18. In the other case, z is a conical limit point. Suppose now that a Z T, (G), that is an unbounded part of o- is projected by .rr into thick, (M) n c (M). It implies thatr (or) accumulates somewhere in this compact and hence, due to Theorem 2.50(VI), z is conical limit point. This completes the proof of GF4. Now we have the equivalence of the first four definitions of geometrical finiteness. To prove equivalence of them to the last one, GF5, we firstly analyze ends of the convex

core C(M) of M = ][i(n+1/G. Due to GF3 and Margulis's Lemma (Theorem 3.1),

we have that non-compact (cuspidal) components of thin,C(M), 0 < s < s(n), are disjoint neighborhoods of the ends of C(M) that correspond to (finitely many) conjugacy classes of maximal parabolic subgroups G1,..., Gm C G.

Let p = oc be a cusp point fixed by such a subgroup Gw C G acting in the half-space R+1. Then the corresponding component of thin,C(M) is as follows: (CG n Voo(G, s)] /Goo = [CG n {('x, xn+1) E Rn x 1[B..F : xn+l > te('x)}] /Goo,

where the parabolic s-neighborhood VOO(G, s) is specified by the function t, (x) (see

Chapter 3, § 3) which satisfies (3.26)-(3.29). Also, if rank GW = k, 1 < k < n -1, we may assume that the convex hull CG lies in a layer of finite width R: {x

+1:IxiI
Therefore, there are precisely Gam-invariant half-spaces {x E IRn+1 : xn+1 > 1/8} C {x E 1i8+ : Xn+1 > 1/8'} ,

that is, horoballs Bs (oo) C Bsi (oo) C H'+1 centered at p = oo with radii 8 and 8', 0 < 8 < 8', such that Bs (oo) n CG c VOO (G, s) n CG C Bs (oo) n CG .

(4.18)

This fact and the finiteness of a number of cusp ends prove the equivalence of compactness of thick,C(M) and compactness of the following subspace of C(M):

CS (M) = [CG \ U Bs(p)]/G,

(4.19)

P

where the union is taken over all parabolic points p of the group G and Bs (p) are disjoint precisely GP-invariant horoballs with sufficiently small radii 8 = 8(s) > 0, which form a G-invariant family.

4.4. Geometrically finite ends and coverings

175

Before finishing the proof of Theorem 4.16, we remark that, due to inclusions (4.18), the orbifolds thick, C(M) and CS (M) are homeomorphic and differ from each other by at most finitely many compacts [CG n (Vp(G, s) \ BS(p))] /Gp.

GF3. Let U1, ..., Urn C IHln+1 U Q (G) be standard disjoint parabolic s-neighborhoods that cover neighborhoods of distinct cusp ends el, ..., e,,, of M(G) (standard cusp ends), and G1, ... , Gm C G be the stabilizers in G of those parabolic cusp neighborhoods. Then, for a given 8 > 0, we take sufficiently small size s, e > 0, of the standard cusp s-neighborhoods (see (4.4)) such that the following inclusion GF5

holds:

p(M(G)\UUi/Gi)=C(M)\Up(UilGi)?CS(M), i

(4.20)

i

where p is the canonical retraction p: M(G) -a C(M) (see (3.60)). Continuity of p, (4.20) and compactness of the complement M(G)\Ui Ui/Gi then imply compactness of Cs (M). Due to the previous discussion in (4.18), this is equivalent to compactness of 8'-thick part of C(M), so GF3 holds.

GF1 = GF5. Due to Lemma 4.10, a geometrically finite group G has at most finitely many conjugacy classes of maximal parabolic subgroups and all parabolic fixed points are cusp points. Because of that, we can consider a G-invariant set {g(Ui) : i = 1, ... , m; g E G} of disjoint parabolic s-neighborhoods for all cusp points of G where m is the number of conjugacy classes of parabolic subgroups. Then the quotient space,

Mc=[(fin+1U0(G))\U{g(Ui): i = 1,...,m; gEG}]/G, is a compact in M(G) whose complement consists of m components Ui/Gi, i = 1, . . . , m, where Gi C G is the stabilizer of Ui in the group G. Obviously, these components Ui / Gi serve as neighborhoods of the standard cusp ends of M(Gi ),

i = 1, ... , m. This completes the proof of GF5 and the whole theorem.

§4. Geometrically finite ends and coverings In this section, we shall give first applications of geometrical finiteness. We start with direct simple consequences of the basic finiteness Theorem 4.16:

Corollary 4.19. Let G C Mbb(n) = Isom J n+t be a geometrically finite group. Then:

1. There are at most finitely many G-orbits of parabolic fixed points; 2. G has at most finitely many conjugacy classes of finite subgroups; 3. The lengths of hyperbolic translations in G (=loxodromic elements) form a discrete subset in R; 0 4. S2 (G)/G = 8M(G) consists of finitely many connected components.

176

4. Geometrical Finiteness

Corollary 4.20. Let G1, G2 C Mob(n) be commensurable discrete groups, that is G, fl G2 is a finite index subgroup in both G1 and G2. Then G1 and G2 simultaneously are or are not geometrically finite.

Now we describe geometrically finite parabolic ends of M(G) which, due to GF5,

are conformally equivalent to the standard cusp ends of (u+ 1 \{oo})/ F where I' c Isom W1 are discrete. As neighborhoods of these cusp ends, we can take the standard parabolic s-neighborhoods US(E)/ F, see (4.4). For parabolic cusps of the maximal rank, that is if I' is virtually free Abelian group of rank n, the boundary of Up(s) in the half-space ][8+ 1 is the horosphere Sm(s) = {x : xn+l = 11. Otherwise, for cusps of non-maximal rank k, 1 < k < n, the boundary of UM (s) is homeomorphic under the F-equivariant vertical projection in 118+1 to a strip L = Rk X B C S, (s) where Rk is the minimal r-invariant subspace and B is a closed (n - k)-ball. Since I' acts on each horosphere S,,o (s) by Euclidean isometries, we can apply the Bieberbach Theorem 2.6 in order to describe all cusp ends of geometrically finite orbifolds M(G).

Namely, in the first case of maximal rankF = n, we have that am (U,, (s)l F) is a closed Bieberbach n-orbifold En finitely covered by a Euclidean n-dimensional torus Tn = S1 x ... X S1. It is clear that the cusp neighborhood Up(s)/ F itself is homeomorphic to En x (0, 1] and is finitely covered by the so-called punctured solid torus:

7-n+1 = Tn x (0, 1] .

(4.21)

An interesting question whether all Euclidean (flat) closed (n -1)-manifolds can be realized as the ends of hyperbolic n-manifolds, can be answered in dimensions n = 3, 4, see B. Nimerishiem [1, 2] and D. Ivansi6 [1, 3]. Namely such flat surfaces are dense in the corresponding Teichmtiller spaces if n = 3. Moreover, all flat 3-manifolds appear as cusps of hyperbolic 4-manifolds. However, in contrast to cusp ends of hyperbolic 3-manifolds whose cusp neighborhoods can be taken as knots/link complements in compact 3-manifolds, not all cusp ends of finite volume hyperbolic 4-manifolds have neighborhoods that are embeddable as surface complements in compact 4-manifolds, see Ivansi6 [1, 3] and Chapter 6, §3. In the second case of rank I' = k < n, the boundary am (U,,, (s)/ I') of a cusp end neighborhood in M(G) is the product Ek x Bn-k of the closed (n - k)-ball Bn-k and a closed Bieberbach k-orbifold Ek finitely covered by k-dimensional Euclidean torus T k. The closure of the cusp neighborhood itself is homeomorphic to Ek x Bn-k x (0, 1 ] and is finitely covered by a punctured solid cylinder (of index k),

en+l,k = Tk X

Bn-k x (0, 1].

(4.22)

Often, even in the case of a torsion-free group G, it is more convenient to have the simplest cusp ends of a Kleinian manifold M(G), that is the punctured solid cylinders and/or tori instead of products of Bieberbach manifolds and open intervals. As the next theorem shows, up to a finite covering of M(G), we can pass to such a situation by using a finite index subgroup of the group G C Mob(n) = Isom H'+'.

4.4. Geometrically finite ends and coverings

177

Theorem 4.21. Any geometrically finite discrete group G C Mob(n) has a subgroup Go C G of finite index such that every parabolic subgroup of Go is free Abelian. In other words, there is a finite covering of M(G) by a manifold M(Go) each of whose cusp ends is either a punctured solid torus or a punctured solid cylinder. Proof. Since geometrically finite groups are finitely generated, we can assume that (up to a finite index subgroup) the group G is torsion free (Theorem 1.13). As in Chapter 3, § 1, we take the rotational part of this group, rot(G). In particular, conjugating each of its parabolic subgroups F to a Euclidean isometry group consisting of elements y, y(x) = V(x) +b where b E 1 \{0}, V E O(n),V(b) = b, we have that the rotational part of y is V, that is we have a homomorphism:

iIr: r -+ O(n),

*(r) = rot(IF) .

(4.23)

Since the group I, is torsion free, this homomorphism induces an embedding ker(*) -+ lin, y (x) = V (x) + b H b, whose image (up to a change of the origin) is a minimal P-invariant subspace Rk C Rn. So ker(4r) is a free Abelian subgroup of rank k in the parabolic subgroup I, C G, and the restriction of the action of the rotational part rot(r) to the subspace Rk is a finite group. The following lemma provides an important inductive argument.

Lemma 4.22. Let r C Isom 1R' be a torsion free discrete group, F a finite nonAbelian group, and q5: r - F an epimorphism. Then the order of the rotational part of ker(o) is strictly smaller than that of the group r.

Proof. The kernel N = ker(ilr) of the homomorphism * in (4.23) and the rotational part rot(r) are connected by the following exact sequence

1--NCI'--+ rot(I) -+ 1. This exact sequence and the epimorphism o: r -+ F induce a commutative diagram with exact rows: 1

N

1 - q5 (N)

> r) rot(r)

>1

> F

>

> F/O(N)

1

In this diagram, the map ¢ is surjective because 0 is epimorphic. Another observation we have from analyzing this diagram is that, if the image of ker(o) under the map

1/r: r --> rot(r) is the whole rotational part, then the image of ker(o) under the composition of maps r -+ rot(r) --* F/o (N) must be F/q5 (N). On the other hand, the image of ker(o) under the composition of maps r -> F -* F/O(N) is trivial. This implies that F = 0 (N), which contradicts to the fact that F is non-Abelian while 0(N) is Abelian. It shows that Vi(ker(4)) is a proper subgroup of rot(r).

4. Geometrical Finiteness

178

Now, continuing the proof of the theorem, let us consider a complete list Fi, ... , Fk

of maximal parabolic subgroups in the group G C Isom H+' that represent the conjugacy classes of parabolic subgroups that are non-Abelian. In each group Fi, we can find elements ai, bi E ri that do not commute. Applying the residual finiteness of the group G (as of any finitely generated linear group, due to Malcev [1]) for each

of its elements gi = [ai, bi] 0 1, we have a finite group Fi and an epimorphism 01: G -a Fi such that 4,i (gi) 0 1. Actually, ker 4i is a finite index subgroup in G, and gi ' ker 4,i. Clearly, such groups Fi are not Abelian because

Fi(at), 4,i(bi)] = 4,t ([ai,bi1) = 0i(gi)

1.

Fbe the product us also consider restrictions of 0 to subgroups Fi, ¢ I r.: ri -) 4, (n), where o (Fi) C F 1, we have epimorphisms onto finite non-Abelian groups is finite. Due to 4i (gi)

(Fi) because F(ai), 0(bi)] = 40(gi) = (01 (gi), ... , (1(g1), . . . , (k(gi))

1

Applying Lemma 4.22, we see that the rotational part of ker(4, I r.) = Fi fl ker 0 is of strictly smaller order than that of the original group ri. On the other hand, any parabolic subgroup of the subgroup ker(4,) C G is conjugate in G to a subgroup contained in some Fi fl ker(q5). These two facts together show that the total order of the (IRk-restrictions of) rotational parts of the parabolic subgroups of the finite index subgroup ker(4,) C G is strictly smaller than that of the original group G. Therefore, continuing this process, after finitely many steps we finally get a finite index subgroup Go C G whose all parabolic subgroups are Abelian, and cusp ends of the manifold M(Go) are either punctured solid tori (4.21) or punctured solid cylinders (4.22).

§5. Geometry of tessellation As we have seen in previous sections, a geometrically finite group G C Isom H" defines a tessellation of the hyperbolic space H' by finite sided (fundamental) polyhedra. On the other hand, the classical Poincare theorems on (compact) fundamental polygons and polyhedra P in the hyperbolic 3-space provides conditions on P which guarantee that the group G generated by isometries pairwise identifying the sides of such P is discrete, and that the G-orbit of P gives us a tessellation of the space. Although there are many ways to establish such conditions for a single polyhedron P in the n-dimensional hyperbolic space, we want to consider this situation in a more general context. Namely, we shall start with a given finite set of polyhedra (maybe, non-convex) with side-pairings. Then our goal will be to determine whether this set can arise from a tessellation of the hyperbolic space HI. In other words, whether this situation can arise from a geometrically finite hyperbolic orbifold if we split it into finitely many polyhedra (by cutting along finitely many geodesic hyper-surfaces) and then consider universal covering of the orbifold.

4.5. Geometry of tessellations

179

1. Aleksandrov's complex and space fillings. To begin with, it is natural to work with arbitrary spaces X" of constant sectional curvature rather than with just the hyperbolic one. Moreover, we prefer to start with a finite set of polyhedra in X" which may be generally non-convex. We have several reasons for that. First, if one starts even with a single convex polyhedron, the natural induction inevitable leads one to gluing several

polyhedra in lower dimensions. Second, as we have seen for geometrically finite groups, it is still an open question whether such a group has a convex fundamental polyhedron. Third, as we have seen for the figure-eight knot and the Whitehead link complements, many of the most interesting manifolds are constructed by using more than one polyhedron. Finally, the non-convex polyhedra that arise in practice can be split up into finitely many convex pieces. Now we start with a construction of filling a space X" by polyhedra, which (in the compact case) is essentially due to A.D. Aleksandrov [2].

Let us consider a finite set of sample polyhedra P1, ..., Pm C X' in a space X" of a constant sectional curvature. Starting with this set, we want to define a locally finite cell complex K" formed by polyhedra each of which is geometrically equal to one of the sample polyhedra. This complex may be infinite, and its supporting space, K" 1, is not supposed to be embedded in Xn. In addition to the complex, we specify a continuous map 7r: I K" I --* X" (we call it the projection), which together with the complex K" satisfies the following conditions. Cl Each polyhedron in the complex K" is isometric to one of the sample polyhedra P1, ... , Pm C X" (possible, some of sample polyhedra are congruent each other). We say that these sample polyhedra generate the complex K". C2 Each (n - 1)-side of a polyhedron P E Kn is a side of exactly one another

polyhedron P' E K" . C3 Any two polyhedra P, P' E K" are joined by a polyhedron chain, that is by a finite sequence of polyhedra in K" such that each two of its neighboring polyhedra are adjacent along an (n - 1)-dimensional side.

C4 A k-dimensional side Qk, 0 < k < n - 2, of a given polyhedron P E K" is a side of another polyhedron P' E K" if and only if these polyhedra are joined by a chain whose each two adjacent polyhedra are adjacent along an (n - 1)-side Q containing the k-side Qk. C5 For any P E K", the restriction of the projection n to P is an isometry.

C6 For two given polyhedra P, P' E K' that are adjacent along a side Qn-1, the images 7r(P) and 7r (P') lie (in a neighborhood of the image 7r(Qn-1)) on different sides of rr(Q"-1). For the convex polyhedra, those images lie on different sides of the plane containing .7r(Qn-1); in general case, this is required only for their intersections with a neighborhood of 7r(Qn-1) It is natural to identify a process of filling the space X" by sample polyhedra from a given finite set (a filling of X") with the defined pair (K", pr). In fact, as it has been shown by Aleksandrov [2], the above six conditions on (K", rr) are sufficient for a complete filling (even for for creating of a tessellation) of the space X" by compact polyhedra P1, ... , Pm. More precisely:

180

4. Geometrical Finiteness

Theorem 4.23. Let P1, ... , Pm C X" be compact sample polyhedra generating a complex (Kn, n) with properties Cl-C6. Then 7r(1K") = X". Furthermore, the projection it is injective if and only if it is injective around each co-dimension 2 side of the complex V. In other words, a complex (K", 7t) with conditions Cl-C6 always defines a filling of the whole space X" by compact polyhedra PI, ..., Pm, and this filling is in fact a tessellation of X" provided the co-dimension 2 side condition (which means that the sum of dihedral angles around each co-dimensional 2 side in K" equals 27r). Here a polyhedral decomposition of a space X" is called a tessellation of X" if any point of X" belongs to at least one polyhedron and distinct polyhedra do not have common interior points. In what follows, we consider only regular (periodic) fillings of spaces X", that is those complexes (K", n) whose side pairings of the sample polyhedra generate a symmetry group of (K", 7t) acting transitively on K" in the sense that each polyhedron P E K" is an image of one of the sample polyhedra. In the case when 7r (I K" I) = X" and our filling is a tessellation of X", we have that this tessellation is also regular, and the union of sample polyhedra serves as a fundamental polyhedron (maybe nonconnected) for the tessellation symmetry group G C Isom Xn. We notice however that Theorem 4.23 is true even for non-regular (non-periodic) fillings (K", 7t) because it does not use any symmetry condition on the complex. Moreover, there are tessellation of Euclidean and hyperbolic spaces that are not regular. A well-known example of such a non-periodic tessellation is the Penrose's "two chickens" tessellation of the plane (by two pentagons, see Berger [1, 1.7.1]); see also Margulis-Mozes [1], Makarov [9], Robinson [1, 2] and S. Stein [1] for other examples in the hyperbolic case and in high dimensions. We will prove a generalization of Theorem 4.23 for regular fillings of X" by unbounded sample polyhedra (Theorem 4.29), which requires some additional assumption. Now, let us assume that (K", 7r) (satisfying Cl-C6) is a regular filling of the whole space X", that is 7r(IK" 1) = X". The (isometric) side pairings for sample polyhedra Pi, ... , P. define an orbifold (9 which has a natural (X", Isom X")-structure. Since 7t (I K" 1) = X", this structure is complete and I K" I covers the orbifold (9. On the other

hand, a complete (X", Isom X")-orbifold O has the form of a quotient X"/G where G C Isom X" is the holonomy group, G = d*(7ti`b(O)). Thus it can be obtained by identifying (n - 1)-sides of a fundamental polyhedron P(G) C X". Therefore, we may consider the natural filling (K", 7r) of the whole space XI = IK"I by one sample polyhedron P1 = P (G) where the projection 7t: I K" I = X" -* (9 is the natural covering map, and this filling obviously satisfies CI-C6. Hence we have: Lemma 4.24. Let (K", 7t) be a regular complex satisfying the conditions C1-C6, and let (9 be the orbifold obtained by the identification of (n - 1)-sides of the sample

polyhedra Pl, ... , Pm C X". Then 7t(JK"I) = X" if and only if (9 is a complete orbifold.

4.5. Geometry of tessellations

181

2. Incomplete hyperbolic structures and tessellations. In contrast to the statement of Theorem 4.23 about space fillings by compact polyhedra, the conditions Cl-C6 are insufficient for filling the whole space X1 if some polyhedra in the filling (K", rc) are unbounded. Actually in the hyperbolic case X" =1HI", n > 2, one can find fillings which are incomplete (they only fill proper subdomains of HI). For the first time, this effect was discovered for fillings of the hyperbolic plane TH[2 by

ideal triangles. Indeed, assuming that such an ideal triangle in the half-plane model of JPA2 has one of its vertices at oo, we can consider an isometric identification of its vertical

sides, which is different from a parabolic transformation (= an Euclidean translation in IIB+) fixing oc. This non-parabolic side-pairing can be done by a hyperbolic translation

having oo as a fixed point, that is by a dilation x H Ax, ). > 1, see Figure 50. It is obvious that such a filling fills only a part of the hyperbolic half-plane {x E R+ xj>0}C1H12.

Figure 50

A similar incomplete filling of the hyperbolic 3-space 1H13 by ideal simplices with

vertices at infinity has been observed by Thurston [1, Ch. 4], when he examined (incomplete) hyperbolic structures on the figure-eight knot complement in S3 (see Example 1.11). All such incomplete fillings (K", r) have the following property of "local topological similarity".

4. Geometrical Finiteness

182

Any n-dimensional polyhedron P defines an (n - 1)-dimensional unoriented cell complex that is isomorphic to the boundary of the polyhedron a P in the category of abstract complexes. Any such (n - 1)-complex is called an n-dimensional abstract polyhedron. We say that two polyhedra P and P' have the same combinatorial type, P - P', if the corresponding abstract polyhedra coincide, i.e. if there exists a oneto-one correspondence between sides of P and P' that preserves dimensions and the incidence relations. Now let (K", 7r) and ('K", rc') be two fillings of a space X", whose sample polyand Pl, ... , P;,, correspondingly. We consider their minimal hedra are Pt, ... , connected subcomplexes K C K" and K' C' K" formed by those polyhedra whose projections adjoin (along some k-dimensional sides, 0 < k < n - 1) the sets m

m

U r(Pi) i=1

and

U7r'(Pi'), i=1

respectively. Then we call the fillings (K", n) and ('K", 7r') locally topologically similar, (K", rc) - ('K", n'), if the subcomplexes K and K' have similar adjacency relations, and this similarity is induced by a homeomorphism

m(IKI) -3. n'(IK'I)

(4.24)

In other words, two fillings with complexes (K", 7r) and ('K", ir') are locally similar if their sample polyhedra have the same combinatorial type, the same side-pairing and if, for any two isomorphic cycles of co-dimension 2 sides in those complexes, the sums of the dihedral angles coincide. One can introduce a natural topology on the set of locally topologically similar fillings of the hyperbolic space lfi[", factorized by hyperbolic isometries of the whole space. Namely, two fillings are close if there exists a continuous map 1H[" IHI" such that it locally coincides with homeomorphisms (4.24) and is close to the identity on compacts in H. Let us describe the space of all incomplete regular fillings (Ka, 9ra) in the case of fillings of the hyperbolic 3-space H3 by ideal octahedra, whose corresponding orbifolds in Lemma 4.24 are homeomorphic to the Whitehead link complement Mw = S3 \L,

see Figure 51.

Example 4.25 (Whitehead link). The fundamental group of the Whitehead link complement is

7ri(M.) = {a, b :

(b-1aba-1)(b-ta-lba)(ba-tb-1a)(bab-ta-t)

= 1}.

(4.25)

We represent the link L,,, as in Figure 51 and create a 1-dimensional complex by adding three oriented segments to L,,, as shown in Figure 52.

4.5. Geometry of tessellations

Figure 51

183

Figure 52

Gluing five 2-cells A, B, C, D and E to this 1-complex as shown in Figure 52, we obtain a 2-complex which splits the sphere S3 into two topological balls bounded by our five 2-cells. The boundary of each of 2-cells A, B, C and D consists of three

oriented segments joined by segments of the link L. For the cell E, we have four oriented segments in aE. So, contracting all the unoriented segments of the link L,,, to points (vertices), we transform these two topological balls into pyramids n and II' with quadrangular bases corresponding to the 2-cell E. We can write:

aII=AUBUCUDUE, arl'=A'UB'UC'UD'UE', where the corresponding 2-cells (A and A', etc) differ only in orientation. Identifying E and E' by the identity map, we may assume that rl U II' is an octahedron whose spherical development is shown in Figure 53, and whose vertices correspond to the segments of the link L,,, in the 1-complex in Figure 52. So we may obtain the complement S3\LW by gluing together the corresponding sides of the octahedron with removed vertices, see Figure 54.

Figure 53

Figure 54

In other words, we represent the Whitehead link complement S3 \LW as a complex consisting of one octahedron, four triangles, three edges and no vertices. It is natural

184

4. Geometrical Finiteness

to represent this octahedron as an ideal octahedron in the hyperbolic 3-space, whose vertices pi, 1 < i < 6, are in the sphere at infinity. Then, since all ideal hyperbolic triangles are isometric, we may identify sides of our octahedron by hyperbolic isometries. In this way, we have a filling of JH[3 by ideal octahedra, that is a complex (Ka, 7r«)

satisfying the conditions Cl-C6, whose orbifold O in Lemma 4.24 is homeomorphic to the Whitehead link complement M. = S3 \L,,,. be the space of isometry classes of (incomplete) hyperbolic structures on the Whitehead link complement M. corresponding to regular octahedral fillings (Ka, 7ra) of H3 locally topologically similar to the tessellation (K,,), Yr.) of 1El3 specified by side pairings in Figure 54. Then the space M. can be embedded into C2 as an open domain.

Proposition 4.26. Let J

Proof. In order to find a tessellation (K,,,, jr) wof ]H[3 by ideal octahedra with the given side-pairing, we remark that this tessellation would correspond to a complete hyperbolic structure on M. having finite volume. Hence, due to rigidity of such structures (Theorem 7.11), this tessellation is unique. On the other hand, in the 3complex representing MU , we have four dihedral angles meeting at each of three edges of the complex. It follows that the regular ideal octahedron P,,, C 1H[3, whose all dihedral angles are right angles, and which exists and is unique due to Andreev's Theorem 2.42, generates such a regular tessellation (Km, 7r,_,) of H3. Now, let Pa C H3 be an ideal octahedron with the side-pairing given in Figure 54.

In order to have injectivity of the projection Ira around edges of the complex Ka generated by the octahedron Pa, we have to have, for any given edge Q of Pa, the sum of dihedral angles at equivalent to Q edges of Pa be 2ir. To describe those octahedra

Pa that satisfy this condition, we split up each Pa into four ideal simplices with the common edge (pl, p3), see Figure 54. We notice that each ideal hyperbolic 3-simplex A C H3 may be characterized (up to hyperbolic isometries of H3) by its dihedral angles at any vertex because each two opposite dihedral angles of 0 are equal. To see the last fact, one can rotate an ideal simplex A by angle it around the geodesic orthogonal to both opposite edges of A (see Figure 55). Furthermore, for any vertex of A at infinity, the sum of the three adjacent dihedral angles of A equals to it because a horosphere centered at this vertex orthogonally intersects a A along a Euclidean triangle. Then, assuming (up to a Euclidean similarity) that the intersection triangle lies in the complex plane C in such a way that its vertices are 0, 1 and a point z E C, Imz > 0, see Figure 56, we can follow W. Thurston [ 1 ] and assign to three pairs of opposite edges in an ideal simplex A C H3

the corresponding complex numbers z, v and w from C+ = {z E C : Imz > 0} such that they label edges in clockwise order if we look from a vertex. It follows that these numbers satisfy the identities:

V = 1/(1 - Z),

w = (z - 1)/Z.

(4.26)

Now, for each of four simplices Oi in our splitting of the octahedron Pa C H3, we assign a triple (zi, Vi, wi) E C.3 of complex parameters in the above way so that the

4.5. Geometry of tessellations

185

Figure 55

parameter zi is assigned to the common edge (p1, p3) of the simplices, see Figure 54. Since the sum of dihedral angles at edges of each equivalence class (either (P1, P3) or one of the oriented edges) equals 2rr, we have the following four conditions on our parameters: Z1Z2Z3Z4 = 1,

W1V4Z1Z2W2V3 = 1

V2W3Z3Z4V1W4 = 1,

V1W2V3W4W3V4W1V2 = 1,

together with eight conditions of the form (4.26). All these conditions are equivalent to the following single relation between the parameters zl, Z2, Z3 E C+: (z1 - 1)(z2 - 1)

=

1 - Z1Z2Z3 (4.27)

Z1Z2Z3

Z3 - 1

It is clear that the (unique) tessellation (K,,,, jr) u,of H3 by ideal octahedra P. with right dihedral angles corresponds to parameters z1 = Z2 = Z3 = i. Using (4.27), we can define the desired embedding, gyp: Je,,, - {(Ka, Ira)} + C2, by the formula sp(a) = (z1 (a), z2(a)) where complex parameters z1(a), z2(a) E C+ are such that the solution Z3 (a) E C of the equation (4.27) also lies in the upper half-plane C+. We finally notice that the polyhedra Pa C I E3 that determine these (incomplete) fillings of IH[3 may be alternatively specified by their vertices p1, ... , P6 as well. Namely, up to hyperbolic isometries, we can fix three vertices of Pa as p1 = 0, P2 = 1, P3 = oo. Then (4.27) gives us that the other vertices p4, P5 and P6 of Pa are any points in C\{0} (the boundary plane of the half-space model of E13) that satisfy the following conditions: P5 E

C+'

(P4 - 1)(P4 - P5)

P6

+'

(P6 - 1)(P6 - P5)

P4

=

_

(4.28)

P5P6 In particular, one can take a one-parameter family (in this 4-dimensional space): P4

P1

P2

P5 E

P3

P4

P5

P6

;

t>0.

(4.29)

4. Geometrical Finiteness

186

3. Poincare-Aleksandrov theorem on fundamental polyhedron. As we have seen in the previous section, the conditions Cl-C6 of Aleksandrov's Theorem 4.23 for a tessellation of X" by compact polyhedra are not sufficient for the completeness of a filling (K, 7r) of the hyperbolic space if a sample polyhedron is non-compact. In other words, we may have a (non-compact) polyhedron P C H" with isometric side-pairings satisfying the edge condition (that is, all fillings of neighborhoods of codimension 2 sides of P by images of P are in fact tessellations), and nevertheless this

polyhedron is not a fundamental polyhedron for the group G C Isom H" generated by those side-pairings. Here we shall present an additional (necessary and sufficient) condition to conditions CI-C6 for a filling (K", 7l) of a space X'1 to be complete. This (localfiniteness) condition may be formulated as follows:

C7 Any compact in the space X" intersects at most finitely many images 7r(P) of polyhedra P E K". We call a filling (K", n) of X" locally finite if it satisfies the condition C7. For bounded sample polyhedra P1, ... , Pm C X" generating a complex K", the local finiteness condition C7 for the filling (K", n) follows from the local finiteness of the complex V. However these two conditions are independent in the case of unbounded sample polyhedra. To study this situation when some of sample polyhedra P1, ... , Pm E K" are unbounded, we may restrict ourselves to the case of hyperbolic spaces.

In the remaining Euclidean case, if (K", ar) is a regular filling of the Euclidean space X" =1[8" by unbounded polyhedra, we may apply the Bieberbach Theorem 2.6 to find a k-dimensional affine subspace Rk, k < n, where the filling (K", jr) induces a filling by compact polyhedra. Since this space 1R7 is invariant with respect to the action of the group generated by side-pairings of sample polyhedra Pi in the filling (K", 7r), the conditions of Theorem 4.23 on compact polyhedra 7r (Pi) fl Xk properly work in the case of unbounded Euclidean polyhedra, too. Theorem 4.27. Let (K", 7r) be a regular f i l l i n g o f t h e hyperbolic space H" b y f i n i t e s i d e d s a m p l e polyhedra P1, ... , Pm C E" satisfying the conditions Cl-C6, and O

be the hyperbolic orbifold obtained by side pairings of these sample polyhedra. Then the filling (K", 7r) and the hyperbolic structure on 0 are complete if and only if the filling (K", 7r) is locally finite, that is, if condition C7 holds.

Proof. Splitting (if necessary) sample polyhedra along hyperplanes, we may assume that all sample polyhedra P1, ..., Pm C H" are convex. Regularity of the filling implies that identifications of these sample polyhedra sides generate the holonomy group G C Isomi" of the orbifold 0 = (Ui 7r(P,))/G. We start with a proof of the sufficiency of condition C7. Sufficiency. One can easily prove that ,-r (I K" I) = I" by using the general fact that, for any proper and open map gyp: Y -* X of a topological space Y to a simply connected

space X, cp(Y) = X. Indeed, due to condition C7 and the local isometry of the

4.5. Geometry of tessellation

187

projection Jr (see C5), we have that this projection is I K" I -+ IHIn is open and has a compact pre-image 7r -1 (C) of any compact C C III", that is r is a proper map. However it is useful to give a direct proof of this fact, which shall show possible

reasons for the incompleteness. So, we assume that the orbifold O = (U1 Pi)/G is incomplete, but any compact C C H" intersects at most finitely many polyhedra g(Pi), g E G, 1 < i < m (more precisely, their 7r -projections). Then the holonomy group G is discrete, and the following sets are G-invariant:

Jr(IK"I)=HK=U{g(Pi)\A(G):gEG, 1
O=HK/G and the natural retraction p: O

and

OG=CK/G,

00 C0C-

,

OG which is induced by the G-equivariant retraction

H" -> C(A(G)) in (3.6). The incompleteness of the orbifold 0 provides us with a geodesic ray in (9 which starts from a point x E (0G, goes to an incomplete end of O and has a finite length (Helgason [1, 1.10]). Due to convexity of C(A(G)), we see that this ray lies in the orbifold OG, and hence 0G is incomplete. Let P* = Ui Pi fl A(G). Due to geometrical finiteness of G, P* is finite. For each point p E P*, there exists a uniquely defined minimal hyperbolic distance between an horosphere Sp centered at p and its images g(Sp) with respect to elements g E Gp fixing p. This distance does not depend on a representative p of the equivalence class p* = [p] E P*/G, and we denote this number by d(p*). If d(p*) = 0 for all p* E P*/G, then we can find a G-invariant family of disjoint horospheres Sp, P E P*. Cutting the horoballs bounded by these horospheres from the sample polyhedra Pi, we obtain an orbifold (with boundary) whose retract Op is a compact suborbifold in (9G. Decreasing the size of the horospheres Sp by a hyperbolic

distance t > 0, we obtain another compact suborbifold Ot C OG, Oo C Ot C OG, which is a t-neighborhood of the orbifold Oo in OG. Therefore, we exhaust the orbifold OG by a family of compacts Ot such that Ot+, is the s-neighborhood of Ot in 00. This contradicts to incompleteness of OG and hence shows that, for at least one

p* E P*, d(p*) > 0. In other words, the incompleteness of the orbifold O implies that some limit vertex p of a sample polyhedron Pi C HI is fixed by a loxodromic element g E Gp C G. However, the C7 condition implies that sample polyhedra Pi (their ir-projections) satisfy the F3-condition (Chapter 2, §3), that is Pi is a part of a locally finite fundamental set of the group G. Hence, due to Lemma 2.51, its boundary a Pi does not contain conical approximation points and, in particular, loxodromic fixed points. This contradiction completes the proof of sufficiency of C7.

4. Geometrical Finiteness

188

Necessity. Let us assume that the orbifold 0 = (U i ir(Pi))/G is complete, that is Tr(I K" I) =E[", and the images g(Um 1 n(Pi)), g E G, cover the whole space IHI".

Due to the local finiteness of the complex K" and the regularity of the filling (K", ir), the holonomy group G of the orbifold 0 is discrete. So, for a fixed point yo E Pi, its images gj (yo) with respect to a sequence of distinct elements gj E G tend to the sphere at infinity aIHE". On the other hand, all sample polyhedra Pi are finitely sided and have at most finitely many different dihedral and polyhedral angles. This shows that the multiplicity of the covering of a point x E 1W' by polyhedra g(ir(P1)),

g E G, 1 < i < in, is finite. Hence any compact C C IHI" intersects only finitely many polyhedra g(Pi ), g E G, and the filling (K",ir) is locally finite. Let us emphasize a fact that we actually proved in the proof of Theorem 4.27:

Corollary 4.28. A regular filling (K", n) of the hyperbolic space IHE" by finitely sided sample polyhedra Pt, ... , P," C ]HI" (and correspondingly, the hyperbolic orbifold 0 = (Um 1 it(Pi))/G obtained by side pairing generating the holonomy group G C IsomH") is complete if and only if the boundary at infinity of the polyhedra 7r (Pi), ... , ir(Pm) does not contain fixed points of loxodromic elements of the holonomy group G.

Now we are ready to describe conditions under which a filling (K", i-r) of the hyperbolic space ]HE" by sample polyhedra P1, ... , Pm C IHE" is in fact a tessellation of H". In the case of a single sample polyhedron P C IHE", such conditions are known as Poincare theorem on fundamental polyhedron.

Theorem 4.29. Let (K", n) be a regular filling of the hyperbolic space IHE" by finitesided polyhedra P1, ..., Pm satisfying the conditions C1-C6. Then this filling is a regular tessellation of H" if and only if it is locally finite (condition (C7) holds), and the projection n : I K" I -+ THE" is an injective map around each (n - 2) -dimensional side of polyhedra in K".

Proof. The first part of the theorem about the completeness of the filling (K", n) (which means that 7(JK" 1) = IEiE") follows from Theorem 4.27. To finish the proof, it is enough to show that the property of the projection zr to be a homeomorphism is equivalent to the condition that it is injective in neighborhoods of (n - 2)-dimensional sides in the complex K". Obviously, it is a necessary condition. Sufficiency of this condition will follow from simply connectedness of H" if we prove that it is a covering map.

To prove that it is a covering map, we define special ball neighborhoods in the supporting space IK"1. Namely, for a point y E IK"I lying inside some polyhedron P E K", this neighborhood is a hyperbolic ball B (y, ry) C P centered at y and having a radius ry > 0. For a point y E I K" I lying on the boundary of some polyhedron

P E K", let Qk C aP be a side of minimal dimension k that contains y. Then, in each polyhedron P' E K", Qk C aP', we take a hyperbolic ball sector centered at y and having such radius ry > 0 that the ball sector does not intersect other sides

4.5. Geometry of tessellations

189

of P' that are disjoint from Qk. Taking the same (minimal) radius ry > 0 for all such ball sectors, we can define a ball neighborhood B(y, ry) C I Kn I as the union of all these ball sectors. This definition is correct because of the local finiteness of the complex K".

Furthermore, for the whole G-orbit {y' = g(y) : g E G} of a point y from a sample polyhedron Pi, 1 < i < m, we can find a uniform radius Ry > 0 of all ball neighborhoods B(y', Ry) C IKn I. This is obvious if y E int Pi, since then Ry = ry. For a point y E a Pi there are at most finitely many G-equivalent points y' = g(y), g E G, lying on the boundary of each sample polyhedra Pj, 1 <_ j < m. It follows from two facts: that all sample polyhedra are finite-sided, and that there are no boundary points of sample polyhedra fixed by loxodromic elements (see Corollary 4.28).

So, due to the finiteness of the set G(y) fl (UT, aPi), we may define: ((IImp

Ry=min{ry': y'EG(Y)n(UaPilll{' i=1

Now, for a point x E ffi[" = ,r (I K" 1), we consider a pointy E n-1(x) fl (UT, Pi) and define the (covering) neighborhood U(x) of the point x as the hyperbolic ball Bn(x, Ry) = zr(B(y, Ry)). We shall prove that for any y' E 7t-1(x) our projection it homeomorphically maps B(y', Ry) onto B(x, Ry) = U(x), which means that it is a covering map.

This fact is trivial if y lies inside a sample polyhedron. In all other cases, the boundary a B (y', Ry) consists of (n -1)-dimensional spherical polyhedra cut out from the hyperbolic sphere of radius Ry > 0 by the ball sectors. These spherical polyhedra form a spherical complex Kn-1 with the properties C1-C4, whose supporting space

IKn-11 is the boundary surface BB(y', Ry). The projection it naturally defines a Sn-1 continuous projection rrn-1: IKn-1I to the boundary sphere Sn-1 of the ball B" (x, Ry) C IHI" because it satisfies condition C5, that is the restrictions of it to polyhedra P E K" are isometries. It also shows that rrn-1 satisfies both the conditions C5 and C6, and hence nn-1(I Kn-11) = Sn-1 because of Theorem 4.23 (which can be proved in the same way by using the above construction of ball neighborhoods B(y, Ry) and induction on dimension, see Aleksandrov [2]). We notice that the projection nn-1 is injective around each co-dimension 2 side in the complex Kn-'1. In fact, each such ((n - 3)-dimensional) side is the intersection of 8B(y', Ry) C IK'I and some (n - 2) -side of the complex Kn that contains the point

y'. So, the injectivity of it around (n - 2)-sides in K" implies that 70-1 is injective around each (n - 3)-dimensional side of Kn-1. Therefore, due to Theorem 4.23, n"-1 homeomorphically projects the supporting space IKn-11 = aB(y', Ry) onto the sphere Sn-1 = aB(x, Ry). This, together with the isometric property of it in the ball sectors of B(y', Ry), gives us that it is a homeomorphism of B(y', Ry) C IKnI onto the ball Bn(x, Ry) C IHI". Since Y' E n-1(x) is an arbitrary point, it shows that Tt : I K" I - IHI" is a covering map and hence a homeomorphism. This completes the proof that our filling (K", n) is a tessellation of I1II".

190

4. Geometrical Finiteness

From Theorem 4.29 and Corollary 4.28, we immediately have:

Corollary 4.30. Let G C Isom 111 be a group generated by isometric identifications

of sides of a finite sided polyhedron P C RP. Then G is discrete, and P is its fundamental polyhedron if and only if, for each maximal set of G-equivalent (n - 2)sides of P, the sum of dihedral angles at these (n - 2)-sides is an integer part of 2n, and neither boundary point z E 8P is fixed by a loxodromic element of G.

4. Rigidity of high-dimensional hyperbolic fillings. Let (Ko, 7ro) be a regular tessellation of the hyperbolic space Hn by finite volume polyhedra F1, ..., Pm. Due to the Mostow rigidity, we cannot deform this tessellation, that is, up to hyperbolic isometries, there are no tessellations of IHIz which are topologically similar to (Ko, no) and distinct from it. However, as Example 4.25 and Proposition 4.26 show, we can deform such 3-dimensional tessellations into tessellations of proper subdomains in H3, that is into incomplete fillings (Ka, 7ra) of H3 which are still topologically similar to

(Ko, no)-tessellation. This is the crucial observation in the proof of the non-rigidity theorem by Thurston [1] (see also Milnor [6]):

Theorem 4.31. Suppose that M = E(3/ r is an orientable hyperbolic 3-manifold which is non-compact but has finite volume. Then there exists an infinite sequence of hyperbolic manifolds, Mj = ]HI3/ hl (F), which have strictly smaller volumes and which approximate the original manifold M as j -* oo in the sense that the homomorphisms hj: F -3 PSL(2, C) = Isom+ H3 tend to the inclusion F C PSL(2, C) in the topology of pointwise convergence. Furthermore, the volumes of the Mj tend to the volume of M.

As the Garland-Raghunathan [1] local rigidity shows, this is not true in higher dimensions n > 4:

For a discrete group G C Isom H' with vol(IEIin/G) < oo, n > 4, there are no homomorphisms hi: G -* Isom HI'1 arbitrarily close to the inclusion.

The next rigidity theorem for fillings of the hyperbolic space Hn, n > 4, gives not only local but even global rigidity of deformations.

Theorem 4.32. Let n > 4 and (K', n') be a regular filling of the hyperbolic space H" by finite volume sample polyhedra P1, ... , P111, which is locally topologically similar to a regular tessellation (K, n) of Hn. Then (K', -t') is also a tessellation of Hn and is isometric to the original tessellation (K, n). Proof. Let 0' and 0 be the hyperbolic orbifolds obtained from the sample polyhedra

... , P,' , and P1, ... , Pm of the fillings (K, n) and (K', it'), respectively. Since (K, n) and (K', n') are locally topologically similar, the orbifolds (9 and 0' are P1',

homeomorphic. In particular, their ends are homeomorphic. Due to Theorem 4.16, all ends E of the (complete, finite volume) orbifold 0 are isometric to standard cusp ends, and their neighborhoods U(E) C (9 are homeomorphic to the products U(E) ^_' En-1 x [0, oo) of closed (n -1)-dimensional Euclidean

4.6. Cayley graphs and geometric isomorphisms of discrete groups

191

orbifolds En-1 and the interval [0, oo). In particular, these neighborhoods are finitely covered by solid punctured tori T"-1 x [0, oo). Supposing that (K', 7r') is not atessellation, we have that (K', 7t') is an incomplete filling of W because, due to the local similarity (K", 7t) - ('Ku, 7t'), the projection 7t' is known to be injective around (n - 2)-sides of K'. Then, due to Corollary 4.28, there is an incomplete end E' of the orbifold (9' which is homeomorphic to a complete

end E of (9 and such that the corresponding vertex p' of a sample polyhedron Pi' is fixed by a loxodromic element of the holonomy group G' of the orbifold (9'. Let G'p, C G' be the stabilizer of p' in G', which is in fact a loxodromic (elementary) group.

Since the set of geodesic rays ending at the point p' is invariant with respect to the action of the loxodromic group G'p this action may be considered as a discrete action of an affine group r c Sim(R1-1\{0}) on a horosphere in H'1 centered at p'. Here the origin in 1Rr-1 is identified with the intersection point of the axis of the loxodromic group G'p, and the horosphere, and the group r is a finite extension of an Abelian group of homothetic transformations in Rn-1\{0}. It follows that there are neighborhoods U (E') C (9' of the end E' which are homeomorphic to the product of the quotient space (R1-1\{0})/ r and the infinite interval [0, oo). At the same time, it is well known (see, for example, Goldman [1]) that any such quotient (Jl _1\{0})/ r is finitely covered by the Hopf manifold S1 X Sn-2. Obviously, for n > 4, finite coverings of the Hopf manifold S1 X Sn-2 and torus Tn-1 are not homeomorphic to each other. This contradicts our previous assumption on a homeomorphism of ends E and E', and hence shows that (K', 7t') is a tessellation. The Mostow rigidity theorem completes the proof.

We remark that one can generalize this rigidity Theorem 4.32 to the case of deformations of a tessellation (K, 7t) of H' by finite sided polyhedra of infinite volume. For that, it is sufficient to add the condition on the tessellation (K, 7t) that the stabilizer

Gp C G of each point p E aP, fl A(G) has a free Abelian subgroup of the rank at least three. Then the above proof is generalized by passing to minimal convex retracts of the corresponding orbifolds.

§6. Cayley graphs and geometric isomorphisms of discrete groups Let 0: G -+ H be an isomorphism of two geometrically finite discrete groups of Mobius transformations of S'1, n > 2 (or, equivalently, of hyperbolic isometries of H1+1) The aim of this section is to describe to what extent the isomorphism 0 can be realized geometrically. By geometric realization we mean that, for some G- and H-invariant subsets X and Y in H'+1 U Sn, there exists a homeomorphism f : X -+ Y

such that f (g (x)) = 0(g)(f(x)) for all g e G and X E X. In this case, we say that f induces 0, and f is G-compatible. As we will see in Chapter 7 for the case

4. Geometrical Finiteness

192

X = Y = Sn, the existence of such a G-compatible homeomorphism f : Sn Sn is essential for Mostow's rigidity Theorem whose claim is equivalent to the assertion that such a homeomorphism f can be chosen as a Mobius transformation. We start with studying combinatorial properties of discrete groups, which will be based on properties of their Cayley graphs.

1. Cayley graphs and group completions. Let G be a finitely generated group and E = {g1, . , gm} some finite set of its generators. Then we define the Cayley graph K(G, E) as a 1-complex whose set of vertices is G and such that a, b E G are joined by an edge if and only if a = bgtl for some g E E. Let I g I be the word norm for g E G (i.e., the minimal length of a word in the alphabet E representing g), and p(r) = 1/r2 for r > 0 and p(O) = 1. Then, for two vertices a, b E K(G, E) joined by an edge, we set . .

d(a, b) = min{p(lat), p(IbI)}.

(4.30)

We may extend this definition (4.30) into a metric on the Cayley graph K(G, E) by taking the shortest paths in K (G, E) between its vertices and assuming that each edge is isometric to an interval. This makes K(G, E) into a metric space which can be completed in the Cauchy sense. Let K(G, E) be the Cauchy completion of K(G, E), whose elements are classes of convergent sequences of vertices in K (G, E). Then we define the completion G of a group G as the following (compact) metric space:

G = G(E) = K(G, E)\K(G, Z).

(4.31)

Each point z E G can be represented as a Cauchy sequence {wi} C K(G, E) and even as a shortest Cauchy sequence, that is a sequence {wi } such that I wi I = i and I Wi 1 wi+l I = 1 for all i E N. To show this, one can write each word wi from the Cauchy sequence in a shortest form and then apply the Cantor diagonal construction to find a shortest ("geodesic") path [id, z) C K(G, E). Although the word norm in the group G depends on a choice of the generating set E, any two word norms corresponding to two generating sets E and E' are commensurable in the usual metric sense. This implies that the group completion G = G(E) does not depend on the used set of generators E in the sense that, for another finite generating set E' for G, the completion G(E') is Lipschitz equivalent to G (E). This fact is due to the commensurability of different word metrics on G and the property that, for a given

natural k E N, there are constants K, K' > 0 such that Kp(r) < p(kr) < K'p(r), for all r E N. Moreover, looking at Cauchy sequences in the Cayley graph, one can derive the following general observations:

(1) Let 1 -- H -` G - F ---k 1 be an exact sequence of groups with H a finite group. Then the group completions G and I, with their metrics d- and dr are Lipschitz equivalent, that is there exist a map f : G -* r and some constants L, L' > 0 such that, for any z, z' E G,

L'dG (z, z) < dr (f (z), .f (z')) <_ LdU (z, z).

(4.32)

4.6. Cayley graphs and geometric isomorphisms of discrete groups

193

(2) Let q5: G -+ F be a homomorphism of groups and a quasi-isometry of word

metrics. Then it induces a continuous map : G -f F. In particular, due to the action of G on itself by inner automorphisms, the group G acts on its completion G. (3) Let IF C G be a finite index subgroup. Then F and G are Lipschitz equivalent.

Proof. We notice that the second claim in (2) follows from the first one because an isomorphism q5: H -+ G is a quasi-isometry of word metrics (*, *)G and (*, *)H (where (a, b) = I a-1b1), that is there are constants C, C' > 0 such that

C'(a, b) H < (4 (a), 0 (b)) G < C(a, b)H ;

a, b E H .

(4.33)

In order to prove the last property, we notice that the inclusion i: P -* G of a normal subgroup of finite index is a quasi-isometry of word metrics (see Wolf [2]). On the other hand, we can assume that our subgroup F is normal because it contains a normal subgroup of finite index in G.

Example 4.33. Let G - Fn, be a free group on n generators, n > 2. Then its Cayley graph is a tree, and the completion G is homeomorphic to a Cantor discontinuum. Figure 57 shows a subset of the Cayley graph for G = F2, whose vertices have the word norm at most four. Obviously, the infinite cyclic group Z has completion 2 consisting of two disjoint points corresponding to the ends of Z.

4 Figure 57

Lemma 4.34. The completion G of a finitely generated free Abelian group G of rank greater than one is a point.

Proof. We will prove a more general claim that G ® H is a point for any infinite groups G and H with finite generator sets E1 and E2, respectively.

4. Geometrical Finiteness

194

To do that, we take E 1 U E2 as a generating set for G ® H and see that the inclusions

of G and H in G ® H are isometries of the corresponding word metrics. Then it is

enough to prove that, for any s > 0, there exist n = nE E N and wn E G ® H such that any g E G ® H with length IgI > n lies within distance s of the vertex W, EK(GED H,ElUE2). Let us specify shortest sequences {wi} and {vi} in infinite Cayley graphs K (G, E1)

and K(H, E2), which define some points z E G and y E H, and an even natural number n such that E n/2 1/k2 < s/4. Then, for any element g = a ® b E G ® H with length IgI > n, we have that either Jal > n/2 or IbI > n/2. In the latter case, we join the vertices g and wn in K(G ® H, E1 U E2) by the following path (see Figure 58):

g=a®b

a-1 w

wn®b

b-1 >

wn,

_1

where a

means a path in the Cayley graph given by writing a-1 wn as a word in shortest form and then taking the steps corresponding to this word, from the base point g E K(G ® H, E1, E2). Obviously, this path [g, wn] has the length at most s.

In the former case of lad > n/2, we choose such a large number m E N that 1/m2 < s/(4Jwn 1al),andjoingandwnbythefollowingpath[g, Wn1 (see Figure 59):

A.

H

H

Wn ®Um aED b

wneb

wn

wn

e

G

G

Figure 59

Figure 58

g=a®b b a®vm a 1

-1

-I

Wn®v,n

vm

wn.

Since the path [g, wn] has the length at most s, we have in both cases that g is within the s-distance of wn. So, G ® H is a point.

2. Cayley graphs of hyperbolic isometry groups. For a geometrically finite discrete Mobius group G, we have a finite sided fundamental polyhedron P = P(G) C ]HIn whose side pairings gi, ... , gm generate the group G. Let E = {g 1, . . . , g,,) and assume that the origin in the Poincare ball model of HI is not fixed by any g E G\{1}.

4.6. Cayley graphs and geometric isomorphisms of discrete groups

195

We define an embedding of the Cayley graph K(G, E) in H, F: K(G, E) -+ H", where vertices g E K(G, E) are mapped to g(0), and edges joining vertices a, b E K(G, E) are mapped to the hyperbolic geodesic segments [a(0), b(0)]. In other words, F(K(G, E)) is the graph that is dual to the tessellation of ffll" by polyhedra g(P), g E G. Obviously, the map F is a G-equivariant proper embedding: for any compact C C JHI", its pre-image F-1(F(K(G, z)) fl c) is compact. Theorem 4.35. Let G C Isom W be a geometrically finite group without parabolic elements (that is, G is a convex co-compact group). Then the map F: K(G, E) y H" is a pseudo-isometry of the word metric (*, *) on K(G, E) and the hyperbolic metric d, that is, there are positive constants K and K' such that

(a, b)/K < d(F(a), F(b)) < K (a, b)

(4.34)

for all a, b E K(G, E) satisfying one of the following two conditions:

either (a, b) > K' or d(F(a), F(b)) > K'. Proof. Since the map F is G-equivariant and proper, it is enough to show that there exist constants K, K' > 0 such that, for all g E G, IgI > K', IgI/K < d(O, g(0)) < K K. IgJ.

(4.35)

Due to finiteness of the generating set E, the right-hand side inequality follows from the triangle inequality, with K = max{d(0, g(0)) : g E E}. To prove the left inequality in (4.35), we may assume that the origin belongs to the convex hull CG = C (A (G)). Due to the Finiteness Theorem 4.16 and non-existence

of parabolic elements in G, we have that PP = P fl CG is compact. Let a be the hyperbolic diameter of Pr,, and C = max{IgI : g E G, d(0, g(0)) < 7a}. Given g E G, we take the geodesic segment [0, g (0)] C CG and split it into disjoint subintervals whose each length is 5a, with a possible exception of the last (shorter) interval. Since each end of these subintervals has the hyperbolic distance to the orbit G(0) at most a, the triangle inequality implies that

IgI < C (1 +

d(0,

g(0))1 J

a

(4.36)

Hence, in the case d(0, g(0)) > 1, we have IgI < (C + C/a)d(0, g(0)). On the other hand, for IgI > C + C/a, it follows from (4.36) that d(0, g(0)) > 1. Therefore, we get the first inequality in (4.35) with C1 K'=max{1,C+, a

K=max

/

C C+-) \ a I

1

,

max{d(O,g(0)) : g E E}

.

4. Geometrical Finiteness

196

Remark 4.36. The above proof (essentially based on Cannon [1] arguments for cocompact groups G C Isom HI) actually proves a more general fact: For a given space X with a path metric d and a co-compact action of a group

G C Isom(X, d), the embedding F: K(G, E) y X is a pseudo-isometry, that is (4.34) holds for some constants K, K' > 0. For geometrically finite groups G C Isom Hn having parabolic elements, the embedding F: K(G, E) y HI is not a pseudo-isometry anymore, however the bounds in the following theorem apply.

Theorem 4.37. For a geometrically finite group G C Isom ]HP` and the natural embedding F: K(G, E) --a IIn, there are constants K, K' > 0 such that, for all g E G with IgI > K', (4.37) 21nIgI Proof. Due to the geometrical finiteness of G (Theorem 4.16), it has at most finitely many conjugacy classes of maximal parabolic subgroups, [Gi],... , [Gm] C G, and there are precisely Gi-invariant horoballs B1, . . . , B. C H'1 whose G-orbits are

disjoint: G(Bi) fl G(Bj) _ 0 for i

j. Removing all horoballs g(Bi), g E G,

i = 1, ... , m, from the convex hull CG C IF, we define a new metric space (Ho, do), where

Ho=C(A(G))\(U{g(Bi):gE G, i=1,...,m}UA(G))

,

and the metric do in Ho is the path metric induced by the hyperbolic metric d. Obviously, d (x, y) < do (x, y), and the inequality becomes sharp if the hyperbolic segment [x, y] intersects some of horoballs g(Bi). We may assume that 0 E Ho. Then, for

g E G, we denote h(g) = d(0, g(0)) and ho(g) = do(0, g(0)). Let P C III be a finite sided fundamental polyhedron, 0 E P, and Po = P fl Ho. Then, since Ho is tessellated by compact polyhedra g(Po), g E G, the argument of Theorem 4.35 (see Remark 4.36) shows that there are constants K, K' > 0 such that,

forgEGwith Igl>K', (4.38)

To compare the hyperbolic metric d and the metric do in 1H10, we may assume that G

acts in the half-space model of HI in R+, where some horoball Bi is the half-space {x : x, > 1) = B,,,,. Then we consider two points x, y E IB+

x=(- t2-1,0,...,0,1), y=( t2-1,0,...,0,1), where t > 0 is the Euclidean radius of the semicircle centered at the origin (hyperbolic geodesic) and joining these points. The (Euclidean) segment [x, y]e joining x and y

in the horosphere 8Bm has length do (x, y) = 2 t2 - 1. On the other hand, direct

4.6. Cayley graphs and geometric isomorphisms of discrete groups

197

calculations (compare Beardon [4, 7.26]) show that the hyperbolic distance between

these points is d (x, y) = 2ln( _t2- 1 + t). This implies that d

2

d(x,)

2-1< t22_

Observing that ho (g) and h (g) differ only if the hyperbolic segment [0, g (O)] intersects a horoball a (BI), a E G, we immediately obtain that ho (g) < eh(a)/2. This and (4.38)

show that, for the above constants K, K' > 0 and for g E G with IgI > K', we have that gI/K < eh(g)'2. In turn, this implies the first inequality in (4.37). The second inequality in (4.37) follows from the right-hand side inequality in (4.38) with the same

K > 0. 3. Group completion and the limit set. The aim of this section is to show that, for geometrically finite discrete groups G C Mob(n) (or G C Isom13["+1), the group completion G is closely related to the limit set A (G). Actually, in most of these cases, the group completion works as an internal definition of the limit set.

Theorem 4.38. Let G C Mob(n) be a geometrically finite discrete group. Then there is a continuous G-equivariant surjective map (DG: G - A(G) which is 2-to-1 onto parabolic fixed points of rank one and injective everywhere else.

Proof. As before, we have an embedding F: K(G, E) y Bn(0, 1) = Hn such that a vertex g E K(G, E) corresponds to g(0) E Bn (we assume that the origin is not fixed by any g c- G\{1}). Also, each edge E = [a, b] C K(G, E) joining two vertices a, b E K(G, E), I a I = k, has the length equal either to kz or to (k}1)Z, depending on b.

Assuming that the constants K, K' > 0 are taken from Theorem 4.37, we take some a E G, I a I = k >_ K. Denoting the Euclidean distance between the origin and a (0) by p (a) = de (0, a (0)) and the corresponding hyperbolic distance by h(a) = d(0, a(0)), we obtain from (1.24) that

h(a) = d(0 , a(0)) = In

1 1

+ p(a)

- p(a)

,

p(a) =

a) _ e h(a)

eh(a) +1

.

(4.39)

Now, since the hyperbolic length of an edge F(E) _ [a(0), b(0)]h is bounded by max{d(0, g(0) : g E E}, there exists a constant C > 0 such that the Euclidean length de(F(E)) of this edge is bounded by C(1 - p(a)2). This, together with (4.39) and Theorem 4.37, implies that ,

2

de(F(E)) < 4C(eh(a) +2+e-h(a))-l < 4C k

,

de(F(E)) < 4CK2 (the length of E in K(G, E)) . Therefore, F is a (G-equivariant) Lipschitz map. It induces a G-equivariant continuous

map F: K(G, E)

1t11

+1

= Hn+1 U Sn whose restriction to G gives the desired

4. Geometrical Finiteness

198

A(G) C S. This continuous G-invariant map CDG is surjective because A(G) is the set of accumulation points of the orbit G(0). Now we shall find out where this map IG is injective. Since both G and A(G) do not change under transition to a finite index subgroups, we may assume that G is torsion free (Theorem 1.13). The geometrical finiteness of G gives us a dichotomy of G into two sets: mapping 4>G: G

G = {w : 4>G (w) is a conical point) U {w : (DG(w) is a cusp point} .

(4.40)

Let z = )G(w) be a conical limit point and {wi} C K(G, E) a shortest ("geodesic") sequence which defines w. We claim that there is a subsequence of {wi} such that {wi(0) = F(wi)} converges to z within a finite distance of a hyperbolic geodesic ending at z. To show this, we form a path v consisting of the union of geodesic segments [wi(0), wi+i(0)] C IH[n+l and converging to z. Then, due to (4.38), the distance between any two vertices a, b E v along this path v is commensurable to do (a, b), the distance in HO.

Now let vi = [0, wi (0)] be geodesic segments in W+1 Since their ends converge to z as i --k oc, the number of vertices of the path v lying between 0 and wi (0) and within a bounded distance of the segment vi tends to infinity. This follows from the fact that v must come near vi (between horoballs, see Figure 60), and either vi crosses a lot of horoballs or the length of vi fl Ho tends to oo. This observation implies that a subsequence of {wi (0)) converges to z within a bounded distance of the geodesic ray [0, z) C IHIn+1

z

Figure 60

4.6. Cayley graphs and geometric isomorphisms of discrete groups

199

We are ready to show that, if w, w' E G and (I)G(w) = z is a conical limit point, then w = w'. To do that, we consider those subsequences in the shortest Cauchy sequences {w1 } and {w' } defining the points w, w' E G whose corresponding images {wz3 (0) } and {wi, (0) } converge to z within a bounded distance of the geodesic I

ray [0, z]. Then we can define a sequence {ak} C G, {ak} C {wt1 } U {wi }, which goes j back and forth infinitely often between these two sequences and is a Cauchy sequence

in K(G, E). Obviously, the point a E G determined by {ak} coincides with w and w, so w = w', and' G is injective on the first set in (4.40). In the second case, we have w E G, whose image 4) G(w) = z is a cusp point, and a shortest sequence {w1 } C G defining w. As before, let v be a path in H0, starting at 0 and converging to z. Since z is a cusp point, the path v lies outside of some horoball

Bz centered at z and precisely invariant for the stabilizer Gz C G. Using the fact that the inclusion iz: Gz --* G is a quasi-isometry in the word metric, we can show (similarly to the above arguments) that w E iz(Gz) where the embedding 11z: GZ -> G is induced by iz. Due to Theorem 4.21, we may assume that GZ is a free Abelian group

whose rank is k = k(z), 1 < k < n. Then, if rank Gz > 1, Lemma 4.34 implies that Gz is a point. Hence OG is injective on the pre-image of the set of cusp points whose _ ranks are greater than one. If rank Gz = 1, then Gz = Z, and Gz = Z consists of two points. So, in that case, the map (DG is at most 2-to-1. To prove that (DG is exactly 2-to-1 in this case, it is enough to prove that the map iz: Gz -+ G is injective. Let g be a generator of Gz. Then two points in GZ are defined by Cauchy sequences {gk} and {g-k}. Introducing a metric in F(K (G, E)) C B'' (0, 1) induced by the Euclidean length of paths, we still have that F is uniformly continuous. Hence, to prove injectivity of O G, it is enough to show that the new distance between vertices gk (0), g-k (0) E F (K (G, E)) does not tend to 0. However, since z is a cusp point of rank one, {g'(0)} and {g-k (0) } go out in different directions from a regular neighborhood in a horosphere SZ C H'+1 of a 1-dimensional horocycle about z, which is hyperbolically isometric to the Euclidean cylinder,

{x=(x',t) ER'l xR+:t=1, Ix'I
4. Geometric isomorphisms of Mobius groups. Here we return to the main problem of this section - to what extent an isomorphism of fundamental groups of two manifolds (or of discrete groups) can be geometrically realized, that is induced by some homeomorphism. This problem is also related to the well-known Nielsen Realization Problem. An affirmative solution to the last problem (for hyperbolic surfaces) follows from Theorem 2.15 and originally was due to Kerckhoff [1] (see also Zieschang [1]): Theorem 4.39. Every finite subgroup of the group of isotopy classes of homeomorphisms of a closed surface S of genus g > 1 can be realized as a group of isometries of a hyperbolic structure on S.

4. Geometrical Finiteness

200

The idea of a geometric realization of group isomorphisms can be traced back to the first result by Fenchel and Nielsen [1] for Riemann surfaces. In the discrete group language, it can be formulated as follows (see Marden [4] and Zieschang-VogtColdewey [1]).

Theorem 4.40. Let G, H C Isom+ Ell be non-cyclic finitely generated discrete groups and gyp: G -->. H an isomorphism such that

(i) 0 is type-preserving, and (ii) 0 (g) has an invariant arc in 0 (H) n a1Hl2 if and only if g has an invariant arc in S2(G) n a1HI2.

Then there exists a homeomorphism f: R2 -+ 1fI2 inducing the isomorphism 0, that

is, f (g(x)) = 0(g)(f (x)) for all g E G and x E IEI[2. Here, the isomorphism 0: G -* H is called type preserving if it carries parabolic elements of G bijectively onto parabolic elements of H. As we will see, this condition is actually the main condition for a group isomorphism to be geometrically realizable. To show this, we shall make use of Cayley graphs and group completions.

Theorem 4.41. Let G, H C MSb(n), n > 1, be non-elementary geometrically finite groups and 0: G -> H a type preserving isomorphism. Then there is a unique equivariant homeomorphism ff: A (G) -+ A (H) that induces 0. Furthermore, f, is a restriction of a Mobius transformation in Sn if 0 (G) = 0. Proof Taking the group completions G and H and applying Theorem 4.38, we have the following diagram of maps:

A(G) <

G

G

+H

1H

A(H),

(4.41)

where ¢ is the G-equivariant homeomorphism of the group completions induced by Let 0GI be a right inverse of the map 4)G, which is compatible with the actions of G on A(G) and G. The map (DG1 exists and is well defined due to Theorem 4.38. Furthermore, Theorem 4.38 shows that 4)Gt is one-to-two on the set of parabolic cusps of rank one and bijective on the complement of this set in A(G). Now we define ff = 4H(DG1 and claim that this is the desired G-equivariant homeomorphism of the limit sets. In fact, this map is obviously bijective. Next, due to compactness of G, we have that, for a given neighborhood U of the one- or two-point set cG1(x), x E A(G), there is a neighborhood V of x such that (DG1(V) C U. This implies that ff is a homeomorphism, obviously G-equivariant. Uniqueness property of ff follows from its continuity, together with the observation that the image f4, (x) of the attractive fixed point x E A(G) of some loxodromic g E G must be the attractive fixed point of a loxodromic element 0 (g) E H, because loxodromic fixed points are dense in the limit set of a discrete groups (Corollary 2.13). The last claim, that fp E Mob(n) if 0 (G) = 0, follows from the Mostow rigidity

Theorem 7.11 because the condition 0 (G) = 0 implies that A(G) = A(H) = Sn, and hence both groups G and H are co-finite. 0

4.6. Cayley graphs and geometric isomorphisms of discrete groups

201

Remark 4.42. There is another approach to Theorem 4.41 (see Tukia [5]), based on a construction of a pseudo-isometry F: CG -+ CH of the convex hulls of the limit sets A(G) and A (H) that induces 0. Here a map F: X -* Y of subsets in the (hyperbolic) space with the metric d is called apseudo-isometry if, for some constants C1 > I and C2 > 0, and for any x, y E X:

d(x, y) C1

- C2 < d(F(x) , F(y)) < Cid(x , y) .

(4 . 42)

Then the existence of f o: A(G) - A(H) follows from the following extension theorem which is essentially due to Efremovich and Tikhomirova [1] (see also Thurston [1, 5.9] and Tukia [5]):

Theorem 4.43. Let X and Y be subsets of II1n+1, X hyperbolically convex, and F: X -a Y be their (Cl, C2)-pseudo-isometry. Then, for X' = cl X f1 aw+l and Y' = cl_Y fl alH[n+l (with closures in }El[n+1), the map F can be extended to a continuous map F: X U X' -* Y U Y' in such a way that Fl x, is an embedding. Furthermore, if d(z, F(X)) is bounded as a junction of z E Y,

then F(X') = Y'. With the additional assumption that aHn+1 = II? U {oo}, the map Flx,nRn

is quasisymmetric if either oo ¢ X' U F(X') or F(oo) = oo; in addition, if F(oo) = oo E X', the map Fix,nlsn is 77-quasisymmetric for some homeomorphism rf: [0, oo) -* [0, oo) depending only on C1 and C2. The canonical G-equivariant homeomorphism ff: A (G) A (H) constructed in Theorem 4.41 (or Theorem 4.43) plays a crucial role in the following two theorems (see Tukia [5]) on extensions of G-compatible homeomorphisms in the discontinuity sets to the limit sets. The first result deals with the compact case.

Theorem 4.44. Let G and H C Mob(n) be geometrically finite groups and 0: G -+ H a type preserving isomorphism. Further, let B C Sl (G) be a G-invariant set whose quotient-space B/ G is compact, and f : B -* 0 (H) be a continuous map inducing 0. Then f and the map ff of Theorem 4.41 define together a continuous map

f:A(G)UB-)-ll

,

j

ifxEB, fo(x), if x E A(G),

which is an embedding if so is f.

If, in addition, oo E A(G) and fp(oo) = oo, then there is a homeomorphism no: [0, oo) -* [0, oo) such that, setting no(p) = rlo(p-1)-1 and r10(0) = 0, no(p) I f (x) - f (z) I

<-

I f (y) - f (z)I :5170 p)If (x) - f (z)I

whenever z E A(G) fl Rn and x, y E (A(G) U B) fl lR' , and ly -zI = plx - zl. Finally, in the general case of geometric isomorphisms, we have the following isomorphism theorem due to Tukia [5]:

4. Geometrical Finiteness

202

Theorem 4.45. Let G, H C MSb(n) be geometrically finite groups in R", n > 1,

and 0: G -* H an isomorphism; for n = 1, assume in addition that 0 is typepreserving. Let f : Q (G) - SZ (H) be a homeomorphism of the discontinuity sets inducing 0. Then the isomorphism ¢ is always type-preserving, and the maps f and ff (from Theorem 4.41) define together a homeomorphism f ofR" inducing the isomorphism rp.

In addition, if n > 2 and S (G) = 0, this f is a Mobius transformation. If n > 2, S2 (G) # 0 and if f is K-quasiconformal for some K > 1, then f is Kquasiconformal, too. If n = 1, f (oo) = oo and if f is locally quasisymmetric (this also includes the case Q (G) = 0), then f IR,, is quasisymmetric.

5. Word and conjugacy problems for hyperbolic groups. Another application of Theorem 4.35 (and Remark 4.36) about pseudo-isometric embeddings of Cayley graphs of convex co-compact groups G C Isom lHl" into IV is the Dehn-Cannon solution of the word and conjugacy problems for such groups. These problems can be stated as follows:

For any pair of words w and v in a finitely presented group G = (E; R) with generators E and relations R, (algorithmically) determine whether the elements that w and v define are equal or different, conjugate or not. These problems were imposed in 1912 by Max Dehn [1, 2], who solved them for the fundamental groups of closed surfaces S,, of genus p > 1:

G=(al,b1,...,ap,bp:R=fl[ai,bi]=1). i

Namely, M. Dehn [1, 2] proved that any non-empty word w in the alphabet E = fat,... , bp} that represents the identity element in G canbe shortened by one of two operations:

(i) deleting a trivial word xx from w, or (ii) for some cyclic permutation R' = A A. B-1 of the relation R, with the length of A greater than the length of B, replacing the subword A in w by the word B. If word w cannot be shortened by these operations, w is called Dehn-reduced. Besides the word problem, this Dehn algorithm also solves the conjugacy problem in the surface group G. Namely, due to this algorithm, we can reduce the problem of recognition whether two words w and v represent conjugate elements in G to the same problem for a pair of Dehn-reduced words denoted again by w and v. Then we have that cyclically reduced words w and v are conjugate in G if and only if there exist cyclic permutations w' and v' of them and a word u of length I u 1 < 4p = K such that w' = uv'u-1 in G. Since the Dehn algorithm answers the question whether these words represent the same element in G, and the set of (conjugating) words u of the length at most K is finite, this gives us an algorithmic answer to the original conjugacy question about the words w and v.

4.6. Cayley graphs and geometric isomorphisms of discrete groups

203

J. Cannon [1] observed that the main idea of the above Dehn algorithm, that is, the

shortening of a non-empty word w representing 1 E G by using relations xx-1 (of the length two) and relations R' (of the length K = 4p), properly works in the case of closed hyperbolic manifolds M". Moreover, as an application of Theorem 4.35, we see that both the word and the conjugacy problems are solvable (by a generalized Dehn algorithm) for any convex co-compact hyperbolic groups G C Isom W geometrically finite groups without parabolic elements): Theorem 4.46. Let G C Isom H" be a convex co-compact group. Then there is an integer K > 0 such that any non-empty word w in generators of G can be shortened by means of relations in G having length at most K.

We call a word w in generators of G a K-reduced word (cyclically K-reduced word), if it cannot be shortened by means of relations in G having length at most K (correspondingly, by using cyclic permutations of these relations).

Theorem 4.47. Let G C Isom H' be a convex co-compact group and K > 0 the constant from Theorem 4.46. Then there is another integer K' > 0 such that cyclically K-reduced words w and v represent conjugate elements in G if and only if there exist

cyclic permutations w' and v' of the words w and v, and a word u of the length I u I < K' such that the words uv'u-1 and w' represent the same element in G.

Scheme of the proof of Theorems 4.46 and 4.47. Both proofs are based on the fact

derived in Theorem 4.35, that the embedding F: K(G, E) y E" maps each of shortest words in K (G, E) into a path in 111 within a bounded distance of a hyperbolic geodesic.

More precisely, for the constants C, C' > 0 from Theorem 4.35, we consider a (C, C')-quasigeodesic path w in K(G, E), that is such a path w that the ends of any of its subpath v C w of the length I v I > C' are mapped by F to vertices xv, yv E F(K(G, E)) with d(xv, yv) > lvI/C. In fact, due to (4.34), any shortest path in K(G, E) is (C, C')-quasigeodesic. Moreover, one can show (Cannon [1, Th. 4]) that there are constants C", L, L' > 0 (dependent on C and C') such that a path w C K (G, E) is (L, L')-quasigeodesic if it is locally (C, C')-quasigeodesic in the sense that each of its subpaths v c w of length I v 1 < C" is (C, C')-quasigeodesic. Now let K > 2max{C", L'} and suppose that aword woflength IwI > K/2cannot be shortened by means of relations of length at most K. Then each subword v C w of length I v I < K/2 is the shortest one. Since C" < K/2, it follows that w is C"-locally (C, C')-quasigeodesic, and hence w is (L, L')-quasi-geodesic if Jwi > K/2 > L'. This implies that the ends F(a) and F(b) of the path F(w) C IEF1 are separated by the hyperbolic distance

d(F(a), F(b)) >

is

1

L

bl

> L'/L > 0.

Therefore we have that a 0 b, which means that w represents a non-trivial element in G. This completes the proof of both Theorems 4.46 and 4.47.

4. Geometrical Finiteness

204

Remark 4.48. One can show that the word and the conjugacy problems are solvable for all geometrically finite groups G C Isom H' . One can use for that a very simple Todd-Coxeter algorithm (see Floyd-Hoare-Lyndon [1] and Apanasov [31, 36]). This algorithm is however very slow; much faster one can be obtained for groups with so-called automatic structures (see Epstein a.o. [1]). This class of groups includes all negatively curved groups in the sense of Gromov [5] (so-called hyperbolic groups) as well as all geometrically finite groups of hyperbolic isometries, see Tatsuoka [1] and Epstein a.o. [1]. We note that geometrically finite groups in the latter class that have parabolic elements are not hyperbolic in the sense of Gromov [5].

§7. Geometrical finiteness for discontinuity set components The aim of this section is to describe connected components of the discontinuity set S2 (G) C Sn = 81HPt+1 of a discrete group G C Mob(n), especially a geometrically finite one. This problem is closely related to a question on quantity and properties of the boundary components of the Kleinian orbifold/manifold M(G) _ [Hn+l U Q (G)] / G, see Chapter 2, § 2.3. First of all, as we indicated in Chapter 2, the properties of connected components in dimension n > 3 may dramatically differ from those in the classical case of dimension

n = 2. Actually, due to Tetenov's [1] remarkable construction (for details, see Ch.3 §2 in Apanasov [36]), we have: Theorem 4.49. For any natural numbers k, ko, n with k >_ 1, k >_ ko > 0, n >_ 3, there

exist Kleinian groups G = G(n, k, ko) C Mob(n) whose discontinuity sets split up into k invariant components each, S2 (G) = S21 U . . U S2k, and the first ko components

are contractible, i.e., icy (E2i) = O for 1 < j < n - 1 and 0 < i < ko. Applying Corollary 2.25, this theorem immediately implies the following important fact:

Corollary 4.50. For any natural k and for any dimension n, n > 4, there exist hyperbolic n-manifolds M = Hn/G, G C Isom][ IIn, whose boundaries at infinity split up into k connected (n -1)-manifolds each, awM = NI U U. . . U Nk = S2(G)/G. Moreover, the inclusion i : Nj C M(G) = (Hn U S2(G))/G of each boundary component o NJ, 1 < j < k, induces a homotopy equivalence of Nj to M.

Nevertheless, for geometrically finite hyperbolic manifolds and groups G C Isom Hn, we shall show (following A. Tetenov [3, 4]) that their properties resemble the classical situation with hyperbolic 3-manifolds and finitely generated Kleinian groups on the plane described by Accola [1] and Ahlfors [1] (although some difference still remains, see Example 4.60). We start with the simplest compact case.

1. Components with co-compact actions. All finitely generated Kleinian groups G C Mob(2) with two invariant components 0o, S21 of the discontinuity set 0 (G) are quasi-Fuchsian. To what extent is this property preserved in dimensions n > 3?

4.7. Geometrical finiteness for discontinuity set components

205

Theorem 4.51. Let G C Mob(n), n > 3, be a Kleinian group whose discontinuity set 0 (G) contains two invariant contractible components S2o and 521 with compact quotients Ni = Q,/G, i = 0, 1. Then M (G) _ [H'+1 U 52 (G)] I G is compact, the group G is geometrically finite, and S2 (G) = 00 U 521.

Proof. Since the limit set A(G) and the property of M(G) being compact are unchanged by passing to a finite index subgroup of G, we can always assume that the group G does not have elliptic and orientation reversing elements. So manifolds No and Ni are boundary components of the manifold M = M(G) and, due to contractibility of components 00 and 521, these three manifolds are K(G, 1)type spaces, that is homotopy groups Jrk(Ni) and 3rk(M) are trivial for k > 2, and 7r1(No) = 7r1(Ni) = 7r1(M) = G. Therefore, embeddings i0: No -* M and i 1: Ni -* M induce isomorphisms of homotopy groups, and due to the Whitehead theorem [1], they are homotopy equivalences that induce isomorphisms of homology groups of manifolds No, N1 and M. Let M be the double manifold obtained by gluing two copies of the manifold M, M and M' = M, along the boundary components No and N1. Then the triad (M; M, M') is excisive in the homology sense, and we can apply an absolute Mayer-Vietoris exact sequence for this triad to calculate homology groups Hk(M), see Dold [1]:

0 -+ Hn+1(M) -dam HH(No)(DHn(N1)

j*

) H,,(M)®HH(M') -`

H,(R) -) ...

Here d* is a boundary homomorphism, Hn (N0) = Hn (Ni) = F1 (M) = Z, and the homomorphism j* maps the element (k, 1) of the direct sum Hn (No) (D H, (N1) = Z ® Z to the element ((k + 1), -(k + 1)) in the group H.(M) ® H,, (M') = Z ® Z. Here, the kernel ker(j*) is a cyclic group consisting of all elements of the group Hn(NO) ® H (N1) having the form (-k, k). This kernel coincides with the grouj Hn+1 (M), and therefore Hn+l (M) = Z. So the (n + 1)-dimensional manifold M is closed. This means that the boundary 3M of M is the union of No and N1, while M itself is compact. Compactness of the manifold M(G) and the GF5-finiteness in Theorem 4.16 imply that G is geometrically finite. Moreover, as we have shown,

(00/G) U (521/G) = No U Ni = 3M(G) = S2(G)/G. This means that Q(G) _ 0o U 521, which completes the proof.

In dimension n = 3, we have an additional information on the discontinuity set: Lemma 4.52. Let 520 C S2 (G) C S3 be an invariant component of the discontinuity set of a Kleinian group G C M6b(3). Then the second homology and homotopy groups of any other connected component 0' C 52(G) vanish, H2(52') = 0 and 1r2(52) = 0. Before we go on to prove this claim, let us check the following general properties of discontinuity set components.

Lemma 4.53. Let G C Mbb(n) be a Kleinian group having an invariant component 52o of the discontinuity set 0 (G). Then, for any other connected component 52' of

4. Geometrical Finiteness

206

S2 (G) and any point z E Sn, the following sets Sn\s2',

aQ'

,

Sn\(s2' U {z})

,

8s2'\{z}

are connected.

Proof. As S20 is G-invariant, 00 U A(G) coincides with the closure, 520, and is connected. Moreover, the union of Sfo with any number of connected components of the complement S' \520 is still connected because the boundaries of these components

are subsets of the limit set A(G). In particular, the set S' \S2' has such a form and hence is connected.

To prove that the boundary aQ' is connected, we take an open neighborhood V of d 2' all whose components have non-empty intersection with a ca'. Then open sets X = S2' U V and Y = (S"\S2') U V are connected, their union X U Y is the whole sphere S" , and x fl Y = v. So the triad (S'; X, Y) is excisive in homology sense, and first terms of its reduced Mayer-Vietoris exact sequence (see Dold [1]) are as follows: H0(V) (jt*,i2 0 HO(X) ® Ho(Y) ti's Ho(Sn) = 0, = HI(Sn) (4.43) where ik* and jk* are homomorphisms induced by the inclusions ik and jk, k = 1, 2, while d* is a boundary operator. Since sets X and Y are connected, homology groups Ho(X) and H0(Y) vanish. This and (4.43) show that Ho (V) = 0, and thus V is connected. As the choice of V is arbitrary, the boundary a Q' is connected, too. Replacing each of the sets in the previous arguments by its intersection with S" \ {z}, we prove connectedness of the set aS2'\{z). On the other hand, act'\{z} is the con-

nected boundary of Sn\(S2' U {z}) in the space S"\{z}. This implies that the set S' \(S2' U {z}) is also connected.

Proof of Lemma 4.52. Due to the Alexander duality theorem (see Dold [1, 8.15]), we have that HO(S3\S2') = H2(S2'). On the other hand, Lemma 4.53 shows that S3\S2' is connected, and hence HO(S3\S2') = 0. So, H2(Sf') vanishes. Now let us suppose that 72 0) # 0. Then, due to the Sphere Theorem 8.6, there exists an embedded 2-sphere S2 _+ Sf' that is non-contractible in S2'. Such a sphere S2 decomposes the 3-sphere S3 into two balls, one of which contains the set S3\S2', while this sphere contracts to a point along the other. This contradiction shows that n2(Sf') = 0. Now we can sharpen Theorem 4.51 in dimension n = 3.

Corollary 4.54. Let G C Mob(3) be a Kleinian group whose discontinuity set S2 (G) C S3 contains two invariant simply connected components 00 and S21 with compact quotients Sgt/G, i = 0, 1. Then M(G) _ [lH[4 U n(G)]/G is compact, the group G is geometrically finite, and 0 (G) = 00 U 521.

Proof. This directly follows from Theorem 4.51 because, due to Lemma 4.52, both components Sfo and S21 are contractible.

4.7. Geometrical finiteness for discontinuity set components

207

2. General case: groups with cusps. Here we consider the case of arbitrary geometrically finite groups G C Mob(n) which may contain parabolic elements. Although the quotient [H"+t U 0 (G)l / G = M (G) and its boundary S2/ G may be non-compact, the main result of this section resembles the co-compact situation in Theorem 4.51.

Theorem 4.55. Let G C Mob(n), n > 3, be a non-elementary geometrically finite Kleinian group having an invariant contractible component 00 C 9 (G). Then the following alternative holds: (i) either 0 (G) = 20 U 01, and 01 is also a G-invariant connected component,

or (ii) additional boundary components of 8M(G) (complemental to S2o/G, always finitely many) are covered by infinitely many non-invariant components of S2 (G)\S2o.

Proof As in the proof of Theorem 4.51, we may assume that the group G has neither finite order elements nor orientation reversing elements. We start with three auxiliary lemmas, where we shall study the role of parabolic subgroups of G.

Lemma 4.56. Let G C Mob (n) be a geometrically finite group with a maximal parabolic subgroup G p C G fixing a cusp point p E S' and having rank Gp = k, k < n, and no C S2 (G) be a G-invariant contractible component. If 0o contains a cusp domain, up fl S2 (G) C 0o, then no = S2 (G) = S"\{p}, and G is elementary. Proof Conjugating G and applying the Bieberbach Theorem 2.6 and Theorem 4.21, we may assume that G acts in 1I8+ 1, p = oo, and G.0 C Isom I[8" is a free Abelian group of rank k < n that leaves invariant k-space IRk c 8118"+++1 where it acts co-

compactly, with lt8k/Gm = Tk a k-dimensional torus. For a cusp domain U = U U fl S2 (G) c no, no \ U is a strong deformation retract of the contractible component

0o. Hence, S2o\U is contractible, as is R" \U containing 2o\U as an open subset. We put M = (ll8"\U)/G,,, and N = (S2o\U)/GO,,. It is obvious that the manifold M is homeomorphic to the product of a closed (n - k)-dimensional ball B"-k and a and, due to the conditions k-dimensional torus Tk. Its boundary 8M is Tk X of the lemma, 8M C N. Since both M and N are K(Gw, 1)-type manifolds, their homology groups coincide. So, considering exact homology sequences for the pairs (N, 8M) and (M, 8M) Sn-k-1

and applying the fact that a M C N, we get that their relative homology groups coincide: H,k(N, 8M) = H,k(M, aM). In particular, we have that H"(N, 8M) = H"(M, 8M) = Z. This means that N is compact and coincides with M. In other 0 words, S2o\U = R\ U, and 0o = R" is the discontinuity set of Gm = G. Lemma 4.57. Let no c S2 (G) be an invariant contractible component for a geometrically finite group G C Mob (n) having a parabolic cusp point p of rank G p = n - 1. If a cusp domain up n S2 (G) does not intersect no then S"\00 is not connected and therefore all the other components of S2 (G)\S2o are not G-invariant.

208

4. Geometrical Finiteness

Proof. As in the previous Lemma 4.56, we assume that p = oo and Goo - 7Gn-1 and, for a cusp domain U = U,, fl 0 (G) C Sn, we put:

M = (R' \U)/Goo D N = Qo/Goo. We have also that the subset N of the manifold M is open in M and has an empty intersection with the boundary aM, while the embedding N C M is a homotopy equivalence. As Hn -I (N) = Z, there exists a compact K C N such that the mapping H,,_1(K) -> H,,_1(N) induced by inclusion is surjective. Let S1 and S2 be two boundary components of M. We shall show that they lie

in distinct components of the complement M\N. If this is not so, aM = St U S2 is contained in some open connected component of the set M\K, and therefore

there exists a simple path y C M\K that connects Si and S2. For this path y, we readily show that Hn_1(M\{y}) = 0. In fact, we can embed the manifold M (homeomorphic to Tn-1 x [-1, 1]) as a torus layer in S. Its complement in Sn consists of two components, and therefore the set (S'\M) U y is connected. This implies that Ho ((S'\M) U y) = 0. So, by the Alexander duality theorem (see, for example, Dold [1, 8.15]), we have that Hn_1(M\y) = 0. We consider the following commutative diagram of embeddings:

K

> M\Y

N

)

M

The homomorphisms of the (n - 1)-dimensional homology groups induced by these embeddings also form a commutative diagram as follows:

Hn-1(K)

xJ Hit-1(N)

'0

B

> Hn-1(M\Y)

Hn-1(M)

In this diagram, X is an epimorphism, 6 is an isomorphism, and cp is a zero map. This provides us a contradiction at once, because (* o cp)(Hn_t(K)) = 0 while (B o X)(Hn_t (K)) = Hn_1(M) = Z. Therefore it shows that M\N is in fact disconnected, and hence Sn\no is either. Due to Lemma 4.53, we then have no G-invariant components different from 0o. Lemma 4.58. Let G C Mob (n) be a non-elementary geometrically finite Kleinian group having an invariant contractible component no C S2 (G). Then each of its maximal parabolic subgroup Gp C G has rank at most n - 1. Proof As before, let G act in IIBn+ 1, S2 (G) C aR+ 1 = Sn, and suppose that the stabilizer Goo C G of a parabolic cusp point oo has rank = n. Considering the

4.7. Geometrical finiteness for discontinuity set components

209

action of G in S'+1 = Rn+1 U {oo}, we have that its discontinuity set is the union fl + 1 U 0 (G) U Rn+1

Moreover, due to contractibility of S2o, we see that a cusp

domain, {x E lI8n+1 : 1xn+1 I > t}, of the cusp point oo in is contained in the open G-invariant contractible set I[8+1 U Qo U Rn+1 This and Lemma 4.56 show that TDn

1 U Qo U IISn+1 = TDn+1 IM+

U S2 (G) U II8n+1 = Rn+1, which implies that Qo

So G must be elementary.

Proof of Theorem 4.55. Due to the finite covering Theorem 4.21 and Lemma 4.58, we may assume (passing to a finite index subgroup of G, if necessary) that each maximal

parabolic subgroup of G is a free Abelian group of rank at most (n - 1). Since G has at most finitely many conjugacy classes of maximal parabolic subgroups, we may enumerate them by their fixed cusp points p 1, ... , ps (at the boundary of a finite sided fundamental polyhedron P C 1[8"++1) and take corresponding cusp neighborhoods

U1, ... , US C llg+ 1 U Q (G) such that G(Ui) fl G(Uj) = 0 for i

j. These

disjoint cusp neighborhoods are projected to disjoint neighborhoods Wi = U1 / G,1, i = 1, . . . , s, of isolated cusp ends of the manifold M(G). Moreover, since all Gi are free Abelian, the closures in M(G) of these neighborhoods Wi are homeomorphic to punctured solid cylinders B"-kj x Tki x (0, 1] where ki = rank Ge,, 1 < ki < n - 1.

Due to the Finiteness Theorem 4.16, the complement Mc = M(G)\ Ui=1 Wi is compact.

Let No = S2o/ G and Not = Mc fl No. Then, due to Corollary 2.25, this component No of the boundary 8M of M = M(G) is homotopy equivalent to M. Now we enumerate cusp ends of M(G) with respect to properties of their neighborhoods Wi in such a way that Wi = Tk' x [0, 1]n-k' x (0, 1] , ki = rank Gpi

f
(4.44)

In the first case of ki < n - 2, we have that Wi fl am are connected and, due to Lemma 4.56, W i fl No = 0. In the second case of ki = n - 1, the boundary W i fl a M is not connected and, due to Lemma 4.56, cannot lie in No:

W1 flaM=Tn-1 x{0,1)x(0,1]¢No. We split the latter case into two subcases according to whether Wi fl No Wi fl No = 0. Namely, let Wi f1 No

,-£0 forr
=0 forq
0 or

(4.45)

So Wi fl No = Tn-1 x {0} x (0, 1] forr < i < q, and Qo fl Ui = 0 for i < q. In the latter case we see from Lemma 4.57 that G does not have invariant components distinct from 00 (this corresponds to case (ii) of the theorem).

4. Geometrical Finiteness

210

Now we shall apply Mayer-Vietoris sequences to evaluate homology groups of the obtained manifolds. To do that, we first observe that (4.46)

H*(Mc, Noc) = H*(M, No) = 0.

This follows from the fact that each of the sets W t fl Mc is a strong deformation retract of W;, so MC is also a strong deformation retract of the manifold M, and this applies

also to the pairs (Mc, Noc) and (M, No). Secondly, for each i-th cusp end, r < i < q, we shall "fill in" the neighborhood Wi as follows (see also Chapter 6, §3, after Lemma 6.39). We start with a general construction for Seifert fibered manifolds X. Namely, changing each collar Sl x [0, 1] in the product X x [0, 1] to a 2-disk, we generally obtain a singular resulting space X. For example, if X is a Seifert fibered homology 3-sphere, the resulting space k is a 4-orbifold whose singularities look like cones over lens spaces (see Fintushel-Stern [1]). However, in the case of S1-bundles f : X -* Y over compact manifolds Y (in particular, in our product case), this construction works without singularities. Namely, formalizing the construction, let us define the space X as

X=C(f)=(Xx[0,l])/^', where (x,O)'-(y,0)4f(x)=f(y). (4.47)

Then locally k looks like the product:

(Bn-1 x S1 x [0, 1])/

Bn-1 x (S' x [0, 1])/(S1 x {01) =

Bn-1

x B2 (4.48)

where, for two sets B C A, A/B means the set obtained from A by collapsing its subset B to a point.

So applying the construction in (4.47)-(4.48), we fill up non-compact punctured solid tori Tn-1 x (0, 1] to solid tori fn-1 = (Tn-1 x [0, 1]) L- = Tn-2 x B2 and obtain a compactification of neighborhoods WW, r < i < q, as Tn-1 x [0, 1]. Here we identify the subset Tn-1 x [0, 1] of the boundary of each of such product with one of the sets Wi fl MC, r + 1 < i < q, where we assume that each of the sets Tn-1 fl Noc is identified with one of Wi fl NoC. Then from MC we get a compact (n + 1)-dimensional manifold M, whose boundary contains the closed component No obtained from the n-manifold Noc by gluing to its boundary (q - r) copies of the solid tori P-1. We have that each of the pairs

(?n-1 x [0, 1], Tn-1 x 101), (Tn-1 x [0, 1], Tn-1 x 101), (MC, Noc) (4.49) is homotopically trivial. Furthermore, the manifold pair (M, No) is the union of the pair (MC, Noc) and the pair q-r

CU j=1

q-r (Tn-1

U (Tn-1 x {0})j

x [0, 1])

,

J

j=1

4.7. Geometrical finiteness for discontinuity set components

211

whose intersection is the pair r CI

I (T n-1 x [0, 1]),

I

j=1

r I (T n-1 x {0})1' j=1 I

/

Therefore we may consider the relative exact Mayer-Vietoris sequence of homology groups (see Dold [1] and Apanasov [36]) for the pairs of triads

(do; Noc,

U(T"-1

x {0})j) C (A?;

Mc, U(T1 x [0, 1])1 j=1

/

and get from (4.46) and the homotopy triviality of the pairs (4.49) that

H*(M,No)=0.

(4.50)

Since the manifold go is closed, and hence H"(No) = 7G, we have from (4.50) that H" (M) = Z. This allows us to show that a M consists of two components. To do that, we consider an exact sequence for the pair (M, aM) as follows:

H,(M) H1(aM) -+ Hl (M) -+ Hl (M, aM) -+ Ho(aM) - Ho(M).

0 -+ H"+1 (M, aM) - HH(aM)

(4 . 51)

Due to (4.50), the homomorphism Hl (8M) --> H1 (M) in this exact sequence is surjective because a )R D No, while H1(N0) = Hi (R). It implies that the kernel of the homomorphism H1 (M, aM) -+ Ho (aft) is trivial, and this homomorphism is in fact a monomorphism. Furthermore, Ho(aM) = Zk where k is the number of connected components of a M, and therefore H1 (M, a M) does not contain finite-order elements. As M is compact, the Poincar6 duality theorem (see Dold [1, Chapter 8, §7]) gives

us that the groups H" (M) and H 1(M, aM) are isomorphic. At the same time, since Ho(M, OM) is trivial, the universal coefficient theorem (see Dold [1, §4 Ch.6]) gives H1(M, 0M) = HI (M-, aM-), so Hl (M, aM) = Hn (M) = Z. On the other hand, we have that Ho (aft) = Z e Z because the homomorphism Ho(aM) -- H1 (M) is surjective and the subsequence in (4.51) is exact:

HI (,Q, aM)

Ho (8 M) --> Ho (R) .

This shows that the boundary of M consists of two connected components, aM = N0 U N1. Let K and L be their disjoint close regular neighborhoods in M. As K and L are Euclidean neighborhood retracts, the tech cohomology for the pair (M\ int K, L)

coincide with the usual ones. Then the Poincare duality theorem (Dold [1, Ch.8]) shows that Hk(M\ int K, L) and H"+1 _k (M\L, int K) are isomorphic.

4. Geometrical Finiteness

212

On the other hand, since pairs (M\ int K, L) and (M-\L, int K) are homotopically equivalent, respectively, to pairs (M, No) and (M, N1), the groups Hk(M, N1) and Hn+1-k(M, No) are isomorphic for any k, and thus

H*(M, 91) = 0.

(4.52)

This, together with (4.50), means that the inclusion Ni C M induces an isomorphism of homology groups.

Now let us suppose that the assertion (ii) of the theorem fails. Then, for each i = r + 1 , ... , s, we have N o fl Wi = Tii-1 x {0} x (0, 1], and Tn-1 x {1} x (0, 1] is the intersection of Ti with one of the other components Nj of the boundary a M. We shall show that there is in fact only one such component, say N1.

Indeed, for i = 1, ..., r, the sets dWi and Wi fl aM are connected. Hence the number of connected boundary components in aM is preserved under the operation of cutting submanifolds Wi out of M. This number of boundary components is also preserved when we replace submanifolds Wi, i = r + 1, ... , s, by compact cylinders Tn-1 x [0, 1]. Therefore, the number of boundary components of M is equal to the number of boundary components of the manifold M, that is aM = No U N1. We have that Mc fl Nl is a strong deformation retract of the second boundary

component of M, N1 C 3M. Each loop in the manifold Ni is homotopic in N1 to a loop in Mc fl Ni, and hence the inclusion Ni C M induces an epimorphism nl (Ni) -+ nl (M). Then, for the natural projection rr: 0 (G) -* 3M, we have due to Lemma 2.23 that the lift n'1(N1) of the second boundary component Ni = aM\No is the second connected component 521, 01 = S2(G)\Q0, which is clearly invariant with respect to G. This completes the proof of the assertion (i) and of Theorem 4.55 in whole.

Directly applying Lemma 4.52, we can sharpen this result in dimension n = 3:

Corollary 4.59. Let G C Mob(3) be a geometrically finite Kleinian group in the 3-sphere S3 having an invariant simply connected component 00 C Q (G). Then:

(i) either 0 (G) = 00, or (ii) c2 (G) consists of two G-invariant components, S2 (G) = 0o U 521, the component Ego is contractible, and 772(521) = 0, or (iii) S2 (G)\S2o consists of infinitely many (but finite modulo G) non-invariant components, and each of them is aspherical in dimension two.

We notice that the simply connectedness of no in the above corollary is essential. This is seen from the following example of G C M6b(3) with two G-invariant (nonsimply connected) components and infinitely many non-invariant components.

Example 4.60. There exist geometrically finite groups G C M6b(3) having two invariant (non-simply connected) components and infinitely many non-invariant ones. Construction. Our group G will be generated by reflections in spheres of from a finite

4.7. Geometrical finiteness for discontinuity set components

213

family E, so it is enough to provide such a family E of spheres o-i C S3 = R3 U fool with the following properties:

(i) For any ai 0 vj, either int of fl int Qj = 0, or these spheres intersect at a rational angle rr/m, m E 7L; (ii) The polyhedron

FE = IE83\ (U int ai) consists of three components Fo, F1, F2, where the closures of the last two

intersect any sphere o" E E (along disks), and there are spheres Q E E disjoint from FO.

Indeed, the group G = GE generated by reflections in spheres a of the family E is the desired group. Its discreteness follows from Corollary 2.39 applied to the extension of GE-action into IHI4, aIHI4 = S3. Due to Lemma 2.22, the discontinuity set of the group G E has two invariant components,

( U (F1))

Q1 = int

and

S22=int( U g(F2)f>

\ gEGE

gEGE

/

and their complement,

S2(G)\(S21 U S22) = int

(U

g(Fo))

\ gEGE

consists of infinitely many non-invariant components because there exists a E E such

that or flFo=0. Now we form the family E from the following nine spheres in ]I83, which are shown in Figure 61.

Q1=

{xER3:x2=0}, o'2={xElli;3:x1=x2}, U3= {xE93:x1+x2=3};

O4 has radius 3 and its center at (0, 0, + 26/3); Q5, o-6, and v7 have radius //3 and their centers correspondingly at (1, 0, 0), (3, 0, 0), and (2, 1, 0); finally, ag and a have radius /3-/3 and centers correspondingly at (2, 0, 0) and (1, 1, //3). To define the balls int Qi bounded by spheres ai, we take those components of the complements II83\oai that do not contain the point Al = (1, 1/2, 10). Then the fundamental polyhedron FE = FO U F1 U F2 consists of three connected components (see Figure 61), where F1 contains the point Al, F2 contains A2 = (3/2, 1/2, -1/2), and FO contains the point AO = (1, 1/2, -10). Direct calculations show that magnitudes of all dihedral angles of these three polyhedra are either n/2, r/3, r/4 or r/6; each of spheres of E E intersects both F1 and F2 along some topological disks, and Fo

4. Geometrical Finiteness

214

intersects neither of five spheres Ei, 5 < i < 9. In Figure 61, the spheres E1, E2 and E3 are taken as transparent planes, and their intersections with balls int ai, i > 4 are shaded.

Figure 61

It is easy to see that all non-invariant components of 0 (G) form the G-orbit of the component no that is the half space,

520 = { x ER 3 111

: x3 < - 7 + 26/3 } = int ( U g(Fo)) 111

\ geGo

where Go C G is the stabilizer of this half space no generated by reflections in spheres E1, E2, E3 and E4, all of which contain sides of Fo.

Notes

215

Although the non-simply connectedness of the G-invariant components Sgt and 522 follows from Corollary 4.59, one can prove this fact directly.

Indeed, let a C F1 be a topological segment connecting the points At and A2, which lie correspondingly in int(FI fl o-4) and int(F1 fl a3). Similarly, let b C F2 be a topological segment connecting the points B1 and B2, which lie correspondingly in

int(F2 fl al) and int(F2 fl o ), see Figure 61. Then we take closed curves a and fi that correspondingly consist of finitely many images of segments a and b under those elements g E G that are compositions of reflections in those sides of the corresponding

polyhedron F1 or F2 that contain ends of these segments. Evidently, these curves a c S21 and fi C 02 are linked in R3, so S21 and 522 are not simply connected. Moreover, images of these loops a and fi under Euclidean translations gm E GE, x2, x3) = (xl + 3m, x2 + 3m, x3), give us an infinite chain of linked loops contained in distinct components S21 and 522, and hence non-contractible in these components. This means that the fundamental groups of both components are infinitely

generated, rank n1 (52i) = oo, i = 1, 2.

Notes Theorem 4.4 combines finiteness theorems of Beardon-Maskit [1] (for convex fundamental polyhedra) and Marden [1] (for Dirichlet polyhedra) in dimension three. The existence of infinite-sided Dirichlet polyhedra for geometrically finite groups (Theorem 4.5 and Lemma 4.6) was discovered in Apanasov [2]; see also Bowditch [1] and, for another proof, Apanasov [36]. The classical definition of a cusp point for Kleinian groups on the plane is due

to Beardon-Maskit [1]. The definition (4.4) was given in Apanasov [13], see also Tukia [5]. Its equivalent reformulation in Theorem 4.8 is based on the generalization of Shimizu's Lemma [1] (Theorem 3.15) in Apanasov [3, 14, 19]. Theorem 4.9 on Beardon-Maskit [1] conditions in higher dimensions is essentially based on the construction of pseudo-Dirichlet polyhedra in Apanasov [31, 34] (see also Apanasov [13]). Since such polyhedra in dimension n = 3 coincide with Dirichlet ones, Corollary 4.12 may be considered as a generalization of Marden's alternative, see Marden [1] .

The proof of Theorem 4.13 on the limit set measure follows Beardon-Maskit's [1] approach for Kleinian groups on the plane. The original proof by Ahlfors [3] for Kleinian groups on the plane has been generalized (with some additional constraints) in Chen [1]. Theorem 4.14 is due to Sullivan [2, 8] and Tukia [3]. The Finiteness Theorem 4.16 generalizes several results providing equivalent definitions of geometrical finiteness in dimension three to higher dimensions. These results are due principally to Beardon-Maskit [1], Marden [1] and Thurston [1] (see also Greenberg [4]). For torsion free groups in higher dimensions, its proof (based on the study of parabolic subgroups and ideas close to Margulis's Lemma) was given by Apanasov [14, 15] (see Apanasov [36]); for the general situation of groups with

216

4. Geometrical Finiteness

torsion, see Bowditch [1] and Hamilton [1]. The latter work contains a proof of Theorem 4.15 as well as a characterization of all geometrically infinite Kleinian groups G C Isom HI whose uniform neighborhoods of the convex cores, U, C (M) C M (G), have finite volume. Although the existence of a finite covering of a geometrically finite hyperbolic manifold having Abelian cusp ends has been claimed by many authors, its first proof (for finite volume hyperbolic manifolds) was given in a recent paper by AravindaFarrell [1]. Our proof of Theorem 4.2 is based on Lemma 4.22 which generalizes the main inductive argument in the finite volume case. This generalization does not use the homology arguments of Aravinda and Farrell and hence is valid for geometrically finite manifolds of variable negative curvature, see Apanasov-Xie [1]. Theorem 4.23 on tessellations of constant curvature spaces by compact polyhedra is due to A. D. Aleksandrov [2]. In the case of tessellations of the hyperbolic plane or 3-space H3 by one polyhedron associated with a discrete isometry group action, this theorem is also known as the Poincare theorem on fundamental polyhedra. Corollary 4.30 of Theorems 4.27 and 4.29 (which are due to Apanasov [23, 25]) is a generalization of the Poincare-Aleksandrov theorem to the case of unbounded polyhedra. For other expositions of the Poincar6 theorem, see Maskit [6], De Rham [1], Seifert [2], Morokuma [1], Mostow [3] and Epstein-Petronio [1]. Theorem 4.32 on the global super-rigidity in Hn, n > 4, generalizes a local rigidity theorem of GarlandRaghunathan [1] and is due to Apanasov [23]. Example 4.25 and Proposition 4.26 on incomplete hyperbolic structures on the Whitehead link complement are essentially due to W. Thurston [1] (see also Apanasov [25]). The notion of the completion of a group and the first of its properties (Lemma 4.34) are due to Floyd [1]. Theorem 4.35 is essentially due to Cannon [1] where it was proved for co-compact groups of hyperbolic isometries; however an analogous fact is true in a more general situation (Remark 4.36). Theorems 4.37 and 4.38 (Apanasov [36]) are generalizations in higher dimensions of results of Floyd [ 1] for Kleinian groups on the plane; they provide a base for the isomorphism Theorem 4.41 for geometrically finite groups. The stronger isomorphism Theorems 4.44 and 4.45 (whose assertions include Theorem 4.41) were originally proved by P. Tukia [5] in a different way, by using the extension Theorem 4.43 which is essentially due to Efremovich-Tikhomirova [1]; see also Tukia [9,18] and Apanasov [47, 50]. The solution of the word and the conjugacy problems for convex co-compact hyperbolic groups in Theorems 4.46 and 4.47 is due to Cannon [1] who proved these results for the basic case of co-compact groups. The results of §5 (in particular, the remarkable Theorems 4.51 and 4.55) are due to A. Tetenov [3, 4]; the present proofs have some changes.

Chapter 5

Kleinian Manifolds

A discrete group G of isometries of the hyperbolic n-space 1H[" acts on the sphere at infinity Sn-1 = alln as a subgroup of the Mobius group Mob(n - 1). This action is discontinuous on IH[n U Q (G) where 0 (G) C Sn-1 is the discontinuity set of G, and one can define the so-called Kleinian manifold/orbifold M(G) as the quotient space

M(G) = [Hn U 2(G)l/G = [IHI"\A(g)]/G.

(5.1)

The natural projection 7L: Hn U S2 (G) ---> M(G) induces a conformal structure on

M(G) which coincides with the hyperbolic structure on int M(G) = ]V/G. The group G is isomorphic to the desk transformation group for the (branched) covering 7r and, if G has no torsion, it is the fundamental group 7c1 (M) - G of the manifold

M(G). In the case of a group G with torsion, M(G) is a conformal orbifold. The boundary BM(G) of M(G) is not empty if and only if the discontinuity set S2(G) of G is non-empty, i.e., if G is a Kleinian group on Sn-1 = 8IH[". In this case, aM(G) = 2 (G)/G. For non-elementary groups G, no component of the boundary &M(G) can be an elliptic, Euclidean or Hopf manifold. In other words, in the case of an oriented 3-manifold M(G), any component of 8M(G) is neither a sphere, nor a sphere with one or two punctures, nor a torus. Our goal here is to describe some geometry and topology of Kleinian manifolds as well as the specific features they have in dimension three and higher.

§1. Basic topology related to Kleinian manifolds It is clear from the finiteness results presented in Chapter 4, that the analysis of the topological structure of geometrically finite Kleinian manifolds M (G) (and their convex cores C(M)) leads to the study of properties of their compact submanifolds. We now state some well-known facts on topology of manifolds M that are important in the sequel.

218

5. Kleinian Manifolds

1. Three-dimensional topology. In what follows in this section (unless otherwise stated) we assume that all manifolds M are three-dimensional. Theorem 5.1 (The Loop Theorem and Dehn's Lemma, Papakiriakopulos [1, 2]). If .£ is a non-trivial loop in aM (a non-trivial element of the group sri(aM)), which is trivial in M, and N C aM is an arbitrarily small neighborhood of f, then there exists a simple loop £' C N also non-trivial in aM and bounding an embedded disk D C M which intersects am along L'. If £ is a simple loop, then t' can be taken equal L. Theorem 5.2 (The Cylinder Theorem, Waldhausen [1]). Let .Li, £2 be disjoint loops

on am that are non-trivial and freely homotopic in M, and let Ni, N2 C aM be disjoint small neighborhoods of them. Then there are simple non-trivial in M loops

.E i C N1 and 4 C N2 that are the components of the boundary of an embedded cylinder C C M, c fl am = 4 U 4. If L1 (or 12) is a simple loop, then we can take it to be 4 (respectively, t').

Theorem 5.3 (Seifert-van Kampen Theorem, cf. Massey [1]). Let S C M be an embedded hypersurface in n-dimensional manifold M such that the induced homomorphism of fundamental groups z1 (S) -+ nl (M) is a monomorphism. If M\S has two components, M1 and M2, then,ni (M) = ni (Mi) * ni (M2) is a free product with the amalgamated subgroup ir1(S). "' (S) If, on the other hand, M\S is connected, then irl (M) = nl (M\S) * is a HNN(S) extension of irl (M\S). it As a particular 3-dimensional case of this theorem we have:

Corollary 5.4. Let M be a 3-manifold with boundary and D C M be a topological disk whose boundary loop in 8M is non-trivial. If M\D has two components, Mi and M2, then Tri (M) = nl (Mi) * ir1(M2) is a free product. If, on the other hand, M\D is connected, then nl (M) = nl (M\D) * Z. Theorem 5.5 (Kneser Theorem, see Hempel [1]). Let M be a compact 3-manifold and each its boundary component, S C am, is incompressible in M, i.e., the homomorphism nl (S) -* ni (M) induced by the inclusion S C M is a monomorphism. If nl (M) - Gi * G2 is a non-trivial free product of the groups G1 and G2, then there exists a two-dimensional sphere embedded in M that splits M into a connected sum of submanifolds M1 and M2, M = M1#M2, where .n1(Mi) - Gi, i = 1, 2. We note that M#S3 is always homeomorphic to M. A manifold M is called prime if any its expression as M = Mt#M2 has M1 or M2 homeomorphic to S3. In a prime

3-manifold, any separating 2-sphere must bound a ball. The only one orientable prime 3-manifold with a non-separating embedded sphere is S2 X S1 (see Hempel [1]). All the other compact, orientable prime 3-manifolds are irreducible, that is every embedded sphere bounds a ball. Theorem 5.6 (The Sphere Theorem; Papakiriakopulos [1], Whitehead [1]). If M is an orientable 3-manifold such that n2 (int M) 54 0, then there exists a two-dimensional

5.1. Basic topology related to Kleinian manifolds

219

sphere embedded in M that represents a non-trivial element of n2 (M). If n2 (int M) _ 0, and 7rl (M) is infinite, then n3 (int M) = 0.

This theorem implies that if a 3-manifold M is orientable and irreducible then 7r2(M) = 0. If, in addition, rrl (M) is infinite, then the universal covering of M is contractible so that M itself is aspherical, that is all homotopy groups rrl (M), i > 1, are trivial.

A 3-manifold M is called sufficiently large in the sense of Waldhausen [4] if M contains an incompressible two-sided surface (that is, the normal bundle to M is oriented) which is not a 2-sphere and whose boundary (if not empty) lies at the boundary 8M of M. A 3-manifold M is called a Haken manifold if it is prime and sufficiently large. We note that, with an exception of the trivial case of a ball, any prime 3-manifold with non-empty boundary is a Haken manifold. Another condition for an irreducible 3-manifold M to be a Haken manifold is that the rank of the first homology group Hl (M) be positive. Another important class of 3-manifolds consists of Seifert fibered manifolds. To define it, we consider pairs (µ, v) of relatively prime integers, the unit segment I = [0, 1] and the closed unit disk B2 C ll which we represent in polar coordinates, B2 = {(r, 0) : 0 < r < 1}. Then we define a fibered solid torus of (u, v)-type as the manifold obtained by gluing two boundary disks of the solid cylinder B2 X I as follows:

P = {(r,9,t)EB2xI

9,

1)=Cr,B- 27r /-1 µ

l /JJJ

In the obtained solid torus P, the fibers S1 are images of segments {x} x I, X E B2, under this gluing, so that each of them but the central one (which is obtained from one segment) consists of µ segments. Up to a fiber preserving homeomorphism of P, we

may assume that g > 0 and 0 < v < µ/2. The integer µ is called the index of the fibered solid torus P. If p > 1, the central fiber is called an exceptional or singular fiber. All other fibers of P are called regular. Now we can define a Seifert fibered 3-manifold (shortly, a Seifert manifold) E as a 3-manifold that is a union of pairwise disjoint simple closed curves (fibers) and such that each fiber r C E has a closed neighborhood U(r) consisting of a union of fibers and homeomorphic under a fiber-preserving homeomorphism to a fibered solid torus P. Such a neighborhood U(r) is called a fibered neighborhood of a fiber T. A fiber r C E is called exceptional or singular if, under a fiber-preserving homeomorphism

of U(r) to a fibered solid torus P, U(r) - P, it goes to the singular fiber of P (if it exists). Otherwise, r c E is called regular. Identifying each fiber of a Seifert manifold E to a point, we define the so-called orbit space (which may be in fact an orbifold). A natural map of E to the orbit space is called the Seifert projection. The union of all regular fibers in any connected Seifert manifold is connected and forms a circle bundle. The universal covering E of a Seifert manifold E without boundary is homeomorphic to one of S3, 1R3 or S2 x R, and hence every Seifert manifold is aspherical unless it is covered by S3 or S2 x R.

5. HIeinian Manifolds

220

Since the fundamental group n1 (E) acts on E preserving the natural foliation, 71 (E) acts on the orbit space X of E, which defines a natural homomorphism n1 (E) -- nirb(X) The kernel K of this map consists of covering translations of E projected to the identity map on X, and is infinite cyclic if E is not compact. Thus, for every Seifert manifold E with an orbifold X as the orbit space, there is an exact sequence

1K71(E)

>

rb(X)--

1

(5.2)

where K is the cyclic subgroup of n1 (E) generated by a regular fiber. The group K is infinite except in the case where E is covered by S3. One can classify Seifert 3-manifolds by using the Euler number e(E), see Neumann-Raymond [1]. We shall define it in the simplest case when a Seifert bundle E is a circle bundle n over a closed surface X. For a non-orientable 77, we define its Euler number e(i7) = 0; for an orientable circle bundle rt over closed surface X, the Euler number e(r7) is the obstruction to the existence a section of 17, which lies in H2(X; Z) isomorphic to Z. Since n1 (X) acts on Z via a natural homomorphism n1 (X) -± Z2, where we identify Z2 with Aut(Z), we can define the Euler number e(77) as a natural number e defined as follows. Let B2 C X be a closed disk with a point x E 8 B2, X' = X\ int B2, M = X' X S1, and a = {x} x S1 and fi = B2 x {e} be generators of n1 (8M), where 8 E S1. Also let t = B2 X S1, and a = {x} x S1 and b = 8B2 X {B} be generators of the boundary torus at. Then an orientable circle bundle r7 over a closed surface X of genus g has the Euler number e = e(r7) if the total space S(g, e) of 7 can be obtained by gluing the solid torus t to M is such a way that the loop a is identified with the loop a and the loop 73 is identified with the loop b ae. The following result explains how the Euler number of a circle bundle alters under finite coverings (the same assertion is true in the general situation of a Seifert fibered manifold where its Euler number can be defined in a different way as a rational number, see Neumann-Raymond [1]).

Lemma 5.7. Let a7 be a circle bundle over a closed surface X with the total space

E, and t a finite cover of E of degree d such that t is the total space of a circle bundle i7 over a surface X. In addition, let the covering X -+ X have degree l and let m denote the degree with which fibers of i3 cover fibers of 17 so that lm = d. Then e(r7) = e(r7)

l/m.

If now M is a Kleinian 3-manifold M (G) for some discrete group G C Isom i3, the following holds:

(1) Alexander's theorem on the unknottedness of a two-dimensional polyhedral sphere into the three-dimensional one (see Bing [1]) implies that M(G) is irreducible. (2) The sphere theorem and (1) imply that M(G) is aspherical. (3) Similarly, (1) implies that M(G) is a Poincare manifold, i.e., each homotopy ball embedded in M(G) is homeomorphic to the Euclidean ball. Therefore,

5.1. Basic topology related to Kleinian manifolds

221

although the Poincare Conjecture has not been proven in dimension three, this does not matter in the study of Kleinian manifolds. Lemma 5.8. Let Mi and M2 be 3-manifolds without boundaries with trivial homotopy groups n2 (Mi) = 7r3 (Mi) = 0, i = 1, 2. If Mi is compact and 7rl (MI) = 7r1 (M2), then there are isomorphisms of the homology groups, Hk (MI, 7L) = Hk (M2, Z) , k = 0, 1, 2, 3, and in particular M2 is also compact.

Proof This follows from the existence of a continuous mapping f : M1 -* M2 that induces the isomorphism 7r1 (MI) -i 7rl (M2) (see Hu Sze-Tsen [1]), together with the Whitehead's theorem (see Spanier [1], Ch. 7, §5) which asserts that this mapping f induces the desired isomorphisms of the homology groups of our manifolds Ml and M2. Then compactness of the manifold M2 follows from the fact that H3 (M2, 7L) ; 0. 0

Theorem 5.9 (The Compact Core Theorem, see Scott [1, 2]). Let M be a 3-manifold whose fundamental group 7r1 (M) is finitely generated. Then 7r1 (M) is finitely presented, and there is a compact submanifold MM of M such that the inclusion induces 7r1 (M). an isomorphism 7r1(Mc)

Let E be the total space of a fiber space, whose base is the circle S1 and whose fiber F is arcwise connected. From the exact homotopy sequence, 0 = 7r2(S) --3 7r1 (F) -+ 7r1 (E) --) 7r1 (S) = Z -* 7ro(F) = 0,

we see that 7rl (E) has the normal subgroup 7r1 (F) with the quotient group Z. If E is a compact 3-manifold, it is reasonable to expect F to be a compact 2-manifold; in particular 7rl (F) would be finitely generated. Now the converse is, to some extent, true:

Theorem 5.10 (Stallings's Theorem, see Stallings [2]). Let E be a compact 3-manifold whose fundamental group 7rl (E) contains a finitely generated normal subgroup G with the quotient group 7rl(E)/ G - Z. Then G is in fact the fundamental group of a 2-manifold F embedded in E. Moreover, if G is not 7L/27L, and if E is irreducible, then E is the total space of a fiber space over S1 whose fiber is the above 2-manifold F.

2. Connected sum of manifolds. In this and the next subsections, we recall some well-known methods of constructing manifolds, some of which allow a direct realization in terms of Kleinian groups and are closely related to the combination theorems, constructions of fundamental polyhedra, etc.

Let M, M1 and M2 be n-manifolds (with or without boundary). Assume that there are n-dimensional topological balls Bi C int M1, i = 1, 2, and embeddings hi: Ni = Mi\ int Bi -* M such that hl(N1) fl h2(N2) = hl(aB1) = h2(aB2) and M = h1(N1) U h2 (N2). Then the manifold M is called a connected sum of manifolds M1 and M2 and denoted by M = Mi#M2. If here M, Mi and M2 are orientable, one requires that the embeddings hi preserve orientation.

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5. Kleinian Manifolds

The construction of manifolds by means of the connected sum (in the geometrically

uniformizable case) can be realized by the Klein combination (see Theorem 5.12, Example 5.13 and Theorem 6.68).

3. Disk sum of manifolds. Let M, M1 and M2 be n-manifolds with boundary and Di a topological (n - 1)-dimensional disk in the boundary of the manifold MM, i = 1, 2, and let f be a homeomorphism between D1 and D2. The manifold M obtained by gluing M1 and M2 according to the mapping f is called a disk sum of the manifolds M1 and M2. In the case of orientable manifolds, one requires that the homeomorphism f preserve orientation. Let G1 and G2 C Isom W be discrete groups acting on 8IH[n = Sr-1 as Kleinian

ones and let G = G1 * G2 be a group obtained by the Klein combination (Theorem 5.12). Then we see directly that the Kleinian manifold M(G) is a disk sum of Kleinian manifolds M(G1) and M(G2). 4. Knot and link complements and universality. As we saw in Chapter 4, neighborhoods of ends of geometrically finite Kleinian manifolds virtually look like "punctured solid flat manifolds" or "punctured solid cylinders". In the particular case of empty boundary, i.e., in the case of hyperbolic n-manifolds of finite volume, end neighborhoods can be finitely covered by the punctured solid torus,

Tn-lx(0,17n-2x{xER2:0
interest, see Chapter 6. For 3-manifolds, the above situation corresponds to the case of 1-dimensional knots and links (in 3-manifolds) whose components T1 S1 are simple loops. We will discuss the hyperbolicity of fundamental groups of such 1-knot (and link) complements in Chapter 6, §3. Now we shall consider a special class of universal knots and links. According to the initial definition of W. Thurston [3], a link L in a closed oriented 3-manifold M was called universal, if every closed oriented 3-manifold N has a link L' C N such that N\L' is homeomorphic to a finite covering of the complement M\L. Actually, due to Thurston [3], the Whitehead link in S3 is universal in the above sense. Here we will however use a stronger definition of universality of knots and links due to HildenLozano-Montesinos [1-3] and Hilden-Lozano-Montesinos-Whitten [1]. These papers contain strong results in this direction and give a new (geometric) approach to 3-dimensional Poincare Conjecture based on the universality, see §3. Namely, a knot or link in S3 is called universal if every closed oriented three-dimensional manifold occurs as a branched covering of S3 whose branch set is the knot or link. Hilden, Lozano and Montesinos [1, 2] found out a lot about universal knots and links, in particular, the figure-eight knot, the Whitehead link and the Borromean rings.

5.1. Basic topology related to Kleinian manifolds

223

5. Dehn surgery. Let L be an (n - 2)-dimensional knot (or an m-component link) in a given n-manifold N, i.e., a locally flat submanifold of N homeomorphic to (n - 2)sphere Sr-2. One can remove a tubular neighborhood of some component of L (whose boundary is homeomorphic to Sn-2 x S1) and sew it back by means of a certain selfhomeomorphism of Sii-2 x S. That procedure is known as Dehn surgery. In particular, for N = Sn, we have a knot K in S" whose exterior X is the closure of the complement of a regular neighborhood of K Si-2 in Sn. It is clear that two isotopic knots (S', Kl) and (Sn, K2) have homeomorphic exteriors X1 and X2. As to the converse in dimensions n > 4 (in contrast to n = 3), a strong finiteness (see Gluck [1], Browder [1], Swarup [1]) takes place. Namely, for n > 4, there are at most two non-equivalent embeddings of a (n - 2)sphere Si-2 in the n-sphere Sn with a given exterior. This finiteness is based on the observation that two self-homeomorphisms of aX Sr-2 x Sl are pseudo-isotopic if and only if they are homotopic. The group of such pseudo-isotopy classes is thus isomorphic to Z2 X Z2 X Z2, where the first two factors correspond to orientation reversal of Sr-2 and S1 respectively, and the third is generated by a homeomorphism

: Sn-2 X S1 --k Sn-2 x S' defined as ri(x, 0) = ('r(0)(x), 0) , where r (0) is the rotation of Sn-2 about its polar Sn-4 by the angle 0. Furthermore, any Dehn surgery on the so-called twist-spun 2-knot K in S4 (see Rolfsen [1] and Chapter 7, §6) always gives S4 itself, see Gordon [1]. Let us be more precise in dimension n = 3 where there are many different Dehn surgeries on a given knot or link L CN. Namely, let Pi be a tubular neighborhood

in N of the i-th component of the link L, Tie = aPi, with ai the meridian (ai = 0 in 71), and bi the parallel on the torus Tie, see Figure 62. We shall consider these meridian and parallel as generators of H1 (Ti , R). Then the generalized Dehn surgery

in M = N\L, with parameters (dl, ... , dm), di = (ai, Pi) E 1[82, is a procedure that gives a space Mdl,

dm by gluing solid tori to M\ (U Pi) in such a way that the meridian

in the glued i-th torus goes over to ai ai + #1b1. If, on the other hand, the i-th link component is not involved, the pair di = (ai, fl) is denoted by oo.

Figure 62

A precise description of Dehn surgery on knots in S3 might be found in CullerGordon-Luecke-Shalen [ 1 ]. Resulting manifolds Md, ,...,dm with infinite fundamental

5. Kleinian Manifolds

224

group will be discussed in Chapter 6 (for the case of finite fundamental groups, see Bleiler-Hodgson [1]).

6. Gluing of polyhedra faces. For a fixed (X, G)-geometry, let us consider a finite family of polyhedra Pi C X, homeomorphic to a ball, and let the faces (i.e., (n - 1)sides) of these polyhedra split in pairs such that any two faces from a pair have a similar combinatorial type. After this, we can glue in pairs the faces of polyhedra Pi by means of orientation-preserving homeomorphisms in such a way that the vertices and k-sides of any face are glued to the vertices and k-sides of the other face. The resulting complex

K is called a pseudo-manifold: neighborhoods of its vertices are cones over some (n - 1)-manifolds. It is orientable if and only if all the gluing homeomorphisms are orientation preserving. This pseudo-manifold becomes a manifold if neighborhoods of all its vertices are topological balls. There is a simple and effective criterion for that:

Theorem 5.11. A pseudo 3-manifold K obtained by face pairing of finite number of polyhedra is a (closed) 3-manifold if and only if its Euler characteristic X (K) is zero.

Proof The necessity of the condition (for any odd-dimensional K) follows from the Poincard duality and the following expression of the Euler characteristic through ranks of homology groups:

X(K) _

(-1)` rank Hi (K) .

(5.3)

Note that this also shows that, for any compact odd-dimensional manifold M with boundary M, we have X (M) = X (0M)/2. T o prove the sufficiency, assume X (K) = 0 and let xl, ... , xm be all singular vertices of K with cone-neighborhoods Ui = Con Si over surfaces Si. Clearly, X (Ui) = 1, and the complement of these neighborhoods in K is a 3-manifold N with aN = Ui Si. So we have: m

X(K) =X(N)+X(UUi) -X(0N) =m-X(aN)/2= i

(1 -X(Si)/2) i=t

For each singular vertex xi, the surface Si is not a sphere, so X(Si) < 2. Hence the equality X (K) = 0 is possible only in the case m = 0. In this case, all vertices xi are not singular, and the complex K is a manifold. As it has been mentioned in Chapter 1, if one takes the gluing homeomorphisms in the above construction from the isometry group of the (X, G)-geometry (e.g., hyperbolic, Euclidean, elliptic, etc.), the corresponding geometric structure can be introduced into the manifold K, see also Chapter 4, §5. In particular, in the case of one polyhedron P, if this polyhedron is a fundamental polyhedron of a Kleinian group G C Mob(n), then one can introduce at least a conformal structure in the manifold K, and K = S2 (G)/G, see also Chapter 6.

5.2. Topological aspects of combination theorems

225

§2. Topological aspects of combination theorems In the previous section, we presented some topological methods to construct manifolds from simple blocks (connected sum, etc). Very often one can realize those constructions geometrically and even algebraically, in the sense that the fundamental group of the constructed manifold can be build from fundamental groups of blocks. Our purpose here is to describe explicitly such splittings of Kleinian manifolds M(G) and the corresponding algebraic structure of Kleinian groups G that occur in such constructions.

1. Free amalgamated products. Here we shall study two distinct but related types of the so-called Klein-Maskit combination theorems for discrete groups, which provide major ways of constructing new discontinuous groups from simple blocks, that is from some discontinuous groups with known properties. In algebra, these constructions are well known as free amalgamated products of groups (in particular, free products) and HNN-extensions of groups, see Lyndon-Schupp [1]. Suppose that we are given two groups, G 1 and G2, with a common proper subgroup H. We also assume that these groups are subgroups of some universal group G of self-homeomorphisms of some space X. Although the main case here is the case of the Mobius group G = Mob(n) acting on the n-sphere X = Sn, one can consider the cases of convergence subgroups G1 and G2 of self-homeomorphisms of the sphere X = Sn or the ball bounded by Sn (including isometry groups of Hadamar manifolds, in particular negatively curved symmetric spaces of rank one). For given groups G 1 and G2, we define a normal form of an element g E (G 1, G2)

as a word g = gn ... gl where either every gk with even k lies in G1\H, and every gk with odd k lies in G2\H, or vice versa. Regarding the forms gn . . . gk ... gl and (gkh)(h-lgk-1) . . . gl as being equivalent, we define a natural identification gn of normal forms. Also we have an obvious multiplication of normal forms by using the juxtaposition of words followed by a possible contraction according to the above equivalence. So, the product of normal forms is equivalent to either a normal form or to an element of the subgroup H. This allows us to define the free product of G1 and G2 with amalgamated subgroup H (or amalgamated free product), G = GI *H G2, as the set of equivalence classes of normal forms with the above multiplication, together

with elements of H. Although a normal form of an element g = gn ... g t E G is clearly not unique, the length I S I = I gn . . gt I = n of g is well defined because equivalent normal forms have the same length.

Every non-trivial element of G = G1 *H G2 has a unique normal form if the amalgamated subgroup H is trivial. In this case we say that G is the free product of groups G1 and G2, and write G = G1 * G2. Klein Combination. The first combination theorem was due to F. Klein [1] who has considered the free product of two discontinuous groups on the plane.

Theorem 5.12. Let {Gi}lEz be a family of (non-trivial) discontinuous convergence groups Gi on the n-sphere Sn having fundamental domains FL and satisfying the

5. Kleinian Manifolds

226

following two conditions

(1) Sn\Fi C Fj for any i

(2) F=int(niFi); 0.

j;

Then the group G = (Gi, i E ,Z) generated by all groups Gi is the free product of these groups, G = *iEr Gi, is discontinuous, and F is its fundamental domain. Proof. As for the free product of two groups G1 and G2, we identify the free product *i Gi with the set of equivalence classes of normal forms gn ... gt (where any adjacent elements gi and gi_1 belong to different groups from our family), with the multiplication given by juxtaposition of words followed by the above equivalence. Every non-trivial element of the free product has a unique normal form. We have to prove that the natural homomorphism

(D:*Gi-->G=(Gi,i EX),

1(gn...gi)=gn...g1,

(5.4)

is an isomorphism, that is (D maps no non-trivial normal form to the identity. It is obvious that no normal form of length one can be the identity in G. Now let us consider a normal form g = gn . . . gi of the length n > I where elements gj are from the corresponding groups Gi,. For any i E I, we denote Xi = Sn \ Fi . Assuming that k ; i1, we obtain the following chain of proper inclusions (due to the first condition of the theorem):

g(Xk) = gn ... g1(Xk) C gn ... g2(Xi1) C ... C gn(Xi,,_1) C Xi So, g(Xk) is properly contained in one of Xi, and hence c(g) is not the identity. Therefore, G is the free product of G1, i E X. Similarly, for any g = gn ... gt E G\{l }, one has

g(F) = g.... gl (F) C gn ... g2(Xi1) C ... C

C Xin

,

and hence g(F) fl F = 0. This shows that the group G is discontinuous and F belongs to its fundamental domain. To finish the proof, we have to show that each ordinary point x E 0 (G) is mapped

by some g E G to a point in F. Assume x ¢ F. Then either x E Xi for some i E x or x E a(Ui Xi). Since x is not a limit point of G, x cannot be a limit point of an infinite sequence of disjoint sets X i . So, even for infinitely many groups G1, x belongs

to some Xi. Moreover, this argument shows that there are at most finitely many sets gj (Xk) which contain the point x. Therefore, it is enough to prove our claim in the case of a free product of finitely many groups Gi, which case is obvious.

Example 5.13. Let G = G1 * G2 * . * Gm, be the free product of Fuchsian groups Gi C Mob(n) which satisfy the conditions of Theorem 5.12 and uniformize closed

hyperbolic n-manifolds Mi, i = 1, ..., m. In other words, we have open balls

5.2. Topological aspects of combination theorems

227

B1, ... , B,n C ][8" sufficiently far apart and invariant for the corresponding groups

Gi, so that Bi/Gi = M. Then G is a Kleinian group with an invariant connected component 0o C 2 (G), S2o C nm 1 ext(Bi), which is simply connected if n > 3. All other components of 0 (G) are G-images of balls B1, and we have U Mn . Here the manifold Mo = S2o/ G may be obtained as the connected sum of manifolds (S"\Bi)/Gi which are hyperbolic manifolds isometric to the manifolds Mi. So MO = MI# - #Mn. In particular, for n = 2, Mo is a surface whose genus is equal to the sum of genera of surfaces Mi. We also mention that, for n > 3, Mo is not hyperbolic because nn_1(Mo) ,-b 0. S2 (G) l G = Mo U Mi U

Another major application of the Klein combination theorem is to an infinite family of block groups Gi, i E X.

Example 5.14. Let E be an infinite family of (n - 1)-spheres o'i C Sn bounding disjoint closed balls Bi. Assuming that the complement of these balls Bi has a nonempty interior FE, we consider the group GE generated by involutions (i.e. elliptic transformations of order two) each of which fixes the corresponding sphere ai E E and maps ext Bi onto int Bi. Each of these involutions generates a group Gi of order two, and GE is the free product of all groups Gi. Theorem 5.12 shows that FE is a fundamental domain of the Kleinian group GE. This approach in particular allows us to construct Kleinian groups G C MSb(n) with discontinuity set components 0o C 0 (G) of a given homotopy type (for example, with given rank nl (0o) = k or

rankn2(S2p) = m). In this case, S2o = FE and 8FF C A(GE). We also note that the proof of Theorem 4.49 (see Ch. 3, §2 in Apanasov [36]) is in fact based on this approach.

Disk sum of Kleinian manifolds. Suppose we are given Kleinian manifolds M(G1) and M(G2) with closed Euclidean disks Di C 8M(Gi), i = 1, 2. In the case when M(GI) = M(G2), we specify that DI fl D2 = 0. Here, by an Euclidean disk D C aM(G) we mean a domain such that each component of it-1(D) c dlH[3 (the preimage of the natural projection n: W U 0 (G) -± M(G)) can be mapped into the unit disk U C R2 by a Mobius transformation. We shall show that there exists a Kleinian group G3 C Isom H3 such that M(G3) is homeomorphic to the manifold obtained from M(Gi), i = 1, 2, by a disk sum, that is by identifying the disks DI and D2. To construct G3, we assume initially that M(G1) # M(G2) (the case of GI = G2 is still possible). We denote by Di a fixed component of rr-1(DI) and find a group G'2 conjugate to G2 (so, M(G2) = M(G2)) such that one of the components of n (D2) coincides with the complement to Di . GI * G2 (see Figure 63) obtained by the Klein Then the group G3 = (GI, G'2) combination from GI and G'2, has the desired property. In the case when M(GI) = M(G2), we specify along with Di a certain component D2 C 7r-1 (D2). It is clear that D1 fl D2 = 0. Let g be a Mobius transformation that transfers the exterior of Di into the interior of DZ, while the groups GI = G2 = G

correspond to the manifolds M(G1) = M(G2) = M(G). Then one can take G3 = (G, g) - G * (g). We note that, if DI and D2 lie on the same component of the

228

5. Kleinian Manifolds

boundary M(G), this process gives a new handle on that component. If the disks Di he on different components of the boundary 8M(G) then the gluing combines these components into a connected component of the boundary of the resulting manifold M(G3), see Figure 64.

Figure 63

Figure 64

In the above, we can assume that the Euclidean disks D1 and D2 are tangent, Dl fl D2 = {p}. Then one can take the mapping g to be a parabolic transformation with the fixed point p* = Di fl D. The effect of gluing disks D1 and D2 then gives a pair of punctures in the corresponding component of the boundary aM(G3). For example, a Schottky group of rank g corresponds to gluing together 2g disjoint disks in the sphere at infinity S2 = aH3, so its Kleinian manifold M(G) is ahandlebody of genus g. For another example, we can consider Fuchsian groups G1 and G2 on the plane that uniformize ni -punctured surfaces of genus gt and $2. Then G3 = G1 * G2 is a functional group having an invariant component no(G3) C aH3 and the surface 920(G3)/ G3 has genus gi + $2 and n 1 + n2 punctures. 2. The first Maskit combination. Here we consider free products G = G1 *H G2 with non-trivial amalgamated subgroups H. In what follows let us assume for convenience that the discontinuity set Q (Gi) C Sn = Rn U {oo} of any given discontinuous group Gi contains the infinity. We can obey this condition by conjugating these groups by suitable transformations g E Mob(n), g(oo) = xi E S2 (G1). By mn(X) we denote the Lebesgue n-dimensional measure for a set X C Sn. Let us start with two auxiliary lemmas which shall be used in the proof of the Maskit combination theorem.

Lemma 5.15. Let G C Mob(n) be a discontinuous group whose discontinuity set 0 (G) contains oo, and Gm = {id}. Also, let rg denote the radius of the isometric

5.2. Topological aspects of combination theorems

229

sphere 1(g) of an element g E G\{id}. Then rg2n

< 00 .

(5.5)

gEG\{id}

Proof. Due to the conditions of the theorem, there is a neighborhood U of oo such that g(U) fl U = 0 for all g E G\{id}. So we can find aball B = B(x, R) centered at a point x with a sufficiently large radius R > 0 such that B contains isometric spheres

of all elements g E G\{id}. Then the images g(B), g E G\{id}, of the exterior B = Sn\B(x, 2R) of the ball B(x, 2R) are disjoint balls inside of the ball B(x, R), whose radii behave like rg /2R as rg -* 0. Since the sum of volumes of these balls g(B) is bounded by the volume of B(x, R), we have that the series (5.5) is convergent. 11

Lemma 5.16. Let G C Mob(n) be a discontinuous group whose discontinuity set SZ (G) contains oo, and H C G a subgroup preserving a Jordan hypersurface S C Rn all whose images g (S), g E G, are bounded. Also, for fundamental domains A and F of groups H and G respectively, let there exist a neighborhood V of S such that V fl A C F. Then, for any coset decomposition G = H U gi H U g2H U ... with gi E G and gi H fl gj H = 0, the Euclidean diameters d (gi (S)) of surfaces gi (S) form the following convergent series: 00

E dn(gi(S)) < 00.

(5.6)

i=1

Proof. Since oo E 0 (G) and its stabilizer subgroup in G is at most finite, we may assume that g(oo) ; 00 for any g E G\{id}. For each such g E G\{id}, let q(g) = g-1(oo) be the center of the isometric sphere I (g). Then the set {q(y) : y E gH} is invariant with respect to H:

q(gh) =

(gh)_1(oo)

= h-1(q(g)).

(5.7)

For each element gi E G, i = 1, 2, ... , from our coset decomposition, there is a unique hi E H such that the point q (gi hi) = hi 1(q (gi )) lies in the fundamental domain A of H. So, one may assume that q(gi) E A. Let Si be the Euclidean distance between q (gi) and the surface S. Then, due to (5.7) and the conditions of the lemma, there is a number io such that q (gio) E A fl V C F. Hence we can choose such

Sp > 0 that, for almost all i, Si > So. Applying Proposition 1.7, we may represent elements gi as gi = Ti zi where Ti E Isom R" and Xi is the inversion with respect to the isometric sphere I (gi) with radius ri > 0. It implies:

d(gi(S)) = d(xi(S)) < d[Zi(Sn-1(q(gi), Si))] These inequalities and Lemma 5.15 complete the proof. Now we have the first Maskit combination theorem:

r2/8o

5. Kleinian Manifolds

230

Theorem 5.17. Let G1, G2 C Mob(n) be discontinuous groups with a common subgroup H, and let the n-sphere Sn split along a hypersurface S C S' into two domains D1 and D2 whose closures D1 and D2 are precisely invariant with respect to H, in G1 and G2, respectively. Let also the following two conditions hold: I. For fundamental domains A, F1 and F2 of the groups H, G1 and G2, there exists a neighborhood V of the surface S such that A fl v C Fi, i = 1, 2. H. For each i = 1, 2, the set A fl Di = D; fl Fi is a proper subdomain in Fi. Then:

a) The group G = (G1, G2) is discontinuous and isomorphic to the amalgamated free product G 1 *H G2.

b) F = F1 fl F2 is a fundamental domain for the group G. c) mn(A(G)) = 0 if and only if mn(A(Gi)) =0, i = 1, 2. d) Each elliptic or parabolic element of G is conjugate in G to an element from G1UG2. Proof. Let 0: G1 *H G2 -> G = (G1, G2) be the natural homomorphism where the image 0 (gn ... 91) of the normal form is the composition gn . gl . In order to prove that G = G1 *H G2, we have to show that 0 maps no normal form g = gn ... gl of the length n > 2 to the identity. For that, it is enough to show that g(F) fl F = 0 and F 0 0. The last claim follows from condition II. Namely, F fl D2 0 0 because it contains a non-empty set F1 fl D2 C A fl D2 = F2 fl D2. Similarly, F fl D1 # 0. Now let x E F, and we assume for definiteness that, in our normal form g = gl, the first element gl is from G1\H. Due to A fl D1 = F1 fl D1 and precise gn

H-invariance of D1 in the group G1, for any x E 51 and gi E G1\H, we have that gi (x) E D2 and gi (x) F. Similarly, for X E 52 and gj E G2\H, we have gj (x) E D1 and gj (x) % F. Therefore, for any x E F, the following takes place: gt (x) E D2 ,

gl (x)' F ;

g2gl (x) E D1 ,

g2g1(x) ¢ F ,

etc.

This shows that g(x) ¢ F, and hence the group G = G1 *H G is discontinuous. Using this fact and conjugating our groups, we may assume that oo E int(F\S), which implies that all surfaces g(S), g E G, are bounded. Now let us prove the inclusions: A f1 V C F C S2 (G) .

(5.8)

The first inclusion directly follows from A n V C Fi, i = 1, 2. To prove the second inclusion, we note that condition II implies that F fl D1 = F2 fl D1 and F fl D2 = Fl n D2. So, for a normal form g = gn ... gl and for any point x E F fl (D1 U D2), there exists a neighborhood U of x such that g(U) fl u = 0 for all but finitely many g E G. To prove the same property for points xp E F fl S, we write obvious inclusions,

F fl S= A fl S C A fl V C F, and take a neighborhood U of xp, U C V. We may consider U so small that all its points are H-equivalent to points from A fl v C F;

5.2. Topological aspects of combination theorems

231

otherwise, we can find a sequence of points xm converging to xo and a sequence {hm} C H such that lim hm (xm) = yo ' V. In this case, the point yo is a limit point of the group H, so it must belong to the surface S, which contradicts to yo ¢ V. Therefore, if U is such a small neighborhood of a point xo E F n S, we have that g(U) n U 0 only for finitely many g E G because g(F) n F= 0 for g E G\{id}. This completes the proof of (5.8). Now, for a normal form g = gn . . . gi E G, we call g positive (correspondingly,

negative) if gi E G1\H (or gi E G2\H). Using this, we consider a fixed coset decomposition of G:

G = H U (U anmH) U (U bnmH) n,m

(5.9)

n,m

where, for a given n > 0, the elements anm > 0 and bnm < 0 are positive and negative normal forms, respectively. So we define:

Tn=Uanm(D1)Ubnm(D2),

Cn=Sn\Tn,

M

C=UCn, T=Sn\C.

(5.10)

n

In order to prove the statements (b) and (c), it is enough to show that, for x E C, there exists g E G such that g (x) E F U A (G 1) U A (G2), and furthermore the set T belongs to A(G) and has zero measure.

Let us describe C1. Since for any x E T1 n D2 there exists gi E G1\H such that g(x) E D1, we see that for any x E G1 n D2 either x E A(G1) or there exists g E G1 such that g(x) E F1 n D2 C F. Similarly, for X E C1 n D1, either x E A(G2) or g(x) E F for some g E G2. Finally, for X E C n S, we see that either x E A(H) C A(G1) n A(G2) or h(x) E A n S= F n S for some h E H. Therefore, for any x E C1, there exists g E G such that g(x) E F U A(G1) U A(G2). For arbitrary n > 1, we see from (5.10) that Tn C T,,_1 and hence C,,_1 C C. Furthermore, for any x E C, there exists a number n such that x E Cn\Cn_1, that is x E Tn_1, and we have two cases: either x = an_1,m(y) for some y E D1 or x = bn-l,m (y) for some y E D2. Assuming for definiteness the first case (the second

one is analogous), we have that the point y is either in Tt or in C1. For y E T1, due to (5.10), there exists a point z E D2 and g E G2\H such that y = g(z). So x = an_1,mg(z) where Ian-1,mgl = n and an_1,mg < 0, which contradicts the fact that x ¢ Tn. This shows that for x E Cn\Cn_1 there are y E C1 and go E G such that go(x) = y. This fact and the property of C1 we have proved above imply that for any x E C there is g E G such that g(x) E F U A(G1) U A(G2). We now characterize the set T. We shall show that each connected component Tnm of the set Tn has the form of either anm (D1) or bnm (D2) and, instead of (5.9) and (5.10), we may write:

G=HU(UcijH), Tn=UTij, i,j

ij

(5.11)

5. Kleinian Manifolds

232

where I ci j I = i and (disjoint) domains Ti j are bounded by disjoint surfaces ci f (S). Namely, for a fixed n, anm (DI) fl bnm (D2) = 0 because of the fact that anm (D1) C D2

and bnm (D2) C DI if and only if n is odd. Next, for two different elements anm

and ank in (5.9), we write anmank = gr... gl where r > 0 and gi, g,- E G1\H. Hence anmank(D1) C D2 and anm(D1) n ank(Di) = 0. Similarly, we get that bnm(D2)flbnk(D2) = 0form 0 k. Moreover, the boundary surfaces of these domains

are disjoint, g(S) fl s = 0 for g E G\H, because this is true for g E (G1 U G2)\H. So, this completes the above characterization (5.11) of sets Tn which are in addition bounded because 00 E F C Cl. Applying Lemma 5.16 to the coset decomposition of G in (5.11), we obtain a convergent series:

dn(Tij) _00YEdn(Tj) < 00.

d"(cij(S)) _ i, j

i, j

Its convergence implies that

lim Y'dn(Tij) = 0.

(5.12)

i-+oo

j

Now, since any point x E T belongs to all sets Ti, i > 1, we have a nested sequence of connected sets Ti, j (i) containing the point x, whose Euclidean diameters tend to zero due to (5.12), limi, d (T , j fii) = 0. Since all elements ci, j ti i are distinct, this fact shows that x is a limit point of G, that is T C A(G).

The obtained description of the sets T and C, T U C = Sn, shows that F is a fundamental domain of the group G. Moreover, the following easy bounds: mn(Tij) < (on

mn(T) < mn(2) _ ./

j

dn(Tj)

and the property (5.12) imply that mn (T) = Mn (T fl A (G)) = 0. So we can prove statement (c) by showing that the complement A(G)\T = A (G) fl C has zero measure if mn(A(Gi)) = 0, i = 1, 2. This fact follows from the countability of G and the following representation based on the above properties of the set C:

A(G) fl c = U g(A(G1) U A(G2)). gEG

Regarding the statement (d), let g* be an element of G conjugate to a given element g E G and such that its length I g* I = m is minimal. Due to this minimality, the number m either is equal to 1 or is even. If m = 1, then g* E Gi U G2, and there is nothing to

prove. So we may assume that m is even and g* > 0 (the case of g* < 0 is similar). Then it is obvious that g*(D1) is properly contained in DI because Di is precisely H-invariant in the group Gi, i = 1, 2. Hence g* has an infinite order, and there are at least two fixed points of g*: one in g* (Dl), and the other in D2. Since these properties are possible only for loxodromic elements, this completes the proof of (d) and of the whole combination Theorem 5.17.

5.2. Topological aspects of combination theorems

233

Punctured disk sum of Kleinian manifolds. As a topological application of the first Maskit combination, we may consider a "punctured disk sum" of Kleinian 3manifolds. For a puncture p in 0M(G), we can define a special (punctured disk) neighborhood, the so-called horocyclic disk D. Such a horocyclic disk D is conformally equivalent to a disk with a puncture, and each component of it-1(D) is a Euclidean disk or a half-plane, invariant with respect to the maximal parabolic subgroup in the group G that is determined by the puncture p. Let M(G1) and M(G2) be two given Kleinian 3-manifolds having punctures p1 in 8M (G 1) and p2 in aM(G2) with horocyclic disks D1 and D2, respectively. Then there exists a discrete group G3 C Isom 1H13 such that M(G3) is homeomorphic to the manifold obtained from M(G1) and M(G2) by identifying D1 and D2. To construct this group G3, we specify a component Di C n-1(D) and denote by Gp a maximal cyclic parabolic subgroup in G1 such that Gp(D*) = D'. Also we take a conjugate

G'2 of the group G2 such that ext(D*) is a component D2 C n-1(D2) and Gp is a maximal cyclic parabolic subgroup in G2 that leaves the component DZ invariant. Then G3 = (G1, G2) G1 G'2, see Figure 65. Gn

Gp

Figure 65

This construction can also be performed for two distinct parabolic punctures pi and p2 at the boundary of one manifold M(G). Namely, for these punctures pi and their disjoint horocyclic disks Di in aM(G), i = 1, 2, we specify lift components D C rr-1(Di) and consider maximal parabolic subgroups Gi C G corresponding to

234

5. Kleinian Manifolds

pi, i = 1, 2. Let g be a Mobius transformation that transforms ext(D') into int(DZ) and conjugates the subgroups G1 and G2, G2 = gGig-1. Then the manifold M(G3) corresponding to the group G3 = (G, g) is homeomorphic to a manifold obtained from M(G) by identifying the horocyclic disks D1 and D2. In particular, for Fuchsian groups G1 and G2 on the plane K2 that uniformize ni-punctured surfaces of genus gi, i = 1, 2, the Kleinian group G3 obtained by gluing one puncture in 8M(G1) to another puncture in 8M(G2) is a B-group. Here Qo(G3)/G3 is a (ni + n2 - 2)-punctured surface of genus (gt + 92). Each of the other two components of 8M(G3) is a n1-punctured surface of genus gj, i = 1, 2, see Figure 65. Gluing two punctures in the same component of 3M(G1), one gets a Bgroup G3, whose surface S2o(G3)/ G3 is a (ni - 2)-punctured surface of genus gi + 1. The other component of aM(G3) comes from 8M(G1), i.e., this is a nl-punctured surface of genus gl. In that way, we can eliminate all the punctures in aM(GI) and obtain a Kleinian manifold M(G3) having only punctured solid tori as its ends.

3. HNN-extensions. Here we shall study a geometric realization of another basic group-theoretic construction, the so-called HNN-extension of groups. Our basic hypothesis is that we are given two groups Go and G1 (which are subgroups of some universal group G of self-homeomorphisms of some space X) with trivial intersection, where G 1 is infinite cyclic with a generator f , and the group Go has two isomorphic subgroups H1 and H2. Additionally we assume that f conjugates H1 into H2, that is,

h r* f hf -1 defines an isomorphism f,,: H1 -> H2. Wedefine anormalformof an elementg E (Go, f) asawordg = f"'. g, f"lgt, where each gk E Go, and gk 1 for k > 1. Here among integers ak, only a,, is permitted to be 0. Moreover if ak < 0 and gk+l E Hl \{ l } then ak+l < 0, and if ak > 0 and gk+1 E H2\ { 1 } then ak+1 > 0. We call two normal forms equivalent if we can get from one to the other by a finite sequence of deletions and insertions of words which we know to represent the identity. Examples of such words are conjugates and

inverses of words of the form fhf-1(f,(h))-1, h E H1. The set of normal forms modulo this equivalence, together with the identity, forms a group G = Go* f where the operation is the juxtaposition of words. This group is called the HNN-extension of Go by f. Notice that the HNN-extension Go* f coincides with the free product Go * (f) if the subgroups H1 and H2 are trivial. This case is the only one when a normal form of g e Go* f is unique. However, even different normal forms of an element g have t h e same length Ig I which is defined as g = Em I am 1. Also, we can define a sign of an element g = f"ngn ... f"1gl as follows. We call a normal form of g positive if an > 0, negative if an < 0, and a null form if an = 0. For a null form, we notice that gn ¢ H2 if an_1 > 0, and gn ¢ Ht if an_1 < 0. Since the equivalence of normal forms preserves such defined sign, we may talk about sign of g as the sign of its normal form. We write g > 0, g < 0, or g ' 0 for these three cases of positive, negative or null-elements. Conditions for an HNN-extension G = Go* f to be geometrically realizable are

5.2. Topological aspects of combination theorems

235

given by the following second Maskit combination:

Theorem 5.18. Let Go C Mob(n) be a discontinuous group with subgroups H1 and H2, and closed domains 51 and D2 bounded by Jordan surfaces S1 and S2 be precisely

invariant in Go with respect to H1 and H2, correspondingly. Let also f E Mob(n) map D1 onto S"\D2, and the following conditions hold: I. For fundamental domains A 1, A2 and FO C Al fl A2 of groups H1, H2 and Go, correspondingly, there exist neighborhoods V1 and V2 of surfaces Si and

S2 such thatDi 0 Vi C F0, i = 1,2. H. Di f1 Di = Fo fl Di, i = 1, 2.

III. F=int(Foflext(D1 UD2)) 00. IV. fHi f-1 = H2. V. g(D1) n"52 = 0 forall g E Go\{l}. Then:

a) The group G = (Go, f) is a discontinuous HNN-extension of Go by f, G = GO* f, and F is its fundamental domain. b) mn (A (G)) = 0 if and only if mn (A (Go)) = 0. c) Elliptic or parabolic elements of G are conjugate in G to elements of Go. Proof. We will provide basic steps of the proof which very closely follows the proof of the First Combination Theorem 5.17. We start with a natural homomorphism

,: Go*f -+ (Go, .f), (D (f angn ... f«'gl) =

.fang.... fa'91, which is an isomor-

phism if and only if no non-trivial normal form lies in the kernel of 4). To prove that G = Go* f, it is enough to show that g(F) fl F = 0 for any non-trivial normal form

g, that is, for g with either gi 0 1 or al 0- 0. This is true because of the following observations: h2 fa(S'\D1) C D2 for a > 0 and h2 E H2; h1 fa(S"\D2) C Dl for a < 0 and hl E Hl; gl (F) fl F = 0 for gl E Go\{1}; gl(D1) C Sn\(D1 U 52 U F) for gl E Go\H1; and finally, gl (D2) C S.\(-D, U D2 U F) for gl E Go\H2. Therefore, G is discontinuous and equal to Go* f. Moreover, similar arguments show that

FCc2(G). Considering a coset decompositions of G modulo either Hl or H2, we have that all representatives of a given coset have the same lengths and signs. Let akm be such a representative from a non-positive coset g H1 of the length k, and bkm a representative from a non-negative coset gH2 of the length k, where k = 0, 1, 2, .... Then we put Tk = U (akm(D1) U bkm(D2)) ,

Ck = Sn\Tk,

m 00

C=UCk, T=S.\C.

(5.13)

k=0

In analogy with Theorem 5.17 (see (5.10)), we want to show that every point in C is a translate of some point in F U A(Go) U fix(f), and that T has measure zero and is

contained in A(G). First, we show that Tk C Tk_1 and Ck_1 C Ck for k > 0. To

5. Kleinian Manifolds

236

do that, we take a point x E Tk and assume (for definiteness) that x E akm(DI) (the case x E bkm(D2) is obviously analogous). We write g = akm = f"kgk . . fa1g1 and set g* = f"kgk ... f«,-1 if ai > 0. Since akm < 0, it follows that gi 0 Hl, and akm(DI) C g*(D2). So Ig*1 = k - 1, and g* < 0. We conclude that x E Tk_1. For k = 0, we take x E Co = S'\ UgEGo g(DI U D2). Then either x E A(Go) or there is g E Go such that g(x) E Fo. So we have g E G such that g(x) E F U A(Go). Now, let xo be a point in C. As we have seen, xo E Ck+1 \Ck for some k >- 0. Then there either exists an element akm with xo E akm (DI), or xo E bkm (D2) for some bkm. Assuming for simplicity the former case (the latter orre is treated analogously),

we consider xl = akm (xo) E Di. If f (x1)

xl, there exists a number a > 0 so that

x2 E Co. Assuming the contrary, we have g" E Go so that x3 = g (x2) E DI U D2. Then we set g* = akm f -" $_1 whose length I g* I is k* = I akm I + a > k because a > 0 and akm < 0. Now we see that, for x3 E D1, g 0 HI and hence g* < 0; similarly, for x3 E D2, g ¢ H2 and hence g* > 0. In both cases, xo = g*(x3) E Tk* which contradicts to our assumption that xo ¢ Tk+1 because Tk* S; Tk+1. Therefore x2 lies in Co, and finally

for any xo E C there exists g c G so that g(xo) E F U A(Go) U fix f. Furthermore, since T C Co and Tk n T = 0 for every k, this implies that every connected component Tkm of Tk is either akm(D1) or bkm(D2). These disjoint components are bounded by disjoint surfaces akm (S1) and bkm (S2) = bkm f (SI ). So we can write (similarly to (5.11)): 00

G=UUcijH1, Tk=UTkj.

(5.14)

j

i=1 j

Normalizing G so that oo E int F and oo V G (SI) and applying Lemma 5.16 for the coset decomposition (5.14), we observe that the following series of Euclidean diameters d converges:

Ed'(cij(S1)) = Ed'(Tij) = EEd'(Tij) < 00. i,j i=0 j i,j This implies that i

lim E d" (T j) = 0, 1100

(5.15)

j

and any point x E T is a limit point of a nested sequence of connected sets Ti, j (1) whose diameters tend to 0. In other words, T C A(G). Due to this fact, we see that every point x E 0 (G) has an image g(x) E F, g E G. This completes the proof of (a). Applying (5.15) in the same way as we have used (5.12) in Theorem 5.17, we see that T has measure m (T) = 0, which implies the claim (b). To complete the proof of the theorem, we only need to prove its claim (c) about non-loxodromic elements of G. For any non-loxodromic element g E G, let g* be a conjugate of g in G with minimal length Jg*J. If Jg*I = 0 then g is conjugate

to an element in Go. In the opposite case of Ig*J > 0, we have a normal form

5.2. Topological aspects of combination theorems

237

g* = fakgk ... f"91 with ak 0. For simplicity, we may assume that a > 0 (the case a < 0 is similar). Then, due to the minimality of Jg*J, the element g* is non-negative, and hence g* (D2) is a proper subdomain in D2. So g* and g are loxodromic, in contradiction with our assumption. This completes the proof of (c) and of the whole Theorem 5.18. We note that although Theorems 5.17 and 5.18 have been formulated for Kleinian groups acting in the n-dimensional sphere S", they can obviously be reformulated for discrete isometry groups in the hyperbolic space H'4-1. Also one can reformulate these theorems for infinitely many combined groups and get G = (G1, G2.... ) and G = (Go, fl, f2, ... ). Then the resulting groups G are either infinite free amalgamated products or infinite HNN-extensions.

Doubles of Kleinian 3-manifolds. As an application of the Maskit combination theorems, let us provide a geometric realization of the doubling of a Kleinian 3-manifold. Let the boundary of M (G) consist of surfaces Si, i = 1, ... , m, for which all the connected components of 7r -1(SZ ), up to Mobius transformations, are Euclidean disks. Then if we exclude the case of a Fuchsian group G, we can obtain a discrete group r C Isom 1HI3 such that M(r) is homeomorphic to the double of M(G) with respect to the surfaces S1, ..,S .-.

To construct that r, we specify a component Di C it-1(S1) and let 11 be the reflection with respect to the boundary circle aDi. In the first step, we consider a Kleinian group G 1 = (G, X1 G x1) which is a free amalgamated product of groups G and 11G11 with the amalgamated subgroup isomorphic to 7t1 (S1). The manifold M(G1) is homeomorphic to the double of M(G) with respect to Si. Each other

component Si, i ; 1, of the boundary aM(G) corresponds to two surfaces Sri, j = 1, 2, on the boundary of the manifold M(G1), whose components n-1(SSj) are Euclidean disks. The next step is to identify the surfaces S21 and S22. For this purpose, we specify

a component D* C 7c-1(S21). Due to our construction, 11(D2) C n-1(S22). Let us denote the reflection with respect to aD2 by 12. Then Z1 X2: D2 ext X1 (D2*), and x1 x2 conjugates the stabilizers in G1 of the components DZ and x1(D2). Therefore we have the situation of the second Maskit combination (Theorem 5.18), and the group G2 = (G1, x112) is a Kleinian group (HM-extension of G1). The Kleinian manifold M(G2) is homeomorphic to the manifold glued from M(G1) by identifying the surfaces S21 and 522. We continue this gluing of the boundary components Sit and St2, i = 3, ..., m, which finally gives the desired discrete group r. Obviously, x1 r x1 = r, and the reflection x1 specifies an anticonformal involution in the (hyperbolic) manifold M(r) that permutes the two sheets in this double of the manifold M(G). This construction does not work only in the case of a Fuchsian group G. In this case, 11 x2 = id. It is evident from other considerations that, in this case, the

238

5. Kleinian Manifolds

double M(G) cannot be homeomorphic to a hyperbolic manifold. Namely, M(G) is homeomorphic to the product S x Sl of the boundary surface and the circle, which

has a non-trivial center in the fundamental group nl(S x S). This is impossible for hyperbolic manifolds, see Chapter 6, § 3. However, assuming that G acts in the allows a conformal 3-sphere 1183, i.e., G C Mob(3), we see that the double M structure because it is homeomorphic to the quotient [S3\A(G)] /G. In fact, this structure is a conformal realization of the (H2 x R)-geometry on the double M__-(G-), compare Chapter 6, §6.

4. Kleinian 3-manifold of a function group. In this section, we shall show how the technique presented in the previous sections applies to a description of oriented Kleinian 3-manifolds M(G) corresponding to geometrically finite groups G C Isom+ H3 with invariant components S2o C 0 (G) C aH3, i.e., function groups. The n-dimensional case will be discussed in § 5 and in Chapter 6, § 7. For a given function group G C Isom+ H3 with an invariant component S20 of the discontinuity set S2 (G), let So = S2o/ G and S1, ..., Sm denote the remaining components of the boundary 8 M (G). Let us remind that, due to Lemma 2.23, function groups are characterized by the property that the homomorphism nl (So) -> r1 (M(G)) is an epimorphism. We shall consider separately some classes of function groups as follows.

Quasi-Fuchsian groups. Let a function group G have two invariant components C 0 (G). Then its Kleinian manifold M(G) is homeomorphic to a surface 0o, layer So x [0, 1], while the group G is quasi-Fuchsian. This is true even for a finitely generated G (its geometrical finiteness is a consequence). To prove this, we note first that 0o and S21 should be simply connected (cf. Accola 0, Theorem 5.1 implies that there exists a loop, non[1]). In fact, if nl (S20) trivial in So and bounding a disk D, embedded in M(G). This disk D separates the

manifold M(G). Otherwise, one could find a loop y C int M(G) having a nonzero intersection index with D and consequently non-homologous to any loop on the surface S1 = S21 I G, S1 fl D = 0. This contradicts to the fact due to Lemma 2.23, that

n1(Si) --* n1 (M(G)) is an epimorphism. Now let N be a component of M(G)\D, not adjoining the surface S1. Then any loop in N is freely homotopic to a loop in S1, and therefore is freely homotopic to a loop in D. It shows that n1(N) = 0. Then we see that each component of n-1 (N) is homeomorphic to N. Therefore, N fl So is a disk, and aD = a(N fl so) is a trivial loop in So. This contradicts our choice of the disk D, so 00 and 01 are simply connected. Now we shall show how to prove that M(G) is homeomorphic to a surface layer So x [0, 1]. We will do this in the compact case, when So is a closed surface of genus g > 1; for details and the non-compact case, see Marden [1]. Namely, we specify a system of simple loops yl, ... , Yep in So such that each y1 intersects only Yi_1 and yi+1 (at unique points and transversally) and is disjoint from the other loops y,, see Figure 66. The surface So \ U y, is a disk. Due to Theorem 5.2, each loop yi corresponds to a

5.2. Topological aspects of combination theorems

239

simple loop y,' c S1, and they bound together a cylinder Ci in M(G). One may assume that Ci intersects only the adjacent cylinders Ci_1 and Ci+1(transversally, along only

one arc in each cylinder). We denote by C = C, x (-s, E) the 6-neighborhood of Ci. Then the surface a(U Cl) is a cylinder, and the complement So\ [So fl (U Cl)] is a disk. Similarly, as U yt bounds a disk in S1, it follows that S1\ [Sl fl (U CL)] is also a disk. Then int(M(G)\ U CI) is a ball, and hence M(G) is homeomorphic to the layer So x [0, 1].

Figure 66

Now let Go be a Fuchsian group that uniformizes a closed surface of genus g > 1. Due to compactness, we can consider a piecewise linear (and hence quasi-conformal)

homeomorphism M(Go) -* M(G). Its lifting in IEI3 U S2 (Go) is again a quasiconformal homeomorphism, which can be extended to a quasi-conformal automorphism f : ff3E3 -_* R3, see Gehring [2]. The restriction of f to the sphere at infinity 8H3 gives a quasi-conformal conjugation of the group Go, whose result is the group

G = fGof-1. B-groups. Let G be a geometrically finite B-group, i.e., G has a simply connected, invariant component S2o C S2 (G). Let Mo be the manifold obtained from M(G) by cutting out all punctured solid cylinders C1, 1 < i < n, that do not intersect the surface So = 92o/ G. By examining the inclusion H1(8Mo) -+ H1 (MO) of homology groups, one can show that 8Mo ; So. As in the case of quasi-Fuchsian groups, it is shown that Mo is homeomorphic to the surface layer So x [0, 1]. Now we shall explain how do cylinders Ci, 1 < i < n, correspond to elements of

G. Due to Theorem 5.2, there exist simple disjoint loops y/ in So, 1 < i < n, that bound, together with loops yi around Ci, cylinders C; in Mo. Here the cylinder C separates Mo if and only if the loop yi separates So. Let N be a component of Mo\ U Ci'

and N* a component of -1(N). Then the stabilizer Go C G of this component N* is a quasi-Fuchsian group, and N is a retract of M(Go). The element in the group G, determined by each of loops yi, is an accidental parabolic element. In other words, this element is parabolic but its action in S2o is alike a hyperbolic element action in

240

5. Kleinian Manifolds

the unit disk; this element of G does not define any puncture in So, see Figure 67.

Figure 67

Free groups. Let G C Isom+ I[il3 be a geometrically finite free Kleinian group (i.e., Schottky or Schottky-type group), and Mo a compact manifold obtained from M(G) by cutting out all (finitely many) punctured solid cylinders. Let us also assume that MO consists of one surface of genus g. As G is a free group, the homomorphism n*: nl (a MO) --* nl (Mo) = G cannot be a monomorphism. Theorem 5.3 shows that there is a simple, non-trivial in 8Mo loop yl c aMo which bounds an embedded disk D1 in Mo.

We assume first that yt does not separate MO, so D1 does not separate Mo. We remove from Mo a neighborhood Di = D1 x (-s, s) of D1 to get a compact manifold M1 whose boundary is a surface of genus (g - 1). Repeating this construction for M1, we obtain the next loop y2 C 3M1 having the properties of the loop yl, and continue the process until, at the g-th step, we obtain a closed ball Mg. The existence of these non-separating aMo loops yl, y2, ..., yg can be proven as follows. Suppose that, for example, all the loops yt separate aMo. There are two possibilities: either the disk D1 splits up Mo into two parts, M1 and M2, different from balls, or 8(Mo\Dl) has two components. In the first case, the common genus of 3M1 U 3M2 is g. Here a simple loop in 3M1 fl aMo separates aMg, i = 1, 2, if and only if it separates aMo. Also, n1(Mo) - nl (M1) * n1(M2), and consequently the groups nl (M1) are free. Therefore, after a finite number of steps (where one takes loops that separate 8Mo), we obtain a manifold M,' whose fundamental group is free and whose boundary is a torus. We thus obtain a contradiction with the fact that a non-trivial simple loop in a torus does not separate it. In the second case, we have two components S1, S2 C 8(Mo\D1). Due to Theorem 5.1, there exists a simple

5.3. Universal groups and Poincare Conjecture

241

non-trivial loop in S1 that separates S1 and bounds an embedded disk in Mo\Dl. One can continue this process to get a submanifold M C Mo that has a component aM as a sphere. That sphere is a component of 0Mo, which is impossible. Thus, we have shown that, if G is a free group, Mo(G) is a handlebody of genus g (a closed ball with g handles). In particular, all free groups G with connected S2 (G)

have this property. Also, if G is a purely loxodromic group then Mo(G) = M(G), and we see that G is a Schottky group (see Maskit [1]). In the general case, one can show that G is a Schottky-type group.

Surface groups. Let a geometrically finite group G C Isom+ IH[3 be isomorphic to the fundamental group of a closed surface S. If Mo(G) is again the manifold obtained from M(G) by cutting out all punctured solid cylinders, then it is homeomorphic to a surface layer S x [0, 1]. The proof is analogous to that in the case of B-groups on the basis of the following two facts: (1) if a subgroup H C G is isomorphic to the fundamental group of a closed surface whose genus does not exceed the genus of S, then H = G; and (2) the group G is not a non-trivial free product of its subgroups. The general case is obtained from the above. Namely, let So = S2o/G. We apply Theorem 5.1 to get pairwise non-intersecting simple loops yi which are non-trivial in So, bound embedded disks Di C M(G), and are not boundaries of rings in So. We take the {y, } loop system as maximal: for each component S' C So\ U yi, the

homomorphism n*: nl (S') -* nl (M(G)) is a monomorphism. Now let N be a component of M(G)\ U Di and N* a component of 7r-1 (N). Then its stabilizer Go is either a B-group, or a free Abelian group of rank two, or a cyclic parabolic or loxodromic group. Here the manifold N is a retract of M(Go). Therefore, in general, G can be represented as a free product of finitely many subgroups which are either B-groups, or free Abelian groups of rank two, or cyclic parabolic or loxodromic groups. The structure of the corresponding Kleinian manifold M(G) is clearly understood by means of the gluings related to corresponding combinations of these subgroups and described in the above.

§3. Universal groups and Poincare Conjecture Studying branched coverings of 3-manifolds in the context of universal knots and links (see § 2), Hilden-Lozano-Montesinos-Whitten [1] proved the following significant sharpening of the universality of the Borromean rings (Figure 13):

Theorem 5.19. Every closed, oriented 3-manifold is a covering of S3 branched over

the Borromean rings and such that the branching indices of the components are (1, 2, 4), (2, 2, 4) and (2, 4). Actually, this result gives a topological uniformization of any closed oriented 3manifold. It is possible to explain this uniformization from the point of view of hyperbolic geometry.

5. Kleinian Manifolds

242

1. Topological uniformization of 3-manifold. A finitely generated discrete group U C Isom H3 is called a universal group if the quotient space IH[3 / U is homeomorphic

to the 3-sphere S3 and if every closed oriented 3-manifold M occurs as the quotient space IHI3/G(M) for a finite index subgroup G(M) of U. The concept of universal groups grew naturally out of the concept of universal links. Namely, if the image of the axes of rotations in U is a knot or a link L in S3, the natural map IH13 / G (M) -+ IHI3 / U S3 is a branched covering with the branching set L, so the knot or link L is universal. Here the group U C Isom IEI3 endows its IlI3/ U with the structure of a hyperbolic orbifold whose singular quotient space S3 set is the knot or link L. This observation and Theorem 5.19 provide a proof of the existence of universal groups:

Theorem 5.20. Let P C IHI3 be the regular hyperbolic dodecahedron with right angles. Then the group U = G(4, 4, 4) C Isom H3 generated by 90°-rotations in the three skew edges of P as in Figure 68, is universal. In other words, every closed, oriented 3-manifold M is homeomorphic (orientation preserving) to the quotient space E[3/G(M), where G(M) is a finite index subgroup of U. This result shows that every closed oriented 3-manifold M is a hyperbolic orbifold which covers the hyperbolic orbifold with the underlying space S3 and the Borromean rings as the singular locus with isotropy groups Z4.

Figure 68

As a direct consequence of Theorem 5.20 and Theorem 2.18, the following holds:

Corollary 5.21. Given closed, oriented 3-manifold M let G(M) be a finite index subgroup of the universal group U C Isom H3 such that M 1H13/G(M). Then the fundamental group Trt(M) is isomorphic to G(M)/F where F is the normal subgroup of G(M) generated by its finite order elements. Particularly, the manifold M % H3/G(M) is simply connected if and only if the subgroup G(M) C U is generated by rotations.

The above results show that we have some "topological uniformization" of closed oriented 3-manifolds. Moreover, this uniformization has "more structure" in the sense

5.3. Universal groups and Poincare Conjecture

243

that it allows us to apply arithmetic methods since the universal group U C ISOIn 1H13

(the hyperbolic dodecahedron group B(4, 4, 4)) is a subgroup of index 120 of an arithmetic tetrahedral reflection group (see Chapter 3, § 9), so U is arithmetic itself. Namely, the following is true (Hilden-Lozano-Montesinos [3, 4]).

Theorem 5.22. The universal group U = B(4, 4, 4) is a subgroup of PSL(2, A) where A is the ring of algebraic integers of the field Q(/-2-, i, \\

-11 2

/J

2. Poincare Conjecture. Some of the mysteries of 3-dimensional topology may be hidden by our lack of knowledge about closed connected simply-connected orientable 3-manifolds. It is the famous Poincare Conjecture asking whether or not such 3manifolds are homeomorphic to the 3-sphere S3. This conjecture is still not proved, see Gabai [4]. There have been many faulty proofs although most of them have not reached print. However, for its generalization in higher dimensions (the so-called Generalized Poincard Conjecture), there have been great successes: S. Smale [1] and J. Stallings proved it in dimension n > 5 as well as M. Freedman [1] has proved the 4-dimensional case. Now one can observe from Corollary 5.21 that, in order to prove this conjecture in dimension 3, we have to check, for all finite index subgroups I' C U C PSL(2, A) of the arithmetic universal group U (with known matrix generators) which are generated by rotations, whether or not the quotient space 1El<3/ P is homeomorphic to S3. This provides a universal geometric base for investigations. On that base, another step toward a possible proof has been suggested to us by J.-M. Montesinos. It can be described as "blowing hyperbolic orbifolds" in the following sense.

3. Blowing hyperbolic orbifolds. Given a closed oriented 3-manifold M homotopically equivalent to the 3-sphere S3, consider the corresponding finite index subgroup G(M) C U generated by rotations such that M is homeomorphic to the hyperbolic orbifold H' / G (M). After that we expect to be able to change the metric in the complement of the orbifold singular locus in such a way that its sectional curvature changes from constant negative -1) through 0 to positive and, finally, to constant positive curvature +1. It is a reformulation of the Poincare Conjecture that in the final situation, the singular locus would have the curvature +1 as well. In other words, the cone structure on the manifold M would become non-singular, and the orbifold representing this manifold M would have a manifold structure, that is M would be homeomorphic to S3. To illustrate this idea, let us consider the case of two-dimensional sphere which can be represented by 1-parametric family of 3-simplex boundaries, see Figure 69. Here, as the family parameter, we take the curvature of boundary triangles of a tetrahedron which takes the values from the segment [-1, 1 ] c R. The initial case is the hyperbolic

tetrahedron with the sides curvature -1, the intermediate one is Euclidean and, in the final case of constant positive curvature of the tetrahedron sides, we see that the sum of angles at each vertex equals 2n. As a result, the singular locus is disappearing and

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the 2-manifold becomes the 2-sphere S2.

Figure 69

As initial steps in developing this idea, Hilden-Lozano-Montesinos [4] have con-

structed a similar family of 3-dimensional orbifolds starting with the initial orbifold B(4, 4, 4) = IHI3/B(4, 4, 4). This orbifold corresponds to the universal group U = B(4, 4, 4) itself and has the hyperbolic metric with sectional curvature -1 outside of its singular locus (the Borromean rings in S3). An intermediate orbifold in the process of changing the metric in the complement of this locus, the orbifold k(2, 2, 2) is an Euclidean orbifold since B(2, 2, 2) acts in R3 as a discrete group of Euclidean isometries (see Example 1.19), B(2, 2, 2) =R3/B(2, 2, 2). Continuing this process of increasing the curvature of the orbifold through making the curvature of the singular locus go down, we have the final situation of the constant positive curvature in the complement in S3 to the Borromean rings. Here the singular locus disappears, and the orbifold B(1, 1, 1) becomes the 3-sphere S3. Blowing the hyperbolic orbifold B (4, 4, 4) to S3, Hilden-Lozano-Montesinos [4] described the family B (m, n, p) of intermediate orbifolds, mainly from the arithmetic point of view. For another (conformal) approach to cone structures on a closed 3manifold, see Theorem 6.28.

§4. Ends of Kleinian manifolds, their compactification and Ahifors's Conjecture We say that a non-compact manifold M can be compactified if there exist a compact manifold M with boundary such that an embedding h: M -* M is a homotopy equivalence and h (int M) = int M. In other words, M can be compactified to a compact manifold M by adding surfaces to the ends. We have seen in Chapter 4, § 3 that any geometrically finite Kleinian manifold M(G) = [ i U cZ (G) ] / G, G C Isom IHIn, n > 2, admits a natural compactification. Namely, each end of such M(G) has a punctured solid flat (n - 1)-manifold En-t x

5.4. Ends of Kleinian manifolds, their compactification and Ahlfors's Conjecture 245

Bn-k-1

[0, oo) or a punctured solid k-cylinder Ek X x [0, oo) as its neighborhood (see Theorem 4.16) and, hence may be compactified by gluing the corresponding flat (n -1)-manifold or k-cylinder at infinity. In other words, the end neighborhood of the form Nn-1 x [0, oo) should be replaced by a compact submanifold of the form Nn-1 x [0, 1]. Some times it is more convenient to see a neighborhood of such a cusp end of a hyperbolic manifold as a regular neighborhood of some codimension 2 submanifold in another n-manifold. In particular, one can always do that in dimension three by considering link complements, see Chapter 6, P. However, it is generally impossible in higher dimensions. Namely, due to Ivansic [2] recent results, it is impossible for some cusp ends of hyperbolic 4-manifolds of finite volume: their neighborhoods appear to be non-embeddable in any compact 4-manifold, see Chapter 6, §3. The problem of compactification of a Kleinian manifold M(G), i.e., describing the end structure, was first considered in 1965 by Lars Ahlfors [3]. He conjectured that such a structure is simple in some sense, see Marden [1]. The first major advance in resolving this conjecture of Ahlfors in a certain class of manifolds (which will be described later) is based on the study by Thurston [1] of geodesic laminations on a surface S admitting a realization in a homotopy class f : S M(G). An important part is played here by Thurston's geometrically tame manifolds (in particular, geometrically finite manifolds). Thurston's research has been completed in a basic case by strong results of Bonahon [1], which we shall expose below.

Geometrically tame hyperbolic manifolds. Let M = Mtp,e] U M[6,,,,,) be the Margulis decomposition of a complete hyperbolic manifold M into s-thin and s-thick parts, see Theorem 3.1. Let s = s(n) be the universal Margulis constant that depends only on n = dim M and has the property that any subgroup of nt (M, x), x E M, generated by loops of length < s, is virtually Abelian (including the trivial case). For this constant s = s(n), we denote the s-thin part M(o,,] by Mt, and the s-thick part M[e,,,,)1 by Mh. The components of the thin submanifold Mt are either Margulis tubes, i.e., quotients of hyperbolic cylinders D(2, r) = {x E lH[n : d(.f, x) < r} by the action of cyclic loxodromic groups Gt C n1 (M) with the axes t, or are parabolic cusps, i.e., quotients of parabolic neighborhoods Vp(s) C lHl (see Chapter 4, §4) by the action of their parabolic stabilizers GP C n1 (M). Neighborhoods of the latter cusps in the hyperbolic manifold M are homeomorphic (up to finite-sheeted coverings) either to the punctured solid tori Tn or to punctured solid cylinders Gn see (4.21, 4.22). Their boundaries in M are homeomorphic to the corresponding Euclidean (n -1) -dimensional manifolds of the form RI-1 / r, where torsion free groups r C Isom li81-1 are discrete. In dimension n = 3, the boundary of the s-thin part Mt is formed from tori, Klein bottles, rings, and Mobius bands. For a hyperbolic manifold M, we denote by MO the submanifold that is the complement in M to the interior of the cusp components of the s-thin part Mt. Lemma 5.23. Let M be an acyclic three-dimensional manifold whose boundary 8M consists of finitely many surfaces of finite type and surfaces with zero Euler characteristic. Also let n1 (M) be a finitely generated group that satisfies the following

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condition: For any non-trivial decomposition 1r1 (M) to a free product 7rl (M) = A * B, there exists a closed curve in aM whose class is not conjugate in 7rl (M) to any element from or B. Then M contains a compact submanifold m, of codimension 0 such that its embedding (Me, Mc n am) -> (M, aM) is a homotopy equivalence.

Here the pairs (N, P) consist of 3-manifolds N and submanifolds P C N, dim P = m < 3. In the case of codim P >_ 1, (N, P) is locally modeled on (118+, R !+n), whereas it is modeled on (1R3, ][8+ x R2) if codim P = 0. In both cases, the topological boundary SP in N is a submanifold of codimension one in N (SP = aP\aN if P is topologically closed ). The proof of this lemma (see Bonahon [1], Lemma 1.2) employs the usual 3manifold topological technique (see §2) and is based on Scott's Compact Core Theorem 5.9: firstly, 7r (M) is finitely presented and secondly, there exists a compact submanifold MM such that the embedding MM -* M is a homotopy equivalence. Theorem 5.24. Let the fundamental group nl (M) of a complete hyperbolic 3-manifold M be finitely generated and satisfy the following condition (B):

For any non-trivial free product decomposition .7rl (M) = A * B, there exists a parabolic element that is not conjugate in.nl (M) to any element from A or B. Then the complement MO to cusp components of Mt C M contains a compact submanifold MM C Mo of codimension zero such that the embedding (Mc, Mc n aMo) -+ (Mo, aMo) is a homotopy equivalence, while the frontier SMc is an incompressible surface (i.e., 7Ct (SMc) -rt (Mo) is a monomorphism). The proof of the first part (see Bonahon [1; Th. 1.4, Lemma 1.5]) directly follows from Lemma 5.23 and Sullivan's cusp-finiteness Theorem 4.3. The second assertion, that 3Mc is incompressible, follows from the following observation. If SMc is compressible, Dehn's Lemma (Theorem 5.1) implies that there is a disk that splits up MM and gives a non-trivial decomposition of the group 7r, (M) to a free product A * B (Corollary 5.4), in which each parabolic element is conjugate to an element either from A or from B. As then 7rl (Me) - 7rl (M) is an isomorphism, the contradiction between that fact and condition (B) shows that 8MM is incompressible.

Waldhausen [4] results on sufficiently large manifolds enable one to show that the submanifold MM C Mo C M is unique up to isotopy in the manifold Mo, see McCullough-Miller-Swarup [1]. We call the submanifold MM the compact core of Mo.

Theorem 5.25. Under the conditions of Theorem 5.24, each component of the complement Mo\Mc contains precisely one end E and one component SE of the boundary SMc. There is thus a one-to-one correspondence between the ends of the manifold Mo and the components of the boundary SMM of its compact core.

5.4. Ends of Kleinian manifolds, their compactification and Ahlfors's Conjecture 247

This theorem (see Bonahon [1], Theorem 1.6) enables one to give a classification of the ends of a hyperbolic 3-manifold M. We say that an end E is geometrically finite if the corresponding component ME, included in Mo\Mc has a geometrically finite structure. In other words, an end E of Mo is geometrically finite if some its neighborhood U(E) does not intersect closed geodesics, or equivalently, 2l(E) does not intersect the minimal convex retract C (M) of M. The end E is called geometrically infinite if it is not geometrically finite. Amongst geometrically infinite ends, Thurston [1] distinguished an important class of geometrically tame ends, where he used his concept of pleated surfaces in a hyperbolic manifold M. Namely, a pleated surface in a complete hyperbolic 3-manifold M is a complete hyperbolic surface S of finite area together with an isometry f : S -+ M that transfers

interior points of any geodesic segment in S to a geodesic segment in M, and each cusp of S into a cusp of the manifold M. Now, let a component UE of the complement Mo\M, (or a component SE of the boundary SMM) determine an end E of the manifold Mo C M. This end E is called geometrically tame if one of the following conditions is satisfied:

(1) the intersection of UE and the minimal convex retract C (M) of the manifold M is compact; (2) the set of pleated surfaces S', homotopic to SE and such that Sh C UE, is non-compact. A hyperbolic manifold M and its fundamental group 7rt (M) are called geometrically tame if Mo\Mc is the disjoint union of geometrically tame ends.

Geometrically tame ends play a key part in research on hyperbolic manifolds. Thurston [1, Ch. 9] has proved the following major facts on them. Theorem 5.26. Let M be a hyperbolic 3-manifold and M be such its covering that Mo has a geometrically infinite tame end E bounded by a surface (SE)t. Then either vol M < oo and a certain finite-sheeted covering of the manifold M fibers over the circle with fibers SE, or the image of the end E in the manifold Mo is an almost geometrically tame end (i.e., up to a compact set, there is its finite covering by a geometrically tame end).

The manifolds in Theorem 5.26 may arise from the examples of geometrically infinite discrete groups G C Mob(2), mentioned in Chapter 4 and constructed by Jorgensen [3] and Jtrgensen-Marden [1]. On the other hand, they are geometrically tame and equal infinite cyclic coverings of finite-volume manifolds that fiber over the circle. Such geometrically infinite manifolds will be used in the last section of this chapter as blocks for the construction of counter-examples to analogues of the Ahifors, Scott and Sullivan finiteness theorems for hyperbolic 4-manifolds. Other examples of geometrically infinite manifolds are given by the hyperbolic 3-manifolds M whose fundamental groups act in the hyperbolic space H3 as degenerate groups, i.e., manifolds M = H / G where G C Isom H3 is anon-elementary Kleinian group whose discontinuity set 0 (G) C a1H13 is connected and simply connected (see Chapter 3).

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The boundaries of Teichmiiller spaces of geometrically finite Kleinian groups mainly

consist of discrete groups that contain such degenerate subgroups. However, the following fundamental result (Thurston [1, 9.6.1]) implies that all these boundary groups G and their Kleinian manifolds M(G) are in fact geometrically tame. Theorem 5.27. Let G C Isom IH3 be a torsion free, geometrically tame group and pi: G -4 Isom H3 its discrete faithful representations preserving parabolic elements and such that the Kleinian manifolds M(piG) are geometrically tame and admit homeomorphisms to the submanifold Mo C M(G) (in the correct homotopy class that preserves cusps and ends). If the representations pi algebraically converge (on generators of G) to a representation p, then the limit manifold M(pG) is geometrically tame and admits a homeomorphism (in the correct homotopy class) to the submanifold

Mo C M(G). Remark. Here it is important to recall the general situation with the algebraic convergence of Kleinian groups pi G to a Kleinian group r = p G. Though in many cases, it was proved that 1E113/ r is homeomorphic to H3/ G (see Thurston [1], Ohshika [2, 3, 5, 8], Canary-Minsky [1], Anderson-Canary [1]), it is not always the case. Indeed, in

Anderson-Canary [2] one can find the first counter-examples of quasiconformal deformations pi G of a geometrically finite group G C Isom H3 algebraically converging to a Kleinian group r = pG with the quotient H3/ r which is not homeomorphic to IE113/G, see also Anderson-Canary-McCullough [1] and Canary-McCullough [1]. For a given hyperbolic 3-manifold M with a geometrically tame end E corresponding to a hyperbolic surface SE C aMa, one can consider a geometric construction (see Thurston [1, 9.5.13]) of a proper homotopy F: SE x [0, oo) -+ M that moves the surface SE to the infinity of the end E. This shows that a neighborhood UE C Mo\MM of this geometrically tame end E is homeomorphic to the product SE x [0, oo), i.e., the end E is a topologically tame end. In other words, these Thurston's results show that all geometrically tame Kleinian

manifolds M(G) (in particular, manifolds M(G) for all groups G in the closure of Teichmtiller space of a geometrically tame Kleinian group r C Isom lE1(3) can be compactified by gluing a corresponding surface SE at infinity to each its end E. Then M(G) is homeomorphic to the interior of a compact 3-manifold and is called topologically tame. These results enabled Thurston [2] to formulate a conjecture that all Kleinian groups G C Isom H3 are geometrically tame. Of course, these groups G should be finitely generated because a simple example of a non-tame manifold can be

taken as M(G) for an infinite free product, G = * Z C Isom H2 C Isom R3, i.e., i EN

a Klein's combination of cyclic loxodromic groups, see Theorem 5.12 and Apanasov [36, Ex. 7.53]). Now we shall describe Bonahon's [1] solution of this problem in the basic case of finitely generated Kleinian groups which are not decomposable into free products.

Theorem 5.28. Let M be a complete hyperbolic 3-manifold whose finitely generated fundamental group 7r1(M) satisfies the following condition (B):

5.4. Ends of Kleinian manifolds, their compactification and Ahlfors's Conjecture 249

For any non-trivial decomposition Jrt (M) = A * B as a free product, there exists a parabolic element Tct (M) that is not conjugate to any element from A and B. Then M is a geometrically tame manifold. Much of the technique used to prove this theorem extends ideas due to Thurston [1, 2-4]. One of the new basic ideas used in Bonahon's proof of this theorem is the concept of geodesic currents in a hyperbolic surface S, which extends the concept of closed geodesics in S. The space of geodesic currents constitutes a completion of the space of homotopy classes of closed curves in S and, for the subspace formed by simple curves, that completion gives the space 9,L(S) of geodesic laminations introduced by Thurston [1]. A lamination L in a manifold M" is a foliation on a closed subset in M. More precisely, if (Ui, cot) is a set of charts covering a closed subset A C M" (the support of L), cpi : Ui -+ R'-k x Rk, then cpi (A f1 Ui) has the form Ilk"-k X B, where B C Rk. Then the changes of charts (transition functions) cpij should be represented as coin (x, Y) = (ft, (x, y), gij (Y)) ,

y E B.

The leaves of a lamination L are defined in the same way as the leaves of a foliation. A lamination whose leaves are geodesics is called geodesic. For a hyperbolic surface S of finite area, any geodesic lamination y in S cannot have the whole surface S as its support, i.e., it is not a foliation in S (moreover, y has zero measure in S which follows from consideration of the Euler characteristic (5.3)). The complement S\y consists of a finite number of domains bounded by closed or infinite geodesics. For a pleated surface S, one can consider bending lines which are those geodesics in S each of whose points has a unique direction of geodesic intervals in the definition

of a pleated surface containing this point. Obviously, the subset y C S consisting of bending lines of a pleated surface S is a geodesic lamination in the surface S. Moreover, the isometry f : S -- M in the definition of a pleated surface transfers all geodesic segments in the complement S\y to geodesics ones in the hyperbolic manifold M. From the viewpoint of the Bonahon's concept of geodesic currents, the notion of a geometrically tame manifold can be reformulated as follows. Let a geometrically infinite end E of a hyperbolic manifold M be determined by a component SE of the boundary 8MM of the compact core in the manifold Mo C M (see Theorem 5.25). Then the end E is called simply degenerate if, for each of its neighborhoods UE, there exists a simple closed curve y C SE homotopic in M to a

closed geodesic y* c UE. A manifold M with finitely generated fundamental group Jrt (M) is geometrically tame if and only if each of its ends is either geometrically finite or simply degenerate. Taking into account the above Thurston's topological characterization of geometri-

cally tame ends, we see that each end E of a geometrically tame manifold M (its neighborhood) is homeomorphic to the product of a surface and the open interval

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250

S x (0, 00). Moreover, for a geometrically infinite tame end E there exists a unique lamination ).. on the surface S that cannot be realized as a geodesic lamination in M and such that there exists a sequence of laminations converging to the lamination ), and whose realizations tend to the end E. This lamination .l is called the ending lamination of E (see Ohshika [3, 4, 6]). Sketch of the proof of Theorem 5.28. Now we shall provide a scheme of the proof of Bonahon's theorem, which consists of proving of the following fact: Let SE be a component of the boundary 8Mc determining a geometrically infinite end E of the manifold MO C M. Then there exists a family (aj) of closed simple loops in SE such that each neighborhood UE of the end E contains closed geodesics ct with sufficiently large j that are homotopic to aj in M (we say that a tend to E as j goes to oo). As the first step in proving this, we construct a set of closed curves aj having that property on SE, but which are not in general simple curves. That part of the proof does not depend on the dimension of M and is substantially simplified if M has no parabolic ends. As these curves aj are not simple, it is necessary to demonstrate that one can select them, as j increases, to be increasingly simple. A more formal condition is that the ratio between the number of double points and the square of the length of these curves

aj tends to zero as j increases. This is obtained from the following extension of Thurston's [1] lemma on the intersection index:

Theorem 5.29. Under the conditions of Theorem 5.28, there is a constant C having the following property:

Given d > 0, let ai and a2 be closed geodesics in M at distances > d from the surface SE and homotopic to two curves at, a2 C SE, where the both homotopies do not intersect SE and approach SE from the same side. Also let each of these geodesics either not intersect Mt or be contained within a certain component of the thin submanifold Mr C M. Then

i(al,a2) <

-I-4,

where i is the intersection index in SE, and 2(a) is the length of the curve a in M. The concept of geodesic currents in SE makes more clear the meaning of the fact

that the curves of on SE become more simple for j -* oo. The space G(SE) of geodesic currents in SE is the completion (in a certain uniform structure) of the set of homotopy classes for closed curves in SE. Introducing a geometrical intersection number for these geodesic currents, which extends the analogous concept for closed curves, we conclude that the space of geodesic currents having zero self-intersection coincides with the space of measurable geodesic laminations. Here, in addition to the definition of such laminations, we introduce a transversal measure which is invariant with respect to translations along lamination leaves. That space of measurable geodesic laminations is also the closure in e(SE) of the set of homotopy classes of simple loops.

5.4. Ends of Kleinian manifolds, their compactification and Ahlfors's Conjecture 251

In the next step, we consider those aj that, after some normalization, converge in e (SE) to a geodesic current a, . Then, due to continuity arguments for the intersection numbers and the estimate in Theorem 5.29, it follows that a. is a measurable geodesic lamination in SE. Here a,, is also the limit of a set of simple loops 1k in SE (with an

appropriate normalization). In practice, only one component f,,, of ac" is the limit for Pk. Actually, 6o, and a,,, coincide. In the last step, it is shown that these simple loops /3k are homotopic in M to closed

geodesic jk that tend to E, i.e., each neighborhood UE of the end E contains all the pk up to a finite number of them. This proves that E is a simply degenerate end and hence, M is a geometrically tame manifold. Corollary 5.30. Let M be a complete hyperbolic 3-manifold whose finitely generated fundamental group G = .ni (M) acts on the sphere at infinity aH3 of the hyperbolic space H3 as a function group (i.e., G has an invariant component S2o of the discontinuity set). Then M is a geometrically tame manifold.

The proof of this fact (Bonahon [1]) is based on the following observation. Let M = H3 / G and S2o C SZ (G) C aH3 be an invariant component for the Kleinian group G. Then, due to Ahlfors's finiteness Theorem 4.1, the complement MO C M to the cusps in M has a geometrically finite end corresponding to the surface SZo/G (of finite topological type). Then it is readily verified that all the other ends of the manifold Mo are such that their fundamental groups satisfy the Bonahon condition (B) in Theorem 5.28. This implies that all ends of M are geometrically tame which completes the proof. Bonahon's theorem has many other consequences. We shall describe only two of them, which are related to the manifold compactification and the Ahifors Conjecture on the limit-set measure for finitely generated Kleinian groups.

Compactification of Kleinian manifolds. Theorem 5.31. Let M be a complete hyperbolic 3-manifold with finitely generated fundamental group rtl (M) and let one of the following conditions hold: (i) for any nontrivial free product decomposition rri (M) = A * B, there exists a parabolic element in .nl (M) not conjugate to elements from A U B, or (ii) .7rl (M) acts on the sphere at infinity aH3 as a function group. Then M is diffeomorphic to the interior of a compact manifold (and is topologically tame).

Proof. The proof follows directly from Bonahon's Theorem 5.28, Corollary 5.30 and the topological tameness of geometrically tame manifolds proved by Thurston [1]. However, Bonahon's technique enables one to give a short proof of the last topological fact based on a result of Freedman-Hass-Scott [1] from the classical 3manifold topology, which does not require refined approximations by pleated surfaces made by Thurston. Due to the finiteness Theorem 4.16, we immediately have:

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Theorem 5.32. If E is a geometrically finite end of a complete hyperbolic n-manifold M, then there exists its neighborhood diffeomorphic to (S, aS) x [0, oo), where S is a compact submanifold of M.

An analogous result is needed also for simply degenerate ends. It is provided by the following fact. Theorem 5.33. Let E be a simply degenerate end of a complete hyperbolic 3-manifold M satisfying condition (i) in Theorem 5.31. Then E is contained in the minimal convex retract C (M) C M of the manifold M. Also, there exists a neighborhood U of the end

E such that the pair (U, u n 8Mo) is diffeomorphic to the pair (SE, aSE) x [0, oo). Proof. Without loss of generality, one may assume that rcl (M) = 7L] (SE), where SE is the component of the frontier 8Mc determining the end E, see Theorem 5.25. In fact, let M be a covering of the manifold M such that7rl (M) = in (SE). Then the component of the complement Mo \MM that contains the end E is lifted diffeomorphically to Mo, and the end E naturally corresponds to the end K of Mo. That end E is simply degenerate (as is E) because the component of the frontier 8MM adjoining it is diffeomorphic to the surface SE C 8M, The latter follows from the fact that (SE, a SE) and (SE, a SE)

have the homotopy type of the pair (VE, 8VE), where VE is the component of the complement Mo\MM containing E. As the minimal convex retract C(M) contains a projection of the corresponding retract C(M) of the manifold M, the end E in MO satisfies the conclusions of the theorem, as does the end f in Mo. Therefore, we can restrict ourselves to the case M = M, when 7tt (M) = in (SE). Also, decreasing of the Margulis constant s = s(n), n = 3, does not change the topological type of the manifold Mo = Mo(s) C M, so we can make s sufficiently small and we can then assume that the distances between the Margulis tubes in Mt = Mt (s) are not less than one. Let us denote by SE the surface obtained from SE by gluing collars St x [0, co] along each component of the boundary 8SE. As the end E is simply degenerate, one can construct a family (pi : SE -* M of hyperbolically simplicial embeddings such that each neighborhood U of the end E intersects at least one of the images oi (SE). Also, the diameter Soi (SE) modulo the thin submanifold Mt is bounded by a constant independent of i, while the images tpi (SE) are contained in the minimal convex retract

C (M) and are pairwise homotopic there. The construction of such embeddings is based on the following "bounded diameter lemma", Bonahon [1]; cf. Thurston [1, 8.8.5].

Lemma 5.34. Let tp: S -* M be an embedding of a compact connected surface S in a hyperbolic 3-manifold M which induces a monomorphism tp*: in (S) - nl (M) and makes a correspondence between components of a S and parabolic elements from

ni(M). Let y C S be a simple closed curve such that tp(y) is homotopic in M to a closed geodesic y*. Then tP is homotopic to a mapping tp': S -} M such that V(y) = y*, tp'(OS) C Mt, and diam& (co'(S)) is bounded from above by a constant C dependent only on e and on the topological type of S.

5.4. Ends of Kleinian manifolds, their compactification and Ahlfors's Conjecture 253

Proof. Here diam6 X of a set X C M is defined as supx,yex dE(x, y), where the semi-metric dE (x, y) is defined by the hyperbolic metric in M modulo the thin part, Mt = Mt(s). In other words, dd(x, y) is the minimum of the lengths of y fl Ma(e) over all curves y = y (x, y) C M joining the points x and y. We note that the assumption n1 (M) = n1 (SE) implies that each curve in SE which is homotopic in M to a curve in the interior of a component ME (= the union of cusp components of Mt intersecting SE) will be homotopic in SE to an end of that surface. Let U be a neighborhood of E. We need to show that U U ME contains at least one tpi (SE). Without loss of generality, one can assume that the frontier 3U in Mo is compact and that U does not intersect parabolic components from Mt\ME. As the action of the group 7r1(M) on the universal covering IH[3 of the manifold M is properly discontinuous, there is only a finite number of Margulis tubes at distances

< c1 from the compact SU. Using induction and the fact that distances between Margulis tubes are > 1, we conclude that there exist at most finitely many Margulis tubes that can be joined with a U by a path whose part going beyond the Margulis tubes has length < c1. Consequently, U contains a neighborhood U' of the end E in Mo such that U' cannot be j oined with 8 U by aath whose length outside the Margulis tubes is < c1. Then, for any i for which tpi (SE) intersects U', we have in fact that (Pi (SE) \ME is contained in U. We also note that the construction of the simplicial embeddings Vi is so that the images tpi(SE) are contained in the minimal convex retract C(M) and are pairwise homotopic there (by "radiation" of their homotopies from M). Therefore, any connected neighborhood Uo of the end E, which does not intersect cpO (SE), is wholly contained in C (M). Hence, for x E Uo, there exists a neighborhood U C int(Uo) of the end E such that Uo\U is connected and contains x. As tpi (SE) is contained in U U ME, each homotopy from tpo to tpi passes at least once through x and therefore lies in C(M). We have thus shown that Uo, and therefore the end E, will be contained in C (M), which was what we set out to prove. Continuing our proof of the last part of Theorem 5.33, namely that a neighborhood in Mo of the end E is a product, we note the following. As E is a simply degenerate end, each component of SSE preserves the orientation of M. So, as n1(M) = n1 (SE), the components of aME are rings (not Mobius bands). Consequently we conclude that each neighborhood U of E contains tpi (SE)\ME. In particular considering the inverse image cpi 1(a ME) we see that, for any neighborhood U of E, 8 SE can be made

homotopic in U n aME. This would be impossible if even one of the components of aME were a Mobius band. To prove that there is a product neighborhood of the end E in MO, we consider a neighborhood U of E. We require the existence of tpi : SE -* M whose image is contained in U U ME. We deform tpi in such a way that the intersections tpi (SE) and aME become transverse. We then take the composition of Vi with an isotopy of the surface SE to produce a homotopy between Vi and VP such that V (a SE) C aME

and tp'(SE) C U U ME. Also, since each component of ME is diffeomorphic to

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254

Sl x R x [0, oo), one can move co (SE) outside the interior of ME and make it contained in U. From this construction of p , we have that cps I sE : SE -+ Mo is homotopic to

an embedding SE -*. Mo, actually to the natural embedding. Now we apply the following lemma to that embedding. This result has been proved implicitly in §2 of Freedman-Hass-Scott [1] in the case aS = 0; the general case is given by passing to doubles.

Lemma 5.35. Let *: (S, 8S) -k (N, 8N) be a homotopy equivalence where S is a compact surface and N is a 3-manifold. If 1/r is homotopic to an embedding then i/r is homotopic to an embedding into a small neighborhood of the image 2/r (S). Using this lemma, we may assume additionally that Cpl (SE) is a submanifold embedded in Mo and contained in U. Hence any neighborhood U of the end E contains Mo, the image of an embedding oI : SE -+ Mo homotopic to the embedding cpo: SE where the homotopy preserves the image of the boundary a SE in a Mo. Without loss of generality we assume that cpi (SE) do not intersect each other and tend monotonically to E in such a way that (SE) lie homotopically between rp'i_1(SE) and cpl+1(SE). As cpl and cp'+, are homotopic, Waldhausen's [4] result shows that (pi_1(SE) and +1(SE) are separated in Mo by a collar diffeomorphic to SE x [i, i + 1]. The union of these collars gives a neighborhood W of the end E in Mo diffeomorphic to the product SE x [0, oo). This completes the proof of Theorem 5.33 and thus that of Theorem 5.31.

Actually, as the following Canary's [4] existence theorem shows a topological description of ends of hyperbolic 3-manifolds is enough: Theorem 5.36. If M is a topologically tame hyperbolic 3-manifold, then M is geometrically time.

Also, as we have seen in Theorem 4.21 for cusp ends of geometrically finite hyperbolic manifolds, such manifolds are finitely covered by manifolds with simplest ends. Here we present a result due to Canary [3] that characterizes all geometrically finite covers of infinite volume, topologically tame hyperbolic 3-manifolds:

Theorem 5.37. Let M = ]H13 / G be a topologically tame hyperbolic 3-manifold of infinite volume, and G be a finitely generated subgroup of G C Isom H3. Then either the covering manifold M = IH[3 /G is geometrically finite, or the submanifold Mo of k

has a geometrically infinite end t such that the covering p: M -+ M is finite-to-one on some neighborhood `U of E. This result is a corollary of the following covering theorem due to Canary [3]:

Theorem 5.38. Let k be a topologically tame hyperbolic 3-manifold which covers another hyperbolic 3-manifold M by a local isometry p: M -* M. If t is a geometrically infinite end of the submanifold Mo C M then either

(a) E has a neighborhood it such that the covering p is finite-to-one on it, or

5.4. Ends of Kleinian manifolds, their compactification and Ahlfors's Conjecture 255

(b) M has finite volume, and there exists its finite cover M' which fibers over the circle and such that if Ms denotes the cover of M' associated to the fiber subgroup then k is finitely covered by Ms. Moreover, if l1I Ms, then M is homeomorphic to the interior of a twisted I -bundle which is doubly covered by Ms. Ahlfors problem on the limit set measure. Another application of Bonahon's theorem is the following strong advance in solving the Ahlfors [2] problem on Lebesgue measure of the limit set A (G) of a finitely generated Kleinian group G. Theorem 5.39. Let G be a finitely generated discrete Mobius group in the 2-sphere that is either a function group or a group which satisfies the following condition: For any non-trivial HNN-extension G = A*C or decomposition G = A *C B into a free product with an amalgamated subgroup C, where C is a finite group (cyclic of dihedral), there exists a parabolic element in G not conjugate to any element

from AorB. Then either the limit set A(G) has zero spherical measure or A(G) = S2. Proof. For a torsion free group G, the claim directly follows from Bonahon's theorem (Theorem 5.28 and Corollary 5.30) and the following theorem of Thurston [1, 8.12.4]) on geometrically tame groups, which extends Ahlfors's theorem on geometrically finite groups [2] (see Theorem 4.13):

Theorem 5.40. If the fundamental group nt (M) of a geometrically tame hyperbolic 3-manifold M acts in S2 as a discrete Mobius group G, then either A (G) has zero spherical measure or A(G) = S2. In the latter case, G acts ergodically in W. Proof. Given a measurable G-invariant set A C A(G), we consider a harmonic function h in JHI3 which is an extension of the characteristic function over the set A. Due to a detailed construction in the proof of Theorem 2.47, such a G-invariant extension is provided by means of the visual Poisson measure which is defined as the volume of the solid angle from which a set in a1HI3 is seen. We have that h is a harmonic function in the hyperbolic space IHI3, i.e., its gradient flow grad h preserves the hyperbolic volume:

Ah = div grad h = 0.

(5.16)

Considering this harmonic function h on the minimal convex retract C(M) = MG C M, MG = [CG\A(G)] /G, and applying (5.16) together with the geometrical tameness of the manifold M (see Thurston [1, 8.12.3]), we have the following property:

inf h = inf h .

C(M)

ac(M)

(5.17)

If A(G) ; S2 then, for all x E 1HI3\HG, we have h(x) < 1/2 because x lies in a certain half-space whose closure does not contain points from A(G). Then we get from (5.17) that

inf (1 - h) > 1/2.

C(M)

(5.18)

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If the set A contains a density point p, h(x) tends to 1 for x -> p. This contradicts (5.18) and thus shows that A has zero spherical Lebesgue measure. Therefore, A (G) has zero measure in S2.

If G contains finite order elements, one can prove Theorem 5.39 by reducing it to the case of torsion free groups by considering a torsion free, finite index subgroup

G' C G (see Theorem 1.13). That reduction, in which A(G') = A(G), is possible because of the following observation. Let M = H3 / G and let Mo C M be the complement of all parabolic components of the e-thin part Mr C M. Then the manifold M contains a compact submanifold MM such that the embedding (Me, MM fl aMo) -+ (Mo, 8Mo) is a homotopy equivalence, while the frontier SMM is an incompressible surface. This fact implies that G' = 7r1(Me) satisfies the Bonahon condition (B) in Theorem 5.28 (compare Hempel [1, Theorem 7.1], where instead of aM we need to take m, fl aM0). Thus, this completes the proof of both Theorem 5.40 and Theorem 5.39. 11

Therefore, the Ahlfors Conjecture on the limit set of a finitely generated Kleinian group G C Mob(2) remains unsolved only for decomposable geometrically infinite Kleiniangroups. Due to Maskit [1], it is true for free Kleinian groups without parabolic elements. One can construct decomposable Kleinian groups as algebraic limits (on generators) of decomposable geometrically finite Kleinian groups, see Maskit [2]. However, even in that case, Kleinian groups without parabolic elements, which are algebraic limits of quasi-conformal deformations of geometrically finite groups, satisfy the Ahlfors Conjecture (Ohshika [4]; see also Canary [4]).

§5. Kleinian n-manifolds and hyperbolic cobordisms Here we shall describe the geometry and topology of (n + 1)-dimensional manifolds

M(G) = (Hn+1 U 12(G)) /G, n > 3, where G C MSb(n) is a geometrically finite group in the n-dimensional sphere S" = a H"+1 whose discontinuity set n (G) has an invariant connected component S2o. Despite many differences with the case of 3-manifolds (see Chapter 2, §2), we shall show that Kleinian (n + 1) -manifolds M(G) for groups G with contractible invariant

components 00 C 0 (G), n > 3, have such properties that their analogy with the 3-manifolds is nevertheless strong enough.

1. Homology cobordisms. As we have seen in § 2, after eliminating some punctures (knots) and removing regular neighborhoods of others, the Kleinian 3-manifold M(G) for a geometrically finite G C IsomH3 having a simply connected invariant component 00 C S2 (G) C aH3 can be converted into a surface layer SS x [0, 1]. It is naturally to ask: To what extend holds the analogy with the surface layer for the

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257

Kleinian (n + 1) -manifold M(G) in higher dimensions n > 3, at least in the compact case? Actually, starting with an (n + 1)-manifold M(G), the analogous procedures give

us, instead of Sg x [0, 1], a certain analogue of the surface layer. For n > 3, one can consider such analogies of the (n + 1)-dimensional layer with various degree of generality: (a) The product of an n-dimensional manifold No = S2o/G and the segment [0, 1].

(b) A manifold M, whose boundary aM consists of two components No and N1 and such that the triple (M; No, N1) is an h-cobordism, and (c) A manifold M with the boundary components No and N1, which is a homology cobordism.

Here we use the following cobordism definition. Let M be a compact manifold whose boundary a M consists of two disjoint connected closed manifolds No and N1, aM = No U Ni. Then the triple (M; No, N1) is called a homology (or homologically trivial) cobordism if all the relative homology groups are trivial:

H*(M, No) = H*(M, Ni) = 0.

(5.19)

The triple (M; No, N1) of compact manifolds with boundary is called a homology

cobordism with boundary if No, N1 C aM, No n Ni = 0, and for the boundary SM = aM\(No U Nl) we have H*(SM, aNo) = H*(SM, aN1) = H*(M, No) = H*(M, N1) = 0.

(5.20)

Replacing in the above definition the requirements (5.19) and (5.20) on triviality of the relative homology groups by that of the corresponding relative homotopy groups, we get definitions of an h-cobordism and an h-cobordism with boundary. We shall see below that a geometrically finite (n + 1)-dimensional Kleinian manifold M(G), n > 3, inherits the surface-layer structure from the case n = 2 in the following form (for simplicity we assume here that M is compact): (1) In the sense of (a), if the group G is a Fuchsian group in S' = alHln+l or its quasi-conformal conjugation in ]Hn+l (and hence in Sn = 8Hn+1) (2) In the sense of (b), if G has two invariant contractible components Qo, S21 C 0 (G) (see Theorem 5.43). (3) In the sense of (c), if G has an invariant contractible component 0o C S2 (G) (see Corollary 5.42). Moreover, it is shown below that there are four-dimensional compact manifolds

M(G) that are homology cobordisms, i.e., they realize (c), but which are not hcobordisms, i.e., do not satisfy (b). Finally, we note that all the above cobordisms have complete hyperbolic structures in their interiors.

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2. Compact core cobordisms. In order to get a reasonable description of Kleinian (n + 1)-manifolds M(G), we have to consider some finiteness conditions on Kleinian groups G C Isom TBln+1 having invariant connected components no C 0 (G). Otherwise Theorem 4.49 and Corollary 4.50 provide Kleinian manifolds M(G) with any

given number k E N of boundary components Ni C 8M(G), i = 1, ... , k, and such that the inclusions Ni C M(G) induce homotopy equivalences of Ni to M(G). Actually, we shall consider geometrically finite groups G (automatically, they are finitely generated) and prove the following compact core theorem by Tetenov, see Tetenov [3, 4]:

Theorem 5.41. Let G be a geometrically finite non-elementary torsion free Kleinian group in Sn, n > 2, having an invariant contractible component no of the discontinuity

set 0 (G) C S' and No = no/ G. Then there is a compact submanifold Mc of the manifold M(G) possessing the following properties:

(a) The boundary 8Mc of this (n + 1)-manifold Mc naturally decomposes to a sum aMc = No, U N1c U (aNoc x [0, 1]) with trivial relative homology and homotopy groups, rr (Mc, Noc) = 0 and H*(Mc, Nic) = 0, i.e., the triple (Mc; Noc, Nic) is a homology trivial cobordism with boundary. (b) Manifold M(G) can be obtained from Mc by gluing up of an open collar SMc x [0, 1) to the frontier 8Mc = 8Mc\8M(G) of Mc in M(G). (c) The components of the collar 8Mc x [0, 1) are homeomorphic to the products Bk x Ti x [0, 1), 1 < k < n -1, where Ti is a closed flat (n - k)-dimensional manifold finitely covered by Euclidean (n - k)-torus T n-k and Bk is a closed k-dimensional ball. From this theorem and Theorem 4.16 we get directly:

Corollary 5.42. Suppose a Kleinian group G is convex cocompact and has an invariant contractible component no c 0 (G) C Sn. Then its compact Kleinian manifold M(G) has two boundary components No = S2o/G and N1 = (S2(G)\S2o)G, and the triple (M(G); No, Ni) is a homology cobordism.

One can also give a direct proof of the latter result without using Theorem 5.41, which on the one hand is simple and on the other casts light on the basic ideas in proving Theorem 5.41.

Proof of Corollary 5.42. The condition that the group G contains no parabolic elements and Theorem 4.16 imply that M(G) is a compact manifold. That manifold and its boundary component No = S2o/G C 8M(G) are spaces of K(G, 1)type. At the same time, the inclusion No C M(G) induces an isomorphism of the fundamental groups 1r1(No) -+ -rl (M(G)) and thus it is a homotopy equivalence: 7r,k(M(G), No) = 0. Then the relative homology groups are trivial, H,k(M(G), No) = 0, as well. Therefore, using Poincare duality and the theorem on universal coefficients (which expresses cohomology via homology, see Dold [1], VI.7.10 and VII.7.2) we obtain that H,k(M(G), 8M(G)\No) = 0, and, in particular,

5.5. Kleinian n-manifolds and hyperbolic cobordisms

259

that Ho(8M(G)\No) = Z. Then aM(G)\No consists of the only one connected componentNi, Ni = (S2(G)\S2o)/G and here H,k(M(G), Ni) = 0. This completes the proof of (5.19), that is (M(G); No, N1) is a homology cobordism.

Proof of Theorem 5.41. Here we use the notations, constructions, and major facts obtained in the proof of Theorem 4.55 which is closely related to this theorem. In particular, we may assume (passing to a finite index subgroup of G, see Theorem 4.21 and Lemma 4.58) that each maximal parabolic subgroup of G is a free Abelian group of rank at most (n - 1). Furthermore, all (finitely many) ends of the manifold M(G) correspond to only "punctured solid cylinders" Wi (see 4.44), and the complement in M(G) to the union of the interiors Wi of these cylinders Wi is a compact manifold,

which we denote by M, These closed neighborhoods Wi of parabolic cusp ends in M(G) are the quotients of cusp domains cl Ui C Hn+1 U 9 (G) by actions of their stabilizers G pi C G (parabolic subgroups fixing cusp points pi). As in Theorem 4.55, the "punctured solid cylinders" W i can be classified and ordered in the following way:

(a) let W1, W2, ... , Wq be those "cylinders" which are homeomorphic to the product of B1 x (0, 1] and a flat (n - 1)-torus T'-I; each Wi intersects two different components of M(G), one of whom is always No = 20/G. (b) let Wq+l, Wq+2, ... , Wq+r be the "cylinders" similar to those in (a), but

_

each disjoint from No. (c) let Wq+r+l, Wq+r+2, ... , Wq+r+S be those "cylinders" that are homeomor-

phic to the product of a flat k-torus Tk and BI-k x (0, 1], k < n - 1, and disjoint from No.

It follows from the proof of Theorem 4.55 that, for a geometrically finite torsion free Kleinian group having a contractible invariant component, there are no other possibilities. Now we obtain a new compact (n + 1)-manifold M:_ We delete from M all the interiors of "cylinders" Ti appearing in (a), (b) and (c). Thus we obtain a compact manifold M, Then we take the "cylinders" W1, W2, ..., Wq of the type (a) (which all are homeomorphic to B 1 x T1- 1 x (0, 1]) and apply the filling-in construction in (4.47)(4.48) to "fill in" non-compact punctured solid tori TI-1 x (0, 1] to solid tori Tn-1 = (Tn-1 x [0, 1]) / Tn-2 x B2. As a result, we obtain a compactification of neighborhoods Wt, r < i < q, as Wi = Tn-1 x [0, 1]. Here we identify the subset Tn-1 x [0, 1] of the boundary of each such product with one of the sets Wi fl Mc, r + 1 < i < q, where we assume that each of the sets Tn-1 fl Noc is identified with one of Wi fl Nos. Then from Mc we get a compact (n + 1)-dimensional manifold M whose boundary contains the closed component No obtained from the n-manifold Not by gluing to its boundary q - r copies of the solid tori Tn-1 Consider what happens with the component No of a M in the course of these transformations. After cutting off all the "cylinders" Wi, this component becomes

_

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5. Kleinian Manifolds

an n-manifold No, = No n aMc with a boundary consisting of q components each homeomorphic to Ti. After gluing the "solid cylinders" Wi to M, each of those components disappears. Thus the manifold No, with boundary turns to a closed n-manifold No, which is a component of M. Then, as has already been proved (see (4.50)), the pair of manifolds (M, N0) is homotopy trivial, and the boundary aM consists of two components. We denote the second one by NI. Moreover, the inclusions go y M and NI y M are homology equivalences (see (4.50) and (4.52)). Now consider the intersections of the manifolds M, No, and N1 with the compact manifold Mc, denoting Ni, = A nMc, i = 0, 1. Then the last homology equivalences, combined with the Mayer-Vietoris exact sequence (see Dold [1]) for pairs of triads

NI n Mc, U(Wi x f 1l)), l (Ii; =1

CM; Mc, U(W x [0, 1])) D i=I

imply (compare the proof of (4.52)) that the pair

(Mc, Nit) = (Mc, NI n Mc) is homology trivial while 7r* (Mc, No,) = 0. In other words, the triple (Mc, Noc, NIc) is a homology cobordism with boundary. Returning to the original manifold M (G) (before taking its finite cover corresponding to a finite index subgroup of G), we see that the boundary of each manifold No,

and Nlc is a disjoint union of q manifolds aWi covered by TI-1 x [0, 1]. It shows the form of the open collars that we have to use in order to reconstruct M(G) from its compact submanifold M, that is the statements (b) and (c).

Regarding the statement (a) for the original manifold M(G), we note that n*(M(G), No) = 0 because both M(G) and No are K(G, 1)-type spaces. So, applying the approach we used in (4.52) for proving that H* (M, N1) = 0 to the case when our manifold is glued from two copies of Mc by identifying their boundary subsets aNoc x [0, 1], we obtain that H*(M(G), N1) = 0. Then the homology triviality of the original pair (Mc, Ni) cfollows from the Mayer-Vietoris sequence for the corresponding pair of pairs of manifolds. Therefore, the original compact manifold Mc is a homology cobordism with boundary. Moreover, its boundary is a union of q manifolds

aWi x [0, 1], and aMc = No, U Nlc U (aNoc x [0, 1]). This completes the proof of the claim (a) and the whole Theorem 5.41. We remark that the proof of Theorem 5.41 (or Corollary 5.42) implies (compare Rourke-Sanderson [1]) that an additional condition of contractibility of two invariant components no, 92, C n (G) turns the above homology cobordism with boundary into an h-cobordism with boundary. In particular, the following theorem applies (compare Theorem 4.51):

Theorem 5.43. Let a torsion free Kleinian group G in S", n > 2, have two invariant contractible components 0o, SZ1 C Q (G) with compact quotient manifolds No =

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261

E20/ G and Ni = Q 1 / G. Then the Kleinian manifold M (G) is also compact, has exactly two boundary components No U No = aM(G), and the triple (M(G); No, Ni) is an h-cobordism, rr*(M(G), No) = rr*(M(G), N1) = 0. It is necessary to note here that the contractibility condition becomes more essential in dimensions n > 3. If dimension n = 2, the condition that a (finitely generated) G has two invariant components automatically implies that they both are contractible and that there are no other ones. However in dimensions n > 3, there are examples (see Theorems 6.74, 6.75 and Example 4.60) of geometrically finite groups which have two non-simply connected invariant components S2o and S21 and an infinite (but finite mod G) set of non-invariant components of the discontinuity set. In the next section, we shall describe a difference between 3 and 4-dimensional hyperbolic cobordisms.

3. Non-trivial 4-dimensional hyperbolic cobordisms. The compact core homology cobordisms (Me; No,, Nlc) in Theorem 5.41 (or cobordisms (M(G); No, N1) from Corollary 5.42 and Theorem 5.43) are automatically trivial in the case of 3-dimensional Kleinian manifolds, that is they are surface layers, see §2. Our goal here is to show that in higher dimensions this is not so. Namely, for this purpose, below we shall construct

geometrically finite groups G C Isom IEl4 without parabolic elements satisfying the conditions of Theorem 5.41 and Corollary 5.42 and such that the Kleinian 4-manifolds M(G) are not trivial cobordisms anymore but are still homology trivial:

Theorem 5.44. There exist compact Kleinian 4-manifolds M(G) (both orientable or not, and with geometrically finite hyperbolic structures in their interiors int M(G) 1B14/G) which are homotopically non-trivial homology cobordisms.

Proof. Due to Corollary 5.42, it is enough to construct geometrically finite groups G C Mob (3) without parabolic elements whose discontinuity sets 0 (G) C S3 consist each of two G-invariant components S2o and S21 where one of them, 520, is a topological

ball and, on the other hand, the second one is not simply connected: Trt (S21) # 0; furthermore, their common boundary, the limit set A (G) N S2 y S3, cannot be a tame 2-sphere.

Indeed, for such a discontinuous group G C Mob(3), Corollary 5.42 shows that (M(G); S2o/ G, E21 / G) is a homology cobordism. However, as S21 is not simply connected while n1 (JEI4 U S2 (G)) = 0, the kernel of the homomorphism n1 (S21 / G) -+

7rl (M(G)) induced by the inclusion is non-trivial. This means that the second relative homotopy group 7r2 (M (G), S21 / G) is non-trivial, and that the cobordism (M(G); S20/G, 01/G) is not an h-cobordism. We shall complete the proof by constructing such "wild" groups G C Mob(3), where we will use a general method explained in the next section. In constructing those "wild" groups below, we shall see that the group G is isomorphic to a certain co-compact Fuchsian group in S3. In other words, G is isomorphic to the fundamental group of a certain closed hyperbolic 3-manifold M3. It is known for

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such a group (Farrell-Jones [1, 3]) that the Whitehead group Wh G is trivial, and also

that Wh2 G = 0, Ko(ZG) = 0, K_m(ZG) = 0 form > 0 and Whm G ® Q = 0 for all m (Farrell-Jones [1]), i.e., G is K-flat. These facts show that there must be nonalgebraic additional conditions on G to provide homotopy triviality of the cobordism M(G) or coincidence of it with a product. This problem is related to the Novikov Conjecture, see Apanasov [42]. 4.

Block-building construction for "wild" groups. We will describe here our

method for the construction of Kleinian groups in space with wild limit sets. It is based on an idea of periodicity in knotting, i.e., equivariant knotting related to the discontinuous action of a group G constructed from a finite number of simple blockgroups. One can see the usefulness of that method in several applications to geometry, topology and analysis in dimension 2, 3 and 4 (see Chapter 7 and also Apanasov [11, 33, 37, 45] and Apanasov-Tetenov [2]). Under block-building we mean a construction of the desired Kleinian group G C Mob (n) from a finite set of block-groups Gi, i = 1, . . . , m by sequential amalgamated products (Maskit's first combinations)

G= *GiGij* Gj*

(5.21)

and perhaps, finally by HNN-extension (Maskit's second combination), where the block-groups Gi are Fuchsian groups acting in corresponding balls Bi C S", where it is possible that B1 = Bm+i. From the point of view of quotient manifolds, we construct a hyperbolic (n + 1)manifold M (G) whose boundary is obtained by conformal gluing of finite number of hyperbolic block n-manifolds with boundary. In the first instance, the block-balls Bi are characterized by one of the following two conditions on their boundary spheres Si = aBi: Si n Sj # 0 if and only if I i - j I = 1(mod m), for the case of free amalgamated products, and and

Si n Sj 0 if and only if Ii - j I = 1, for the HNN-extension case. Moreover, the intersections Si n Sj ((n - 2) -spheres or single points) are invariant for the amalgamated subgroups Gi3 = Gi n Gp Then due to some additional non-intersection condition concerning a fundamental domain of the group G and a (UT, Bi ), the discontinuity set 0 (G) of the constructed group G has an invariant component 0 C S'z with nontrivial (infinitely generated) fundamental group 7rl (0) and, consequently, the limit set L (G) = an is a wild "Julia surface". For instance, it can be either an everywhere wild 1-knot Sly S3, or a wild 2-knot S2 y S4, or an everywhere wild 2-sphere S2 _+ S3. In the next two sections we shall show how this method works in the last case, where it simultaneously gives the above mentioned construction of non-trivial hyperbolic 4cobordisms M(G) as well as constructions of counter-examples for famous finiteness problems for Kleinian n-manifolds.

5.5. Kleinian n-manifolds and hyperbolic cobordisms

263

5. Quasi-symmetric, everywhere wild Fox-Artin spheres. Here we shall apply the above block-building method for a construction of desired geometrically finite groups

G C M6b(3) with non-trivial homology 4-cobordisms M(G). It will complete the proof of Theorem 5.44 in Section 3. For the construction of the pertinent blockgroups Gi in (5.21), we will use standard (generated by reflections) Fuchsian groups in block-balls Bi C S3 related either to the unit hexagonal prism or to the unit truncated icosahedron, see Figures 70 and 71.

V Figure 70

Figure 71

5.1. Construction of a finite set of generator spheres. Let us consider a right prism P in it of height 17 whose base is a polygon formed by the union of 55 congruent regular hexagons with unit sides and disjoint interiors. These hexagons cover the whose vertices are the centers of the extreme hexagons, regular triangle with side see Figure 72. Let us enumerate all the hexagons line by line so that the three extreme ones have numbers 1, 10 and 55, and the central one has the number 31.

Figure 72

5. Kleinian Manifolds

264

We divide the prism P into (55 x 17) small hexagonal prisms P (k, n) of unit height

enumerated by pairs (k, n), where k, 1 < k < 17, is the "floor" number of P(k, n) and n, 1 < n < 55, is the number of the hexagonal projection of P (k, n) in the base. Then we form (in the "building" P) three disjoint apartments Pg, Pr and Pb of green, red and blue colors, respectively. As rooms of these apartments, we take the unions of small prisms P (k, n) with the number-pairs (k, n) from the sets in the Diagrams 1-3. (18,14) (17,14) (16,14) (15,14) (14,14)

(18,42)

(17,44) (17,48) (17,47) (17,42)

(16,44) (15,44) (14,44) (13,14) Blue apartment, Pb: (13,44) (12,14) (12,44) (11,44) (11,14) (10,44) (10,14) (9,44) (9,14) (8,14) (8,44) (7,14) (7,44) (6,14) (6,23) (6,31) (6,38) (6,44)

Diagram 1 (18,33)

(17,33) (17,26) (17,17) (16,17) (15,17)

Green apartment, Pg:

(14,17) (13,17)

(12,17) (11,17)

(9,42) (9,37) (9,31) (9,24) (8,42)

(7,42) (6,42) (5,42) (4,42) (3,42) (2,42) (1,42) (0,42)

Diagram 2

(9,16)

(10,17) (9,17)

5.5. Kleinian n-manifolds and hyperbolic cobordisms

265

(12,28) (12,29) (12,30) (12,31) (12,32) (12,33) (12,34) (11,28) (10,28) (10,28) (9,28) (8,28) (7,28) (6,28)

Red apartment, Pr:

(7,34)

(6,20) (4,13)

(1,14) (0,14)

(11,34) (11,34) (10,34) (9,34) (8,34)

(6,12) (5,12) (4,12)

(6,34) (5,34) (4,34) (3,34) (2,34)

(3,13) (2,13) (1,13)

(1,34) (1,33) (0,33)

Diagram 3

On the 0-th and 18-th floors, six extra rooms are adjoined to the apartments Pg, Pr and Pb. These are truncated (semiregular) icosahedra and are labeled P(0, 14),

P(0, 33), P(0, 42), and P(18, 14), P(18, 33), P(18, 42). The new rooms are obtained from the icosahedron in Figure 73. They are attached to the existing first and 17-th floors along hexagonal bases.

Figure 73

Those truncated icosahedra with regular pentagonal and hexagonal sides and with

unit edge length are obtained from the Euclidean regular isocahedron with edge length 3; see Figure 73. Since the radii of the circumscribed and inscribed balls for the icosahedron are 3

3

and

2( 1)

(

+1

_

-(3+_) 4

5. Kleinian Manifolds

266

respectively, the radii of the inscribed and circumscribed balls for our truncated icosahedron correspondingly are

r=

-(3+41 5 - )

Now let Si be the spheres of radius

and R = r2 + 1 =

58 418

(5.22)

/3 with centers at vertices of rooms P (k, n), 0 <

k < 18, forming the apartments Pgr, Pred and Pb. If two such spheres Si and Sj intersect, their centers are the adjacent vertices of some polyhedron P (k, n), and their intersection angle is n/3. Denote by B(k, n) the ball concentric with the room P(k, n) and of radius 1-1/12,

if 1 < k < 17, and

79 I- 275 8

if k = 0, 18.

(5.23)

Its boundary sphere S(k, n) is orthogonal to each of the spheres Si whose centers are the vertices of P (k, n). Next, we replace the polyhedral rooms P (k, n) in our apartments by the round rooms B(k, n). In doing so we obtain the linked tubes pictured in Figure 74.

green

Figure 74

We note that we are going to use the balls B(k, n) as the block-balls in the blockbuilding construction in Section 4, where we will also use an HNN-extension by a hyperbolic transformation h E M6b(3) having the following isometric spheres:

I =I(h)=(x 1183: IDh(x)I=1)=S2(z,Q), P = I(h-t) = S2(z', Q),

5.5. Kleinian n-manifolds and hyperbolic cobordisms

267

where z, z' and Q will be determined later. This hyperbolic transformation h is the composition of reflections with respect to the sphere I and the plane which bisects the segment joining the centers of I and I'; it has two fixed points p E int I and q E int P. Furthermore, our union of apartments is a fattening Uh of the knotted period dh of a wild Fox-Actin arc d invariant under the cyclic group (h), see Figure 75.

Figure 75

Now we shall finish the forming of a finite family e of spheres Sj, with the following properties:

(1) The union of the annuli Xi = 8B1 fl ext I fl ext I' fl (fl ext Bj) is covered by the interiors of Sj. j#` (2) For each k, j either Bk fl Sj= 0 or Sj is orthogonal to the sphere BBk; the same is true if I (h) and I (h-1) are taken instead of BBk. (3) If sin Sj# 0 the exterior dihedral angle (i.e., the one containing oo) between the spheres equals Jr/m for some integer m. (4) If Sl fl Si # 0 then there exists a common annulus Xk for which Xk fl Si

0

and XkflSj :A0. (5) There is a one-to-one correspondence between the spheres Sj E e which intersect the sphere I (h) and the spheres Sp E G which intersect I (h-1), such that Sp = h(Sj). In other words, we form a finite bubble cover of our apartments boundary 8 Uh with good dihedral angles between the bubbles and right angles between the bubbles and the spheres BBk, a cover that respects the h-periodicity. Taking into account

5. Kleinian Manifolds

268

the Andreev-Thurston rigidity of circular coverings of the sphere S2 (Thurston [1, Ch. 13]), there are only a few such families. Note that the interiors of the spheres Si with the centers at the vertices of P (k, n) do not cover the entire boundary aUh, i.e., they do not completely cover the spherical annuli Xj C S(k, n). Still uncovered are the hexagonal, quadrangular and pentagonal domains on these annuli corresponding to sides of P (k, n). Let us cover each hexagonal domain on Xi C S(k, n) by the union of the interiors of seven spheres orthogonal to the sphere S(k, n). Six of these spheres have equal radii, they are orthogonal to the spheres Si, and they intersect at the angles n/3. The seventh sphere intersects the six others at the right angles and does not intersect the spheres Si with the centers at vertices of P(k, n); see Figure 76.

Figure 76

Figure 77

We cover each quadrangular domain on Xi C S(k, n) by the interiors of five spheres orthogonal to the sphere S(k, n). Four of these spheres have equal radii; they are orthogonal to the spheres Sj and thus intersect at the angle n/3. The fifth sphere intersects the four others at right angles and does not intersect the spheres Sj; see Figure 77. We do similarly in the case of the pentagonal domains.

We complete the construction of the desired family e if we construct our two isometric spheres I = I (h) and I' = I (h-1). They have a common radius Q and centers z and z' lying on the axis L of the prism P such that I and I' are orthogonal to the spheres St of radius J/3 with centers at the vertices of the truncated icosahedra P(0, n) and P(18, n), n = 14, 33, 42, which are nearest to z and z', respectively. For this purpose, let us consider the three straight lines L(0, n), n = 14, 33, 42 (respectively, L(18, n), n = 14, 33, 42) which pass through the centers of the truncated icosahedra P(0, n), n = 14, 33, 42 (respectively, P(18, n)) and also through

5.5. Kleinian n-manifolds and hyperbolic cobordisms

269

the centers of its slanting hexagonal sides adjoining to the lower sides of P(0, n), (respectively, the upper sides of P (0, n)) and facing the axis L of the building P. The points z and z' are the intersection points of the axis L with the straights L(0, n) and L(18, n), n = 14, 33, 42, respectively. In this case the angle V between L(0, n) (or L(18, n)) and axis L is given by

/2r = 2/(3 + N_), i.e., q

= arctg

2 .

Therefore, for the distance between z and z' we get

17+2(r+2V/tggyp) = 17+2 15+x(3+.)/2.

(5.24)

The orthogonality condition for the intersection of I and I' with the spheres Si nearest to z and z', respectively, tells us that Q =

(493 - 81)/24

(5.25)

is the radius of the isometric spheres I = I (h) and I' = I (h -1). Direct calculations show that the resulting hyperbolic transformation h E Mob(3) and the constructed finite covering e of the boundary a Uh by the spheres Si have all the properties (l)-(5).

5.2. The block-building for the group G. Denote by GS, Gr and Gb the Mobius groups generated by reflections with respect to the generator spheres Sj E C whose interiors intersect our green, red and blue apartments, respectively. These groups are discrete due to property (3) of the family e and the Poincare-Aleksandrov theorem on fundamental polyhedra (see Theorem 4.29 and Corollary 4.30). Here the last theorem should be applied for the extensions of these (Mobius) group actions on

S3 to the isometry actions in the hyperbolic 4-space H4. For this extension, the hyperplanes corresponding to the generator spheres Sj bound a convex hyperbolic polyhedron with the same dihedral angles that Si and Sj have in S3. The first Maskit combination in S3 (Theorem 5.17) shows that these groups are amalgamated free products. In fact, they are of the (5.21)-type, where every amalgamated subgroup Gi.l = GL n Gj is generated by reflections with respect to spheres from e whose interiors cover the circlebil = aBinaB;,Gi(Bi) = Bi,Gj(By) = Bj. The condition, crucial in the combination Theorem 5.17, on the precise invariantness with respect to the subgroup Gi3 in Gi (correspondingly, G1) of one of the two closed topological balls with common boundary sphere, follows from the fact that the hyperplane in H4 that contains the circle biJ at infinity (orthogonally) intersects only those spheres from C that intersect the balls Bi and BJ and whose interiors cover the circle b1. Since every two spheres Sj, Si E C that cover distinct apartments do not intersect each other, and due to the Klein combination Theorem 5.12, we have that the discrete

270

5. Kleinian Manifolds

group Go generated by reflections in the spheres of the family C is the free product of three groups: (5.26) Go = Gred * Gblue * Ggreen

Now, let us consider the extension G of the group Go by the hyperbolic translation h c- Mob (3) whose isometric spheres were described in Section 5.1, G = (Go, h) .

(5.27)

Actually, this extension is an HNN-extension of Go because the transformation h conjugates some isomorphic subgroups H, H' C Go. To prove that, let us denote I (h) fl d = {x1, x2, x3} and I (h-1) fl d = {xi, x2, x3}, where h(xi) = x and these six points are placed on the Fox-Artin arc d in the following order (see Figure 75): xl, x2, x3, xi, x?, x3. We also denote by Hi (respectively H'), i = 1, 2, 3, the subgroups of Go generated by reflection in those spheres of the family e whose interior balls cover the circles I (h) n aBj (respectively, the circles I (h-1) fl aBj,). The points xi (respectively x') are the centers of the disks on the spheres I and I' bounded by these circles. Now we set

H=H1*H2*H3, H'=H1*H2' *H3 CGo.

(5.28)

As above, extending the group action in the hyperbolic 4-space, we see that the closed balls in k3 bounded by the isometric spheres 1(h) and I (h-1) are precisely invariant in the group Go with respect to the subgroups H and H', respectively. Here we use the property of h that its isometric spheres I (h) and I (h-1) intersect (and are orthogonal to) only the generator spheres Si E C with centers at the vertices of the truncated icosahedra P (0, n) and P (18, n), n = 14, 33, 42, that are nearest to one of the centers z and z' of the isometric spheres. Thus, we have the following presentation of the group G: G = (Greg * Gblue * Ggreen, h

: hHih-1 = H', i = 1, 2, 3) .

(5.29)

Now we claim that the group G in (5.29) is the desired group:

Lemma 5.45. The group G C MSb(3) defined in (5.29) is a geometrically finite Kleinian group isomorphic to a discrete co-compact group r C Isom H3. Its limit set A (G) is a wildly embedded in S3 2-dimensional sphere which splits S3 into two G-invariant discontinuity set components, one of them being a quasiconformal ball while the other one is non-simply connected. Proof. First of all, we shall to clarify algebraic properties of G and to show that SZ (G) is the union of two G-invariant components. To do that, we split the set e into three disjoint subsets Cred, Cblue and egreen which consist of those spheres SI that intersect correspondingly our red, blue and green apartments. Then, instead of the generating reflections in (5.29) contained in the group GLed, we take their conjugations by the

5.5. Kleinian n-manifolds and hyperbolic cobordisms

271

element h, i.e. the generators of the group hGredh-1 which are reflections in spheres from the set h(Cred). The standard technique of Tietze's transformations (see Tietze [1], Lyndon-Schupp [1, § 2, Ch. 2]) shows that the presentation (5.29) can be rewritten in new generators in the following form:

G = (Gblue * hGredh-1 * Ggreen, h HZ

:

hH3h-1

= H3) .

(5.30)

H1

In other words, the presentation (5.30) means that the group G is an HNN-extension by the element h of the group Go generated by reflections in spheres from the new family C* = G''blue U Cgreen U h (Cred)

Due to the second Maskit combination Theorem 5.18 applied to (5.27), we see that the spherical polyhedron F C S3, F = fl{ext Si : Si E C} fl ext I fl ext r is a fundamental polyhedron for the group G. It has four connected components: an unbounded polyhedron Fw and three bounded ones, Fred, Fgreen, and Fblue. Extending all 2-spheres supporting this polyhedron F to hyperplanes in H 4, we obtain a finitesided convex polyhedron F4 C 1114 which is fundamental for the action of G in W.

This shows that G is a geometrically finite group without parabolic elements. Now one may change F C S3 to another fundamental polyhedron for G, F* = F00 U int (Fblue U Fgreen U h(Fred)) = Foo U FO,

(5.31)

where F* has only two connected components: Foo and a bounded connected polyhedron Fo. We note that Fo contains the segment (x3, x3) of the Fox-Artin are d and, due to Andreev's classification of hyperbolic 3-polyhedra (Theorem 2.41), it has the combinatorial type of a compact hyperbolic polyhedron FO' C H3 with the same dihedral angles (later we will find this polyhedron in a constructive independent way). Using the fact that each component, FO or F00, of the fundamental polyhedron F* has side identifications by elements of G and applying Lemma 2.24, we see that the domains

SZo = U{g(Fo) : g E G} and 01 = U{g(F00) : g E G}

(5.32)

are G-invariant components of the discontinuity set S2 (G), and 52 (G) = 520 U 521. This completes the first part of the proof. In the second step, we shall construct (by using an idea similar to "bendings", see Chapter 7) a quasi-symmetric homeomorphism ,f

:

o = Qo U A(G) -} B

(5.33)

that maps the closure of the first component 52o to a closed round ball B C R3, where f is quasiconformal in Qo and compatible with the action of the group G. To do that, we have to define a quasiconformal homeomorphism fo: FO -+ FO'

of the bounded component FO of the fundamental polyhedron F* in (5.31) onto a compact hyperbolic polyhedron Fo C B - H3, with the sides corresponding.

5. Kleinian Manifolds

272

Let us consequently enumerate the balls int I (h) and int I (h-1) and all those balls B(i, j) or h (B(i, j)) whose union contains the polyhedron Fo, as follows:

Bo = int l (h),

B1 ,

B2, ..., Bk-1

,

Bk = mt I (h-1)

,

where each of the circles bi = bE_1,i = aB1_1 n dBi, i = 1, ... , k, is orthogonal to a generator-sphere Sj from the set C * if and only if Si n bi 0. Then, in the first step, we define two quasiconformal "bending" self-homeomorphisms pl and rok of S3 preserving the balls B1, ..., Bk_I and the set C* and transferring aBo = I (h) and aBk = 1(h-1) into 2-spheres orthogonally intersecting a BI and a Bk-1 along the circles b1 and bk, respectively. To construct the bending o1 (cPk is similar), we may assume that the balls Bo and B1 are half-spaces with boundary planes aBo and aB1 and such that b1 = {x E 1R3 : XI = x2 = 0} is their intersection line. From our construction of the group G, we have that the dihedral angle of the intersection Bo n B1 has a magnitude a, 0 < a < .7r/2, and there exists a dihedral angle V1 C 1R3 with the edge b1 and magnitude where 0 < < n/4 and a < r such that Vi contains all the spheres in C* disjoint from b1. Let us assume the natural complex structure in the orthogonal to b1 plane 1182 = {x E 1183 : x3 = 0}. Then the quasiconformal homeomorphism (P1: S3 --i S3

can be described by its restriction to this plane C = 12 (where - r < arg z < sr is the principal value of the argument of z E C) as follows, see Figure 78:

'P1(z) =

z

if IargzI>-Jr-

z.exp(i(2 -a))

ifIa - argzI <

z.exp(i(2 -a)(1+ r z+,))

if

-a <argz

BO

f1= compression

'rN

'

(5.34)

z . exp (i (2 _a)(, - n Z2, )) if a + < arg z < 7r -

fl = rotation N

fl = expansion

Figure 78

\

/

5.5. Kleinian n-manifolds and hyperbolic cobordisms

273

We remark that cpl = id in V1 and hence it is the identity on all spheres Si E e* disjoint

from b1 = aBo n aB1. Also all spheres Sm E (* intersecting b1, and the exterior dihedral angles of their intersections with other spheres Sj are still invariant with respect to cpi. On the other hand, the restriction of cpI to the dihedral c-neighborhood of the disk aBo n B1, that is to the spherical lens with the edge b1, is a rotation, so

it transfers the side To n aBo of the polyhedron F0 which is a regular hyperbolic hexagon with angles .n/3 (in the Poincard hyperbolic metric in 2-disk aBp n B1) to a regular hyperbolic hexagon with angles ir/3 in the hyperbolic 2-plane cpl (a Bo n B1) (in the Poincare model of the hyperbolic 3-space in the ball B1). Similarly, in each of subsequent steps related to balls Bi_1 and Bi with the common boundary circle bi = a Bi _ 1 n a Bi, we construct a quasi-conformal bending homeomorphism Soi that transfers Bi U Bi _ 1 onto the ball Bi and which is conformal in dihedral

'i-neighborhoods of the spherical disks aBi\Bi_1 and aBi_1\Bi. Namely, let Bi and Bi_1 be half-spaces whose boundary planes aBi and aBi_1 contain the origin and intersect along the third coordinate axis bi = {x E R3 :X1 =X2=01 C 1R3 at an angle a, 0 < a < .rr, and let be a fixed number such that 0 < < 7r12, 0
(Pi (Z) =

- zr-

z

ifIargzl

z exp (i (ir - a))

if 17r - a - arg z1 <

z exp (i(7r - a)(1 - ngz2 ))

if a - n -{-

zexp(i(r-a)(1 I-t

)) if

-n <argz
Si

Figure 79

< argz < n -

(5.35)

5. Kleinian Manifolds

274

Note that in each i-th step, 2 < i < k - 1, we reduce the number of balls Bj by one. The constructed quasiconformal homeomorphisms Vi have the properties:

(1) coj = id in a neighborhood of spheres from the family G* that are disjoint from bi and intersect some balls Bj, j > i. cpt transfers all spheres from G* to spheres or(2) The composition cpicoi_1 thogonal to the boundary spheres of some balls Bj, i < j < k - 1, where all intersection angles between these spheres do not change. . cpl -images of the regular hexagonal sides so = Ton 8 Bo and sk = Fo fl 8 Bk (regular hexagons with angles it/3 in the Poincare hyperbolic

(3) The cpkcpi 1 _ 1

.

metrics in the disks 8Bo fl BI and 8Bk fl Bk_1) are regular hexagons with angles n/3 in the corresponding hyperbolic planes in the balls BZ and Bk_1; note that all such regular hyperbolic hexagons are isometric. Finally, we define our desired quasiconformal homeomorphism fo : To -+ Fp, Fo C cot of our bending homeB = Bk_1, as the restriction of the composition RR-i

omorphisms coj on the fundamental polyhedron Fo c SZo of the group G. The properties (1)-(3) of the constructed bendings cpj show that the following diagram is commutative.

so=1(h)flFo h sk=I(h-1)flFo foi

j.fo h'

fo(so) C Fo

(5.36)

.fo(lk) C Fo

Here h' E Isom 1H13 is a Mobius automorphism of the ball Bk_ 1 represented as a composition of two reflections with respect to hyperbolic planes in TllI = Bk_1. First of these planes contains the side fo (so) of Fo, while the second one is the bisector plane for the hyperbolic segment j oining the centers of the regular hyperbolic hexagons fo(so) and fo(sk) which are isometric to each other. This fact and the property (2) of the bending homeomorphisms cpj allow us to define a co-compact group G' C Isom 1H13 generated by reflections in those sides of the polyhedron Fo C ffil3 that are different from the sides fo(so) and fo(sk) and by the hyperbolic translation h' pairing the sides fo(so) and fo(sk) of Fo (which are orthogonal to other sides of this polyhedron). Similarly to the construction of the group G, we define G' as the group generated

by reflections with respect to the hyperbolic planes from the set e' = fo (C*), that is, the supporting planes for the polyhedron F. Then an application of the second Maskit combination Theorem 5.18 shows that the group G' is an HNN- extension of its subgroup Go C G' by the hyperbolic translation h' C Mob(Bk_1) = Isomiii[3 in (5.36). The property (2) of bendings cpj (invariance of dihedral angles of Fo) clearly shows that the subgroup Go is isomorphic to the subgroup Go C G in (5.30), Go*_ - Gblue FI'2

hGredh

t

Ggreen H

5.5. Kleinian n-manifolds and hyperbolic cobordisms

275

Applying the presentation (5.30) we conclude that the geometrically finite groups G C M6b(3) and G' C Isom ]Hl3 are isomorphic and have no parabolic elements. Moreover, extending the mapping fo: Fo -+ Fo compatible with the actions of the groups G and G' upon the images G(Fo), that is to the whole component S2o, we obtain a quasi-conformal mapping f : S2o -+ B conjugating the conformal actions of the groups G and G'. In other words, for our isomorphism x : G -+ G' and for any element g E G, we have f g f -1 = x (g) E G'. Now it remains to extend the mapping f to a homeomorphism f of the closed domains, f : S2 -+ B. This extension f can be taken as the usual continuous extension. The fact that the obtained map is a homeomorphism follows from the isomorphism Theorems 4.41 and 4.44 applied to the isomorphism X : G -* G'. So the map f is the sought-for homeomorphism (5.33). It is quasiconformal in S2o and quasisymmetric in the boundary aS2 = A (G). This implies that f is quasisymmetric in the closed domain S2o, see Aseev [2], Cor. 3.2. Finally, to complete the proof of Lemma 5.45, it suffices to show that the topological sphereA(G) = a S2o = 001 = i-1(S2) is widely embedded in 1183. This follows from Alexander's theorem on unknottedness of any polyhedral two-dimensional sphere in R3 (see Bing [1, VIR.4]) and the fact that 7r1(S21) 0. Indeed, consider a non-trivial loop £ in the unbounded component & C S21 of the polyhedron F (see Figure 74) and assume that it is contractible in 01 in (5.32). Then Dehn's lemma (Theorem 5.1) shows that there exists a disk D C S21 such that

a D = 2. As D is compact, it is covered by a finite number of polyhedra gi (F..), gi E G. On the other hand, as the loop t is not trivial in 1183 \d we have D r) d # 0. Therefore, there exists go E G such that go (F,,,) fl d 0 0. This contradiction completes the proof of Lemma 5.45 and thus completes our construction of the group G (up to a torsion free subgroup of finite index, with the same limit set) in the proof of Theorem 5.44, see Figure 80.

Note that the set of points near which the topological sphere A(G) C S3 is widely

knotted is a dense subset in A(G). This follows from the density in the limit set A(G) of the G-orbit G ({x, y}) of the ends x, y of the arc d which are fixed by the hyperbolic element h E G, see Corollary 2.16. So we immediately have an everywhere wild quasisymmetric embedding (p = f -1 of a closed 3-ball B into 1183, see Figure 80: R3 of a closed round Corollary 5.46. There are quasisymmetric embeddings rp: B ball B C 1183 which are quasiconformal inside B and cannot be extended to an embedding of any neighborhood of any boundary point x E a B.

276

5. Kleinian Manifolds

Figure 80

§6. Finiteness problems for Kleinian n-manifolds Here, in addition to the construction of non-trivial hyperbolic 4-cobordisms in the previous § 5, we shall describe another application of our block-building construction. As the block-groups Gi acting in block-balls Bi, all those applications use some finitely generated groups isomorphic to surface bundle groups in Isom H3. Resulting groups G C Isom l1i4 in the form (5.21) will provide counter-examples to the natural generalizations for Kleinian groups in the n-sphere S", n > 3, of all finiteness theorems for Kleinian groups on the plane (see Chapter 4, § 1). Namely, we shall present finitely generated Kleinian groups G C Isom H4 which realize the following pathologies:

(a) A Kleinian 4-manifold M(G) with finitely generated fundamental group G, which has no compact core Mc whose inclusion MM C M(G) induces a homotopy equivalence; compare the Scott compact core Theorem 5.9.

5.6. Finiteness problems for Kleinian n-manifolds

277

(b) A finitely generated Kleinian group G which is not finitely presented; compare Theorem 5.9.

(c) A conformal 3-manifold No with infinitely generated fundamental group n1 (No), which is a connected component of the quotient c2 (G)/ G, S2 (G) C S3, for a finitely generated Kleinian group G C Mob(3); compare the Ahlfors finiteness Theorem 4.1. (d) A Kleinian 4-manifold M (G) with finitely generated fundamental group G C Isom 1H14, which has infinitely many cusp ends; compare the Sullivan finiteness Theorem 4.3. (e) A finitely generated Kleinian group G C Mob(n), n > 3, whose conformal nmanifold N = Q/ G has an infinite topological type (whose (n -1)-homology group Hn_1(N, Q) has infinite rank); compare the Ahlfors finiteness Theorem 4.1. (f) A finitely generated Kleinian group G with infinitely many conjugacy classes of finite subgroups. (g) Two geometrically finite Kleinian groups G1, G2 C Mob (n), n > 3, whose finitely generated intersection subgroup r = G1fG2 isnotfinitelypresented; compare Chapter 4, § 1.

Now we start with a construction of one of those pathologies, by presenting a finitely generated Kleinian group G C Isom ]H14 having an invariant component S2oof the discontinuity set with infinitely generated fundamental group of the quotient S2o / G.

For this construction, we use our block-building with four block-groups Gi. Each of them acts in a corresponding 3-ball Bi C S3 = 81H1'1 (and in its exterior Bi = S3\-B,) as a discrete subgroup of Isom H3, while Bi are mutually tangent unit round balls in R3 whose centers lie in the plane {x E I[83 : X3 = 0}, see Figure 81. As in (5.21), the group G will be obtained by sequential amalgamated free products of "adjacent" block-groups Gi and Gj with Abelian rank 2 amalgamated subgroups Gii = Gi fl Gi = Z G Z. As the block-groups Gi (finitely generated, isomorphic to each other), we shall use discrete subgroups in Isom H3 isomorphic to the fundamental group of a surface bundle over S1 with a complete hyperbolic structure. To define such surface bundle groups G1, let us consider a regular hyperbolic ideal octahedron P0 in the ball B1 =1H13 with right dihedral angles and two of whose vertices

at infinity are tangent points p1 and P2, {p1} = B1 fl B2 and {p2} = 91 fl B4. Due to Thurston [1] (see also Example 4.25), there is a finite index torsion free subgroup F C Isom 1H[3 of a finite extension of the group generated by reflections in sides of Po by four elements which are automorphisms of of this group of order 3, such that the hyperbolic 3-manifold M = 1H13/ F is homeomorphic to the Borromean rings complement (compare Example 1.19 and Figure 12) and fibers over S1 (Rolfsen [1]). Let c be a normal subgroup of r which corresponds to the fiber of M, and P C 1H13 B1 be the Dirichlet polyhedron Do(F) of the group r centered at the center 0 of the ball B1 and having p1 and p2 as its vertices at infinity. Note that P is the bounded component of the isometric fundamental polyhedron for the natural Mobius action of r

278

5. Kleinian Manifolds

in 1R3; the unbounded component P* C Bi of this polyhedron is symmetric to P with respect to a B1. The polyhedron P * is the Dirichlet polyhedron Do, (r) C Bi = H3 centered at oo.

J2 S2

Figure 81

Let S1, S2 C 1[83 = S3 be two extended planes tangent to the balls Bi, i = 1, 2, 3, 4,

at their tangent points pj and such that P1 E S1 and P2 E S2. We may think about Si \{ pi }, i = 1, 2, as horospheres in the hyperbolic 3-space model in Bi . These horospheres are invariant with respect to maximal parabolic subgroups Hi c r with fixed points pi, i = 1, 2. We claim that they are precisely invariant with respect to their stabilizers H' in a finite index subgroup r' of r and, moreover, the Dirichlet polyhedra of these parabolic stabilizers satisfy the following compact inclusions: H3\Do(Hi') C=

i g` j,

i, j = 1, 2.

(5.37)

This follows directly from the Malcev's [1] result on the residual finiteness of a finitely

generated linear group r, which states that for a given non-trivial element y E r there is a finite index subgroup r' C r which does not contain y. Namely, due to discreteness of r, it has only finitely many elements Yk E Hi whose Dirichlet planes (2.37) intersect either the horosphere Sj \ { pj} or the sides of the Dirichlet polyhedron

5.6. Finiteness problems for Kleinian n-manifolds

279

Doc (H3), j ; i; i, j = 1, 2 (perhaps, at infinity 3B1). The chosen finite index subgroup r' C F has a normal subgroup c' C (D n F' and an element y E F' such that r' = (V, y). Then we define a finite index subgroup I' of r, (5.38)

H = 1J2 fl F' and, for some Y E C1" and n E Z, Then we consider horoball neighborhoods Vi C Bi

where

1112 of the horospheres

S i \{pi } bounded by some horospheres Qi c Bi . Here 0-i U (pi) is a 2-sphere in R3 tangent to B1 at the point pi, i = 1, 2, see Figure 82.

S2

Figure 82

We shall find a finite index subgroup i c I' such that its action in the closed horoballs Vi C Bi is the action of parabolic groups Hi = Hi n F, i = 1, 2. Actually, due to (5.37) and the Klein combination Theorem 5.12, the group

ft =(H1,H2)=Hi *H2CP

(5.39)

is the free product of parabolic subgroups, that is, H is a Schottky-type group. As a fundamental polyhedron for the group H, we may take the intersection DCO(H1) n DOO(H2) of Dirichlet polyhedra for subgroups Hi. Clearly we have that this polyhedron has a compact intersection

X = (oi u Q2) n Doo(H1) n Doo(H2) C BI* -= H'

5. Kleinian Manifolds

280

with the boundary horospheres Qi = 8 Vi. Hence there exist at most finitely many elements yk E F such that yk (X) fl X 0. A direct application of locally extended residually_finiteness of such finitely generated groups (Scott [5] ) gives us a finite index

subgroup r C r such that

H C I' and y (X) f1 X= 0,

y E F.

(5.40)

There are two facts crucial for this application of Scott's result. First, our subgroup H is geometrically finite due to the Klein combination Theorem 5.12. Second, our original octahedron Po C H3 has all dihedral angles equal to. r/2. Finally, we have four groups Fi, i = 1, 2, 3, 4, acting in the balls Bi:

r1=r,

r2=J1r1J1, 1'3=J2J1r1J1J2,

(5.41)

r4 = J2F1 J2 = Ji J2 Ji r1 J1 J2J1

Here Ji are reflections in the extended planes Si tangent to B1 at pi, and the groups Fi have normal subgroups corresponding to the normal subgroup (D1=(D

(5.42)

Using the property of F that the horoballs Vi (neighborhoods of S=\{pi}) are precisely invariant with respect to the maximal parabolic subgroups with fixed points

pi, i = 1, 2, we can apply the block-building construction (5.21) where the blockgroups are ri and amalgamated subgroups rid = ri fl rj are the maximal parabolic subgroups fixing tangent points Bi fl B. Here we use the first Maskit combination Theorem 5.17 with the extended planes Si as the splitting surfaces. As result, we obtain the following Kleinian group G C Mob(3):

G = (171 * r2) r12

* (173 * 174) r14*r23 r34

(5.43)

where

F12 = Ht ,

r14 = H2,

r23 = J1H2J1,

1734 = J2H1 J2 .

(5.44)

Lemma 5.47. The Kleinian group G in (5.43) has a finitely generated normal subgroup F C G. Their discontinuity set n (G) = Q (F) has a connected component S20 which is invariant for both groups F and G and whose quotient manifold N(G) = 00/ G is the interior of a compact manifold.

Proof. First of all, we remark that the groups (free amalgamated products) G1 =

rl * r2 and G2 = 1'3 * 1'4 are non-degenerate B-groups, that is geometrically r12

r34

finite groups with invariant contractible components S2i C 52 (G1) and 522 C S2 (G2), respectively. This follows from the first Maskit combination (see details in the proof of Theorem 6.72).

5.6. Finiteness problems for Kleinian n-manifolds

281

Next, the manifolds N (Gi) = Qi / Gi are homeomorphic to the interior of a surface bundle over S1. In fact, each of them is the torus sum of two hyperbolic 3-manifolds (with respect to tori at their infinity). In particular, N (G 1) is the torus sum of M(I'1) = 1HI3/ F1 and M(F2) = IHI3/ r2 along the tori at infinity that both correspond to the parabolic subgroup H1 = F12 = r1 n I'2. More exactly, N(G1) is obtained from

Ni = [Bi \F1(Vi)] / I'1 = M(F1)\(V1/H1) and

N2 = [B2\F2 (Ji(Vl))] / F2 = M(r2)\(J1(Vl)/H1) by gluing their boundary tori via the homeomorphism

cp: 8N1 -1 W2,

aNi = aN2

(S1\{pi}) /Hl

Sl x S1.

(5.45)

Each of the manifolds Ni is a surface bundle over S1 whose fiber corresponds to the normal subgroup (Di < F1 in (5.42), and the gluing homeomorphism p in (5.45) preserves the bundle structure because it is covered by the identity homeomorphism of the plane Si. Thus N(G1) is a surface bundle, and the subgroup F1 = ((D 1, J1t'1J1) a G1

(5.46)

corresponds to the fiber of N (G 1) and is a normal subgroup in G i with G1 / Fi = Z. Applying (5.38), we also have that G1 = (Fl, ) where c E H2. Each surface bundle M(Fi) admits a natural compactification by adjoining a torus to each parabolic cusp end. The same is true for the glued manifold N(G1), hence the normal subgroup F1 a G1 is finitely generated. Now, rewriting (5.43) in the form

G=G1 * G2,

G12=G1nG2=F14*123=H2*J1H2J1,

we finally claim that the geometrically finite group G C M6b(3) has an invariant component SZo C S2 (G), and its finitely generated subgroup F = (F1, J2 F1 J2) is normal.

The first part of this claim, geometrical finiteness, follows from the first Maskit combination Theorem 5.17. Correctness of this combination follows from the fact that the closed upper half-space S2 C l[82 bounded by S2 is precisely invariant in G i with respect to the amalgamated subgroup G12 C G1:

G12(S2)=S2;

g(S2)nS2 =o for gEGl/G12.

(5.47)

One can see this fact from the precise invariantness of the closed horoballs V2 C Bi and J1 (V-2) C BZ with respect to subgroups H2 c rl and J1 H2J1 C F2, respectively. Denoting W = v2 n D,,,, (H2) n J1 (v 2 n D.(H2)), we obtain an even stronger fact of the precise invariantness of a neighborhood U of S,+, where U = U{h(W) : h E H2 * J1H2J1 = G12).

5. Kleinian Manifolds

282

Therefore, the first Maskit combination shows that G is geometrically finite andhas an invariant component S20 C 0 (G). This component S2o is the union of polyhedra g (P), g E G, where P is the closure of the unbounded component of the intersection of fundamental polyhedra P (G i) fl P (G2) (compare Lemma 2.24). Simultaneously we see that the manifold N(G) = S2o/ G admits a natural compactification by adjoining

tori to each of (finite many) cusp ends. We obtain this manifold N(G) from the manifolds N(G1)\(S2 /G12) and N(G2)\(J2(S2 )/G12) by gluing them along the compact surface S = [S2\A(G12)] /G12 of genus 2 which is a connected sum of two tori. As another consequence of the first Maskit combination, we note that the subgroup

F = (Fl, F2) C G is a free amalgamated product of its generating subgroups:

J=G12f1F,

(5.48)

where the extended plane S2 is precisely invariant with respect to the subgroup J in gFg-1 F. To prove the last assertion on the subgroup F = (Fl, F2) that = F for any g E G, we have to verify that g1F2gi 1 C F and g2F1g2 1 C F for any gi E G1 and g2 E G2.

For some elements fl E F1 and f2 E F2 and integers m and n, we have from (5.42) and (5.46) that gl = fl . m and 92 = f2. n where E H2 C G12 = G1 fl G2 and Gi = (Fi, ). Hence it follows that: 1 = f1F2f1 1 C (F1, F2)

g1F2g1 1 =

g2F1g2

1

=

= F,

1 = f2F1f2 1 C (F1, F2) = F.

Now we see that J = G12 fl F is a normal subgroup of G12. Thus the proof of Lemma 5.47 is completed by the observation that

A(G) = A(F),

A(G12) = A(J),

(5.49)

which is true because of the fact that the limit set of a normal subgroup of a discrete group coincides with the limit set of the group itself.

Theorem 5.48. Let F = (Fl, F2) be the finitely generated normal subgroup of the geometrically finite Kleinian group G C M6b(3) in (5.43) and 00 C 0 (F) be the Finvariant component. Then the fundamental group 7r1 (N (F)) of the quotient manifold

N(F) = S2o/F is not finitely generated.

Proof. Step 1. Let p: 0o - S2o/G = N(G) be the natural covering. Then the reflection J2 in the extended plane S2 commutes with the action of G in S2o and is projected by p to the involution 'C2: N(G) -> N(G). From the proof of Lemma 5.47, we see that the fixed-point set of this involution i2 is the surface S of genus two, S = [S2\A(G12)] /G12. Similarly, let f2: N(F) -+ N(F) be the projection of J2 by

5.6. Finiteness problems for Kleinian n-manifolds

the natural covering q: 520

283

52o/F = N(F) and r: N(F) -a N(G) be a regular

covering with the deck-transformation group G/F. So we have the commutative diagram:

(520, S2\A(G12)) q (N(F), S) ' (N(G), S) 1.12

J f2

I T2

(5.50)

(cZo, S2\A(G12)) q . (N(F), S) > (N(G), S) Here S = r-1(S) = q (S2\.A (G12)). Clearly S is the fixed-point set for the involution z2. We claim that S is a connected surface. To prove that, it is enough to verify the following two facts:

(i) G1(S2)=F(S2); (ii) G is generated by F = (Fl, F2) and in (5.42). Indeed, we have G 1 = (Fl, ) where E H2 preserves S2+. Thus the orbits G 1(S2 ) and F1 (S,+) coincide. To prove the second fact, for a given element g E G, g = gl, g2, ... , gm with some g1 E G1 U G2, we obtain g = f k (for some f E F and k E Z) due to representations gl = E (Fl, ) U (F2, ), fi E F1 U F2.

Step 2. Let us prove that G/F = Z. As we have seen in the proof of Lemma 5.47, the surface S splits the manifold N(G) into two parts whose closures are N(G)- = N(G1)\(S2+/G12) and N(G)+ = N(G2)\(J2(S2 )/G12). Let N(F)be the manifold r-1(N(G)-). We know that this manifold N(F)- coincides with the complement N(Fi) \ (S2 /(G12 n Fl)) in the manifold N(F1). Therefore the covering r: N(F)- -* N(G)- is nothing but the restriction of the infinite cyclic covering N(F1) -+ N(G1). Letus remark that both manifolds N(G)- andN(F)- donotcontain fake cells since they are covered by subdomains of the 3-sphere. Also the cusp compactification N(G)* of the manifold N(G)- contains the surface S of genus 2 as a boundary component. Hence this compact manifold N(G)* (homotopy equivalent to N(G)-) is not a surface bundle over S1. Step 3. Here we shall prove that the fundamental group N(F)- is not finitely generated. Supposing the contrary, we have an exact sequence (due to Step 2),

1 -- nt(N(F) ) --> nl(N(G)) = ri(N(G)*) -> Z -* 1, where xl (N (G)-) is a non-Abelian group, and the manifold N (G)- does not contain projective planes and is in fact irreducible because the manifold N(GI) is covered by the contractible domain 521, see Lemma 5.47. Applying Stallings's Theorem 5.10 on fibering 3-manifolds over S' to the compact 3-manifold N(G)*, we have that N(G)* should be a surface bundle over S1. This contradiction to Step 2 shows that 7rl (N(G)-) is not finitely generated.

Step 4. Finally, it remains to prove that nl (N(G)) itself is not finitely generated. For that, let us consider the universal covering n : 9 -+ N (F) with the deck-transformation

5. Kleinian Manifolds

284

group 11 = i1 (N(F)) and a lift i2: N -+ N of the involution i2 in N(F) where N(F)- = N(F)/1 . We have 11 = i2f 12, so the 2-extension II' = (II, i2) acts discontinuously on N. Due to Theorem 2.18, the fundamental group of the quotient N/ I1' = N(F)- is isomorphic to the quotient of the group II' by its normal subgroup generated by all finite order elements. As we already know, this normal subgroup

of II' has index 2 and In (N(F)-) is not finitely generated. Evidently, the group ,rt (N(F)) itself is not finitely generated. This completes the proof.

Corollary 5.49. There exists a torsion free Kleinian group F C M6b(3) with an invariant component Qo of the discontinuity set 0 (F) C S3 = aH4 such that F is finitely generated but is not finitely presented, and its hyperbolic 4-manifold M = H4/F has no compact core Mc homotopy equivalent to M.

Proof As the group F in the statement, we may take the finitely generated group F C M6b(3) constructed in Lemma 5.47 and Theorem 5.48. What we have to prove is the absence of any finite representation for F. Due to (5.48), we have that F is a free amalgamated product of subgroups F1 and F2 with amalgams J = G12 f1 F. Moreover, J and F are normal subgroups of the groups G12 and G, respectively. Here G12 C G has a precisely invariant extended plane S2 C Il83, and the surface [S2\A(G12)] /G12 = S is a compact surface of genus 2. Using (5.49) and taking a restriction of the covering r in (5.50), we see that the regular covering rs :

[S2\A(J)] I J -> [S2\A(G12)] /G12 = S

is infinite because the map rs is just a restriction of the infinite regular covering N (Fl) -* N (G 1). This can be shown similarly to Step 2 in the proof of Theorem 5.48. Then the Ahlfors finiteness Theorem 4.1 shows that the Kleinian group J C Mob(S2) is in fact infinitely generated and has infinitely many defining relations. Due to a theorem of B. H. Neumann [1], if a group is generated by a finite set of elements but cannot be defined by a finite set of relations in these generators, it cannot

be defined by a finite set of relations in any system of generators. Hence our group F = F1 * F2 is infinitely presented, and its quotient 4-manifold M = W/F cannot J have a compact submanifold MM homotopy equivalent to M (compare Theorem 5.9 in dimension three). 0

Remark 5.50. Modifying the construction of the groups G and F a G in Lemma 5.47, one can consider block-balls Bi, 1 < i < 4, whose boundary spheres aB1 are mutually intersecting along circles. In such a way, it is possible to construct groups F and G having all the properties of the groups in Theorem 5.48 but without parabolic elements. Another modification of the block-building construction (5.43) used in the proof of Theorem 5.48 deals with a geometrically finite Kleinian group G' C M6b(3) and its

5.6. Finiteness problems for Kleinian n-manifolds

285

finitely generated free normal subgroup F' < G', G'/F' - Z, such that the following properties hold:

(C-1). The group G' = (Fi, F2, F3, F4) is generated by block-groups r' = hir?hL 1 acting in the block-balls B' = hi (Bi) as subgroups of Isom 1H13 and obtained from (iso-

morphic) finite-index subgroups F0 of the block-groups Fi in (5.41) by conjugations by some transformations hi E Mob(3).

(C-2). Let H2 =

(S) be the maximal parabolic subgroup of the group I'1 = r

with the fixed point P2 E 8 B1 and the generator from (5.42), and Fo be a finite-index subgroup of F1 such that:

S E F0 ,

r° = (q'0, n) ,

(D°

4 r° .

(5.51)

Here n E Z, a normal subgroup (De is finitely generated, and isometric spheres I (y)

of all elements y E Fo\{S, 6-1, 1) do not intersect the plane in the half-space S2 tangent to the isometric spheres I(S) and I(3 -1) (compare (5.40)). Then, changing r, in (5.41) by Fo, one obtains finite-index subgroups r9 c Fi with the following maximal parabolic subgroups:

H°= H1nr°=JiH°J1, H°= H2nFo, J1 H2 ii =J2J1H2J1J2, 12H°J2. (5.52)

(C-3). The conjugating transformations hi E Mob(3) in (C-1) are:

h2 =h3 =id,

h1(x) =rc(hc(x) -pl)+p1,

h4 =J2h1J2,

p1 = B1 n B2.

(5.53)

Here rc is a positive number, 0 < rc < 1, and the hyperbolic transformation he E Mbb(B1) satisfies the following two conditions (see Figure 83):

(i) h1 commutes with the action of the parabolic group Ho. (ii) Isometric spheres I (Y1) and I (Y4) are disjoint for all non-trivial elements Y1 E F1 = h1F?hi 1 and Y4 E F4 = h4F4h41 except for two pairs of the following elements 31 = h18hi 1 81 1 =

,

84 = h4J28J2h41 ,

h13_1hi 1, S41

=

and h4J28-1

(5.54)

where the tangent points q and q',

I(S1)nI(84)=q,

1(Si1)nl(S41)=q',

belong to the extended plane S2, see Figure 83.

(5.55)

5. Kleinian Manifolds

286

J2(Pl)

B' 3

J2

S2

S1

Figure 83

Now let us remark that the group G' satisfying these conditions (C-1)-(C-3) is a free amalgamated product of its block-subgroups F,,

G'=r' *r2*r3*I'4, rig

where

r23

rs4

r= fl r' and I'12 = H?, r23 = J1H0J1, r34 = J2H?J2. One can

prove this splitting similarly to (5.43) by applying a week combination theorem of Maskit [12, VII-A10] instead of Theorem 5.17. Furthermore, G' = (F', *) where F' 4 G', and the point q in (5.55) is the fixed point of a parabolic transformation t = 841 .81 which generates a maximal parabolic subgroup in G' with a precisely C F', m E Z, are maximal invariant cusp-domain. Hence, the groups non-conjugate parabolic cusp subgroups of the finitely generated group F' a G'. This shows the absence of Sullivan's cusp finiteness theorem in higher dimensions (compare Theorem 4.3):

Theorem 5.51. There exists a finitely generated torsion free Kleinian group F' C Isom Hn, n > 4, whose Kleinian n-manifold M(F') has infinitely many cusp ends, i.e., F' has infinitely many conjugacy classes of maximal parabolic cusp subgroups. 11

Remark 5.52. Considering the group F' C Isom H4 as a discrete subgroup of Isom ffi[n, n > 4 (via the natural inclusion), one has the corresponding Kleinian n-

manifold M1 (F') = (En U c2(F')) /F' where S2 (F') C Sn-1 = all. Then, for

5.6. Finiteness problems for Kleinian n-manifolds

287

the boundary of this manifold, Nii-1(F') = Q(F')/F', the (n - 2)-dimensional homology group of Nii-1(F') has infinite rank:

rank

(Hn-2(N"-1(F'),

Q)) = oc

(5.56)

This shows that the analogue of the Ahlfors finiteness theorem does not hold in any dimension n > 3: the quotient manifold cZ (G) / G of a finitely generated Kleinian group G C Mob(n) may have an infinite topological type.

Remark 5.53. One can modify the construction of the group F' C M6b(3) in Theorem 5.51 to find a finitely generated Kleinian group F" a G" C Mob(3) which has infinitely many conjugacy classes of finite order elements (for geometrically finite groups, it is impossible due to Corollary 4.19). For this purpose, it is enough to replace the condition (5.55) in (C-3) to the condition of intersection of the isometric spheres I (31) fl I (84) and I (61 1) fl I (341) along circles with equal rational dihedral angles n/k where k E N is sufficiently large. In fact, the elliptic subgroups

mE7L,

are non-conjugate in F", where G" _ (F", *). It seems very likely that another modification of the construction of the group F' C Mob(3) in Theorem 5.51 may bring an affirmative solution to the following our conjecture. Conjecture 5.54. There exists a finitely generated torsion free Kleinian group G C Isom)Eiln, n > 4, whose Kleinian n-manifold M(G) has infinitely many connected components of the boundary aM = SZ (G)/G. In other words, such a finitely generated Kleinian group G C Mob(n) would show that the analogue of Ahlfors finiteness theorem fails in higher dimensions n > 3 not only in the sense of a possibility of an infinite topological type of a component in the quotient manifold 0 (G)/ G but also in the sense of the quantity of its connected components.

Now we can apply all of the above constructions of "wild" finitely generated subgroups F of geometrically finite Kleinian groups G C Mob(n), n > 3, (or the Bowditch-Mess [1] construction) to present two geometrically finite Kleinian groups G1, G2 C Mob (n), n > 5, whose finitely generated intersection subgroup G1 fl G2 is in fact that group F with any of those wild properties (a)-(g), for example, either not finitely presented, or with infinitely many conjugacy classes of parabolic or elliptic subgroups, or whose Kleinian manifold has an infinite topological type, etc. Such constructions are based on the following basic technique due to ApanasovBrady-Susskind [1]. We notice that the constructed finitely generated subgroups F

of geometrically finite Kleinian groups G C Mob(n), n > 3, with the pathologies (a)-(g) satisfy the following short exact sequence 1 -+ F -+ G --- 7L -+ 1.

(5.57)

288

5. Kleinian Manifolds

Here G C Mob(n), n > 3, is a geometrically finite Kleinian group, 0: G -)Z is an epimorphism (related to fibering of a manifold over the circle), and F is the kernel of the epimorphism 0. Now we consider the natural isometric actions of all those groups in the hyperbolic spaces In+1 C lien+3, and define our first Kleinian group G1 as the above group G.

In order to define the second group G2 we first pick an irrational rotation a c Note that a fixes the subspace Hn+1 pointwise, and commutes with the group G1. Then for any given g E G, we compose this hyperbolic isometry with aO(9). The resulting elements generate the Isom(]HIn+3) about the subspace lH[n+1 C Hn+3

group G2 C Isom(lH[n+3)

Since the irrational rotation a commutes with G1, we have that the group G2 is isomorphic to G1, and the orthogonal projections to the invariant subspace 1B[r+1 C fn+3 of its Dirichlet and pseudo-Dirichlet polyhedra centered at a point y E Hn+1 are the corresponding Dirichlet and pseudo-Dirichlet polyhedra of the group G C Isom(ffi[n+1) Therefore G2 is geometrically finite (which also independently follows from Theorem 4.16). This definition clearly implies that G 1 fl G2 = F, which finishes our construction. It is worth to note that the Mobius group generated by elements of both groups G1 and G2 is not discrete because of its infinite order elliptic elements. Future applications of the block-building construction, as well as other properties of Kleinian n-manifolds will be presented in Chapters 6 and 7 in connection with conformal structures on n-manifolds and varieties of discrete faithful representation of hyperbolic groups.

Notes For another proof of Theorem 5.11, see Seifert-Threlfall [1]. Theorem 5.12 is essentially due to F. Klein [1]; for another proof in the case of two factors, see Maskit [12]. Our proofs of the Maskit combination Theorems 5.17 and 5.18 mainly follow the original scheme of Maskit [5] for Kleinian groups on the plane. The description in §2 of topological aspects (for Kleinian 3-manifolds) of combination theorems is due to an impressive machinery built up by Marden [1]. The term "universal group" first appeared in Hilden-Lozano-Montesinos-Whitten [1], where the first such group has been discovered. The process we called "blowing hyperbolic orbifolds" was first described for Borromean orbifolds by Hilden-Lozano-Montesinos [4]. The central results of §4, Theorems 5.30 and 5.41, solving the conjecture of Ahlfors [2] on the limit set and the conjecture of Thurston [1] on ends of hyperbolic 3-manifolds in the basic case of manifolds M with either non-decomposable or functional 7r1 (M) are due to Bonahon [1]. An implication of the positive solution of this Thurston Conjecture toward the Ahlfors Conjecture (Theorem 5.40) was first observed by Thurston [1, 8.12.4]. Theorems 5.36-5.38 on topologically tame manifolds are due to Canary [3, 4].

Notes

289

The central result for our description of topology of high-dimensional hyperbolic manifolds, the Compact Core Theorem 5.41, is due to Tetenov [3, 4] (see Apanasov [36] for another proof). For the compact case (Corollary 5.42), see Apanasov-Tetenov [2]. The effect of non-trivial hyperbolic homology cobordisms in Theorem 5.44 was discovered in Apanasov-Tetenov [2]. Other applications of our so-called "blockbuilding construction" in § 5.4, which is a crucial component for constructions of pathological hyperbolic groups, are in § 5 and Chapter 7 (Apanasov [33, 37]) as well as in § 6 (where the results are due to Kapovich and Potyagailo [1] (Theorems 5.48 and 5.51), Kapovich [4] (Remarks 5.52 and 5.53) and Potyagailo [1] (Corollary 5.49)). The "wild" hyperbolic groups constructed in § 6 show the absence of Ahlfors's [1], Scott's [1, 2] and Sullivan's [5] finiteness theorems as well as non-finiteness of the set of conjugacy classes of maximal finite subgroups in finitely generated Kleinian groups in dimensions more than three. For a different approach to these pathological phenomena, see Bowditch-Mess [1]. The construction of geometrically finite groups with such "wild" intersection subgroups is based on Apanasov-Brady-Susskind [1].

Chapter 6

Uniformization

§1. Classical uniformization The concept of uniformization for Riemann surfaces and more general manifolds is a milestone in the classical function theory. It has led to several developments in different branches of mathematics. In topology, one usually regards the concept of a covering space as directly arising from uniformization. In fact, as simple examples demonstrate, the essence of uniformization of multi-valued analytic functions on a

manifold M is such that, in order to make such an uniformization, one needs to consider the universal covering M of M where the corresponding functions are singlevalued because M is simply connected. In geometry, the classical uniformization goes substantially beyond the usual covering space theory. The special role here is played by the group of Mobius transformations. Namely, constructing the covering spaces leads to consideration of domains in Euclidean space and corresponding discrete groups of their automorphisms. For suitable chosen domains, these automorphisms are Mobius

transformations forming discontinuous groups G with invariant simply connected components 0 of the discontinuity set. Then, the original manifold M is conformally equivalent to the quotient space S2/G, and a multi-valued analytic function f on M becomes single-valued in 0 = M (in terms of the universal conformal covering jr: S2 -). S2/G, that is the function f o n). There is no complete solution of the uniformization problem, except in the classical complex one-dimensional case, where two classes of non-elementary Mobius groups have been used. The Klein-Poincare uniformization theorem for Riemann surfaces states that

each conformal structure on an orientable surface S is induced by a constant curvature Riemannian metric.

In the basic case of negative Euler characteristic, X (S) < 0 (in particular, if the genus of S is greater than one), one has the class of Fuchsian groups G on the plane preserving the unit disk B C R2, B = S, and the Klein-Poincare metric has negative curvature and defines a hyperbolic structure on the surface S = B/G. In the exceptional cases of the sphere and the torus, one has elementary Mobius groups

6.2. Modem concepts of uniformization

291

G on the plane, and the Klein-Poincare metric has zero or positive curvature and specifies either a spherical or a Euclidean structure on S. In these cases, the surface S is presented in the form S/ G where S is either the sphere R2 = 1182 U {oo} or the plane 1[82, and the discrete group G = nl (S) of motions of S is either trivial or a cyclic group generated by some Euclidean translation, z i-+ z + a, a E R2\{O}, or else it is a free Abelian group of rank two, generated by two Euclidean translations, z H z + al,

and z H z + a2, by linearly independent vectors al and a2. In all three cases, the natural projection n : S -+ S/ G is a covering and uniformizes all (multi-valued) functions f on S because the functions f o in are single-valued functions in the domain S c Y. Another uniformization approach, involving Kleinian groups which act discontinuously in some domain 0 C 1W2, is due to P. Koebe [1]. Koebe uniformization principle asserts that, for a covering Riemann surface S, S -* S, homeomorphic to a planar domain 0 C 1W2, there exists a conformal homeomorphism f : S -+ 0 such that the group G(S, S) of deck transformations of the covering S goes over to a Kleinian group G C Mbb(2), G = f G(S, S) f -1. So one has that S = S2/G, and the problem of (analytic) uniformization reduces to the topological problem of finding all planar S of a given Riemann surface S. In particular, one such solution of coverings S the uniformization problem for Riemann surfaces is given by the Koebe theorem on cuts. The corresponding planar regular covering S -± S of a closed Riemann surface S of genus p > 1 is defined by the free normal subgroup G of the fundamental group 7r, (S) = (al, b1, . . . , ap, by : 111
§2. Modern concepts of uniformization As we have seen above, any 2-surface S admits a geometric (X, G)-structure which is in fact either spherical, or Euclidean, or (in the basic case) hyperbolic. In other words, its universal cover X is either the sphere S2, or the plane R2, or the hyperbolic plane 11112,

while G = Isom X is the isometry group of X. Such an (X, G)-uniformized surface S is locally isometric to X via its development d: S -* X, and its fundamental group nl (S) acts discontinuously on X via the holonomy homomorphism d*: irl (S) -* G; furthermore, S = X/ G. In this way, we come to the uniformization problem for general manifolds M, which is the same as that of studying (X, G)-structures on M where the (X, G)-geometry satisfies the identity condition (IC) that elements of G are uniquely determined by their action on any open subset in X, see Chapter 1, § 5.

292

6. Uniformization

Now we consider some specific realizations of this concept.

1. Gromov's length spaces and convexity. One cannot solve the general uniformization problem by using only homogeneous (X, G)-geometries as it was done in the surface case. In fact, even among 3-manifolds, one can find manifolds that have no locally homogeneous geometries. That is why M. Gromov [2] suggested to study the so-called length spaces generalizing Riemannian manifolds and associated with the concept of hyperbolicity, see Chapter 4, § 6.5. Let X be a complete metric space. Then a length structure in X is given by a metric and a length function on curves, such that the distance p (x, y) between any two points x, y E X is equal to the length of the shortest curve (supposed to exist) joining these points. We can define straight segments, rays and straight lines in a length space X as (isometric) maps respectively from [a, b], R+ and R into X, which preserve the distance function and hence are always embeddings. Here a locally isometric map X is called a geodesic. Continuing in this way, one can define the ideal boundary R BMX, the Busemann function and horospheres. Furthermore, it is possible to classify isometries of X as elliptic, hyperbolic or parabolic, where hyperbolic elements have invariant geodesics (axes), and each parabolic one has a unique fixed point at 8X, compare Chapter 1, § 3. Then the following fact plays an important role in linking the geometry of a length space X with its fundamental group 7rl (X): Let X be a compact length space and X its universal covering with the induced length structure. We suppose here that X topologically is a polyhedron or a more general space admitting covering space theory. Then X is coarse quasi-isometric to 7rl (X) with the natural word metric, that is, there exists a quasi-isometric map f : nl (X) -* X such that f (nt (X)) is s-dense in X for some s > 0. As examples of length spaces, one can consider the following.

(1) X C R" is a piecewise smooth polyhedron which carries the metric d(x, y) induced from R'. We can consider the induced length metric on X, p (x, y), as the length of the shortest curve on X joining x and y. Since, for any topological circle X = St, we have that D(X) = sup,,,yex p(x, y)/d(x, y) > 2, in general it follows that the condition D(X) < 2 implies that X is simply connected, see also Gromov [2]. (2) The natural length structure of a quotient-space X = Y/ G of a Riemannian manifold Y by a group G of its isometries (with torsion) has richer structures than that of Y. (3) Let X be a Riemannian manifold with boundary. Its length structure is more complicated than that of a Riemannian manifold without boundary. In fact, shortest curves between interior points can touch the boundary (as we have seen in the proof of Theorem 4.37 for X =1EIo C IEI", the complement of a set of horoballs). Now, among all length spaces, we consider convex spaces, that is those spaces X whose distance functions are convex. In other words, X is called convex if for any two geodesic segments, x: [a, b] - X and y: [c, d] -* X, the distance function

6.2. Modem concepts of uniformization

293

dist(x(t1), y(t2)) is convex on [a, b] x [c, d]. A space X is called locally convex if it can be covered by open sets Ui such that the distance function dist(x, y) is convex in each of them. As an example, one has that a Riemannian manifold X is locally convex if and only if its sectional curvature is not positive. Moreover, it is convex if nl (X) = 0. Adapting classical arguments of E. Cartan [1] to the case of length spaces, we have a generalization of the Cartan-Hadamard theorem: Theorem 6.1. A simply connected locally convex length space is convex. Every convex

length space is straight (in the sense that any two points can be joined by an unique geodesic segment) and hence contractible. In the context of length spaces, the uniformization problem can be formulated as the following questions. G-I Let X be a K (1r, 1) -type n-manifold. Does there exist a locally convex space (non-compact but preferably finite dimensional) homotopy equivalent to X?

G-II Let X be an n-manifold. Does there exist a convex length space X and a discrete group G C Isom X such that X/G is homeomorphic to X? G.III Are there convex length spaces which are topological manifolds other than the Euclidean space W2?

As we have seen in §1, there are (affirmative) answers to the first two questions in dimension two. In dimension three, the Thurston theory [1-6] (see also J.-P. Otal [1]) provides an affirmative answer to the first question for a large class containing all known examples of 3-manifolds. Regarding the second question for the basic class of hyperbolic 3-manifolds, the following geometric and topological rigidity takes place (see Gabai [4, 5]):

Theorem 6.2. Let N be a closed, orientable hyperbolic 3-manifold containing an embedded hyperbolic tube of radius (log 3)/2 = 0.549306 about a closed geodesic (or satisfying the so-called insulator condition). Then: (i) If f : M --> N is a homotopy equivalence where M is an irreducible 3manifold, then f is homotopic to a homeomorphism. (ii) If f, g: M --), N are homotopic homeomorphisms, then f is isotopic to g. (iii) The space of hyperbolic metrics on N is path connected.

Also, due to Farrell and Jones [4], if f : M -+ N is a homotopy equivalence between closed manifolds and N is a hyperbolic manifold of dimension at least 5, then f is homotopic to a homeomorphism.

2. Thurston's piecewise uniformization of 3-manifolds. As we have seen in Theorem 1.16, there are only eight maximal homogeneous three-dimensional Riemannian geometries, and each of them uniformizes either Seifert fibered manifolds or nontrivial torus bundles over S1 or, in the basic case, hyperbolic manifolds. One can easily find 3-manifolds that carry no complete locally homogeneous Riemannian metrics. For example, any connected sum of closed 3-manifolds but ][8P3#RP3 has

6. Uniformization

294

no locally homogeneous geometry. However, there is a canonical decomposition of 3-manifolds (along embedded 2-spheres and 2-tori) into simple pieces which, due to Thurston's theory, probably are "geometric." We formulate this as the following Thurston's Geometrization Conjecture: Conjecture 6.3. The interior of any compact 3-manifold can be canonically split up into pieces such that each of them carries one of the eight homogeneous geometries in Theorem 1.16.

This conjecture is nothing but the claim that all canonically obtained pieces of a given 3-manifold (after gluing 3-dimensional balls to their boundary 2-spheres) are in fact geometric, that is, they are carrying some (X, G)-structures. To describe the canonical decomposition of 3-manifolds, we start with embeddings of tori into a 3-manifold.

We call a 3-manifold M (homotopically) atoroidal if each map f : T2 -. M of a 7r, (M), 2-torus T2 into M, which induces a monomorphism f*: 7r1(T 2) = Z ED Z is homotopic to a map T2 -- W. In other words, an irreducible 3-manifold M with incompressible boundary a M is atoroidal if any subgroup of 7r1 (M) isomorphic to Z ® Z is conjugate in 7rl (M) to the fundamental group of some component of aM. The torus shell T2 x [0, 1] and the "thickened" Klein bottle (T2 x [0, 1])/762 (see Example 1.4) are simplest examples of atoroidal 3-manifolds. A fundamental role of incompressible, 2-sided, embedded tori in an irreducible 3manifold M became clear after results of Jaco and Shalen [1] and Johannson [1] which provided a homotopy-classification of Seifert fibered manifolds in M. Basic facts here

are given by the Mapping Theorem (Jaco-Shalen [1, Ch. III]) and its corollary, the Splitting Theorem (Jaco-Shalen [1, Ch. V]) which we formulate as follows.

Theorem 6.4. In a Haken 3-manifold M, there are pairwise disjoint 3-dimensional Seifert fibered submanifolds E1, , Z. C M whose boundary components are incompressible in M, and the following holds. Each map f : S M of a Seifert fibered manifold S with non-cyclic fundamental group ri (S) that induces a monomorphism

f,,: nl (S) y nl (M) is homotopic in M to a map whose image of S is contained in some E1.

Moreover, if the inclusions Ei C M are not homotopic to maps Et -- Ej, i, j = 1, ..., m, then the family E1, ... , Em C M is unique up to ambient isotopy. This theorem implies the existence of a canonical system of incompressible tori in a 3-manifold M (Jaco-Shalen [1, V.5.1]):

Corollary 6.5. In a given Haken 3-manifold M, there exists a canonical family {T1, ... , Tm} of pairwise disjoint incompressible Lori (unique up to ambient isotopy) such that:

(i) each component of the complement M\ tJ, Ti is either a Seifert fibered manifold or an atoroidal manifold; and (ii) the family is minimal, that is, no its proper subfamily has the property (i).

6.2. Modern concepts of uniformization

295

Now we are ready to go through a reasonable canonical process of decomposition of a compact 3-manifold M into finitely many simplest blocks which appear to be "geometric".

In the first step, we represent a given M as a finite connected sum of prime man-

ifolds, M = #i Mi, see Theorem 5.5, where prime manifolds Mi are obtained by cutting M along embedded separating 2-spheres, and then adding a 3-ball to each of the obtained boundary spheres. Although the family of separating spheres is not unique up to ambient isotopy, the involved blocks Mi are unique if M is orientable (Milnor [1]). We note that for non-orientable MO, the manifolds MO #(S2 x Sl) and MO #(S2 x S1) are homeomorphic, and hence one cannot uniquely define blocks Mi for non-orientable case. In the second step, we need to consider compact orientable prime 3-manifolds M. Here we have three classes: (i) manifolds M with finite 7rI (M); or (ii) reducible manifolds M with infinite 7r, (M); or (iii) irreducible manifolds M with infinite 7r1(M). Each M with finite 7r1 (M) is either a 3-ball or is closed and hence is covered by a homotopy 3-sphere (Hempel [1]). For such manifolds, Thurston's Geometrization Conjecture can be reformulated as follows:

Conjecture 6.6. Each closed orientable 3-manifold M with finite 7r] (M) possesses the spherical geometry (modeled on S3). Therefore, in addition to Poincard Conjecture (see Chapter 5, § 3), this conjecture also asserts that all free actions of finite groups on S3 are conjugate to orthogonal actions. Algebraically, due to a recent result of R.Lee [1], finite fundamental groups of 3-manifolds are isomorphic to subgroups of the orthogonal group 0(4), but the action of such groups on S3 is unknown. Although some progress on such finite actions has been made, this question is still open, see discussion in Bass-Morgan [1], Myers [1], Scott [6] and a recent promising approach by H. Rubinstein [2]. There is only one 3-manifold in the class (ii), the Hopf manifold S2 X S1 which has (S2 x R)-geometry (Hempel [1]). All other 3-manifolds are in class (iii). These compact orientable irreducible manifolds with infinite 7r1 (M) are aspherical due to the Sphere Theorem 5.6. Such a manifold M may or may not have boundary. If the boundary a M is not empty, it is either compressible or not. Applying the Loop Theorem 5.1 (similarly to Chapter 5, § 2), one can find finitely many embedded 2-disks that split M in blocks with incompressible boundaries. Although the family of splitting disks in M is not unique up to isotopy, the obtained blocks which are not 3-balls are uniquely determined up to isotopy in M. Therefore, the problem is now reduced to the case of compact orientable irreducible 3-manifolds M with infinite Irl (M), which either are closed or have incompressible boundary. To continue our decomposition process, we split M along a finite family of disjoint incompressible 2-sided embedded tori T I ,. .. , T,,, C M, which is provided by

296

6. Uniformization

Corollary 6.5 and is unique up to an isotopy in M. Obtained as a result of this splitting

process pieces are the final blocks of our canonical decomposition of an originally given 3-manifold M. These blocks are conjectured to admit geometric structures and are either Seifert fibered 3-manifolds or have no embedded incompressible tori except possibly those parallel to the boundary. Since all Seifert fibered 3-manifolds admit geometric structures (see Scott [6]), we now only need to consider compact orientable irreducible 3-manifolds M with infinite ]ri (M) all whose embedded incompressible tori are parallel to 3M. Such M may or may not be a Haken manifold. Here we can apply a recently proven important fact that the Seifert Fibered Space Conjecture is true (Gabai [1, 2], Casson-Jungreis [1]):

Theorem 6.7. Let M be a compact orientable irreducible 3-manifold with infinite 71 (M). Then M is a Seifert fibered manifold if and only if 7r1 (M) contains a cyclic normal subgroup.

Scheme of proof. The problem can be reduced to Theorem 2.16 which shows that convergence groups on S1 are isomorphic to Fuchsian groups in the unit open disk B2 C J 2. This reduction was done by G. Mess [1] as follows. If J< ..7rl(M) is a cyclic normal subgroup and M is closed, then one can show that the covering space of

M with the fundamental group J is B2 x S1. Then the action of I' = irl (M)/J (via covering transformations) on B2 X S1 can be projected to a vaguely defined action on B2. It turns out that this action is coarse quasiisometric in the sense of Gromov (see §2.1) either to the Euclidean plane or to the hyperbolic plane. In the former case, I' is isomorphic to the fundamental group of an Euclidean orbifold. In the latter case, r induces a well defined convergence group action on S1 = alEI2 and, due to Theorem 2.16, F is isoinorphic to the fundamental group of a hyperbolic orbifold. Thus rrl (M) is isomorphic to the fundamental group 7r1(N) of a Seifert fibered manifold N which (like M) is closed and aspherical since M and N are both covered by 183. Now, due to homotopy equivalence of M and N, M and N are homeomorphic (Scott [7]).o

Due to a result of Scott [4], if Z ® Z C nl (M) for an orientable irreducible 3manifold M then either M contains an incompressible torus or irl (M) has a cyclic normal subgroup. This fact and Theorem 6.7 imply the following Torus Theorem (for the first proof, see Feustel [1, 2]): Theorem 6.8. Let M be an orientable irreducible 3-manifold and Z ® 7G C 7rl (M). Then M either is Seifert fibered or contains an incompressible torus.

Since, in the case of Haken manifolds, Thurston has proved his Geometrization Conjecture (see the next section), we can apply Theorems 6.6 and 6.7 to reduce the class

of manifolds under consideration to closed non-Haken manifolds M whose infinite 7r1 (M) have no infinite cyclic normal subgroups. For this class, the Geometrization Conjecture 6.3 can be rewritten as follows: Conjecture 6.9. Each closed, orientable, irreducible, atoroidal, non-Haken manifold M with infinite irl (M) which does not contain an infinite cyclic normal subgroup is hyperbolic.

6.2. Modem concepts of uniformization

297

There is a related conjecture (see Hempel [2]) which asserts that if M satisfies these conditions then there is its finite covering Mo which is Haken. If one knew this to be the case, it would imply that Mo admits a hyperbolic structure. Then the manifold M itself is homotopy equivalent to a hyperbolic manifold N, and it (M) acts isometrically and co-compactly on the hyperbolic space ]H[3 due to the following result by Gabai [3]:

Theorem 6.10. A closed irreducible 3-manifold M is homotopy equivalent to a hyperbolic 3-manifold N if and only if M is finitely covered by a hyperbolic 3-manifold Mo. Sketch of proof of necessity. We have that 7r1 (M) is a torsion free finite extension of in (Mo) C Isom IHI3. Then the Mostow rigidity Theorem 7.12 shows that the action

of nl (M) on IH13 is homotopy equivalent to an isometric action, so 7rl (M) - r c Isom 1H[3, and M is homotopic to a hyperbolic 3-manifold N =H3/ I' since M and N both are K(n, 1)-type manifolds.

Now, applying the Gabai rigidity Theorem 6.2, we have that such a homotopy equivalence M --+ N = 1HI3 /7tt (M) is homotopic to a homeomorphism provided some condition on N which holds for all known hyperbolic 3-manifolds. If one does not assume that our non-Haken manifold M is finitely covered by a Haken manifold, then essentially nothing can be said about M in Conjecture 6.9. 3. Four-manifold decomposition and homology three-spheres. The most successful and coherent account of uniformization of 3-manifolds is given by the Thurston's geometrization approach whose main idea is to decompose manifolds into simple pieces and then uniformize them separately. This motivates one to apply this fruitful decomposition idea in dimension four, and then make use of 4-dimensional geometric structures, classified in Chapter 1, § 7, to the "elementary pieces". The first question one needs to answer is about a reasonable canonical decomposition of 4-manifolds M (along a finite collection of disjoint 2-sided hypersurfaces Si C M) into "prime" geometric blocks. Though no 4-dimensional analogue of Jaco-Shalen-Johannson's Splitting Theorem 6.4 is known (except for the case of manifolds of negative curvature, see P. Kropholler [1] and Z. Sela [2]), we may consider possible splittings along cusps of corresponding non-compact geometric blocks. Since locally homogeneous, finite volume noncompact 4-manifolds with negative curvature may only have either real or complex hyperbolic structure (i.e., the universal cover is either the (real) hyperbolic space I[14 or the complex hyperbolic space CIHI2), it is natural to expect that one class of splitting closed 3-manifolds consists of 3-dimensional flat and Ni13-manifolds modeled on the Heisenberg group je3, which appear at cusp ends of such negatively curved 4-manifolds, see Chapter 1, § 6, Chapter 4, § 3, Apanasov [50] and Apanasov-Xie [1]. Similarly, if a noncompact geometric block does not have negative curvature, we have to deal with its cusp ends. In this case we have to concentrate on the following six geometries, see § 1.6: S2 x 1HI2,

1R

X ]HI2, ]HI2 X 1H[2, ThI3 x R, SL 1152

x I[5, 1E4.

6. Uniformization

298

Here, since any lattice in a Lie group must meet the radical in a lattice it follows that the cusp ends are also flat in the mixed Euclidean cases JR2 x H2, H3 x R and SL218 x JR. Similarly, the cusp ends of S2 x 1H[2-manifolds are S2 x 1[8-manifolds. Also, since the elements of PSL2((C) corresponding to cusps of finite area hyperbolic surfaces are parabolic, the cusp ends of F4-manifolds are Nil3-manifolds, and the cusp ends of H2 x 112-manifolds are So13-manifolds, see Shimizu [1]. Considering the above cusp structures of two adjacent geometric blocks, Hillman [2] determined the possible combinations of geometries that appear in the geometric blocks of a 4-manifold with proper geometric decomposition (along above cusp hypersurfaces). This result implies that the class of 4-manifolds with non-trivial geometric (cusp) decomposition is quite limited (contrary to the 3-dimensional case):

Theorem 6.11. If a closed 4-manifold M admits a geometric (cusp) decomposition then either (1) M is geometric itself,, or (2) M has a 2 -fold cover which is an (1112 x S2)-manifold; or (3) the geometric blocks of M have geometry 1HI4, H3 x R, R2 x 1H12; or (4) the geometric blocks of M have geometry C1H[2 or 1F4; or (5) the geometric blocks of M all have geometry 1112 x H2.

If M is in one of the last three classes then it is aspherical and X (M) > 0. Now we shall describe another natural splitting of 4-manifolds along integer homology 3-spheres which generally are not homeomorphic to the 3-sphere S3. We shall also see clearly another significant difference of the four-dimensional case related to

the existence of different smooth structures on 4-manifolds, see Freedman [1] and Donaldson [1].

The main object of our interest here is going to be a closed oriented 4-manifold X. This manifold will be either a topological manifold or a smooth one ( with C°Ogluings). To avoid the group theoretic problems arising from the fact that any finitely presented group can occur as the fundamental group of a (smooth) closed 4-manifold, we assume our manifolds are simply connected. Most of the classical invariants for 4-manifolds are encoded by the intersection form Qx. This form is an integral unimodular symmetric bilinear pairing given by the cup product: Qx: H2(X; 7L) ®H2(X; 7G) -+ Z. By picking an integral basis in H2 (X, Z), we identify the intersection form Qx with its matrix. The unimodularity of the form means exactly that this matrix has determinant

±1. In smooth case, the cup product on de Rham cohomology can be described as (a, fi) H fx a A P, where a and P are closed differential 2-forms representing de Rham cohomology classes. From the intersection form, one can determine its rank (which is the second Betti

number b2 = rank H2(X; Z) = b+ + b-), its signature = o (X) = b+ - b- (where b: are the dimensions of the ±-eigenspaces of Qx over 18), and its type which is even

6.2. Modern concepts of uniformization

299

if Qx (x, x) = 0 modulo 2 for all x, and odd otherwise. We call a manifold X definite, indefinite, even, odd, etc if so is its intersection form QX. The operation of direct sum of forms is parallel to that of connected sum of manifolds X#Y: the intersection form of the manifold X#Y (which is smooth if so are X and Y) is Qx#Y = Qx ® Qy.

Standard Examples. The standard examples of (smooth) 4-manifolds and their intersection forms are: S4

Q=

CP2

Q = (1)

(CP 2

Q = (-1) p(l) ® q(-l))

#PCP2#gCP2

Q=(01

S2xS2

O)

=H

Q=2E8®3H

K3

Here ?P2 is the complex projective space CP2 with the orientation opposite to the canonical one, K3 is the Kummer surface in (3.68), and E8 is the following unimodular (8 x 8)-matrix:

-2 1

E8 =

1

0

0

0

-2

1

0

0

1

0

0 0 0 0 0

0 0 0 0

1

0 0

1

-2

1

0 0 0 0

0

1

-2

1

0

0 0 0

1

-2

0

1

-2

1

0

0

-2

0 0

0 0

0 0

0

0 0

0 0 1

1

0 0

-2

0

The intersection form is a fundamental homotopy invariant of 4-manifolds due to Whitehead [3]:

Theorem 6.12. The homotopy type of an oriented simply connected smooth 4-manifold is completely determined by its intersection form. Moreover, the h-cobordism classes of closed simply connected smooth 4-manifolds

are in fact in one-to-one correspondence with their homotopy types because of the following result of S. Novikov [1] and C. T. C. Wall [1]:

Theorem 6.13. Closed simply-connected smooth manifolds of dimension four are h-cobordant if and only if their intersection forms are isomorphic. The fundamental work of M. Freedman [1] establishes the intersection form as a fundamental topological invariant:

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6. Uniformization

Theorem 6.14. Any integral unimodular form can be realized by a simply connected topological 4-manifold. Moreover, the form, together with one extra piece of data (the Kirby-Siebenmann obstruction in Z2, which is always zero for smooth manifolds or for even forms), determines the manifold up to a homeomorphism. In particular, using this result of Freedman, one can show that any closed simplyconnected smooth manifold with an indefinite odd intersection form is homeomorphic to a connected sum, #b+CP2#b-(CP2 ,

of b+ copies of (CP2 and b--copies of (CP2. The situation is more complicated for even and definite forms. It has been known for 40 years that some forms cannot be realized by a smooth simply connected manifolds. A classical theorem of V. Rokhlin [1] asserts:

Theorem 6.15. If the intersection form of a smooth closed 4-manifold is even then its signature is divisible by 16. For instance, there is no smooth manifold whose intersection form would be E8 just because E8 is even and sign E8 = -8 is not divisible by 16. However, by Freedman's Theorem 6.14, there exists a topological manifold with such intersection form. So, the most intriguing questions in smooth 4-topology which remained unanswered until 1980's were:

I. Determine the unimodular, symmetric, bilinear integral forms that are realizable as the intersection forms of closed simply-connected smooth four dimensional manifold. H. Given a closed simply connected smooth 4-manifold M, determine all possible distinct smooth structures on M. This is the point where S. Donaldson's work [1] started in 1983. He proved that, for a smooth definite manifold X, its intersection form QX is diagonalizable over integers, which eliminates "non-standard" definite forms. For instance, the form E8 ® E8 cannot

occur as the intersection form of a smooth closed 4-manifold, even though is not prohibited by the Rokhlin Theorem 6.15. In Donaldson [2], some restrictions on even indefinite forms to be realizable by smooth 4-manifolds were obtained. In particular, it was shown that neither 2E8 ® H nor 2E8 ® 2H can be the intersection form of a smooth closed manifold. On the other hand, we saw above that QK3 = 2E8 ® 3H, and hence neither of summands E8 or H can be split off by means of connected sum. Further progress towards the problem (I.) was recently made by using invariants of N. Seiberg and E. Witten, see Witten [1], Kronheimer-Mrowka [2], Taubes [1, 2].

Thus, there exist many topological 4-manifolds that are not smoothable. On the other hand, if a topological 4-manifold admits a smooth structure, this structure need not be unique. Moreover, the uniqueness fails in the worst possible way: there are manifolds which admit infinitely many smooth structures, see e.g. Friedman-Morgan [1].

6.2. Modern concepts of uniformization

301

Example 6.16 (Splitting of blown-up Kummer surface). Let X be a blow-up of the Kummer surface K3 in (3.68), that is X = K3#(CP2. The intersection form Qx of

XisQx=2Es®3H®(-1),so rank Qx=23andsign Q=-17 with b+=3 and b- = 20. Since the form Qx is odd and indefinite, it is similar over integers to the form 3 (1) ® 20 (-1) . Therefore, due to Theorem 6.14, X is homeomorphic to the connected sum #3CP2#20(CP2. On the other hand, these two smooth manifolds are not diffeomorphic. This can be seen by using the Donaldson polynomials, an invariant

of smooth structures introduced in Donaldson [3]. For both X and these polynomials are explicitly computed (see e.g. Kronheimer-Mrowka [1]) and are correspondingly equal to the homogeneous components of the following Donaldson

#3CP2#20U2,

series:

Dx(x) = eQx(x,x)12 . cosh Qx(E, x), and D#3cP2#20CP2(x) = 0,

where E E H2(X; Z) is dual to the generator of H2((CP2; 7G). As we have seen above, if a 4-manifold splits as a connected sum, its intersection form is a direct sum of the intersection forms of the summands. What is known about the converse? Obviously, it is true in the category of topological manifolds and is a direct consequence of Theorem 6.14. Now let us assume that M is a simply connected smooth closed 4-manifold whose intersection form QM decomposes as QM = Qx ® Qy, where X and Y are smooth 4-manifolds (with boundary). Notice that any integral unimodular bilinear form can be realized as the intersection form of a compact smooth 4-manifold with boundary a homology sphere, see W. Browder [1]. M. Freedman and L. Taylor [1] proved in this case that M can be represented as a connected sum M = X#E Y, where the gluing

takes place either along an ordinary 3-sphere E N S3 ( and then it is an ordinary connected sum, X#Y ), or along a homology 3-sphere E with

H*(E;7L) = H*(S3;Z),

(6.1)

where, generally speaking, rrl (E) # 1. Also one should mention that though X and Y are no longer closed manifolds, the intersection forms Qx and Q y are well defined on the second cohomology and are unimodular due to the cohomology condition imposed on E (because of (6.1) and the Poincar6 duality).

Example 6.17 (Splitting of K3-surface). The Seifert fibered homology sphere E (2, 3, 7) (see an account of homology spheres below) embeds in the Kummer surface (3.68), splitting it into two submanifolds with the intersection forms E8 ® 2H and E8 ® H, correspondingly. This embedding is described in Looijenga [1] and Fintushel-Stern [1]. One can keep decomposing this manifold as in Fintushel-Stern [2], and finally split it into five pieces:

K3 = X1 UE Yl U_E/ Y2 UE' Y3 U_E X2.

(6.2)

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302

Here the intersection forms of X1 and X2 equal Es, and those ones of YT's, i = 1, 2, 3,

equal the form H in the above Standard Examples. The homology spheres E and 1 in this decomposition (6.2) are correspondingly the Poincare homology sphere E(2, 3, 5) = S3/ r(2, 3, 5), F(2, 3, 5) C 0(4), and the Seifert fibered homology sphere E (2, 3, 7) (see Example 6.22). Also the minus sign in (6.2) means the change of orientation. However it seems that the obtained five blocks in the above splitting of K3-surface cannot support any of homogeneous 4-geometries classified by Filipkiewicz [1] and (from the point of view of Kahler structures) C. T. C. Wall [2, 3], see Chapter 1, § 7. In particular, it is known that one of their boundary components, the Seifert fibered homology sphere E (2, 3, 7) has an SL2 R-structure and can be represented as the quotient of S3\Si by "conformal" action of a discrete subgroup of PU(1, 1) C

PU(l, 2) on the round sphere S3(0, 1) C C2 (in the conformal structure induced there by the complex hyperbolic geometry in B2 (0, 1)). This means that E (2, 3, 7) has the so-called spherical Cauchy-Riemannian structure whose natural extension in dimension four is the complex hyperbolic geometry in the complex ball B2 (with the Bergman metric, see Apanasov [50], Apanasov-Xie [1], Goldman [5]). Nevertheless, neither of YY blocks in (6.2) (with the intersection form H) can support a complex hyperbolic structure because each of them has two compact boundary components. In fact, in a sharp contrast to the real hyperbolic case, the boundary of any compact complex hyperbolic manifold M(G) is connected, which follows from the KohnRossi's [1] analytic extension theorem, see Epstein-Melrose-Mendoza [1]. One more example of a 4-manifold decomposition is the following splitting of elliptic surfaces.

Example 6.18 (Splitting of elliptic surfaces). Let V be a minimal complex elliptic surface fibered over the 2-sphere, with the Euler characteristic being nonzero. Due to R. Gompf [1], this elliptic surface V can be split into two pieces along an embedded Seifert fibered homology sphere E(2, 3, 6n - 1), where the number n is uniquely determined by V. The "small" piece of this decomposition, the nucleus, has the Euler characteristic 3 and encodes the main information about V. Most of the ambient manifold V lies in the complement of the nucleus, and is diffeomorphic to the Milnor fiber of E (2, 3, 6n - 1). That is, this submanifold is diffeomorphic to the locus of { (x, y, z) E

C3 : x2 + y3 +

z6n-1

= s } C BC

in the closed 6-ball in C3 centered at the origin. The importance of splittings along homology spheres is further emphasized by the following result of R. Matveyev [1] (see also C. Curtis and W. Hsiang [1]): Theorem 6.19. Let U be a smooth 5-dimensional simply-connected h-cobordism with boundary a U = M1 U M2. Let f : M1 -* M2 be the homotopy equivalence induced

6.2. Modem concepts of uniformization

303

by U. Then: (1) There are decompositions

Mi = M#EW1,

M2 = M#E W2

such that

i2* o it*1 = f*: H2(Ml)

H2(M2)

where il* and i2* are the maps induced in the second homology by embeddings of M into M1 and M2 respectively, W1 and W2 are smooth compact

contractible 4-manifolds, and E = a W1 = a W2 = 8M is a homology sphere. (2) These decompositions may be chosen so that W1 is diffeomorphic to W2.

As we have seen above, the homology 3-spheres are of central importance in studying smooth 4-manifolds, so they deserve a closer look. We give account of the subject following D. Eisenbud and W. Neumann [1]. From now on, by a homology sphere we mean a closed oriented 3-manifold E with H* (F,; 7L) = H* (S3; Z). Sometimes,

it will be convenient for us to deal with links in homology spheres rather then the homology spheres themselves. By a link L = (E, K) = (E, S1 U U. . . U Sm) we mean a pair consisting of a homology sphere E and a disjoint union of embedded circles S1, . . . , Sm c E. Empty links are also allowed; they are just homology spheres. If L = (E, K) = (E, Si U . . . U Sm) is a link, we denote a closed tubular neighU N(Sm). We also denote by ml, fi borhood of K in E by N(K) = N(Si) U the standard meridian and longitude of the link component Si. These are the pair of oriented simple closed curves in a N (Si) which are determined (up to isotopy) by the following homology and linking conditions:

mi - 0,

.£i

L(mi, Si)

Si

1,

in H1(N(Si)), L(ti, Si) - 0.

Here f(*, *) denotes the linking number in E. Let L = (E, K) and L' = (E', K') be links and, for chosen connected components S C K and S' C K', let N(S) and N(S') be tubular neighborhoods with the standard meridians and longitudes m, 2 C 8N(S) and m', f' C 8N(S'). Then one can form a new manifold,

E" = (E \ int N (S)) U (E' \ int N (S'))

,

by pasting along the boundaries, with matching m to t' and m' to f. It is easily seen that E" is a homology sphere. We call the link (E", (K - S) U (K' - S')) the splice of L and L' along S and S'. A link L = (E, K) whose exterior E\ int N(K) admits a Seifert fibration is called a Seifert link. A simple link is defined to be a link L = (E, K) with the property that the complement E \ N(K) is irreducible, and any incompressible torus in E\N(K)

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304

is boundary parallel, that is E \ N(K) is atoroidal. By Thurston's hyperbolization Theorem 6.29, this is equivalent to saying that the link complement E\K admits a complete hyperbolic structure of finite volume, except possibly the situation when K = 0 and E is not sufficiently large (see Conjecture 6.9). The splitting Theorem 6.4 implies that any irreducible homology sphere E can be expressed as the result of splicing together a collection of Seifert links and simple links,

and moreover, the minimal way of doing this is essentially unique. In order to give a precise formulation, we first need to describe a "desplicing" operation, following Eisenbud-Neumann [1]:

Theorem 6.20. Let L" = (E", K") be a link and T2 C E"\K" an embedded torus. Then the linkL" is the resultofa splicing operation of links (E, K) and (E', K') along

components S C K and S' C K' such that T2 = 8N(S) = 8N(S'). Furthermore, this splicing is uniquely determined up to reversing the orientations on both S and S.

In the light of the defined desplicing, one can interpret the splitting Theorem 6.4 as follows.

Theorem 6.21. Let E be an irreducible homology 3-sphere. Then there exists a minimal collection {Ti } of disjoint incompressible non-boundary parallel embedded tori in E such that if one desplices E along all these tori, as in Theorem 6.20, each resulting link is either a Seifert link or a simple link. Moreover, such a collection of embedded tori is unique up to an isotopy of E. The next examples describe three main classes of homology 3-spheres.

Example 6.22 (Seifert homology spheres). An empty Seifert link is just a Seifert fibered homology sphere. A complete classification of such homology spheres is given in Neumann-Raymond [1]. It is shown there that, for any pairwise coprime integers al, ... , an, each ai > 2, there exists a unique Seifert fibered 3-manifold whose (unnormalized) Seifert invariants are {0; (al, bl), ... , (an, bn)} and satisfy the equation b`

i=1

ai

=1.

(6.3)

The obtained manifold is denoted by E(al, ...,an). This is a homology sphere, and the equation (6.3) provides it with a canonical orientation. For example, Seifert fibered homology sphere E (2, 3, 5) is just the Poincar6 homology sphere. E (2, 3, 5) is the only one known homology sphere with finite fundamental group 7ri (E), and one can introduce a spherical geometry on it by representing the fundamental group 7rl (E) as an orthogonal group r(2, 3, 5) C 0(4) acting free on S3; then E(2, 3, 5) = S3/ 1(2, 3, 5) can be obtained by identifying the opposite sides of a spherical dodecahedron whose dihedral angles are 27r/3 (see e.g. Krushkal-Apanasov-Gusevskii [3]). We saw other examples of Seifert homology 3-spheres while decomposing the K3surface in Example 6.17 and an elliptic surface in Example 6.18. Due to J. Milnor [2]

6.2. Modem concepts of uniformization

305

(see also Raymond and Vasquez [1]), all of these Seifert homology spheres (with infinite fundamental groups) have SL2-structures whose holonomies can be also realized by representations in PU(1, 2). In particular, the Seifert homology sphere E (2, 3, 7) is diffeomorphic to the quotient S3 / 1' (2, 3, 7) where r (2, 3, 7) acts free in S3 = 8 Bc'

as a discrete subgroup of PU(l, 1) C PU(l, 2) whose action in B.2fl (C x 0) is the action of the (2, 3, 7)-triangle subgroup of SL2 R = PU(1, 1), see Figure 10. Example 6.23 (Graph homology spheres). A homology sphere whose pieces in the splice decomposition are all Seifert fibered, is called a graph homology sphere. Such spheres were first introduced by Waldhausen [3] and then completely classified by Siebenmann [2] and Eisenbud-Neumann [1]. Some of these graph homology spheres have CR-structures induced by free discrete PU(l, 2)-actions of their fundamental groups on S3 = 8B2, see Apanasov-Xie [1].

Example 6.24 (Hyperbolic homology spheres). Numerous examples of closed hyperbolic homology 3-spheres, that is empty simple links, can be produced as follows. Let (S3, K) be a one-component simple link. Then, due to the hyperbolization Theorem 6.29, the knot complement S3\K has a complete hyperbolic structure. On the other hand, for any knot K C S3, each closed 3-manifold E = S3 ± n K obtained from S3\K by Dehn (1, ±p)-surgery along K is a homology 3-sphere. Hence, according to Theorem 6.42, for all but finitely many numbers p E 7G, the 3-manifolds

E = S3 ± p K obtained from S3\K by Dehn (1, ±p)-surgery along K are hyperbolic homology spheres. At the same time, hyperbolic knots form the main subclass of knots in S3. For example, any twist knot in Figure 84 with either m full twists, m > 1, or (2m - 1)-half twists, m > 2, is hyperbolic, see Thurston [1] and Example 6.43.

Figure 84

We refer the reader to N. Saveliev [1-11] for more details on homology 3-spheres and the Saveliev's invariant based on the Floer instanton homology groups. Due to a recent Saveliev's [9] solution of an Atiyah's problem, these Floer instanton homology groups of Brieskorn homology spheres E (p, q, r) can be expressed in terms of their branching sets over the 3-sphere, or more exactly, in terms of the Casson's ?.-invariant

of E(p, q, r) and the signature of the Montesinos knot k(p, k, r) C S3, see also Saveliev [11]. It is further shown by Saveliev [10] how Floer homology is related

306

6. Uniformization

to other invariants of knots and 3-manifolds, among which is the well-known 11invariant defined for graph homology spheres (from absolutely different points of view) by W. Neumann [1] and L. Siebenmann [2], and the Jones polynomial. Also an essential progress is made in proving the homology cobordism invariance of the Saveliev's v-invariant, see Saveliev [10]. 4. Conformal uniformization. As the classical Klein-Poincare theorem shows, studying conformal structures on a 2-manifold M is equivalent to studying its geometric structures which are either hyperbolic, or spherical, or Euclidean. Here and in the case of Koebe uniformization of such conformal structures on M, we emphasize the special role played by the group of Mobius transformations whose subgroups also represent the isometry groups of these geometries. In several respects, it comes close to the more elementary idea of the development used in differential geometry for locally homogeneous structures (see Chapter 1, § 5), although conformal geometry is not Riemannian. Here by a conformal (= conformally flat) n-manifold/orbifold M we mean an n-dimensional manifold/orbifold modeled on (S", Mob(n))-geometry, see Chapter 1, § 2. Fixing a conformal structure c on M, i.e. a pair (M, c), means choosing a maximal conformal (in fact, Mobius) atlas on M. For n = 2, this coincides with the notion of projective structures on surfaces because (S2, Mob+(2)) = (CP 1, PSL(2, C)). For n > 3, the notion of conformal structures coincides with the notion of conformally flat structures because of the Liouville Theorem 1.6. A homeomorphism f : (Ml, cl) -+ (M2, c2) of two conformal orbifolds is called

conformal if the conformal structure f*c2 on Ma obtained by lifting the structure c2 coincides with the original structure cl; in other words, a conformal map is a morphism in the conformal category, that is a (S", Mob(n))-map. Using the standard notions of the development map d: M -* S" and the holonomy homomorphism dk:7rirb(M) -+ Mob (n) with the holonomy group G = d*(irl(M) (see Chapter 2, § 2), we can define uniformizable conformal structures on M. Namely, a conformal orbifold (M, c) is called uniformizable, if its holonomy group G C Mob(n) is a Kleinian coup with a component n of the discontinuity set 0 (G) C S" such that S2 = d(M), and (M, c) is conformally equivalent to 0/G with the natural conformal structure induced by the natural projection 0 - S2/G. Conformal structures are more flexible and universal than locally homogeneous structures which often are rigid in dimension n > 3, see Chapter 7. Moreover, conformal approach unifies different geometric structures (which allow conformal models). For example, spaces of constant sectional curvature (i.e., hyperbolic, spherical and Euclidean) have natural conformal structures induced by their geometries. In the next sections, we will describe many other classes of conformally uniformizable manifolds. Utilizing the Thurston idea on splitting a manifold into uniformizable simple pieces, we have formulated (see Apanasov [30]) the following conjecture toward the conformal uniformization:

Conjecture 6.25. Let M be a compact n-manifold which can be "canonically" split into finite volume geometric pieces all of which differ from the product of a flat (n -1)-

6.2. Modern concepts of uniformization

307

manifold and the interval and whose interiors possess conformal structures. Then the interior of M (or its finite covering) possesses a conformal structure. Furthermore, if the conformal structures of the geometric pieces are uniformizable then the common conformal structure is uniformizable as well.

In dimension three, due to the known canonical decomposition and the progress in solution of Thurston Geometrization Conjecture 6.3, one can reformulate this conjecture in the following stronger form:

Conjecture 6.26. Let M be a compact 3-manifold whose canonical splitting along 2-spheres and 2-tori does not include Sol- and Nil-manifolds. Then the interior of M (or its finite covering) possesses a uniformizable conformal structure. In the next sections, we will provide a strong evidence for this claim. Another approach to conformal uniformization of 3-manifolds can be made by using the socalled conformal cone structure.

To define such cone structures, we fix an angle a, 0 < a < 27r, and define a Euclidean lens of angle a as the intersection of two closed round balls at the dihedral

angle a if a < jr, as a closed round ball together with a circle in the boundary if a = n, and as the complement in S3 = R3 U {oo} of the interior of a Euclidean lens of angle 27r - a if a > -r. Then an a-cone 3-sphere S3 is said to be the quotient of a Euclidean lens of angle a by a rotation in S3 about the edge of the lens that identifies the two boundary half-spheres of the lens. Now we define a conformal cone structure with cone angle a on a 3-manifold M as a singular conformal structure on M such that each point in M has a neighborhood conformally homeomorphic to

an open set in a-cone 3-sphere S. In other words, a conformal a-cone structure is the structure of a conformal 3-orbifold whose all local groups are generated by rotations by angle a, see Chapter 1, § 8. As we have already seen in Chapter 5, § 3, the topological uniformization of all closed oriented 3-manifolds (Theorem 5.20) actually introduces structures of hyperbolic orbifolds on such 3-manifolds, with local groups generated by 90°-rotations. Since the hyperbolic geometry is conformally realizable, we immediately have:

Corollary 6.27. Any closed orientable 3-manifold M has a conformal 2 -cone struc-

ture, and its holonomy group is a Fuchsian group G - arorb(M), G C Mob(3), co-compactly acting in a round ball B C S3.

Due to a recent result of F. Luo [3], we can vary the cone angle a, 0 < a < 2n, of singularities of such a-cone conformal structures on M:

Theorem 6.28. Given any positive a < 27r, any closed orientable 3-manifold M has a conformal a-cone structure. Furthermore, if the given cone angle a is -for some natural number n, then the holonomy group of this a-cone structure is a discrete subgroup G,,, C Mob(3).

The singular set of such conformal orbifold structures (M, ca) on a given 3manifold M forms a link in M whose number of components may become arbitrarily

308

6. Uniformization

large as a tends to 2n. This makes it difficult to use this approach in order to find a non-singular conformal structure on M.

§3. Hyperbolization of manifolds and hyperbolic volumes In this section, we give a topological description of complete hyperbolic manifolds of finite volume and some their invariants. Geometrically, these manifolds are complete connected Riemannian manifolds with constant negative sectional curvature and with finite volume, or equivalently, quotients IIF/G of the hyperbolic space IH[" by free actions of lattices G C Isom IHIV. Most manifolds in dimension at most three are hyperbolic, which is due to the Poincare-Klein uniformization theorem (§1) for 2manifolds and the Thurston Geometrization Theory in dimension three.

1. Topology of hyperbolic 3-manifolds. As we have seen in § 1, Chapter 5, any hyperbolic 3-manifold M is irreducible, aspherical (r (M) = 0 for i > 2) and is a Poincare manifold, i.e. each embedded homotopy ball B C M is homeomorphic to a Euclidean ball. Furthermore, if nl (M) - G C Isom H3 is a non-elementary group then its center is trivial, and the fundamental group.nl (S) of any boundary component S c 8M is non-Abelian. If, in addition, the manifold M is compact, its fundamental group nl (M) has no free Abelian subgroup Z ® Z of rank two.

In addition to these necessary conditions on hyperbolic manifolds M, we note that most hyperbolic 3-manifolds are sufficiently large and thus are Haken manifolds. However, there are also "small" hyperbolic manifolds among them, for example, the

manifold obtained by an identification of the opposite sides of the regular hyperbolic dodecahedron with dihedral angles 2n/5, see Seifert-Weber [1] or KrushkalApanasov-Gusevskii [3]. A great achievement of W. Thurston [1-6] is his discovering that these necessary topological conditions on a 3-manifold M are somewhat sufficient for the existence a complete hyperbolic metric on M. More precisely this fact is formulated as the following Thurston hyperbolization theorem: Theorem 6.29. A closed Haken 3-manifold M admits a complete hyperbolic structure if and only if M is atoroidal.

This Thurston hyperbolization revolutionized low-dimensional topology since its discovery in 1978. However, although parts of the proof appeared over the next decade

(cf. Morgan [1]) a complete proof had not been published until two papers by Curt McMullen [1, 2] filled the last gap, see also McMullen [3, 4, 6]. Remark 6.30. The basic idea of Thurston's proof that an atoroidal Haken 3-manifold is hyperbolic starts with the fact that such a manifold can be inductively built up from finitely many 3-balls by gluing together incompressible surfaces in the boundary. So one starts with a finite collection of disjoint 3-balls which can certainly be given a hyperbolic metric. At the inductive step, one has a hyperbolic manifold and a gluing

6.3. Hyperbolization of manifolds and hyperbolic volumes

309

map on an incompressible surface in the boundary. The goal is to find a hyperbolic structure on the 3-manifold that respects the gluing map whenever the quotient is atoroidal. The set of possible hyperbolic structures on the 3-manifold is in one-toone correspondence with the set of conformal structures on the boundary. By using quasiconformal deformations, this gluing problem can be reformulated as a fixedpoint problem for a map of the Teichmtiller space of the boundary to itself. Here the McMullen results give a proof of the key fact that the proposed Thurston's iteration of the map converges. A reader can find complete details of this in recent works of J.-P. Otal [5, 6], see also Otal-Paulin [1]. Thurston's Hyperbolization Theorem 6.29, together with Jaco-Shalen-Johannson decomposition results (Theorem 6.4 and Corollary 6.5), give a strong evidence toward the Structural Conjecture 6.3 (Thurston [2]):

Theorem 6.31. The Structural Conjecture 6.3 is true for all Haken 3-manifolds and for all compact irreducible 3-manifolds with non-empty boundary. A nice class of examples of 3-manifolds (in Theorem 6.31), for which the Thurston

Conjecture 6.3 is true, is formed by knot complements in S3. For any such a knot K C S3, that is an embedding Sl c S3, its complement S3\K is homeomorphic to the interior of a compact 3-manifold M whose boundary is a torus. Among knots K C S3, one can distinguish torus knots, that is, knots which can be placed on an ordinary torus T2 C S3. As an example, Figure 85 shows a torus knot K C 1[83 of type (2,5).

Figure 85

With any non-trivial knot K C S3, one can associate a family of other knots, called satellites of K. These K-satellite knots are obtained by non-trivial embeddings

S' y N(K) of a circle in a small solid torus neighborhood N(K) of the knot K. Here, we call such an embedding non-trivial if it is not isotopic to the knot K itself S3 is called and is not contained in a ball inside the solid torus N(K). A knot St a satellite knot if it is a satellite of a non-trivial knot K C S3. Since the complement S3\K of a knot K is always a prime manifold, and its torus decomposition (along tori in Corollary 6.5) is nontrivial if and only if K is a satellite knot, we immediately have the following corollary of Theorem 6.29 (see Thurston

6. Uniformization

310

[2]), which was first conjectured by R. Riley [1] on the base of his (historically the first) constructions of hyperbolic structures on knot and link complements: Theorem 6.32. A given knot complement S3 \K has a geometric structure if and only if the knot K is not a satellite knot. Furthermore, this complement has a hyperbolic structure if and only if the knot K is not a torus knot.

2. Hyperbolic manifolds of finite volume. Hyperbolic 3-manifolds in Theorem 6.32, in particular the figure-eight knot complement in Example 1.11, provide first examples of non-compact hyperbolic manifolds of finite volume. Due to the Mostow rigidity Theorem 7.12, such hyperbolic manifolds M, vol M < oo, are completely determined by their fundamental groups zr1(M). Moreover, since these manifolds are geometrically finite due to Theorem 4.16 (GF2), one can describe them as follows:

Theorem 6.33. A complete hyperbolic n-manifold M of finite volume is the union of a compact n-submanifold whose boundary (if any) consists of finitely many closed flat (n - 1)-manifolds Ei, and finitely many (if any) submanifolds each of which is homeomorphic to the product of a closed flat (n - 1)-manifold Ei and the interval [0, oo).

Due to the Gauss-Bonnet Theorem, volumes of hyperbolic even-dimensional manifolds Mn only take a discrete set of volumes, vol(M") = Cn . x (Mn), where X (Mn) is the Euler characteristic and Cn is a constant depending on the dimension. In the case of 2-manifolds, it follows that the area of a hyperbolic surface S is equal to -7rX (S). Moreover, in dimension n > 4, one has even stronger rigidity due to the following theorem by C. H. Wang [1] (compare super-rigidity Theorem 4.32):

Theorem 6.34. For a given dimension n > 4 and a constant C > 0, there are at most finitely many pairwise non-homeomorphic (= non-isometric) complete hyperbolic n-manifolds M with vol(M) > C. In particular, in dimension four due to recent results by Ivanic [1, 2] and RatcliffeTschantz [1], the set of hyperbolic volumes {vol M4} is the set of the positive integral multiples of 4.rr2/3. Moreover, at least "half" of those numbers (even multiplies) can be realized by presenting a specifically constructed series of finite volume hyperbolic 4-manifolds, see Theorem 6.51. The case of dimension three is exceptional due to the Thurston non-rigidity Theorem 4.31; here the set of hyperbolic volumes is not discrete anymore. However, it is still possible to describe those hyperbolic 3-manifolds whose volumes are bounded by a constant. To do that, we define the space 3e of all hyperbolic 3-manifolds with the topology of geometric convergence defined as follows.

We call two manifolds M, N E .3e a-close to each other, s > 0, if there exists a diffeomorphism thickeM -+ thickeN of s-thick submanifolds which is s-close to an isometry. More formally, we say that manifolds Mi C 3e converge to M E 3?

if, for some xi E Mi, X E M, and for any numbers 8 > 0 and R > 0, there exist io = io(8, R) E N and mappings fi,

fi:Mi(R)={yEM1:di(xi,y)<_R}-*M,

fi(xi)=x, i>io,

(6.4)

6.3. Hyperbolization of manifolds and hyperbolic volumes

311

such that (i) fi (Mi (R)) D M(R - S) _ {y E M : d(x, y) < R - S} (ii) (1 - S)di (Y, z) < d(fi (Y), fi (z)) _< (1 + S)di (Y, z) for ally, z E Mi (R).

It is clear that, with respect to this topology in the space 3C, the volume function, vol: 3f -+ I[8+, is continuous. Moreover, the following Jorgensen description of subspaces Sec = {M E 3C : vol M < C) applies, see Thurston [1, Ch. 5]:

Theorem 6.35. The continuous map vol: 3e - R+ is proper, that is each sequence {Mi } C Sec, C > 0, has a convergent subsequence. Furthermore, for a given C > 0, there is a finite set {Ml, ... , Mk} C Sec such that each manifold M E Mc can be obtained from one of Mi, 1 < i < k, by a Dehn surgery. Corollary 6.36. Given C > 0, there is a (hyperbolic) link Lc C S3 such that each 3-manifold M C Sec can be obtained by a Dehn surgery on the link complement S3\Lc. It is natural to try to establish an analogue of the above situation for higher dimensions: when may we think of a noncompact hyperbolic (n + 1)-manifold M of finite volume as a complement of a closed codimension-k submanifold A inside a closed (n + 1)- manifold N? If indeed we may do so, i.e. M = N\A, we will say that M is a codirnension-k complement. Here we shall present interesting answers to that given by a recent work of D. Ivan9i6 [1, 2]. We start with a clear observation that whether M is a codimension-k complement depends only on the ends of M, that is, whether they look like a regular neighborhood of some manifold. So let us assume that M is embedded in a closed (n + 1)-manifold

N so that M = N\A, where A is a codimension-k closed submanifold of N that has as many components as there are ends of M. We assume that A has a tubular neighborhood in N, that is, A has a neighborhood in N that is a k-disk-bundle over A. Let B1, ... , B1z be the components of A and Ei x [0, oo) the end of M that corresponds to Bi. Due to Theorem 4.16, each Ei here is a compact flat manifold, and the compact part of M is M = M\(El x (0, oo) U . . . U Em x [0, oo)) whose boundary is E1 U U Em. The interior of k is homeomorphic to M, so we will abuse language and say that each Ei bounds M. Note that the assumtion that A have as many components as M has ends allows us to avoid a situation when k = 1, m = 2 and M is bounded by two homeomorphic flat manifolds E -- El = E2, while a closed N in which M will be a complement of E is obtained by gluing El and E2 together. Now we can use the following Farrell-Hsiang's [4] topological characterization of flat manifolds (for n = 4, see Quinn [1]; and for n = 3, apply the Waldhausen's theory of sufficiently large manifolds combined with a theorem of Heil [1]): Theorem 6.37. Let F" be a closed aspherical manifold such that r1 Fn is isomorphic to ,rl En, where E" is a closed connected flat manifold. If n = 3 assume, in addition, that Fn is irreducible. Then F" is homeomorphic to En.

6. Uniformization

312

Its application shows that if M is a codimension-k complement, then every bound-

ary component of M must be an Sk-1-bundle over some manifold, see Ivansic [3, Pr. 1.1]:

Lemma 6.38. Let B be any component of A, which is irreducible when n = 3, and E x [0, oo) be the end of M corresponding to B. Then E is an S1 -bundle over B. However, flat manifolds can be S1'-bundles only for k = 1 or k = 2, see Ivan"sic [3, Pr. 1.2]:

Lemma 6.39. Let E be a flat manifold that is a fiber bundle over some manifold with fiber S1, the 1-dimensional sphere. Then

(i) The number l must be 0 or 1. (ii) When l = 1, the map i,, : .7r1 S1 -- n1 E induced by inclusion of a selected fiber, is S1 --+ E, is injective and the image i*(ir1S1) is a normal subgroup

of riE. Conversely, let every boundary component E of M be an S°- or S1-bundle over some n- or (n - 1)-manifold A, respectively. Then it is easy to construct a closed (n + 1)-manifold N so that M = N\A. Considering one of the ends E x [0, oo) C M and the component B of A corresponding to that end, let p: E -+ B be the bundle projection. Then construct the disc bundle P = E x [0, 1] U B/((x, 1) ^- p(x)): this willbe a compact (n+ 1)-manifold with 8 P = E and we will have E x [0, oo) = P \ B. By gluing P, in any chosen way, to M along E we will have `filled in' one end of M. We can do the same for every end of M. Thus, the deciding factor that makes M a complement is whether each of boundary components of M is an SO- or an S1-bundle. Therefore, detecting when flat manifolds are SO- or S1-bundles is the primary question. Complete answers are given in Ivan3ic's [1, 3]. In particular, for S1-bundles it is due to the following Ivangic's [3, Th. 3.1] criterion and its corollary, see also Vasquez [1] for direction (ii)=(i):

Theorem 6.40. Let E = R'/ G be a compact flat n-manifold, where G is a discrete subgroup of Isom Rn. Then the following are equivalent:

(i) E is an St-bundle over some base manifold B. (ii) There is an element f G so that the cyclic group (f) is a normal subgroup of G and for every g E G, gk E (f) implies g E (f ). Furthermore, if n 0 4 and (i) holds, the manifold B is homeomorphic to a flat manifold. For n = 4 we get the same conclusion if we assume, in addition, that B is irreducible.

Corollary 6.41. Let E = R/ G, G C Isom ]t8", be a compact flat n-manifold that is an S1-bundle over a manifold B. If f E G, f (x) = x + v, is the element that corresponds to the loop given by a fiber, and g(x) = Ux + a, U E 0(n), is any element of G, where U # I and a E ker(U - I), then Uv = ±v and a 4 v. Of course, this would not be interesting if all flat manifolds had this property. However, some flat 3-manifolds turned out not to be S1-bundles which creates an ob-

6.3. Hyperbolization of manifolds and hyperbolic volumes

313

stacle (that exists in higher dimensions as well) to generalizing the idea of considering hyperbolic 3-manifolds as link complements. One can employ the above criterion to show that starting with dim E = 3 there are flat manifolds in each dimension that are not St-bundles over any manifold; several such examples are given by Ivansic [3, §4]. This means that already some hyperbolic 4-manifolds may not be codimension-2 complements. Indeed, in the last few years several authors have constructed a number of examples of noncompact hyperbolic 4-manifolds, see Ivansic [1, 2], Niershiem [2], Ratcliffe-Tschantz [1], among which, as it turns out by Ivansic [1, 3], there are plenty of manifolds that are and that are not codimension-2 complements. Continuing with hyperbolic 3-manifolds, it is natural to ask the converse to the above Jorgensen Theorem 6.35: how many 3-manifolds Md,,...,dk obtained from a hyperbolic finite volume manifold M by Dehn surgery (with parameters (dt, ... , dk) E x S2, see Chapter 5, § 1) on its ends are hyperbolic? We notice that if these S2 x manifolds Md..... dk are hyperbolic then, due to Theorem 4.31, they become close to the original manifold M as parameters (dl, ... , dk) of Dehn surgery tend to (oo, ... , oo) in the product topology of S2 x x S2, S2 = 1[82 U {oo}. In fact, this provides an answer based on Corollary 6.36 and the following result of Thurston [1]:

Theorem 6.42. Let M =

S3 \L

be the complement to a k-component link, which admits a complete hyperbolic structure. Then there exists a neighborhood U c S2 x . . . X S2 o f the point (oo, ... , oo) such that, f o r any ( d 1 , . . . , dk) E U, the manifold Md1.....dk admits a complete hyperbolic structure.

Example 6.43. As an example, we consider Dehn surgeries on one component of the Whitehead link, that is, (oo, d2)-surgeries on the manifold Mw = S3\LW from Example 4.25, see Figure 51. The obtained manifolds are the complements of the twist knots shown in Figure 86 (compare Figure 84).

tll\

Figure 86

6. Uniformization

314

The first among these knots is the figure-eight knot whose complement has a hyperbolic structure due to Example 1.11. Due to Thurston [1], all other knot complements are also hyperbolic. Continuing (generalized) Dehn surgery, now on the figure-eight knot complement M, one obtains closed irreducible 3-manifolds Md, which are in fact hyperbolic for all but six parameters d = (a, $) E 1R2 C S2 (Thurston [1]). The six non-hyperbolic surgeries are as follows:

(1) the trivial surgery with d = (1, 0); it gives M(1,o) = S3;

(2) the (1,1), (2,1) and (3,1)-surgeries give a Seifert fibered manifold and two sufficiently large ones;

(3) d = (0, 1) gives a torus fibering over S1; (4) d = (4, 1) gives a graph-manifold. Similarly there are, in fact, plenty of examples of orientable noncompact hyperbolic 4-manifolds M of finite volume that are codimension-2 complements. Every example constructed in Ivansic [2] is such, and so are the orientable examples from Niershiem [2] with ends that are S1-bundles. However, as the following results of Ivangic's [1, 3] show, if we want a hyperbolic 4-manifold M to be a codimension-2 complement inside a particular closed 4-manifold N, say the 4-sphere S4, then the number of possible candidates is very limited.

Proposition 6.44. Let M be afinite-volume noncompact hyperbolic (n + 1)-manifold, n > 3. If M = N\A, where A is a codimension-2 submanifold of a closed (n + 1)manifold N, then x (M) = x (N).

Proof. Due to Lemma 6.38, every boundary component Ei of the compactification M of M is an S1-bundle over a component Bi of A, i = 1, ... , m. Let Pi denote a2disk-bundle neighborhood of Bi inside the manifold N. Then N = M U P1 U U Pm. U Pimplies The Mayer-Vietoris sequence for M and P1 U m

m

m

x(MU(UPi))=X(M)+x(UPi)-X(Mn(UPi) i=1

i=1

i=1

This gives m

m

X(N)=X(M)+X(Pi)-X(Ei) i=1

i=1

However, since every Ei is a flat manifold and every Pi a disk bundle over a flat manifold (so Pi is homotopy equivalent to Bi ), we get x (Pi) = X (Ei) = 0 for all i = 1, ... , m. This implies X (M) = x (N).

6.3. Hyperbolization of manifolds and hyperbolic volumes

315

Theorem 6.45. Let M be a finite-volume noncompact hyperbolic 4-manifold embedded inside a compact 4-manifold N as a codimension-2 complement, i.e. M = N\A. Then A is a disjoint union of tori and Klein bottles. Furthermore, for a given N, there are only finitely many manifolds M with those properties. In particular; there are at most finitely many noncompact hyperbolic 4manifolds that are codimension-2 complements inside S4.

Proof In the notation of proof of Proposition 6.44, every Bi is a flat 2-manifold due to the last statement of Theorem 6.40. Thus, every Bi is a torus or a Klein bottle. Due to Proposition 6.44 and Gauss-Bonnet's theorem,

Vol(M) = 47x2/3 X (M) = 47x2/3 X (N) . However, the Wang Theorem 6.34 asserts that there exist only finitely many hyperbolic

manifolds of dimension > 4 with a given volume. We see that any hyperbolic 4-manifold candidates M for codimension-2 complements inside S4 must have the Euler characteristic X (M) = X (S4) = 2. The only explicitly given examples of manifolds M with this property that the author is aware of are two hyperbolic 4-manifolds obtained by Ivan9i6 [1, 2] from a convex hyperbolic 4-polyhedron by way of two different side-pairings. Further work may yet identify these as codimension-2 complements inside the 4-sphere S4. Since a finite volume hyperbolic manifold M E M is completely determined by its fundamental group 7r1 (M), the volume of hyperbolic manifolds is a topological invariant providing a single measure of complexity of these manifolds. To make this more precise, one can use the following Gromov's invariant.

Let X be a topological space and H,k(X, R) its real singular homology. Taking a k-chain C = Ei ai of where vi is a continuous map of the standard k-simplex Ak into X, we define a norm of C as 11C11 = >i lai I. Then the Gromov norm on Hk(X, R) is defined as III II = inf 11z 11 where infimum is taken over all singular cycles z representing ," E Hk(X, IR). Now, for a closed orientable manifold M, we define II M I I as the Gromov norm of the fundamental class of M. So obtained Gromov

invariant of M has a property that, for a map f : M -- M', IIMII >_ Idegree(f)I - IIM'II .

In particular, since the n-sphere Sn allows maps f : S' -3 Sn of degree at least 2 for n > 1, the Gromov invariant II Sn II vanishes; however II S° II = 2. A relation between hyperbolic volumes and the Gromov invariant is given in the following Gromov theorem, see Thurston [1]. Theorem 6.46. Let Vn be the hyperbolic volume of a regular simplex in the hyperbolic n-space with vertices at infinity. Then vol Mn = Vn II Mn II for any closed orientable hyperbolic n-manifold Mn. Here we notice that the volumes V, of regular ideal simplices in IEIIn are in fact the upper bounds for volumes of all (ideal) hyperbolic n-simplices. In dimension n = 2,

316

6. Uniformization

it is trivial because the area of any ideal hyperbolic triangle equals >r. In dimension

n = 3, this fact is due to Milnor [5, 6] and, for n > 3 it is due to Munkholm [1]. Furthermore, always Vk < Vk_t /k - 1. For example, V2 = 7r, V3 = 1.0149416... .

Due to Thurston [1], Gromov's Theorem 6.46 can be sharpened as follows:

Theorem 6.47. Let f : M -a N be a nonzero degree map between closed orientable hyperbolic 3-manifolds. Then vol M > I degree(f) I vol N and, if the equality holds, the map f is homotopic to a covering map which is a local isometry. Furthermore, for hyperbolic manifolds M and N such that M N\Lfor a non-empty link L C N, vol N < vol M. Now the Jorgensen and Gromov Theorems 6.35 and 6.47, together with the existence Theorem 6.42 imply the following Thurston [11 description of the set of volumes of hyperbolic 3-manifolds:

Corollary 6.48. For any C > 0, the volume function vol: 3e -3 IR+ has a finite preimage vol-t (C). Furthermore, the set vol(e) C R is well-ordered, and its order type is w'. We note that here co' is the countable ordinal which describes the order type of polynomials (of finite degree) in the symbol co with natural number coefficients, ordered by the limiting order of the values when bigger and bigger integers are substituted for cv. This means that there is some lowest volume vt E vol(3e), then some second volume v2, and so on until the first accumulation point of these volumes v0 , which is the smallest volume of a manifold with one cusp. Then there is a next highest volume vG,+t, and so on until the second accumulation point v,,,2 which is the smallest volume of a hyperbolic manifold with two parabolic ends, etc. It is an intriguing question about the structure of the set vol(.), ein particular, what is it possible to say about manifolds with minimal volumes vi, v, v,2, etc., and about manifolds in finite sets vol-t (C), C E R+. Many methods and invariants can be used to obtain some partial answers to these questions. They include arithmetic lattices, geometric and topological constructions, Chern-Simons invariant, Hamiltonian mechanics and the so-called complexity of 3-manifolds. In particular, as we showed in §11, Chapter 3 due to calculation of volumes of Fibonacci manifolds, one can find pairs of closed and non-compact hyperbolic 3-manifolds or pairs of arithmetic and non-arithmetic hyperbolic manifolds that have equal volumes, see Corollary 3.52. For concrete hyperbolic manifolds, many such questions can often be answered by use of the powerful computer program "Snap-Pea" created by J. Weeks (see, for example, Weeks [2] and Sakuma-Weeks [1, 2]). Here, we start with the question about equal volumes by showing that the size of finite sets vol-1(C), C E R+, cannot be uniformly bounded:

Theorem 6.49. For any given natural number N, there are convex bounded and unbounded polyhedra PN C H3 of finite volume that are fundamental polyhedra for at least N different hyperbolic 3-manifolds.

6.3. Hyperbolization of manifolds and hyperbolic volumes

317

Proof. First, we consider the case of non-compact hyperbolic manifolds, which is easier. To construct an arbitrarily large number of manifolds that have the same fundamental polyhedron, we start with a "sufficiently large" polyhedron P C lHl whose projection on 81HE3 = ][82 (in the half-space model) is shown in Figure 87.

8 C

c

h-1

d-1

Figure 87

h d

Figure 88

It is a fundamental polyhedron for a subgroup G in the Picard group SL(2, Z[i]), generated by (k + 2) parabolic elements. The boundary M(G) consists of 2k trice-punctured 2-spheres which are parallel to totally geodesic surfaces Si, S' in WIG, S1, Sil N S2 \ { p l , P2, P3 } . Pairwise identifying these Si and Si' by either parabolic or loxodromic hyperbolic isometries, that is identifying the corresponding components of a P fl aH3 by means of transformations f, g, h.... shown in Figure 88, we obtain different cofinite HNN-extensions Gj C SL(2, Z[i]) of the group G. The groups Gj uniformize the complements Mj = S3 \Lj of different (hyperbolic) S3\Lj, and all these hyperbolic manifolds Mj have the same links Lj, 1E113/Gj fundamental polyhedron Pk obtained from P by cutting off hyperbolic half-spaces bounded by the hyperbolic planes which universally cover the surfaces S1, Si' C 1H[3 / G.

For k = 1, one has three such links shown in Figure 89. First of them is the Borromean rings, and it corresponds to parabolic gluings. Two other links (corresponding to loxodromic gluings) can be obtained by half-integer twists along the indicated disk

6. Uniformization

318

spanning a component of the Borromean rings.

Figure 89

For k > 1, the parabolic gluing of Si and Sd corresponds to a cyclic n-fold covering of the Borromean rings shown in Figure 90, where the central point indicates a component of the link passing through oo.

Figure 90

Replacing one of the parabolic gluings of Sj and Sj by a loxodromic one gets a link obtained from this chain by a half-integer twist along a twice punctured disk spanning a link component, see Figure 91 where k = 4.

'01

Figure 91

Therefore, in such a way, we obtain N = 3k different links all whose fundamental polyhedra (and hence, hyperbolic volumes) coincide. Clearly, all these links are different, so Gj and Gj, are not isomorphic.

6.3. Hyperbolization of manifolds and hyperbolic volumes

319

To prove the assertion of the theorem for closed manifolds, one can use surfaces allowing large symmetry groups. Namely, for a given natural number k, one can construct a closed orientable hyperbolic 3-manifold MN, N = 4k, which has a totally geodesic surface SN C MN with I Isom Sn, I > N. The manifold MN can be obtained by side pairings of the compact hyperbolic polyhedron PN C H3 whose spherical projection (to 31H13) is shown in Figure 92.

Figure 92

The polyhedron PN has right dihedral angles at edges Ai fl Cj, A; fl Bj, v fl B,, w fl Ci for i, i= 1, ... , 2N, and at those edges C, fl Cj and Bi fl Bj that are marked in Figure 92 by the right angle sign. The rest of the edges Ct fl CJ and B, fl Bj have

6. Uniformization

320

the dihedral angles equal to 7r/2N. Finally, all dihedral angles at edges Ai fl Aj are 7r/3.

The sides of P are pairwise identified as indicated in Figure 92. Namely, a j : B3 j+1 -+ B3j+6N+1 ,

$j:B3j--*B3j+2,

Y j C2j+1 - C3j+6N+1 ,

j = 0, 1, ..., 2N -1;

3: C3j- C3j+2,

184N: B12N -+ B2,

34N: C4N

C2

j=1,2,...,4N-1;

and

r: V -3 W

are hyperbolic translations. The rest of the identifications are screw-translations with order two rotational parts:

j=0,1,...,2N-l; mj:A3j-+A12N-3j+4, j=2,3,...,N;

Cpj:A3j+2-),A3j+6N+2,

m1:A3-+A1,

h1: A3j+3N+4 -+ A9N-3 j

rj : A3 j+4 -+ A6N-3 j , sj:A6N+3j+4-+A12N-3j, 0
To obtain 2N different closed hyperbolic manifolds MN, one can change the translation is V -a W to screw-translations ti each of which is the composition of r and a rotation

Ul around the axis oft (the symmetry axis of PN) by the angle 7ri/2N, 1 < i <_ 2N. These screw-translations correspond to different isometries (rotations) of the totally geodesic surface SN C MN obtained from the sides V and W of PN by the above gluings, see Apanasov-Gutsul [2]. Though the set of volumes of hyperbolic 4-manifolds is much smaller than that of volumes of hyperbolic 3-manifolds, we still have an analogue of Theorem 6.49, see Ivangic [1, 2]:

Theorem 6.50. Given any number N, there exist more than N non-homeomorphic, non-compact, complete hyperbolic 4-manifolds of finite volume that share the same fundamental polyhedron in lEl4. In particular, they have the same volume.

The proof of this theorem is by constructing polyhedra in 1El4 with different side-pairings and utilizing the Poincar6-Aleksandrov polyhedron Theorem 4.29 (Corollary 4.30) to see that identifying paired sides yields complete hyperbolic 4-manifolds. The manifolds are then distinguished by how many ends they have. Since each end of the manifold corresponds to a cycle (an equivalence class under identification by

side-pairing) of ideal vertices of the polyhedron, it is enough to count classes of ideal vertices for each of the constructed side-pairings and to show that we arrive at different numbers for different side-pairings. Therefore, the resulting manifolds are non-homeomorphic, because they have different numbers of ends. It turns out that each of the manifolds may be obtained by taking two basic manifolds, cutting them along a two-sided totally geodesic embedded 3-manifold and stringing several of these together by gluing them along the isometric cuts. A by-product of that construction is an (even) part of a proof of the following fact independently proved by Ratcliffe-Tschantz [1]:

6.3. Hyperbolization of manifolds and hyperbolic volumes

321

Theorem 6.51. The set of all volumes of hyperbolic 4-manifolds consists of the posi-

tive integral multiples of 42/3.

3. Matveev's complexity of manifolds. Due to Corollary 6.48, it is possible to enumerate all hyperbolic 3-manifolds of finite volume. It can be done by using the Matveev [3, 4] invariant of 3-manifolds M called the complexity of M, c(M). Every compact connected 3-manifold M collapses (after deleting a ball, if 8M = 0) to a codimension one C W-subcomplex Q C M, called a spine of M. Among such 2-dimensional CW-complexes, we distinguish special spines. Each of them is characterized by the property that each of its points has a neighborhood homeomorphic

either to a disk or to a cone over a circle with 2 or 3 radii from its center. In other words, these neighborhoods look like those in Figure 93.

Figure 93

If a compact C W-complex P can be embedded in some special spine, we call it

almost special. A point v E P of an almost special spine is called a vertex if the link of v is homeomorphic to the circle with 3 radii, that is a neighborhood of v is homeomorphic to the last complex shown in Figure 93. Now we can define the complexity c (M) of a compact 3-manifold M as the minimal number of vertices of its almost special spines. One can easily check that there are only three closed orientable irreducible 3-manifolds with zero complexity c (M) = 0. Namely they are S3,RP3 and the lens space L3,1. There are only two such manifolds with complexity c(M) = 1, both are the lens spaces L5,2 and L4,1. Moreover, due to Matveev [5], this topological invariant c(M) distinguishes closed irreducible 3-manifolds M up to finite sets because, for a given k E N, there are at most finitely many closed irreducible 3-manifolds M with c(M) = k. Furthermore, c(M) is additive with respect to the operation of connected sum, that is c(Ml#M2) _

c(MI) + c(M2). Now, to answer the above question about simplest hyperbolic manifolds, it is useful to consider closed manifolds and non-compact manifolds of finite volume separately. The latter case (after compactification) involves compact 3-manifolds with boundary consisting of tori. Simplest examples of such manifolds are so-called graph manifolds (Waldhausen [3]). All of them can be obtained by gluing some number of blocks homeomorphic to the products B2 x S1 or N2 x S1, along their boundary tori

6. Uniformization

322

x S I; here B2 and N2 are the closed 2-disk and such a disk without interiors of disjoint subdisks, see Fomenko and Zieschang [1, 2]. T2

S1

For complexity c(M) < 3, we have a complete answer to the question about simplest non-compact hyperbolic 3-manifolds (Matveev [5], Fomenko-Matveev [1]):

Theorem 6.52. All connected orientable irreducible atoroidal 3-manifolds M with complexity c(M) at most three and whose boundary consists of tori are graph manifolds, except for two manifolds Q 1 and Q2 with complexity two and nine manifolds Qi, 3 < i < 11, with complexity 3. Figure 94 gives regular neighborhoods of 1-skeletons ("Z S1 US1 with two common

points) of the simplest special spines of Q1 and Q2. Such a neighborhood of a 1skeleton of a special spine determines this spine and hence the corresponding manifold

itself. So one can see that the interior of the second manifold Q2 is homeomorphic to the figure-eight knot complement S3\K (see Figure 7 and Example 1.11), and the interior of the first manifold Q i is homeomorphic to the complement L5,1 \ K of a knot in the lens space L5,1, which can be obtained by (5, 1)-Dehn surgery on one component of the Whitehead link (Figure 51).

The other exclusive manifolds Qi, c(Qi) = 3, allow a similar description. In fact, each of 1-skeletons of their simplest special spines is homeomorphic to one of the graphs shown in Figure 94. The interiors of these manifolds Qi, 3 < i < 11, are homeomorphic to knot complements either in the sphere S3, or in the projective space IRP3, or in the lens spaces L3,1, L5,1, L6,1, L7,2, L9,2; in particular a knot K c S3 corresponding to one of Qi is 52, and another such knot in S3 has a projection with 12 intersections.

Figure 94

Since a graph manifold cannot have a complete hyperbolic structure, only 11 manifolds Qi, 1 < i < 11, among all compact orientable irreducible atoroidal 3manifolds of complexity at most three, may be hyperbolic. In fact, due to FomenkoMatveev [1], it is the case, that is there are exactly eleven (orientable, non-compact, finite volume) hyperbolic manifolds M with complexity c (M) at most three. Although it is unknown which non-compact orientable hyperbolic 3-manifold has the minimal volume vu,, a good candidate for that is the manifold int Q2 homeomorphic to the figure-eight knot complement in the 3-sphere. Its complexity is minimal, c(Q2) = 2, and due to A. Reid [1, 2], this is the only one arithmetic hyperbolic knot in S3.

6.3. Hyperbolization of manifolds and hyperbolic volumes

323

Among non-orientable hyperbolic manifolds, the Gieseking manifold MG (obtained by gluing sides of the regular ideal simplex in 1HI3, see Example 1.10) has the minimal complexity, c(M) = 1. This leads to a conjecture that this manifold has the minimal volume, vol MG = V3 = 1, 0149416... , among all non-compact hyperbolic manifolds. In fact, this is true, see Adams [3]. To answer the question about simplest closed hyperbolic manifolds, one can apply the following Matveev [5] result which was originally proved in Fomenko-Matveev [1] by using a computer: Theorem 6.53. All closed orientable 3-manifolds M with complexity c(M) at most eight are graph manifolds. It follows that the complexity c(M) of a closed oriented hyperbolic 3-manifold must be at least 9. Moreover, due to Fomenko-Matveev [1], among all closed hyperbolic manifolds obtained by Dehn surgery from the simplest non-compact hyperbolic

manifolds Qt, 1 < i < 11, there are only two manifolds M1 and M2 whose complexity is 9: the manifolds M1 = (Q1)5,-2 and M2 = (Q2)5,1 obtained from Q1 and Q2 by (5, -2) and (5, 1)-Dehn surgeries, respectively. Regular neighborhoods of 1-skeletons of the simplest special spines of the manifolds M1 and M2 are shown in Figure 95.

(L5,1 \ K)5,-2 Figure 95

The second manifold M2 was initially conjectured by Thurston [1, 2] to be a candidate for the hyperbolic manifold of the smallest volume v1. The manifold Ml, which was discovered later by Fomenko-Matveev [1] and Weeks [1], turned out to have the smaller volume. So it is natural to expect that the volumes of M1 and M2 are the minimal ones among all orientable hyperbolic 3-manifolds:

v1 = vol M1 = 0.94272... < vol M2 = 0.98139... = V2.

(6.5)

As a partial argument toward this conjecture, one has the fact that both M1 and M2 are arithmetic (Chinburg [2]) and hence play an extremal role. We refer the

324

6. Uniformization

reader to a work of Gehring-Maclachlan-Martin-Reid [1] which combines geometric and arithmetic ideas toward the study of extremals in various geometric problems in Kleinian groups and shows discreteness and arithmeticity of many of these extremals that are two-generator subgroups of PSL2 C. Among those extremal problems, there are problems of the minimal hyperbolic volume, the Margulis constant (see Chapter 3, § 1 and Gehring-Martin [4]) and the minimal distance between elliptic axes (and fixed points) of the group elements, see Derevnin [1, 2], Derevnin-Mednykh [11 and Gehring-Martin [5, 6]. We note that, as one expects for extremals, arithmetic manifolds/orbifolds are somewhat rare among hyperbolic ones in 3e because their volumes form a discrete subset in R (Chinburg [1]). Moreover, the FomenkoMatveev-Weeks manifold Ml has the smallest volume among orientable arithmetic hyperbolic 3-manifolds, see Chinburg-Friedman-Jones-Reid [1].

§4. Uniformizable conformal structures First study of conformal manifolds (= conformally flat Riemannian n-manifolds) in dimensions n > 3 has been undertaken by N. Kuiper [1-3]. In these papers, there

was originally defined such basic tools as the development map d: M -+ Sn and the holonomy representation dk: nl(M) -* Mob(n) of a conformal structure on a manifold M (see Chapter 1, §5). In particular, the following facts about the simplest classes of conformal structures were proven.

Theorem 6.54. Let M be a closed conformal n-manifold (n > 3) with finite fundamental group 7ct (M). Then its development d: M -* Sn is a homeomorphism onto the sphere Sn, and M is conformally equivalent to the quotient Sn/G with the natural conformal structure, where the holonomy group G = d,, (ict (M)) freely acts on Sn by orthogonal transformations. Theorem 6.55. Let M be a closed conformal n-manifold (n > 3) whose fundamental group Ttl (M) is infinite Abelian. Then its development is a homeomorphism onto either Rn or Rn\{0}. In the first case, the holonomy group G = d,K(nl (M)) is Euclidean, G C Isom Rn, and M is conformally equivalent to a flat n-torus T n = Ilgn /fin In the second case, the group G consists of Euclidean similarities, G C Sim Rn, and M is conformally equivalent to a Hopf manifold St X Sn-1 In both cases in the last theorem, the holonomy group G of a conformal manifold M consists of Euclidean similarities. It turns out to be a crucial condition for Theorem 6.55 to hold. It follows from the following claim which was first formulated by Kuiper [3] and proven by Fried [1] and Goldman [1]:

Theorem 6.56. Let a closed conformal n-manifold M (n > 3) be also a similarity manifold, that is its holonomy group G = d*(7rt(M)) C SimR'. Then the development map d: M -} Sn is a homeomorphism onto either Rn or IRn\{0}.

In the first case, G is a finite extension of a rank n free Abelian group 7Z' of

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325

Euclidean translations, and M is a flat manifold finitely covered by a Euclidean torus

T"=R"/Zn. In the second case, G is a finite extension of an infinite cyclic group Z generated by a dilation g, g(x) = AA(x), A > 1, A E O(n), and M is a virtually Hopf manifold, that is, it is finitely covered by a Hopf manifold S1 X Sr-1. It is natural to require some properties of considered manifolds to be satisfied up to their finite coverings, which correspond to finite index subgroups of the fundamental group 7r1(M). We say that a group G virtually has a property "P" if some finite index subgroup G' C G has this property "P". In this way, we get virtually Abelian, nilpotent, solvable or polycyclic groups, respectively. In the case of Mobius groups, these classes have the following clear description due to Goldman [1]:

Lemma 6.57. Every virtually solvable group G C Mob(n) either is conjugate to a subgroup of O(n + 1) or has a finite index subgroup G C G conjugate in Mob(n) to a group of Euclidean similarities of l[8n.

This allows to generalize Theorem 6.56 as follows (Goldman [1], Karnishima [1]):

Theorem 6.58. Let M be a closed conformal n-manifold (n > 3) whose fundamental group nl (M) is either virtually nilpotent, or virtually solvable, or virtually polycyclic of rank at most three. Then M is finitely covered by either the n-sphere Sn, or a flat torus Tn = R'/7.n, or a Hopf manifold S1 x Si-1 = (R' \{0})/7L, respectively. To apply this result to 3-manifolds, we can consider some special classes of them, with regard to the canonical splitting of 3-manifolds, see §2. Namely, let us consider those 3-manifolds M that are "prime" pieces under such a splitting, are not covered by S1 X S2, and whose fundamental groups are infinite and virtually solvable. Then we obtain a trichotomy of this class into three subclasses, regarding to properties of the fundamental group Jrl (M), as follows. (E) .r 1(M) is virtually Abelian; (N) 7r1(M) is virtually nilpotent but is not virtually Abelian; (S) Jrl (M) is virtually solvable but is not virtually nilpotent. The class (E) consists of conformal manifolds finitely covered by Euclidean tori T3. All 3-manifolds from the class (N) are covered by non-trivial fiber bundles over S1 whose fibers are tori T2, and the attaching maps are Dehn twists on T2. Furthermore, such (N)-manifolds have Ni13-geometry, see Chapter 1, §6. Such manifolds can be obtained as quotients of the Heisenberg group by its discrete isometry groups (due to Evans-Moser [1] combined with Scott [7] in the non-Haken case). The class (S) consists of 3-manifolds covered by non-trivial T2-bundles over S1 whose attaching maps are hyperbolic maps with eigenvalues e` and a-t; so they have So13-geometry, see Chapter 1, §6. The above classification, together with Theorem 6.58, provides us with a complete description of conformal structures on T2-bundles over S1 (Goldman [1]): Corollary 6.59. Let M be a 2-torus bundle over the circle. Then M admits a conformal structure if and only if the attaching map of this bundle is periodic.

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326

In other words, 3-manifolds from the classes (N) and (S) cannot be conformal.

1. Almost uniformizable conformal structures. Studying geometric structures on manifolds, we are in a category where the covering space theory is valid. This makes it natural to call an (X, G) -structure on a manifold M uniformizable if its development d: M -+ d (if) C X is a covering map and the holonomy group d * (nl (M)) C G acts freely and discontinuously on d(M). We call an (X, G)-structure almost uniformizable if it satisfies only the first condition, that is if its development d is a covering map. The first general result on uniformizable conformal structures was obtained for CP1-structures on surfaces (Gunning [1], Kra [1]). It asserts that, for a projective structure on a compact Riemann surface S of genus more than one, the following statements are equivalent:

(i) D = d (S) C; _ (ii) the development d: S -a D C C is a covering; (iii) the holonomy group G = d*nl(S) C PSL2(C) acts discontinuously in D, which is a connected component of the discontinuity set 0 (G). In the general case of projective structures on S with surjective developments

d: S --+ C, d(S) = C, the maps d are not coverings and the holonomy groups G = d,k (nl (S)) are not discrete anymore, see Maskit [8]. Analogous non-uniformizable conformal structures exist in higher dimensions, in particular on n-manifolds admitting hyperbolic structures. One can construct such non-uniformizable structures from hyperbolic ones by "bending" them along totally geodesic hypersurfaces by angle 2n (see Goldman [2] and Chapter 7, §3). However,

even in these cases, one can restrict the development map to a subdomain in the universal cover M of the conformal manifold M (preimage of the normality domain of the holonomy group G) where this map is in fact a covering. This observation known as the uniformization theorem for geometric structures is due to KulkamiPinkall [1], and is based on the notion of a boundarypoint of the development map d introduced by N. Kuiper [2]. We shall prove this uniformization theorem in the case of conformal structures where we can simplify both the statement and its proof as follows.

Let G C Mob(n) be a group, not necessarily discrete. We call G properly discontinuous at a point x E S" if, for a neighborhood U C S' of x, the set Z(U) = {g E G : g(U) n u ; 0} is compact. The set O (G) of all such points x E Sn where G acts properly discontinuously is called the proper discontinuity set. Using the method in the proof of Theorem 2.12, one can easily show that the complement of the proper discontinuity set, L(G) = Sn\O(G), called the proper limit set of G, coincides with the closure of the set of fixed points of loxodromic elements g E G, and hence it equals the subset of points x E Sn approximated by the orbit G(z), for some (any) point z E 1lIIn+l,

ajn+l = Sn

A point x E Sn is called a boundary point of the development d: M -* S' of a conformal structure (M, c) if it is the end point y (1) of a continuous curve

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327

y: [0, 1) --+ D = d(M) with the following properties: (i) There exists a continuous d-lift y : [0, 1) -> M, d (y (t)) = y (t) fort E [0, 1); (ii) The continuous curve y cannot be continuously extended at t = 1. Let B(d) C S" be the set of all boundary points of the development d: M -p S".

Theorem 6.60. Let M be a closed conformal n-manifold (n > 3) with the development d: M -* S" and the holonomy group G = d*7r1(M) C Mob(n), and let Oo be the union of those components of the proper discontinuity set O (G) that intersect

D = d(M). Then d I d-1

foal = do: d-1 (00) --> Oo

(6.6)

is a covering map.

To prove this uniformization theorem, we need the following fact about boundary points:

Lemma 6.61. Let M be a closed conformal n-manifold with the holonomy group G = d*ni (M) C Mob(n). Then B(d) C L(G). Proof. Let p E S' be a boundary point of the development d: M --} S" and y : [0, 1) -> d (M), y (1) = p, be a continuous curve with its continuous d-lift y : [0, 1) -- M nonextendable at the end 1. Since M is closed, there exists a fundamental (open) domain F C M for the action of the decktransformation group G(M, M) = G, G = 7r1(M), such that F = cl,K(F) is compact. Since the development d is a local homeomorphism, we see then that there are infinitely many different elements gi E G whose images gi (F) intersect the curve y. Otherwise, the curve y could be continuously extended to the end 1 of [0, 1). We recall that, due to Corollary 2.1, Mobius groups are convergence groups. Hence,

in the holonomy group G = d*G, we have an infinite sequence {gi} (gi = d*gi, after a renumeration) which either uniformly converges to a Mobius transformation g = Jim gi or degenerates in the sense of (2.2), that is, for some xo, yo E S",

Jim gt 1(x) = xo lim gi (y) = yo , and 1->oo

i>oo

(6.7)

uniformly in compact subsets of S"\{xo} and S"\{yo}, respectively. We would like to show that only the latter degenerate case (6.7) is possible. Namely, supposing that lim gi = g E Mob(n), we may assume that the boundary

point p E B(d) is not an accumulation point for the sets gi (d(F\F)). Let ti E (0, 1) be such parameters that y (gi) E y fl gi (F), i E N. Then there are a neighborhood U (p) c S" of the point p and an integer N such that

U(p)flgi(F\F)=0 for i >N, and y(t)C U(p) fort>ti(N).

(6.8)

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However, due to the choice of {gi}, the curve y ([ti(nr), 1)) intersects the boundary of gi (F). This contradiction with (6.8) finally shows that Jim gi cannot be a Mobius transformation, and hence (6.7) holds. Since the development d is a local homeomorphism, one may assume that the

point xo E Sn in (6.7) does not lie in d(F\F). Then (6.7) implies that gi (d(F\F)) converges uniformly to the point yo. Thus the boundary point p E B (d) is in fact the point yo in (6.7), that is a proper limit point of the holonomy group G. This completes the proof of the lemma. Proof of Theorem 6.60. Due to Lemma 6.61, the proper discontinuity set O (G) does not contain boundary points of the development d. Therefore, the map do: d-1(0) -4 Oo in (6.6) is a local homeomorphism, and any continuous path in Oo can be lifted to a continuous path in d-1(Oo). So do is a covering map.

A direct application of the uniformization Theorem 6.60 and Theorem 6.56 on conformal similarity manifolds shows that all conformal manifolds with non-surjective developments are almost uniformizable:

Theorem 6.62. Let M be a closed conformal n-manifold (n > 2) with the infinite fundamental group nl (M). Then the following assertions are equivalent: (a) the development map d is not surjective: D = d(M) # Sn; (b) the development map d: M -* D C Sn is a covering map. is obvious because rrl (Sn) = 0, Sn is compact, and Proof. The implication M is not compact because I ni (M) I = oo. If the boundary aD consists of either one point {x} or two points {x, y}, one can assume that x = oo, and D C R. In both cases, the holonomy group G = d,k ri (M) consists of Euclidean similarities. Hence (b) follows from Theorem 6.56. Now suppose that aD contains at least three points. Then the holonomy group G (preserving aD) is not elementary, and one can see (as in Theorem 2.12) that G has loxodromic elements, there is no common fixed point, i.e. fix(G) = 0, and the proper limit set L(G) belongs to aD because L(G) is the closure of the loxodromic fixed points (see Corollary 2.13). Hence d is a covering map because of Theorem 6.60.

In contrast to projective structures on surfaces in dimension n = 2, an almost uniformizable conformal structure on an n-dimensional manifold M, n > 3, may be non-uniformizable because its holonomy group G = d iri (M) may act non-discontinuously on D = d (M). It is shown in the following example. Example 6.63 (Non-uniformizable, almost uniformizable (M, c)). Let M = Sg x S1 be the product of a close orientable surface Ss of genus g > 1 and the circle S1. This manifold M has an (IH[2 x 118)-structure. Namely, since g > 1, we may assume that 2ri (Sg) C Isom 111[2. Then 7r1(Sg) x Z - .nl (M) naturally acts by isometries on IEi[2 x lag so that Z acts on R as a cyclic group generated by a translation h, h(t) = t + a

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329

for a fixed a E R. Obviously, M

(112 x I[8)/Orl (Sg) x Z), and we have a conformal structure ca on M induced by the constructed (1H12 x 118)-structure with the development

da: M 1HI2 x R -a 1183\{1I8 e1} C 1183, where el = (1, 0, 0) E 1183. It is clear that for values a E R non-commensurable with jr, i.e., if a/..7t ¢ Q, the holonomy group Ga = d* (Yrl (Sg) x Z) C M6b(R3\Re1) is not discrete, but its restriction to R R. el is. In fact, the subgroup d, ({ 1 } x Z) is an infinite elliptic subgroup generated by a rotation about the line R R. el by an irrational angle. Therefore, for a E Il with a/ir V Q, conformal manifolds (Sg x S1, ca) are non-uniformizable. However, they are almost uniformizable because the development d* : M -a 1[83\{1[8e1 } is a covering map, see also Theorem 6.62.

We remark that there is another sufficient (non-necessary) condition on conformal

manifold (M, c) to be ahnost uniformizable. Namely, instead of posing the nonsurjectivity condition on the development d: M -± S", one can set restrictions on the holonomy group G C Mob(n)R as follows (Kulkarni-Pinkall [1]): Theorem 6.64. Let M be a closed conformal n-manifold, n > 3, whose holonomy group G = d*rrl (M) C Mob (n) has the connected proper discontinuity set 0 (G) with finitely generated nl (0 (G)). Then M is almost uniformizable. Here we remark that the last condition on the finiteness of the rank of ,t(0(G)) in fact means that either 0 (G) is simply connected or nl (0 (G)) is cyclic. Otherwise, assuming rank 7r, (0 (G)) > 2, one would immediately have that irl (O (G)) is infinitely generated (due to arguments similar to those in Theorem 2.3). 2. A criterion of uniformization. The structures constructed in Example 6.63 play an exclusive role because all other almost uniformizable conformal manifolds are in fact uniformizable. We formulate this result in dimension three: Theorem 6.65. Let M be a closed conformal 3-manifold different from manifolds admitting either spherical or (IHI2 x R)-structures, that is, neither r1 (M) is finite, nor M is finitely covered by a product Sg x S. Then the following assertions are equivalent:

(a) the development d: M -* S' is not surjective: D = d (M) # S"; (b) the development d is a covering map; (c) the holonomy group G = d*itl (M) acts discontinuously on D.

Proof. Due to Theorem 6.62, we have the equivalence (a)q(b). Hence it is enough to prove the equivalence (a).

(c).

(c)=(a). Supposing that D = d(M) = S3, we have from the discontinuity of the G-action on D that G is finite, and hence M admits a (S3, O(4))-structure. This contradicts the conditions of the theorem, so D 0 S3, and d is non-surjective. (a)=(c). Similarly to the proof of Theorem 6.62, we may assume (due to Theorem 6.56) that the complement S3 \ D has at least three points. Hence, if the holonomy

group G is discrete, its limit set A(G) belongs to aD (Corollary 2.13, (iv)), and G acts discontinuously on D.

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Therefore, if the assertion (c) were false, the group G would be not discrete. Then, due to Theorem 1.12, the only possible (alternative) cases would be as follows:

(1) G is dense in Mob(3); (2) for some point x E S3, G(x) = x; (3) for some different points x, y E S3, G({x, y}) = {x, y}; (4) for some round circle l C S3, G(l) = 1; (5) for some round sphere L C S3, G(L) = L; (6) up to a conjugation in M6b(3), G C 0(4). Now we are going to show that all these cases are in fact impossible. Obviously, the latter case is impossible because M is not a spherical manifolds. Suppose the case (3) holds. Then, up to a conjugation and a subgroup of index two, we may assume that x = 0, y E oo and G(0) = 0, G(oo) = oo. Since the case (6) is impossible, there is a loxodromic element g E G fixing 0 and oo. This implies

that {0, oo) does not lie in D. Otherwise, for z E S3\D, lim g' (z) E {0, oo} C D, which contradicts to G-invariance and openness of D. Assuming that oo ¢ D, we have (due to Theorem 6.56) that the development d is a homeomorphism compatible with the G-action. The contradiction of this to non-discreteness of G shows that (3) is impossible. Analogous arguments eliminate the case (2), too. In the case (1), due to the transitivity of the Mob(3)-action on S3, we have for

x E S3\D and y E D that some sequence {gj (x)}, gl E G, converges to y. This contradicts to G-invariantness and openness of D, and eliminates (1).

In the case (5), up to a conjugation and a 2-index subgroup, we may assume that G preserves L = {x E 1183 : X3 = 0) and both half-spaces Iil and R3 . Let I' C M6b(2) - Mob(L) be the subgroup obtained by restricting the G-action to L. Clearly, the groups G and I' simultaneously are or are not discrete. Appealing to other cases, we may assume that the group r does not preserve any round circle, or a point, or a two points set. Hence, if IF were not discrete, it would be dense in Mob(2). As in the case (1), from the transitivity of the M6b(2)-action in 1R2 and in 1l one has that, for any points x, y E 1183, there is a sequence {gi } C G such that lim gi (x) = y. This fact, together with the G-invariantness and openness of D, shows that the only three

cases may be possible: either D =1R+, or D = R3, or D = S3, where the latter case has already been eliminated. In cases when D is a half-space, d is a homeomorphism because it is a covering map, and D is simply connected. Since a homeomorphism cannot be compatible with a non-discrete G-action, it shows that (5) is impossible. To eliminate the remaining case (4), we need the following fact.

Lemma 6.66. Let M be a conformal closed 3-manifold whose development d maps

M onto D = {x E R3 : x2 + x3 0 0}. Then M has an (H2 x R)-structure and is finitely covered by the product Ss x St of the circle and a closed orientable surface Sg of genus g > 2. Proof. We introduce a conformally flat Riemannian metric on the complement D =

6.4. Uniformizable conformal structures 1[83\{i[8

331

el } to the first coordinate axis in R3 as follows:

ds2D =

dx2 + dx2 + dx3 2 x2 + x2 2

(6 .9)

3

Then the isometry group of the obtained Riemannian space (D, dsD) coincides with the group M6b(R3\{118 el}) C M6b(3) of conformal automorphisms of D. Now let Y be the half-space, Y = {(x, r, (p) E I[83 : r > O}, with the following Riemannian metric: 2 dx2 + dr2 2

ds p =

r2

+ drp

,

( 6 . 10)

and let q: Y -* D be the map: q (x, r, gyp) = (x, r cos gyp, r sin gyp) .

(6.11)

Obviously, the map q is a covering map with the cyclic deck transformation group generated by 2n-translation y,

y: (x,r,(p) H (x,r,(p+27r). Due to (6.9) and (6.10), the metric in Y obtained by lifting the metric dsD in D by the covering map (6.11) coincides with the metric dsY. Hence q .is a local isometry. This, together with D = d (M), implies that the manifold M with the Riemannian metric lifted from (D, dsD) by the development d, is isometric to the space (Y, dsY). Therefore, the original conformal manifold M is conformally equivalent to an (11112 x R)-manifold, N.

As the foliation of 112 x 118 by the lines {x} x R is invariant for the action of Isom(1H12 x 118) = Isom(1H12) x Isom(R), it is clear that this foliation projects to a

foliation of our closed (112 x R)-manifold N by circles, so N is a Seifert fibered manifold. Let r C Isom(1H12 x R) be a discrete group acting freely on 11112 x R, and N = (1H12 x R) / r. Then, for the natural projection r -+ Isom 1H12 with the image ro, the orbifold X =1H12/ ro is the orbit space of this Seifert fibration. Since N is closed,

X is compact and has a negative Euler characteristic, X (X) < 0. This and the fact that the Euler number e (q) of the Seifert bundle structure Yl on the (11112 x R)-manifold

N equals zero show that there is a finite covering M of the closed manifold M whose induced Seifert bundle structure n is a trivial circle bundle, that is M = Sg x St where Sg is a closed surface with X (Sg) < 0.

Case (4). Now we assume that the holonomy group G C Mob(3) of the manifold M preserves a round circle l C S3 which can be taken as I = {x E II83 : X2 = X3 = 0}. Let H be the half-plane {x E R3 : x3 = 0,x2 > 0}, aH =1, and

,M+(H) = {g E Mob+(3) : g(H) = H, g preserves orientation on H} M+.(H) C 4((l) = {g E Mob+(3) : g(l) = l} .

6. Uniformization

332

Up to a 2-index subgroup, we also may assume that Gll = r C Mob+(1). Then, since the cases (2) and (3) have been already eliminated, there are the only two possibilities for a non-discrete group G: either r is dense in Mob+(1) = M+ (H), or r is discrete.

Suppose that the first case takes place, r = Mob+(1). Then D = d(M) c X = R3\l because D 0 S3 (Lemma 6.61). Moreover, D X due to Lemma 6.66, and there is a point z E X\D. Since Mob(l) acts transitively on H and r is dense in M6b(1), we have analogously to the case (1) that, on each circle S(a, r) centered at

a = (al, 0, 0) E l\{oo},

S(a,r)={xER3:Ix-al=r,xl=al}, there exists a point p = p (a, r) which does not belong to D. On the other hand, the density of r in Mob+(1) implies that there are two elliptic elements el , e2 E r of infinite orders which do not commute: el e2 ; e2el (Greenberg [1]). Each of these elements ei, i = 1, 2, pointwise fixes some circle S(ai , ri) C R3\1,

ai E l\{oo}, and can be lifted to an element hi = riei E G where ri is a rotation about 1. Clearly, the action of hi on the circle S(ai, ri) coincides with the rotation ti, i = 1, 2. Hence, if ti has infinite order, the closure of the orbit {r (pi) : n E Z) of the point pi = p (ai , ri) (which does not belong to D) coincides with the circle S(ai , ri ). Therefore S(ai , ri) fl D = 0, and hence D = 0 due to the transitivity of the Mob+(1)-action on H and the density of I' in Mob+(1). Absurdness of that shows that the groups (rl) and (r2) have finite orders, say, n1 and n2. Setting ui = hn`, i = 1, 2, we have due to commutativity of ei and ri that the elements ui = en` are elliptic, have infinite orders, and u 1 u2 0 u2u 1. It follows that the group (u1, u2) generated by them is dense in Mob+(1) - 4(+(H). Applying this fact for any point x E R3 \q U D), we see that the whole half-plane HX, a HX = 1, which contains the point x, cannot intersect the domain D. So D = d(M) is either disconnected, or simply connected, or empty. However, the development d is a covering map, and hence D must be simply connected, and d must be a homeomorphism compatible with the (non-discrete) action of G. Since this is impossible, we pass to the remaining case of the discrete group F. Let A(I,) C l be the limit set of the discrete group r = Gl1 C Mob+(1). Then D fl A(P) = 0 because D ; S3 (Lemma 6.61). If D S3\A(F), then the quotient D/G of the open in S3 \A (G) domain D by the action of G is non-compact, which contradicts to compactness of M. Therefore D = S3\A(F), and two cases are possible: either D = R3\l (i.e., A(r) = 1), or A(r) is a discontinuum, and D is simply connected. Due to Lemma 6.66, the first case is impossible. In the second case of nl (D) = 0, we have that our covering map d: M -- D is a homeomorphism compatible with the action of G. This shows that G must be discrete, which eliminates the last case (4) and completes the proof of the Theorem 6.65.

Remark 6.67. Similar arguments can be applied in dimensions n > 4. Here all closed conformal n-manifolds M with non-surjective developments d: M D C S" are

6.4. Uniformizable conformal structures

333

either uniformizable, that is they have discrete holonomy groups G C Mob(n), and M = DIG, or otherwise, their non-discrete holonomy groups G allow the following characterization, see Kamishima [1]: Suppose that there is a positive dimensional closed subgroup g C Mob(n) such

that G C g and the proper limit sets L(G) and L(g) coincide. Then the identity component go of 9 is compact, and g /go is infinite and has no torsion free normal solvable subgroups. Furthermore, there exists a g-invariant k-dimensional sphere Sk, 1 < k < n - 2, which contains L(G).

3. Uniformization of connected sums. As we have seen in Chapter 5, §2, the free product of two Kleinian subgroups G 1 and G2 in Mob (n) can be realized as a Kleinian

group G = Gi * G2 C Mob(n) provided some geometric conditions on G1 and G2. Since this Klein combination corresponds to the connected sum M = M1#M2 where M1, M2 and M are correspondingly uniformized by G 1, G2 and G, it is natural to ask whether it is a general situation, that is whether the converse is true. This question has the following affirmative answer.

Theorem 6.68. A connected sum M = M1#M2 of conformally uniformizable nmanifolds, n > 2, admits a conformally uniformizable structure such that the inclusions of the n-disk complements in M1 and M2 into M are conformal.

Proof. Let Gi C Mob(n), i = 1, 2, be the holonomy groups of the manifolds M, acting freely and discontinuously in Gi -invariant open connected subsets Qi C S", S2i C 2 (Gi ), and Mi - M1/G. Let also G = GI * G2 be the abstractly defined free product. We shall find an open connected domain 52 C S' where the group G acts freely and discontinuously as a Mobius group, and the quotient 52/ G is diffeomorphic to the given connected sum M = M1#M2.

We denote ri = 7r1(Mi), i = 1, 2, with projections pi : r -* Gi having kernels (i = ker pi. Then the connected coverings S2i -+ Mi can be described by two sets of loops {aJ}jEJ, C Ml and {af }jEJ2 C M2 whose homotopy classes [a ] normally generate (Di, so that each lift of a loop a is a loop in S2i. Now let

K = (Mt U [0, 1] U M2)/ - be a CW-complex where the point 0 is identified with a point in M1, and the point 1 is identified with a point in M2. Obviously, 7rl(K) = F1 * 172, and we have a canonical projection p : 7r1 (K) -> G = G1 * G2 whose kernel I is normally generated by {aj)J EJ1 U {c }jEJ2. Now we can consider the covering k of K defined by the normal subgroup 4) < 71 (K). The complex K can be constructed out of copies of 521 (as many as I GI G 11),

IGI copies of interval I = [0, 1], and I G/ G21 copies of 522. All these copies are connected in such a way that: (i) each copy of 01 is connected by copies of I to IG1I copies of 522; (ii) each copy of Q2 is connected by copies of I to I G21 copies of 521; (iii) no copy of 01 is joined to a copy of S22 by more than one copy of I.

We now thicken the segment I = [0, 1] in the complex K, remove the interior and consider a conformal homeomorphism o of Sr-1 x [0, 1] which preserves the equa-

334

6. Uniformization

torial sphere Sn-1 x {l/2} and maps Sr-1 x {0} to Sr-1 x {1}. In other words, the homeomorphism o- is a conformal involution preserving an arbitrary small round sphere and swapping its interior and exterior. Here we assume that Si-1 x [0, 1] is conformally embedded in S" as a spherical layer. As a result, we obtain the connected sum M = M1 #M2 so that M has a conformal structure which restricts to the prescribed conformal structures on the parts of Mi and M2 that lie in M, see Kulkarni [1, § 7]. We do the corresponding thickenings together with the corresponding gluings in the covering complex k and, as a result, obtain a conformal manifold 52 which covers the conformal manifold M = M1#M2 with the covering group G = G1 * G2. Now we can conformally embed 52 into S'. Namely, we consider the inclusion of one copy of 521 with I G i I disjoint round disks removed, into S'. Then, in each hole of this copy we can (conformally) insert a copy of 02 (with I G2 1 disjoint round disks removed) to which 01 is attached in Q. In each hole of this copy of 02 we can (conformally) insert a copy of 521 (with I G 1 I holes) to which that copy of S22 is attached in S2 etc. In all these

steps, we use the same conformal gluings or as in the connected sum M1#M2, that is each c is a conformal involution preserving a round sphere Sr-1 C S" and swapping the components of S" \ Sn-1. Due to the rules (i)-(iii), we can continue this process to obtain a conformal embedding of 0 in S". Since every conformal transformation defined on a connected open subset 0 C S" is a restriction of a Mobius transformation (Theorem 1.6), we can now regard the deck transformation group G = G1 * G2 as a

subgroup of Mob(n), and M - 0/G. Remark 6.69. The above proof also works properly in the case of non-uniformizable conformal structures on the summands M1 and M2 to produce a conformal structure on M1 #M2.

One can use the connected sum of conformal manifolds to obtain conformal manifolds which are not almost uniformizable.

Example 6.70 (Non almost-uniformizable conformal manifolds). Let Ml be a conformal (uniformizable) 3-manifold different from S3, and M2 one of the conformal

3-manifolds Ma, a E R, a/,n 0 Q, constructed in Example 6.63, that is Ma is a closed 3-manifold with (1H12 x R)-geometry whose natural conformal structure is non-uniformizable but almost uniformizable. The conformal manifold M = M1#M2

has the development d : M -* S3 onto the whole sphere S3. This together with Theorem 6.62 implies that the conformal structure on M is not almost uniformizable. One can also observe that the fundamental group nl (0 (G)) of the proper discontinuity set O (G) of the holonomy group G of M is infinitely generated because there are infinitely many disjoint G-images gi (L(G2)) C S3 of the proper limit set L(G2) of the holonomy group G2 of the (11112 x ]R)-manifold M2, L(G2) S1. This shows that the condition on rr (0 (G)) in Theorem 6.64 is essential.

In the next two sections we shall discuss conformal uniformization of manifolds obtained by torus sum of conformal manifolds. We note that if an analogue of Theo-

6.5. Conformal uniformization of "flat" connected sums

335

rem 6.68 were true for such torus sum, it would solve (in dimension 3) the Conformal Uniformization Conjecture 6.26 because, as we will see, all "prime" geometric pieces obtained by the canonical splitting of a 3-manifold have finite coverings with uniformizable conformal structures.

§5. Conformal uniformization of "flat" connected sums Let M1 and M2 be compact n-manifolds whose boundaries consist of flat manifolds EL and E?, i = 1, ..., k, respectively. In addition, we assume that the interiors of M1 and M2 admit uniformizable conformal structures and that the corresponding boundary components Et and E? are diffeomorphic. Then we can consider a flat connected sum, that is a closed manifold M = M1# flatM2 obtained by gluing each of the (flat) boundary components El to the corresponding component EL , i = 1, . . . , k. We are going to investigate the problem whether the closed manifold M1#flat M2 admits a (uniformizable) conformal structure. In particular, such a problem can be considered

for a double M of a given manifold M with flat boundary components. We start with a basic example of such a double for the Whitehead link complement M,,, = S3 \L W , see Example 4.25 and Figure 51.

1. Double of the Whitehead link complement. Let G1 be the finite index subgroup of the Picard group PSL2(Z[i1) that is generated by the following three matrices:

91 =

2]

[o

g2=[0 1],

93

=[-il 101]'

Extending the natural (Mobius) action of G1 from the extended complex plane C (identified with K2 x {0} C 1183) to the 3-space 1[83 , we define an inclusion G 1 C Mob (3)

as a discrete subgroup. Then the limit set of G1 is A(G1) = R2 x {0} C

]f83, its

discontinuity set S2 (G1) =1183\(1182 x {0}) splits up into two half-spaces, and an open fundamental polyhedron F (G 1) C 1183 can be taken as the intersection of the exteriors of the isometric spheres I (g3) and I (g3 1) with the following fundamental polyhedron

for the stabilizer H = (g1, 92) of oc in G1, see Figure 96:

A={xE1183:-1<x1
6. Uniformization

336

homeomorphic to M.

Figure 96

Now we define another discrete group G2 C M6b(3) by conjugating the group G1 action by a vertical translation y, y(x) = x + 3e3. Obviously, the group G2 = yGiy-1 has y(F(Gi)) = F(G2) as a fundamental polyhedron. Furthermore, G1 fl G2 = H = (91, g2), and the 2-sphere S = {x E 1[83 : X3 = 21 is precisely invariant for H in both G1 and G2 groups. Therefore the groups G1 and G2 with the common subgroup H = G1 fl G2, precisely invariant half-spaces D1 = {x E I[83 : X3 > 2}

and D2 = {x E R3 : x < 2}, and the fundamental polyhedra F1 = F(Gj), F2 = F(G2) and 0 satisfy all conditions of the combination Theorem 5.17. Hence the group G = (G1, yGly-1) is in fact the free amalgamated product G1 *H yGiy-1. Moreover, due to Lemma 2.22, the Kleinian group G has an invariant component, Qo C {x E 1[83 : 0 < x3 < 3}, and all the other connected components of 0 (G) are G-images of half-spaces 01 =1183 and 02 = {x E 1[83: X3 > 3}. Therefore, 2(G)/G = (c2o/G) U (c21/G1) U (22/G2) Here the (hyperbolic) manifolds 01 / G and 122/ G2 are homeomorphic to the Whitehead link complement M,u = S3 \ Lshown in Figure 51, and the conformal manifold S2o/G = Mo is homeomorphic to the link L complement shown in Figure 97.

Figure 97

6.5. Conformal uniformization of "flat" connected sums

337

We obtain this link complement from Mw by removing a regular neighborhood of a component of the Whitehead link Lw C S3 and taking the double of the obtained 3-manifold with boundary with respect to its boundary torus T2 = S1 X S1. Moreover, since the reflection xo in 1R3 with respect to the plane {x E R3 : X3 = 3/2} conjugates the groups G1 and G2 (see Figure 99), the manifold Mo has an orientation reversing conformal involution co: Mo -* MO whose set of fixed points is the incompressible (embedded) torus T2 = (S\{oo})/H C Mo. The complement Mo\ fix(cp) splits up into two 3-manifolds homeomorphic to the Whitehead link

complement M. We notice that the manifold Mo is neither hyperbolic nor universal. The first claim

follows from the fact that the incompressible torus T2 = (S\{oo})/H C Mo is not parallel to any boundary tori of the link L in Figure 97. Non-universality of the link L is due to an observation by J. M. Montesinos that Mo is a four-fold branched covering of a torus link complement in S3. This covering is shown in Figure 98 as the composition of two-sheet coverings pl and P2.

C

lool 1-1

P2

Figure 98

Now we shall finish our construction of a closed 3-manifold M,,, which is a double

of the Whitehead link complement M. = S3\Lw, by identifying the boundary tori 8N1 and 8N2 of regular neighborhoods Nl and N2 of two components of the link L, see Figure 97. This topological construction has a geometric realization based on the

338

6. Uniformization

second Maskit combination Theorem 5.18.

Figure 99

Namely, the group G = Gi *H yGiy-1 with Qo/G = Mo N S3\L has two conjugacy classes of maximal parabolic subgroups. The first class is represented by

the stabilizer in G of the origin pi = 0, Gpt = {g E G : g(O) = 0}. The second class is represented by the stabilizer subgroup GP2 C G of the point p2 = 3e3. These parabolic subgroups are isomorphic to each other and moreover, they are conjugate by a conformal transformation f = 10Z1. Here xo is the reflection with respect to the plane S\{oo} = {x E R3 : X3 = 3/2}, and xi is the reflection with respect to the sphere Si = {x E 1R3 : Ix - e3/41 = 1/4}. The sphere S1 bounds a ball B1 which is precisely invariant with respect to the parabolic subgroup Gpt in G. The image sphere f (Sl) = S2 bounds a ball B2 which is precisely invariant with respect to the parabolic subgroup GP2 in G, see Figure 99. One can easily check that the subgroups H1 = Gp, and H2 = GP2 in G and the balls B1 and B2 satisfy all the conditions of Theorem 5.18. So, the group G. = (G , f) c Mob (3) is a discontinuous HNN-extension of G by the transformation f . Its discontinuity set 0 (G,,) has an invariant component n,, C 0o, and the closed manifold S2,,/ G, is homeomorphic to the double Mw of the Whitehead link complement Mu, = S3 \L,,.

2. Non-degenerate B-groups. In the general situation of a connected sum M = Ml#flatM2 of two manifolds along their flat boundaries, we can apply Theorem 6.68 on connected sums to reduce the problem of conformal uniformization to the case

when both compact manifolds M1 and M2 are irreducible and hence are K(n,1)type manifolds. In this case, the holonomy groups Gi = d , (In (Mi)), i = 1, 2,

6.5. Conformal uniformization of "flat' 'connected sums

339

of uniformizable conformal structures ci on int Mi are geometrically finite and have invariant contractible components Qi of their discontinuity sets n (Gi) C S" such that (int Mi, ci) are conformally equivalent to S2i / Gi. Such Kleinian groups Gi C Mob(n) are called non-degenerate B-groups in S" (compare § 2.2 of Chapter 2). In particular, if Ml and M2 are in addition atoroidal 3-manifolds, Theorem 6.29 implies that they have uniformizable conformal structures on int Mi induced by (unique) hyperbolic metrics of finite volume, hence the holonomy groups Gi preserve round balls Qi C S3. In this case, the basic example of the double of the Whitehead link complement shows a method of conformal uniformization of such a connected sum

M = Ml#flatM2. In the general situation, we have the following uniformization theorem. Theorem 6.71. Let Ml and M2 be compact n-manifolds with boundary whose interiors have uniformizable conformal structures cl and c2 with non-degenerate B-groups

G 1, G2 C Mob(n) as the holonomy groups, and M = M1# flatM2 be a flat connected sum of Ml and M2 with conformal gluings of the boundary flat manifolds. Then M admits a uniformizable conformal structure c whose holonomy group G is a non-degenerate B-group. Furthermore, there are conformally embedded fiat (n -1)manifolds Ei C M, i = 1, . . . , k, such that (M \ Ui Ei, c) is conformally equivalent to (int M1, cl) U (int M2, C2).

Before we go on to prove this theorem, we describe the so-called "parabolic combination" of B-groups, which preserves the class of non-degenerate B-groups.

3. Parabolic combination. Let G1, G2 C Mob(n) be two non-degenerate B-groups with Gi -invariant contractible components S2i c 0 (Gi ), and Ni = S2i / GZ, i = 1, 2. In addition, we assume that the quotients Ni are noncompact, and all their ends ej and ej are pairwise conformally equivalent. Due to Theorem 4.16, all the ends e are parabolic cusp ends, and the equivalence condition means that the corresponding disjoint standard parabolic neighborhoods U of these e are such that hj U?, j = 1, . . . , k, for some conformal homeomorphisms hj. Due to Lemmas 4.56 4.58, the maximal parabolic subgroups G 1 C G 1 and GJ C G2 that correspond to each pair of ends ei 1 and e have ranks equal (n -1) and are conjugate by lifts hj E Mob(n)

of the hj, hjG h 1 = G G. Moreover, the standard parabolic neighborhoods U are universally covered by "horoballs" B! C S2i with disjoint GZ-orbits of closed horoballs, ) : g E G/Gi }, g{g(Bi

(Bj) fl g'(Bj,) = 0 for [g] # [g'],

(6.12)

and each of whose boundary spheres 8 B contains the corresponding parabolic point pj fixed by the maximal parabolic subgroup Q., i = 1, 2 and j = 1, ... , k. Therefore,

for j = 1, ... , k, the lifts hj of conformal homeomorphisms hj : U -- U? have the

6. Uniformization

340

following properties: (6.13)

We also notice that all transformations h j and parabolic subgroups G and G commute

with the reflections JJ1 and ./] with respect to the boundary spheres a B and a B? , correspondingly, in the following sense: Jj'g=gJJ`

forgEGj',

h J JJ1

i = 1 , 2, j = 1... , k;

= J?hj I = I .... k . ,

(6.14)

Now we can define a parabolic combination of two non-degenerate B-groups G, and G2 as the following process producing a Kleinian group G C Mob. In the first step, we define a free amalgamated product (see Figure 100):

Go = G1 * ftG2f1 1,

f1 = h,_1Ji E Mob(n) .

(6.15)

GfI

Figure 100

Here, due to (6.13) and (6.14), the groups G1 and f1G2 fi 1 have a common (amalgamated) subgroup G1 = G1 fl f1 G2 fi 1 and, together with the (n -1)-sphere aBi, satisfy the conditions of the first Maskit combination Theorem 5.17. Hence the obtained group Go C Mob(n) is a Kleinian group with an invariant connected component 00 c S2 (Go). Arguments similar to those in the proof of Theorem 6.68 show that the component S2o can be obtained by sequential gluings of (conformal

6.5. Conformal uniformization of "flat" connected sums

341

images of) copies of 02 \ G2(Bi) to copies of S21 \ G1(Bi) (and vice versa), along horospheres g(aBi \ {pi }) and g'(aBi \ {pi}) where g E G2/G1 and g' E G1/Gi. The constructed in (6.15) group Go has (k - 1) pairs of conjugacy classes of maximal parabolic subgroups represented by the groups G,

and

fl

J,- EMob(n),

(6.16)

which are conjugate by the MObius transformations fj = h j 1 J1 2h j J 1, j = 2, ... , k. Each of the transformations fj maps the exterior of the ball B] onto the ball fl (B ),

which in turn is precisely invariant in Go with respect to the parabolic subgroup fl G fI 1 in (6.16). So we can apply the second Maskit combination (Theorem 5.18) and obtain the desired group G as the following HNN-extension of Go by the elements f2, , fk: G = (G1 *Gi f1G2f1 ', f2, ..., .fk) (6.17)

Theorem 6.72. Let G C Mob (n) be a group obtained by the parabolic combination (6.17) from two non-degenerate B-groups G1, G2 C Mob(n) with Gi-invariant contractible components Qi C E2 (Gi), i = 1, 2. Then G is a non-degenerate B-group. Proof. Since all HNN-extensions in (6.17) and the free amalgamated product in (6.15) were done by using the combination Theorems 5.17 and 5.18, the group G is a geometrically finite Kleinian group and has an invariant component Q c 0 (G). In order to show that this component is contractible, we notice that it is obtained by gluing (conformal) copies of the following two domains S29 C Sn : E29

= ci \Gi (B 1 U ... U Bk),

i = 1, 2.

(6.18)

Each domain 09 is the complement in S2i to the Gi -invariant infinite family of disjoint closed horoballs g(B J ), g E Gi. Obviously, the domains S2,9 are contractible domains

because of the natural retraction of each such horoball B.i to the parabolic fixed point see Figure 101. pj E a

pi,

Figure 101

6. Uniformization

342

Now we notice that the component 0 is obtained as the result of a process similar

to that in the proof of Theorem 6.68. Namely, we start with 0? and attach to it, along each of the horospheres au, g (BJ) , g E G 1, j = 1, ... , k, copies of SZ2 (up to conformal transformations). Then, into each hole of these copies of 522, we attach (along corresponding boundary horospheres) conformal images of 520, etc. Thus, we have an increasing sequence of contractible open domains in Sn because, in each gluing step, we glue two contractible domains along an horosphere isomorphic to lRn. Since the component 0 C S2 (G) is the limit of this increasing sequence { `gym}, M. C Dm+I, of contractible domains, it is contractible itself, compare Brown [1].

Proof of Theorem 6.71. Let GI and G2 be non-degenerate B-groups uniformizing conformal structures on the interiors of manifolds M1 and M2, and S2i and 522 be their

contractible invariant components of the discontinuity sets. Then we may identify compact manifolds MI and M2 with the quotients

20IGI = M1,

2/G2 = M2,

(6.19)

U Bk) are obtained from 0,9 in (6.18) by attaching all boundary horospheres, and the boundary components of Mi are (n - 1)-dimensional flat manifolds where S29 = SZi \Gt (Bi U

i=1,2, j=1,...,k.

(6.20)

Now, applying the process of parabolic combination (6.17), we construct a new Kleinian group G C Mob (n) which is a non-degenerate B-group due to Theorem 6.72. Obviously, the G-quotient of the contractible G-invariant component S2 C Q (G) is

a closed manifold obtained from M1 and M2 in (6.19) by conformal gluings hj of the boundary flat components (6.20), and the inclusions Mi C M = MI #flatM2 are conformal embeddings.

Remark 6.73. Theorem 6.71 provides a sufficient condition for a closed 3-manifold M canonically decomposed into two hyperbolic 3-manifolds MI, M2 of finite volume to have a uniformizable conformal structure which induces on blocks Mi the given complete hyperbolic metrics (in the way shown in § 8).

§6. Conformal uniformization of Seifert manifolds As we have already seen, five 3-dimensional geometries that allow conformal models are S3, ][83, H3, IH[2 x l[i; and S2 x R. On the other hand, manifolds with Nil and Sol geometries have no conformal structures. Therefore, among Seifert 3-manifolds, we have to study only those which have the §-L2-geometry.

6.6. Conformal uniformization of Seifert manifolds

343

Let E be a closed orientable Seifert 3-manifold whose orbit space X admits a hyperbolic structure. Then E may be finitely covered by a circle bundle E over a closed hyperbolic surface k of genus > 1. If the Euler number e(E) of this St-bundle is zero then its total space is homeomorphic to the product k x S' and has an (1I2 x R)-

structure. Namely, as in Lemma 6.66, one has that k x S1 = (R3 \ R et) IG where G C PSL2(R) C M6b(R3\1[8 - el) is a torsion free discrete subgroup uniformizing

the surface X, X = R2 /G. So we may assume that the Seifert manifold E is finitely covered by a circle bundle S(g, e) over a closed surface 5g of genus g > 1 and its Euler number e(E) is at least 1. In this case, the S1-bundle S(g, e) and the Seifert manifold E allow the SL2geometry, see § 6, Chapter 1. However, the invariant Riemannian metric on SL2 (IR) is not conformally flat, and we cannot conformally uniformize E in the way similar to the above. Nevertheless, we shall prove the following uniformization theorem: Theorem 6.74. Let E be a closed Seifert fibered 3-manifold with a hyperbolic orbit space X. Then there is a finite covering E -+ E admitting an uniformizable conformal structure, that is E is virtually conformally uniformizable. In order to construct this uniformization, we start with a concrete 3-manifold E.

Lemma 6.75. Let S(12, 1) be the total space of a circle bundle over the closed orientable surface X of genus 12 with the Euler number e = 1. Then there exists a torsion free discrete group G = G(12, 1) C M6b(3) such that 0 (G)/G is homeomorphic to S(12,1).

Proof. To construct the group G = G(12, 1), we shall construct its fundamental polyhedron P C S3. It will be homeomorphic to the solid torus S1 x B2, and its sides will be spherical annuli. As a matter of fact, the sides of P lie on round spheres Si in J3 whose interiors Bi, i = 1, ... , 48, cover the double-twisted band L shown

in Figure 102 in such a way that B1 f1 BJ 0 0 if and only if Ii - j I = 1 mod 48. Additionally, spherical annuli bi = 8B1 \ (BL-1 U Bi+t) should be gi-equivalent to bi+2 by some gi E Mob(3), i = 1, 2 mod 4. To build our family {S1 }, we fix a number 6 = n/8 and take the first sphere Si = S2(Cl, r) as the sphere of radius r = tan(0/2)/ cos(0/6) centered at the point C1 E JR3 with coordinates (1, 1-tan(9/2), 0). Then the spheres S2, S3, S4 and S5 are obtained by (counter-clockwise) rotating the sphere S1 about the vertical line passing through the point (2, 1, 0), by angles 0, 20, 3 0 and 49, respectively. Similarly, the spheres 548, S47, S46 and S45 are obtained by (counter-clockwise) rotating S1 about the vertical line passing through (0, 1, 0), by the same angles, see Figure 103. It is easy to see that the centers of the spheres S4 and S5 lie on the first coordinate axis, and the exterior dihedral angles bounded by any two neighboring spheres equal

0/3. We also see that each of the side-pairing transformations gt : b1 -a b3 and 92 : b2 -* b4 can be taken as the composition of the inversion with respect to the sphere Si, i = 1, 2, and the reflection with respect to the vertical plane which is the symmetry plane for the corresponding pair of spheres. Similarly, we define the

344

6. Uniformization

side-pairings g23 : b45 -+ b47 and $24 : b46 -* b48. All of them preserve the xy-plane.

Figure 102. Ball covering of double-twisted band

Figure 103

Now we pass from the xy-plane to the xz-plane and define the next set of spheres. Namely, the spheres Si, 6 < i < 17, are obtained by clockwise rotating S5 about the line passing through the point (2, 0, -1) and parallel to the second coordinate axis, by angles 0, 29, ..., 129, respectively. Similarly, the spheres S18, S19 and S20 are

6.6. Conformal uniformization of Seifert manifolds

345

obtained by (counter-clockwise) rotating the sphere S17 about the line passing through the point (0, 0, -1) and parallel to the y-axis, by angles 0, 20 and 30, see Figure 104.

Figure 104

To finish the construction of the family {Si }, we rotate all 24 spheres we already defined, around the y-axis by the angle Yr. Clearly, the balls Bi = int Si cover the band, and all dihedral angles bounded by neighboring spheres equal 0/3. We also have 24 side-pairings gi : bi -- bi+2 as compositions of the inversions with respect to Si and reflections with respect to the mirror planes for the pairs of spheres Si and Si+2,

i = 1, 2 mod 4. Finally, we define the solid torus P C S3 and the group

G C Mob(3) as follows: 48

P=nextBi, G=(gi, i=1,2mod4, 1
(6.21)

i=1

We notice that all edges of P (= intersections of neighboring annuli bi fl bj, I i - j I = 1 mod 48) are G-equivalent, and the sum of all dihedral angles in P equals 480/3 = 2ir. Furthermore, every point x0 E a = b1 fl b48 is fixed by the product of 12 commutators, 45

n(x0) =

JI i=1

Lgi+1, gi 1](x0)

i=1 mod4

= Lg46,45g1] ... 1g6, g5 ] - 1g2, gl 1](x0) = x0

(6.22)

6. Uniformization

346

Using the properties of P and G in (6.21) and naturally extending P C S3 into the hyperbolic space E3[4, aH4 = S3, as a convex polyhedron P bounded by planes Hi, 8IR[i = Si, with sides pairwise identified by the generators gi of the group G (acting in ffl[4 by isometries), we can apply the Poincare-Aleksandrov theorem on fundamental polyhedron. Namely, due to Corollary 4.30, we have a tessellation G(P) of 1H14, and hence our group G is discrete, a P fl S3 = P is its fundamental polyhedron in S3, and we have the following presentation of G with 24 generators: 45

fl

G = (gi; i = 1, 2 mod 4, 1 < i < 46 I

[gi+l, gi 1] = 1).

(6.23)

i=1 i=1 mod4

Now, let xo be one of the two points in bo fl b48 fl L, so that (6.22) holds. Since all generators gi E G map the arcs bi fl L onto the arcs bi+2 fl L, we clearly see that all images of the point xo with respect to the elements of G are represented by initial subwords of the word Il in (6.22), that is the points g1(xo), ... , [g2, gj 1](xo), g5[g2, 91 1](xo), ... ,

[g6, 95 1][g2, 911](xo), ... , n(xo) = xo

(6.24)

lie on the same boundary curve of the band L as xo does. So, pairwise joining those points in (6.24) that lie in a common annulus bi by the arc yi C bi fl L, we obtain a simple closed curve y = Ui yi c a P which is invariant under the side-pairing of P, see Figure 105. Obviously, the loop y c a P has the linking number 1 with the central curve of the band L, or, equivalently, the linking number of y and the central circle of the solid torus S3 \ P equals 1. We also notice that y is homotopic on the torus aP to the curve a ,B where a = b1 fl b48 and the homotopy class [0] of the loop ,B C 8 P generates the kernel of the homomorphism r1(aP) - 7r1 (P), provided appropriate choices of orientations on the above loops.

gi

gi+1 Figure 105

Now we are ready to show that the Kleinian group G = G(12, 1) C M6b(3) conformally uniformizes the circle bundle S(12, 1) over a closed orientable surface X

6.6. Conformal uniformization of Seifert manifolds

347

of genus 12, with the Euler number e = 1, that is that S(12, 1) = 0 (G)/G = PI G. Indeed, let T C int P be a torus which is parallel to the boundary aP, F be the bounded component of P \ T, and 7r : P --+ PIG be the natural projection. Then, due to (6.23), the manifold M = Tr (F) is homeomorphic to the product X' x Sl where X' is obtained from a closed oriented surface X of genus 12 by removing a disk B2, and the manifold T = 7r (P \ F) is a solid torus homeomorphic to B2 X S1. The manifold 0 (G)/ G can be now obtained by gluing the solid torus T to the boundary torus of

M X' x S1 in such a way that the generator a = {x} x St of K1(T) is glued to the loop a' C 8M, a' = 7r(aT), and the loop 8B2 x {0} C am -- aB x S1, 0 E S1, is glued to the loop y' = .rc (yT) C a T. Here aT and yT are the loos on the torus T C P which are parallel to a, y C a P. Hence our gluing of M and T in fact produces the circle bundle S(12, 1) over X with the Euler number e = 1, see § 1, Chapter 5 for its definition. Remark 6.76. It is possible to adjust the above construction of G(12, 1) c Mob (3) to produce a similar Kleinian group G(10, 1) C Mbb(3) which conformally uniformizes the circle bundle S(10, 1). One can do that by decreasing the number of spheres in our family {Si }, from 48 to 40.

Proof Namely, we can replace 16 spheres Si, 13 < i < 28, centered at points on the xz-plane by only 8 new spheres, 13th through 20th. To do that, we fix the intersection circles S12 fl S13 = 81 and 528 fl s29 = 32, and consider a new sphere S13 with radius R = tan(9/2) 4 ttan2(9/6) centered at the point (2 - 2 tan(9/2), 0, -2), so that it intersects the sphere S12 along the same circle 81. Then, similarly to the spheres 514, S15 and S16, we obtain new spheres S1'4, S15 and 516 by clockwise rotating the sphere S13 about the line which is parallel to the y-axis and passes through the point (2, 0, 0) (instead of (2, 0, -1)), by angles 6, 29 and 39, respectively. Four additional spheres Si', 17 < i < 20, are obtained by counter-clockwise rotating the sphere

516 about the line which is parallel to the y-axis and passes through (-2, 0, 0), by angles 0, 20, 30 and 49 = 7r/2. Notice that 5210 fl S29 = 82 = S28 fl S29, and hence we can renumerate the new chain of spheres as {Si'} 1
annuli bi and bi+2 where gl: Si - 51+2, i = 1, 2 (mod 4) are obtained as compositions of inversions in Si' and reflections in the mirror planes for pairs (Si , 51+2). In this way, we have that all properties of the family {Si } are still true for the new family {S}, but the sum of all dihedral angles of the polyhedron P is now less then 27r. To make the sum of dihedral angles equal 27r, we can continuously increase the radius r of the original spheres S1, so that the sum of the dihedral angles becomes equal

to 27r. Indeed, clear computations show that the new dihedral angle 9(r) between spheres Si (r) and Si+1(r), the radius p (r) of circles 81 (r) and 82(r), and the radius R(r) of bigger spheres Si(r), 13 < i < 20, are as follows:

p(r) =

r2 - tan2(9/2),

R(r) = r2 -{- 3 tan2(9/2),

6. Uniformization

348

0(r) = 2 arctan

'/_r2 - tan2(0/2) tan(0/2)

In particular, it shows that there exists ro, tan(0/2)/ cos(0/6) < ro < 1/2, such that the angle B (ro) equals it/2, and all non-neighboring spheres Sj (ro) and Sj'(ro),

Ii - j I > 2 mod 40, are disjoint.

Figure 106

Since, for r = rp, the sum of all dihedral angles is more than 31 9(ro) > 21r, there exists a value r',

tan(9/2)/ cos(0/6) < r' < ro < 1/2,

such that the sum of all dihedral angles of the solid torus P = P(r) equals 27r. This completes our construction of the Kleinian group G(10, 1) C M6b(3) which conformally uniformizes the circle bundle S(10, 1). Corollary 6.77. Let g and e be integer numbers, 0 < e < (g - 1)/9. Then there is a torsion free Kleinian group G = G(g, e) C M6b(3) which conformally uniformizes the total space S(g, e) of the circle bundle over the closed orientable surface Xg of genus g whose Euler number equals e, that is S(g, e) = 2 (G)/ G. Proof. Let I' be a subgroup of index j in the group G(10, 1) constructed in Lemma 6.75 and Remark 6.76. Applying Lemma 5.7 to the covering

mr)/r -* n(G)/G = S(10, 1) and the Riemann-Hurwitz formula to the j-sheet covering Xg -* Xto of Riemann surfaces of genera g = 10 and g, that is 2g - 2 = j (2g - 2), we immediately have that g = 9j + 1, and that r = G (9j + 1, j) uniformizes the circle bundle S (9 j + 1, j)

6.6. Conformal uniformization of Seifert manifolds

349

with any given Euler number j > 1. Hence, to complete the proof, we only need to construct the group G(9e + k, e) C Mob(3) for any given integer k > 1. Let B C R3 be the halfspace whose (vertical) boundary plane contains the line Q = {z = y = 0} and is orthogonal to the xy-plane, and h E Mob(3) be the following hyperbolic transformation, see (6.23): 29

h=

11

[9i+1, gi 1] = [g331, g34] . [g371, 9381.

i=1 i=1 mod4

(6.25)

The transformation h (and the whole free group F4 = (g33, g34, 937, gas) ) preserves the disk in the xy-plane bounded by the circle C centered at (0, 1, 0) with radius /- (r')2 + tan2(0/2), so that C is orthogonal to the eight spheres Si(r), 33 < i < 40, constructed in Remark 6.76. Moreover, using the fact that the closed half-space B C IR3 is precisely invariant in G(10, 1) with respect to its subgroup generated by h, we can apply the combination Theorem 5.17 to show that the group G(10,1) is the free product of its free subgroups F4 and G'(8, 1) C G(10, 1), with the amalgamated cyclic subgroup (h).

Now we can choose a subgroup I, C G(10, 1) of any prescribed index e > 0 such that it contains the group F4. Hence, due to the above arguments, the group F = G(9e + 1, e) can be obtained by the first Maskit combination (Theorem 5.17) from its free subgroups F4 and G'(9e - 1, e), G(9e + 1, e) = F4 *(h) G'(9e - 1, e).

(6.26)

Using this decomposition of the group G (9e + 1, e) into the free amalgamated product,

we can construct the group G(9e + 1 + k, e) with e >_ 1 and k > 1 by replacing the subgroup F4 = (g33, g34, g37, 938) in (6.26) by a free (Fuchsian) group F4+2k C Mob(2) C M6b(3) with the following properties: (i) F4+2k preserves the disk in the xy-plane bounded by the circle C; (ii) (h) C [F4+2k, F4+2k];

(iii) the half-space R3 \ B is precisely invariant in F4+2k with respect to its subgroup (h). Clearly, such a group F4+2k exists and, together with the group G' (9e -1, e) in (6.26), satisfies the conditions of the combination Theorem 5.17, with the cyclic amalgamated

subgroup (h). Therefore the group G C Mob(3), G = (F4+2k, G'(9e - 1, e)) = F4+2k *(h) G'(9e - 1, e),

(6.27)

uniformizes the circle bundle S(9e - 1 + (2 + k), e) = S(9e + 1 + k, e). This bundle can be obtained by gluing the boundary tori of the products X'+k x S' and X9e_1 x S1 Here each surface X' is obtained from the closed orientable surface XJ of genus j by

6. Uniformization

350

removing an open disk B2, and the gluing is similar to the gluing in the definition of a circle bundle with the Euler number e, see § 1, Chapter 5.

Proof of Theorem 6.74. Let E be a closed Seifert manifold with a hyperbolic orbit space X. We may assume that E is orientable and has non-zero Euler number. Then, due to (5.2), we have an exact sequence:

1-+7G-*

nl(E)4F-+ 1,

(6.28)

where F is a discrete co-compact subgroup of Isom 3H[2. This implies that there is a finite-index torsion free subgroup Fo c r such that 1312/ I'o is a closed orientable surface of genus g > 10. Due to the exact sequence (6.28), the q5-preimage group Go C n1 (E), Go = 0-1(I'o), has the following presentation, with e # 0:

(al,b1,...,ag,bg,cI Taking d = ce, we then obtain that the group G C Go, G = (at, b1, ... , ag, bg, d I

[al, bi] .... [ag, bg] = d) ,

is a finite index subgroup of Go, and hence of 7rl(E). The subgroup G C irl(E) defines a finite covering Eo -+ E where Eo is a circle bundle over a closed orientable surface of genus g at least 10 with the Euler number e(Eo) = 1. Hence, due to Corollary 6.77, Eo admits a uniformizable conformal structure.

§7. Torus sums of conformal structures Let M1 and M2 be compact 3-manifolds with boundary toriT1 C a M, and T2 C 0M2, whose interiors admit uniformizable conformal structures (M1, ci) with parabolic ends

corresponding to tori Ti and T2. Then, as we have seen in § 5 (Theorem 6.71), a torus sum M = M1 Uh M2 obtained by gluing M1 and M2 by a homeomorphism h : T1 -+ T2 admits a uniformizable conformal structure (M, c) if the gluing map h is isotopic to a conformal map of the boundary tori. Hence in general, in order to find a conformal structure on Mi Uh M2, we need to deform conformal structures on M1 and M2 so that the restrictions of the holonomy homomorphisms dl and d2 of new structures to the boundary tori T1 and T2, respectively, give conformally conjugate discrete subgroups of the holonomy groups of M1 and M2. Although in general (in particular, for hyperbolic manifolds) it is an open problem whether such deformations are possible, it can be done in many cases, as it is implied by the following result.

Theorem 6.78. Let M be a closed Haken 3-manifold with unsolvable fundamental group such that the canonical splitting of M into hyperbolic and Seifert submanifolds does not involve gluing hyperbolic manifolds with hyperbolic or Euclidean ones. Then some finite covering k of M admits a uniformizable conformal structure.

We illustrate the main idea of the proof of this uniformization theorem due to Kapovich [1] by two examples which explain two basic gluings of block manifolds.

6.7. Torus sums of conformal structures

351

Example 6.79 (Gluing Seifert manifolds). Let Ml = X1 x S1 and M2 = X2 X St, where Xi, i = 1, 2, are compact oriented surfaces of genera gi > 0 with connected S1. The product structure on Mi introduces a natural "meridianboundaries 8Xi longitude" basis in Trl(aMi), see § 1, Chapter 5. Let us consider a manifold M = Ml U f M2 obtained as a torus sum of manifolds Mi and M2 via a homeomorphism f : 8M1 -+ 8M2 of the boundary tori. In the natural bases of rri (3M, ), i = 1, 2, the homeomorphism f can be represented by a matrix A E GL2 (Z),

A=

(ail a12)detA = -1, a21

a22

a21 = 1.

(6.29)

Then we consider Kleinian groups G1 = G(gl, Ia221), G2 = G(g2, jail l) C Mob(3) which conformally uniformize the total spaces of the circle bundles S(gl, Ia221) and S(g2, Jail I). Due to Corollary 6.77, such groups exist for sufficiently large genera gi and g2. Conjugating the groups G1 and G2 by Mobius transformations, we may assume that their actions on S3 are such that the complements of their fundamental polyhedra Pl and P2 (which are unknotted solid tori in S3) are isotopic to the link shown in Figure 107.

Figure 107

Now we can apply the Klein combination Theorem 5.12 and see that

G = (G(gi, 1a221) , G(g2, jail 1)) = G(gi, 1a221) * G(g2, Jai 11) is a Kleinian group, and P = Pi fl P2 is its fundamental polyhedron. Let T C int P be a torus isotopic to both boundary components a P1 and a P2, and let PO C Pi and P° C P2 be the corresponding components of the complement S3 \ T. Then the quotient manifold N = S2 (G)/ G = P/G can be obtained as follows:

N = {(P1 \ PO)/G1} U {(P2 \ P°)/G2} Pz M1 Uh M2.

(6.30)

T

Looking at the gluing map h : a M1 -+ 8M2 in (6.30), we see that the S 1 -fiber in 8Ml is identified by h with the loop on aM2 that wraps once around aM2 and Jaii I times along the S 1-fiber of aM2 C X2 X S 1 (assuming an appropriate choice of orientations).

Similarly, the S1-fiber in aM2 C X2 x S1 is identified by h with the loop on aMi

6. Uniformization

352

that wraps once around 8M1 and 1a221 times along the Sl-fiber of aM1 c 1 x Sl. Hence, the associated with h matrix B E GL2(7L) has the form: B

_

-(

fall

1 ± alla22

1

fa22

Conjugating the groups G1 and G2 by orientation reversing Mobius transformations makes an appropriate choice of the orientation, and then B becomes equal to the matrix A in (6.29). This shows that the manifold N in (6.30) is homeomorphic to the original torus sum M = MI U f M2. We remark that this construction works in the general case of gluing Seifert manifolds along boundary tori, as well. Indeed, one can always find a finite covering of a given torus sum M of Seifert manifolds such that the corresponding coefficients a21 of the gluing matrices A are equal to 1 for every pair of adjacent manifolds (as in the example above). Using such gluings, one can conformally uniformize finite coverings of graph-manifolds, see Kapovich [1] for details. For a general account on geometric uniformization of graph-manifolds, in particular for conformal and isometric uniformization, we refer reader to Buyalo-Kobel'skii [1, 2].

The second type of gluings of block-manifolds obtained by the canonical decomposition of a closed 3-manifold is a gluing of hyperbolic and Seifert manifolds along their boundary tori. Although a complete hyperbolic structure on a finite volume manifold M is rigid, there are deformations of such a manifold M which correspond to Dehn surgeries along its boundary tori, that is to homomorphisms 4t : nl (M) -m PSL2 (C) - Isom+ 1H13 with discrete images, see Thurston's nonrigidity Theorem 4.31. Obtained in such a way closed hyperbolic manifolds M, _ 1H13/ I',, ri = 0i (irlM), have simple closed geodesic loops aj which are universally covered by the axes of loxodromic subgroups Ply C ri whose preimages 0i 1(rl;) are parabolic subgroups in n1 (M). In other words, of 1(F) are the fundamental groups of the boundary tori of the original manifold M. For sufficiently small sneighborhoods U6(aj) C M, (homeomorphic to solid tori) of the geodesic loops aj, their complement Ml \ 1Jj UE (aj) is homeomorphic to the manifold M. We can use such complements as conformal block-manifolds glued to boundary components of Seifert manifolds. We illustrate this idea by the following construction. Example 6.80 (Gluing Seifert and hyperbolic manifolds). Let N be a closed hyperbolic manifold, N = H3/ G 1, nl (N) - G1 C Isom ]H[3, and a C N be a simple closed geodesic corresponding to a (pure) hyperbolic subgroup I,,, C G1 with invariant axis

a C ]E[3 and a hyperbolic generator g, We notice that Fa = (ga) preserves any hyperbolic half-plane Ha C ]H[3 bounded by the geodesic a. For a sufficiently small s > 0, an s-neighborhood UE (a) C N of the loop a is a solid torus, and the interior of the complement Ml = N \ U,(a) is a hyperbolic manifold of finite volume, see Kojima [1].

6.7. Torus sums of conformal structures

353

As the second manifold M2, we consider a Seifert manifold with one boundary torus, M2 % S x S1, where S is a hyperbolic surface of genus r > 1 with one boundary component (at infinity). We can glue M1 and M2 along their boundary tori as follows. We may assume that the group G1 acts in S3 as a MObius group preserving the unit ball B3 = B3 (0, 1) B13, and its hyperbolic subgroup ra C G 1 preserves around circle y C II82 x {0} c R3

orthogonal to the boundary sphere S2(0, 1) = aB3. In other words, the la-axis a is the intersection y fl B3, and we define the hyperbolic half-plane Ha C B3 as the intersection of the round disk B2 = B3 fl (II 2 x {01) and the disk A in the plane 2 x {0) bounded by the circle y, see Figure 108.

Figure 108

Let C (S) be the convex core of the surface S, see (4.12) and Theorem 3.19. Deforming the hyperbolic surface S, we may assume that the geodesic boundary ,B = 8 C(S) has the same length as that of the geodesic loop a C N:

length(a) = lenght(f).

(6.31)

Let Us (i) be a S-neighborhood of the geodesic 6 in the surface S. We can choose a uniformization of the surface S by a discrete torsion free group G2 C Mob(A) C M6b(3) acting by hyperbolic isometries in the disk A = H2 = S such that the geodesic ,6 is covered by the are A n a B3. Moreover, we choose the number S > 0 (the size of S-neighborhood of P) so that

arccos(1/ cosh(S)) = aresin(1/ cosh(s)) .

(6.32)

This condition implies that the surface X = C(S) \ Us (,B) is universally covered by a subdomain X C A which is contained in Ha fl UE (a) and whose boundary curve aX lying at the distance S from the geodesic ,B = A fl aB3 (in the hyperbolic metric on A) is in fact the curve 8UE(a) fl Ha = aUE(a) fl 8X. For the choice (6.31) and

6. Uniformization

354

(6.32) of the uniformizing group G2, we have that its action in S3 has the following properties: (i) the circle 8 A (which is orthogonal to 8 B3) is G2-invariant;

(ii) ga belongs to G2, and (ga) lies in the commutator subgroup [G2, G2], i.e., it corresponds to nl(0C(S)) C 7ri(S). Now, if we define a homogeneous Riemannian metric on a circle S1, then the product Riemannian metric on X x S1 induces a uniformizable conformal structure (as in Lemma 6.66). Choosing an appropriate length of the circle S1, we may assume that the product X x S1 is covered by a subdomain S2 of the discontinuity set 0 (G2). The part S2 \ 3A of the domain 0 can be obtained by rotating the universal covering space X C An B3 about the circle 8 A; in particular, SZ C UE (a). Hence we can apply the first Maskit combination (Theorem 5.17) to the groups G1 and G2 with common subgroup ra = (ga) and obtain a new Kleinian group G C M6b(3), G = (G1, G2) = G1 *r,, G2.

The group G conformally uniformizes a manifold M which is obtained by gluing together the Seifert manifold M2 = X x S1 and the hyperbolic manifold M1 of finite volume along the boundary tori. We notice that the above conditions (6.31) and (6.32) allowing a conformal gluing of boundary tori of M1 and M2 are made possible by appropriate deformations of the original manifolds. Since the condition of the pure hyperbolicity imposed on the subgroup I'a is very restrictive, the construction should be altered to cover the general case, see a scheme in Kapovich [1]. Namely, one needs to replace the block group G2, corresponding to the trivial circle bundle over X, by a group obtained by small deformations from the groups G(r, e) constructed in Corollary 6.77.

Remark 6.81. In order to prove the Conformal Uniformization Conjecture 6.26, one needs to figure out how to uniformize torus sums of hyperbolic manifolds - in addition to the uniformization results in § 4-6. There is no need to consider the case of gluing hyperbolic and Euclidean manifolds. Indeed, there are only two (orientable) Euclidean 3-manifolds with boundary. Those are the torus shell T2 x [0, 1] and the thickened Klein bottle (Example 1.4), the latter being also double covered by T2 x [0, 1]. Hence, any torus sum H U f E of hyperbolic H and Euclidean E manifolds has a double covering which is obtained by gluing together two copies of the hyperbolic manifold H. So we can apply Theorem 6.71 if the attaching map f is isotopic to a conformal one.

In the general case of gluing two hyperbolic manifolds M1 and M2, one can first deform M1 and M2 to get closed hyperbolic manifolds Mi and M2 with removed solid tori which are s- and 8-neighborhoods of simple geodesic loops a C Mi and P C M2, see Example 6.80. Then the problem of conformal gluing of their boundary tori 8M,' -* 8M2 can be reduced to the problem of cone deformations of conformal structures on Mi and MZ along the geodesics a and ,B, compare Chapter 7, § 4. If this

6.8. Canonical Riemannian metric on conformal manifolds

355

were possible, we would glue the deformed manifolds in such a way that the manifold M = MI U f M2 would be conformally equivalent to the quotient SZ/G where (up to conjugation of groups G1 and G2)

G = (GI, G2) = G1 *r G2

(6.33)

is a free amalgamated product of the groups G1 and G2 uniformizing Mi and M2, respectively, and the amalgamated subgroup F is conjugate to both loxodromic groups

(ga) C G1 and (gf) C G2. Here ga and g# correspond to the geodesic loops a C MI and ,B C M2. Schematically, this gluing and a geometric realization of the amalgamated free product (6.33) are shown in Figure 109.

Figure 109

§8. Canonical Riemannian metric on conformal manifolds The aim of this section is to present a smooth canonical Riemannian metric km on a given conformal manifold (M, c) such that it induces the original conformal structure c.

6. Uniformization

356

The main idea for our construction has two sources. First of all, it is based on a method of the so-called maximal round ball coverings of open subsets in the n-sphere

S', or equivalently, maximal round ball coverings of the conformal manifold M. The second source is the Kobayashi [1] construction of biholomorphically invariant pseudo-metrics on complex-analytic spaces. Namely, we replace the Kobayashi notion of holomorphic chains of 1-dimensional complex discs by the chains of "round balls" in a conformal manifold, that is chains of the hyperbolic n-spaces up to the conformal

development d : Al -- ) S'. 1. Maximal ball coverings. Let D be an open domain in the n-sphere S" whose boundary aD contains at least two points, JaDI = card(aD) > 2. We shall show that there is a canonical stratification of such a domain, which is invariant with respect

to the group Mob(D) of Mobius automorphisms of D and all whose strata have hyperbolic structures. To construct such a stratification, we introduce the notion of a maximal ball in D.

A round open ball B C D is called maximal if B is a maximal ball in D with respect to the inclusion. Later in Theorem 6.82, we shall prove that the boundary sphere a B of a maximal ball B contains at least two boundary points, i a D n a B I = card (aD n 3B) > 2. We denote the family of all maximal balls in a given domain D C S" by 2 (D). Given a maximal ball B C D, let B' C B denote the convex hull C(aB n aD) of the common boundary subset aB n aD, with respect to the Poincare hyperbolic metric in the ball B E D (D), see § 6, Chapter 3. We call this B' the hyperbolic stratum in a maximal ball B, see Figure 110.

Figure 110. Hyperbolic stratification of a domain

Theorem 6.82. Any open domain D C S" with at least two boundary points is covered by maximal balls B E 2(D). Moreover, given a point z E D, there is a

unique maximal ball B(z) E 2(D) such that z E B' = B'(z) C B(z), and the following alternative holds for any z, z' E D: either B'(z) = B'(z')

or B'(z) n B'(z') = 0.

(6.34)

Proof. Let us fix a point z E D and consider all increasing sequences b(z) _ (Bi) of open round balls Bi C D containing it. In particular, for z oo, we may

6.8. Canonical Riemannian metric on conformal manifolds

357

take b(z) = {B' (z, ri)} which consists of balls centered at z with increasing radii ri+1 > ri > 0. Then, given such a sequence b(z), the union

Bb = U{Bi : Be E b(z)} C D is not conformally equivalent to the whole space Rn since the boundary 8 £ contains at least two points. Hence Bb is a round ball. We may assume that a Bb fl aD # 0. If this boundary subset of 8 Bb has at least two distinct points, the ball Bb is a maximal ball containing the point z E D. Otherwise I a Bb fl a £ I = 1, and we consider all other increasing sequences b'(z) _ {Bl } and the corresponding open round balls By C £D such that Z E By and aBb' fl

8£) # 0. Along this way, let us define a special "ball-neighborhood" U(z) C £ of the point z E `a0 as the union of all open round balls B containing the point z. Here we may assume without loss of generality that z = 00 ED. In this case, the complement of any ball B, z = 00 E B, is a closed ball in Rn, and hence the complement of U(z),

DZ=Sn\{B:ZEB}, z=oo,

(6.35)

is the intersection of closed Euclidean balls. Therefore DZ is a closed Euclidean convex subset in 1[8n, 8) C Dz. Let B(z) C Rn be a closed round ball of minimal

radius that contains the set D. We claim that this ball B(z) is unique. Indeed, for any two closed round balls B1, B2 C Rn which contain Dz, the intersection lens B1 fl B2 is contained in a round ball with smaller radius than that of B1 and B2, see Figure 111.

Figure 111

Let us denote by B (z) the complement of the unique closed ball h (z), DZ C B (z).

It is clear that z E B (z) C U (z) C £, and B (z) is a maximal ball from the family 2 (0D) containing at least two boundary points, I a £ fl a B I > 2. Furthermore, for the

358

6. Uniformization

convex hull B'(z) C B(z) of the subset 8B(z) n d l) in the Poincard hyperbolic metric in the maximal ball B(z), that is for the hyperbolic convex hull in BZ of a subset of 8 B (z) of extreme points (in the sense of the Krein-Milmann theorem) of the Euclidean convex set Dz, we have that

z E B'(z) C B(z) E S(JO).

(6.36)

To prove this fact, it is enough to prove that the hyperbolic convex hull k(z) in the round ball int(B(z)) contains the center i of the unique ball B(z). Here the center z is the image of the point z = oo under the reflection x with respect to the boundary

sphere 8B(z) = 8B(z) where x(B'(z)) = B'(z). In fact, if the last assertion were false, that is z E B (z)\B'(z), the hyperbolic convex hull B'(z) would not intersect a Euclidean hyperplane P passing through the point z, see Figure 112. Then it is easy to see that, moving the ball-center i closer to B'(z) along the ray orthogonal to P, we will obtain a round closed ball Bo, Dz C Bo, whose radius is less than the radius of the ball h (z). This contradicts the definition of the ball h (z) of a minimal radius and hence completes the proof of (6.36).

Figure 112

Therefore, given a point z E 1 we have a maximal ball B(z) E S(D), Z E B(z), such that z E B'(z) C B(z). Applying the simple fact from conformal geometry (see Figure 113) that, for any intersecting maximal balls B1, B2 E S(D), Bt n B2 ¢ 0, their hyperbolic convex hulls Bi and BZ do not intersect, we have the uniqueness of the

obtained maximal ball B(z), Z E B'(z), that is the alternative (6.34). This completes the proof. The notion of maximal balls in domains can be naturally extended to the notion of maximal balls in conformal n-manifolds M by using the development map d: M -a M.

6.8. Canonical Riemannian metric on conformal manifolds

359

In the case of a uniformizable conformal manifold M, we call an open subset in M a maximal ball if the development image of its universal lift is a maximal ball in d(M) C Sn, that is in a component of the discontinuity set Q (G) of the holonomy group G C Mob(n). In the general case (of non-uniformizable manifolds), one needs an additional condition that the development map d factors through the covering

projection n: M -> M. Among manifolds with this property, we consider those whose development d has at least two boundary points, I B(d) I > 2, see § 4. We call such manifolds non-elementary. It follows from Lenuna 6.52 (see also Theorem 6.58) that elementary conformal manifolds are either spherical or flat. Now we obtain the notion of maximal balls in a non-elementary conformal manifold M as open subsets corresponding via the development map d to maximal balls in the proper discontinuity set 0 (G) C S' of the holonomy group G of M (instead of the discontinuity set S2 (G) which may be empty for non-uniformizable manifolds). Since the family £ (.)) of maximal balls in a domain 0 C Sn is invariant with respect to Mob(D), Theorem 6.82 immediately implies: Corollary 6.83. Any non-elementary conformal manifold M has a canonical Aut(M)invariant stratification by hyperbolic strata.

Figure 113

Clearly, the constructed hyperbolic strata B' in a domain Z C Sn may have any dimension, 1 < dim B' < n, see Figure 113. Let £s(D) C £(D) be a subfamily of those maximal balls B C £ whose hyperbolic strata have full dimension, that is whose boundary sphere 8 B is the sphere of lowest dimension containing the subset 8 B n 8 D. We call such balls B strictly maximal. Examples of the products a)n-1 x IR C Il8n or solid ellipsoids in 1W2 show that the family 8S (.)) can be empty, and hence there are conformal manifolds M with infinite fundamental group, which have no strictly maximal balls because these domains (e.g., ellipsoids) may be components of their holonomy groups.

360

6. Uniformization

However in dimension two, in the main case of finitely generated Kleinian groups

G C Mob(2), the family 2S(G) of strictly maximal disks in the discontinuity set 0 (G) has the following remarkable properties.

Theorem 6.84. Let G C Mob(2) be a finitely generated Kleinian group whose limit set A (G) does not lie in a round circle. Then: (i) The discontinuity set 0 (G) is covered by strictly maximal disks from the family .S (G), finite modulo the action of G. (ii) The disjoint union

S2'=U{B': B'CBEBS(G)}

(6.37)

is the open set of bijectivity of the map p = 61 c which is the restriction to S2 (G) of the natural retraction p: H3 -> CG. Furthermore, plc' is a conformal homeomorphism: given a connected component B' C 0', the restriction P I B' to this component is the right-angle rotation in S3 around the pointwise-fixed 1-sphere aB. (iii) The boundary aCG of the convex hull CG = C(A(G)),

aCG = cl (U {p(B') : B' C B E SS(G)})

(6.38)

is the closure of the disjoint union of the family of totally geodesic pieces p(B'), finite modulo G. Proof. Passing to a finite index torsion free subgroup (Theorem 1.13) which has the same discontinuity set, we may assume that G has no torsion. So G acts freely on the boundary H* = hn CG homeomorphic to S2 (G), and due to the Ahlfors finiteness Theorem 4.1, H*/ G consists of finitely many surfaces of finite topological type. In other words, the hyperbolic area Area(H*/ G) is bounded, cf. Epstein-Marden [1]. Due to the well-known Krein-Milmann theorem, the Euclidean convex hull C(X) of a set X C Rii+1 is the convex hull of a subset E C X of extremal points, and its boundary aC(X) is an n-manifold which consists of n-polyhedra whose vertices are extremal points of X. Using this fact and the projective Klein model of the hyperbolic space IHIn+1, we see that the boundary H* of the convex hull CG consists of hyperbolic

totally geodesic n-polyhedra whose vertices he at infinity aH'+1 So, in dimension n = 2, H* is the union of geodesic polygons lying on hyperbolic planes (support planes of CG) and bounded by disjoint infinite geodesics whose ends are limit points of G. From now on, we call such geodesics, that is geodesics which do not belong to interiors of totally geodesic (n - 1)-dimensional pieces of the boundary H*, the bending geodesics of H*. Clearly, the set of bending geodesics is G-invariant and corresponds to a geodesic lamination L C H*/G whose geodesics are either infinite or closed. Taking the double of H* / G which consists of hyperbolic surfaces, we have that its Euler characteristic equals 2X (H* / G). This and the Gauss-Bonnet formula for these

6.8. Canonical Riemannian metric on conformal manifolds

361

hyperbolic surfaces imply that Area ((H*/G)\L) = Area(H*/G), see Thurston [1, § 8.5]. Therefore, the lamination L has zero measure, Area(L) = 0, and hence H*/G is the union of totally geodesic domains Di bounded by closed or infinite geodesics. Obviously, each such domain Di has Area(Di) > n where it is the area of an ideal hyperbolic triangle. This, together with the fact that Area(H*/G) < oo, implies that the number of the geodesic domains Di is finite. Each such geodesic domain Di is covered by a domain (bounded by geodesics) in a support plane Hi C 1H13 whose boundary circle 8 Hi has at least three limit points. It follows that the circle 8 Hi bounds an open round disk Bi C 0 (G), and it is a strictly maximal disk. Clearly, we obtain in such a way the whole family Ss (G) which is thus finite modulo the G-action because of the finiteness of domains Di.

To show that Q (G) = U {B : B E ms'(G) }, we take a point z E Q (G). Since

Area(L) = 0, there is a number i such that the composition of the retraction H* and the natural projection n: H* -+ H*/G maps z into the clop: S2(G) sure Di. If n(p(z)) E int Di, we may assume that p(z) lies inside a totally geodesic face of the convex hull CG. Since the restriction of p to this open side is a homeomorphism (rotation pi by n/2 around the corresponding circle 8Di at infinity), the point z lies inside the strictly maximal disk Bi bounded by 8 Di.

If n(p(z)) E 8Di, the point p(z) lies in a bending geodesic y C H*, and two cases are possible. In the first case, when y is the common boundary geodesic for two geodesic sides Di, Di C H*, it is clear that the point z belongs to one of the strictly

maximal disks Bi or BJ, see Figure 113. Otherwise 8Bi = BBB, and the limit set A (G) lies in a round circle, which contradicts the conditions of the theorem. In the second case, the bending geodesic y is the limit of an orbit g,,, (Di ), gm E G,

of a totally geodesic side of H*. Here we may assume that y is the limit of two sequences gm (Di) and gk (Dj) converging to it from two directions, where one of the sequences maybe constant (if y is a boundary geodesic of Di or Dj ). Then, replacing these totally geodesic sides of H* by the corresponding strictly maximal disks Bi and Bj, we have limit points xi, yi E 8Bi and xj, y, E BBB, whose g,,,- and 9k-images converge to the limit ends of the geodesic y = (x, y),

x = lim gm (xi) = lim gk (Xi), m-+oo

k-*oc

y = lim gm (Yi) = lim gk (Yj ) m-aoo k-+ oo

Therefore, either the point z lies in some gi0-image of one of strictly maximal disks

Bi or Bi, or else the round disks B = lim,,, w gm (Bi) C S2 (G) and Bj* = limk,0 gk(Bi) C cZ(G) have the common boundary circle containing the limit set A(G). The latter case contradicts the theorem assumption, which completes the proof of (i). The claims (ii) and (iii) clearly follow from the properties of the canonical retraction P, P = 71 S1 (G) (see § 6, Chapter 3), and the definition of strictly maximal disks Bi as the disks at infinity 8 ]H13 bounded by the boundary circles a Hi of the universal covering

planes Hi = p(Bi) of the corresponding totally geodesic components it p(BI) of

6. Uniformization

362

(H*/G)\L, see Figure 114.

D

Figure 114

We notice that, due to Theorem 6.82, we have a well-defined natural map:

c: )

(,D, £(D)) ,

c(z) _ (z, B(z)).

(6.39)

As in Theorem 6.84, it is useful to look at this map from the point of view of the convex

hull CD of the boundary 81 in the hyperbolic space Hn+1, 8H'+' = S. Here we have a natural retraction p: IEII"+1 U 8H'+1 --> CD along geodesics orthogonal to hyperplanes supporting the convex hull, that is the map p such that p(x) is the point of 8C.0 "nearest" to x. The restriction of this retraction p to £0, p = ; JJ D is not bijective. On the other hand, for any given maximal ball B E 2(i)), the restriction PI B, : B' -* 6C.0 is a conformal homeomorphism. Namely, it is the n/2-rotation in Sn+l around the pointwise fixed sphere 8B.

We note that it is possible that p(B') = p(B2) for maximal balls B1 , B2 C 1 with disjoint strata Bl fl B2 = 0, see Figure 113. However, in the case of strictly maximal balls Bl E SI ()), we have (as in Theorem 6.84(ii)) that the map

p':

U int Bi --> aC,o, B i E,`Bs (,D)

obtained by restricting p to U lint B= : B; E 2 (D)), is a conformal homeomorphism. Here the latter case corresponds to support planes in IE1ln+l containing totally geodesic n-dimensional pieces of the pleated n-manifold 8C,D C Hn+1 and the former one corresponds to support planes in 1Ei[n+1 containing singular geodesic pieces of aCD. In this sense, the hyperbolic half-spaces bounded by supporting n-planes HZ C 1EIIn+1 whose spheres at infinity are the boundary spheres of the maximal balls B (z) E 2 (G), roll along the boundary of the convex hull CD C k1(n+1 In this rolling, the geodesics

in Hn+1 which join the marked points z E £), z E B'(z) C B(z), and the points

6.8. Canonical Riemannian metric on conformal manifolds

363

p(z) E 8Cs are orthogonal to the hyperplanes HZ C ]Hln+1, see Figure 113. In that Figure 113, the retraction p maps the shaded lens to the bending geodesic y = (x, y) C JI3 in the boundary aC,0. We note that in the Klein projective model of Hn+1 in the ball B C I8n+1, the above rolling is the usual rolling of half-spaces along

the Euclidean convex body Cv C B. Such our description shows that we have a well defined map t of the set 2*(M) of marked maximal balls (i.e., the set of pairs (z, B(z))) to the unit tangent bundle T1IHIn+1 of the hyperbolic space IHIn+I,

t: B. (D) = (D, 2(A) ----) T1W+1

(6.40)

Indeed, for any pair (z, B(z)), we have a unique geodesic ray joining the points z and p (Z) E 8 CD, which defines a unit tangent vector to ]EIIn+1

The map t in (6.40) allows us to define a metric in the space B*(D) of marked maximal balls and to study metric properties of the map c in (6.39). It is well known that the development (rolling) of the boundary of a Euclidean convex body locally has the Lipschitz property. As a result of this fact, we have that the map c: z --) (z, B(z)) in (6.39) is a locally Lipschitz map. This and arguments in Corollary 6.34 imply: Lemma 6.85. Let M be a non-elementary conformal n-manifold. Then with respect to the metric in the space of marked maximal balls induced by the map tin (6.40), the map c: M -+ £* (M) is locally Lipschitz.

2. The Kobayashi conformal metric. Let M be a conformal n-manifold which is neither spherical nor flat, that is M is "hyperbolically conformal". In this section we shall define a smooth canonical Riemannian metric in M which induces the given conformal structure on M.

For this purpose, we adapt the well-known Kobayashi's [1] approach for biholomorphically invariant intrinsic pseudo-metrics in complex-analytic spaces to the conformal (Mobius) category, on the base of our method of maximal ball coverings of conformal manifolds developed in the previous section. As a matter of fact, instead of Kobayashi's holomorphic chains of 1-dimensional complex disks in a complexanalytic space that connect given pairs of points, we shall consider conformal chains of the hyperbolic n-spaces JHI in a conformal n-dimensional manifold M. Naturally, these hyperbolic spaces are taken as conformal round balls in M, that is those subsets B of a given hyperbolically conformal manifold M whose images d(B) under the development d: M -+ M are open round balls in the (proper) discontinuity set 0 (G) C Sn

of the holonomy group d*(lr1M) = G C Mob(n), see Theorem 6.60 and Lemma 6.61. We start with a definition of a conformal pseudo-distance km on a conformal manifold M (which resembles the Kobayashi complex-analytic pseudo-distance).

Definition. Given two points p and q in a conformal manifold M, we choose: 1) points p = po, P1, ... , Pk-1, Pk =gin M; 2) points al, . . , ak, b1, ... , bk in HI; and 3) conformal mappings f, : HI -* M, i = 1, ..., k, of the hyperbolic n-space W (taken as the unit ball Bn (0, 1) -1 In with the Poincar6 hyperbolic metric .

6. Uniformization

364

d) into the manifold M such that fi (ai) = Pi-1 and fi (bi) = pi for i =

1,...,k. Then, for each choice of points and mappings thus made, we consider the sum of the hyperbolic distances in En: (6.41)

and define km (p, q) as the infimum of the numbers (6.41) obtained in this manner for all possible choices.

It is easy to verify that the continuous function kM: M x M -± II8 satisfies all pseudo-distance axioms, and it is a distance if km (p, q) > 0 whenever p ; q. The most important property of the conformal pseudo-distance is given by the following fact, the proof of which follows right from the definition. Proposition 6.86. Let M and M' be conformal manifolds and f : M -* M' a conformal mapping. Then kM(P, q) ? kM' (.f (P), f (q)) for

p, q E M .

Corollary 6.87. Every conformal homeomorphism f : M with respect to conformal metrics on M and M':

M' is an isometry

kM(P,q)=kM,(f(p),f(q)) for p,gEM. Directly from the definition (due to the triangle inequality), we obtain that the conformal pseudo-distance km is the "largest" pseudo-distance on M for which every

conformal mapping f : IiI' - M is a contraction: Proposition 6.88. Let k(*, *) be a pseudo-distance on a conformal n-manifold M such that k(f (a), f (b)) < d(a, b) for every conformal mapping f : Hn -* M and

for a,bEW1.Then km (p,q)>k(p,q)for all p,gEM. As the conformal Kobayashi pseudo-distance is defined in a manner similar to the distance function on a Riemannian manifold, it is compatible with coverings:

Theorem 6.89. Let M be a conformal manifold with a covering r: Al --) M, and let

points p, q E M and p, q E M be such that ir(p) = p and n(q) = q. Then km (p, q) = inf kM(p, q) ,

(6.42)

9

where the infimum is taken over all q E M such that Jr(q) = q.

Proof. Since n : M --+ M is distance-decreasing due to Proposition 6.86, we have kM(p, q) < inf4 kM(p, q). Assuming strict inequality, we have for some s > 0:

kM(p, q) + s < inf kfi(p, 4). 9

(6.43)

6.8. Canonical Riemannian metric on conformal manifolds

365

Then, by the definition of kM, there are points al, ... , ak, bl, . . . , bk E Hn and conformal chain-mappings fi : ffi[n -> M, i = 1, ... , k, such that p = fl (al), fl(bl) = f2(a2), ... , .fk-.1(bk-1) = .fk(ak), .fk(bk) = q, and k

d(ai, bi).

kM(p, q) + s >

(6.44)

i=1

Let us lift the maps fi to conformal maps fi : Hn -a M in such away that .7r o fi = fi for

i = 1, .. . , k, and p = f1(ai), fi (bi) = fi+1(ai+1) for i = 1, ... , k - 1. Therefore, if we set q = fk(bk), then 7r (4) = q and k

d(ai, bi)

kM(P, 4)

(6.45)

i=1

Now we see that the sum of the inequalities (6.45) and (6.44) leads to a contradiction to our assumption (6.43), which completes the proof.

Remark 6.90. The property (6.42) of the pseudo-metric km on a conformal manifold

M allows us to study its covering M instead of M itself. So we can pass to the proper discontinuity set 0 (G) C Sn where the development d is a covering due to Theorem 6.60. In particular, applying Theorem 6.89 together with Theorems 6.51 and 6.82, we immediately have that the function km becomes a distance function (kM(p, q) > 0 whenever p ; q) if the manifold M is hyperbolicaly conformal, that is if its development d has at least two boundary points x, y E B(d) in Sn or, equivalently, M is neither spherical nor flat. The converse is also true: if a conformal manifold M is elliptic or Euclidean then the pseudo-distance km degenerates. Indeed, given points p, q E 1R', we may consider a sequence of round balls Bn (p, Ri) centered at p with unbounded radii Ri. This implies that the hyperbolic distances di (p, q) in the i - th balls tend to zero, and hence the pseudo-distance kRn is trivial.

3. Conformal metric km as a Riemannian metric. Here we shall introduce a differential form FM : TM ® TM R+ on the tangent bundle of a given conformal n-manifold M (which is neither spherical nor flat, see Remark 6.90), which induces the Kobayashi conformal metric km:

kM(p, q) = inf

FM,

for

p, g E M,

fy

where the infimum is taken over all differentiable curves y in M joining the points p and q. More exactly, we shall prove the following:

366

6. Uniformization

Theorem 6.91. Let M be a non-elementary conformal n-manifold, that is M is neither spherical nor flat. Then the Kobayashi conformal metric on M is a complete twice differentiable a.e. (with Lipschitz derivatives) Riemannian metric which induces the original conformal structure on M.

Proof. Let FB be the quadratic form which defines the Poincare hyperbolic metric in the unit ball B'(0, 1). Then, for a given element of the tangent bundle of W = B", that is forgiven x E B"(0, 1) and E IR", 211 F(xJt) B - 1_Ix12

(6.46)

For a given conformal manifold M, we define the metric form FM on the tangent 1[8+, as follows. For a given element (x, ) E TM bundle TM, FM : TM ® TM of the tangent bundle of M, we set

'Fm (x, ) = inf FB (y, 0 , where the infimum is taken over all y E B" (0, 1) and

(6.47)

E II8", and over those conformal

mappings f : B'(0, 1) -+ M that satisfy two conditions:

f(y)=x,

d f (y)

=4.

(6.48)

In particular, if M is the ball BR = B'(0, R) C I[8" of radius R > 0, (6.46) and (6.47) imply that the form FBR agrees with the Poincare hyperbolic metric (of constant sectional curvature -1) in the ball BR:

FBR (x,) = R2_Ix12' We shall now show that, for all (x, t) E TM and Y E B"(0, 1), there exist a conformal mapping f : B"(0, 1) -+ M and E R" such that (6.48) holds. Indeed, as in Remark 6.90, we may consider (instead of M) the development-domain D = d(M)\B(d) c S" whose boundary (the proper limit set of the holonomy group) contains at least two points, I 8 £ I > 2, because of the assumption on M. This

domain ,) contains a round ball B" (x, r) centered at x with some radius r > 0. For a given E R", let hg: B"(0, 1) -+ BI (x, r) be a Mobius transformation such that h (0) = x and Dh (0)et = I. Then we define the transformation f : B'(0, 1) £ as the composition f (z) = h1 o gy(z) of h and the hyperbolic translation gy E M6b(B"(0, 1)) in the unit ball B"(0, 1) which maps a given point y to the origin, see (1.29). Clearly, the transformation f = h gy satisfies (6.48). Since the group of Mobius automorphisms acts transitively in the ball B'(0,1), and its elements are isometries in the Poincare hyperbolic metric FB, we notice that

6.8. Canonical Riemannian metric on conformal manifolds

367

the pointy E B" (0, 1) in the definition (6.47) can be chosen arbitrarily and kept fixed. In particular, choosing y = 0, we can rewrite (6.47) as follows: FM (x, ) = inf (c I

:

E R' , D f (0) _ for a conformal mapping f : B' (0, 1) -± M, f (0) = x} . (6.49)

It is easy to see that FM (x, ai') = I a I FM (x, ) for any a E R. In terms of the development-domain eD C S", the geometric meaning of the definition (6.49) is as

follows. Given a point x e 2), we should consider an open ball B C 2), x E B, whose hyperbolic metric is minimal at the point x. So, directly from the construction of the canonical maximal ball B(x) C 2) in Theorem 6.82, we see that our desired

ball B C 2) is the unique maximal ball B(x) E 6(.D) constructed in the proof of Theorem 6.82. In other words, the ball B C 2) is that unique ball B(x) in 2) whose convex hull B'(x) C B(x) of the boundary subset aB(x) n a2) contains the point x. Therefore, the metric form F,qj for the development-domain 2) C S" may be defined in such a way that, for any canonical hyperbolic stratum B' C 2), B' C B E

2 (D), F.IB/=FBIBF.

(6.50)

We notice that (6.50) provides a direct definition of the metric form FF (and hence of FM) because ,D and M are canonically stratified by hyperbolic strata, see Theorem 6.82 and Corollary 6.83.

In order to understand how smooth the Riemannian metrics Fg and FM are with respect to changes of the base points x or, equivalently, of the canonical hyperbolic strata B', we need the following estimates.

Lemma 6.92. Let B be a maximal ball in the development-domain 0 C S", dB its Poincar6 hyperbolic distance function, and B' C B the hyperbolic convex hull in B of the boundary subset a2) fl 8B. Then F'B(x, )

< Fm (x, ) < FB(x, ) cosh(dB (x, B')) -

(6.51)

Proof. In the trivial case of B = B(x), the claim (6.51) is true (with equalities) due to (6.50). So we may assume that B ; B(x). In this case, we shall prove an inequality slightly stronger than that in (6.51). Namely, we change the distance dB(x, B') there to a smaller distance dB(x, Hx), dB(x, Hx) < dB(x, B'), where Hx is the hyperbolic (n - 1)-plane in the ball B whose (n - 2)-sphere at infinity is the intersection aB fl 8B(x). Figure 115 shows the intersection of the balls (half-spaces) B = ll and B (x) where a B fl a B (x) is pictured by the straight line. We also notice that the magnitude a of the dihedral intersection angle of B and B(x) is less than 7r/2,

6. Uniformization

368

0 < a < r/2. This follows directly from the assumption that x E B fl B'(x).

Figure 115

Let us consider a straight segment orthogonally joining the point x and the intersection (n - 2)-sphere 8B fl 8B(x). Let r (x) be the Euclidean length of this segment, and ,B > 0 the angle between this segment and the hyperbolic plane Hx C B which

is orthogonal to 8B. Clearly, 0 < a < 0 < r/2. Also it is a well-known fact in the hyperbolic geometry that the hyperbolic distance dB (x, Hx) between a point x and a plane Hx is as follows (cf. (1.15) or Beardon [4; 7.20.3]):

cosh(dB (x, Hx)) =

1

cos,8

This implies:

FB(x, ) coshdB(x, Hx)

I4I/xn

ICI

coshdB(x, Hx)

r(x) cos,8 coshdB(x, Hx)

<

IV

=FB(x)(x,

r(x) cos($ - a) <

ICI

r(x) cos,8 =FB(x,

which completes the proof of Lemma 6.92.

)=FF(x, )

6.8. Canonical Riemannian metric on conformal manifolds

369

Now, using Lemma 6.92 and Lemma 6.85 about the locally Lipschitz map c in (6.39) which maps x to a marked maximal ball c(x) = (x, B(x)) E £*(.)), we can prove the regularity of our Riemannian metric. Indeed, the function 1 / cosh(t) has zero derivative at t = 0. Hence the form FD is differentiable due to Lemma 6.92. Moreover, its derivative is locally Lipschitz because the map c: £ --> 2* (D) is locally Lipschitz. It is a well-known fact that mappings with Lipschitz continuous derivatives are twice differentiable almost everywhere, see Reshetnyak [2; §2.6, Ch. 1, Cor. 2]. Hence we have the desired smoothness of our metric. Its completeness follows from the smoothness and the coincidence with the complete hyperbolic metric in the canonical totally geodesic strata B' C B E S(D). This completes our proof of Theorem 6.91. 4. Sullivan theorem on convex hulls. Let N be a uniformizable conformal n-

manifold with the development map d : N --* Sn and the holonomy group G C Mob(n). Then N is a boundary component of a Kleinian manifold M(G) = (Hn U 0 (G)) / G covered by a component 0 C S2 (G), N = S2/ G, and is homotopy equivalent to a pleated boundary component NN of the convex core C(M) C M. Here C(M) is the minimal convex retract of the hyperbolic manifold IH[n+l/G given (with the retraction p) as follows, see Theorem 3.19 and (3.60):

p : Hn+1/G --> C(M) = [C(A(G))/A(G)]/G. Without loss of generality, we may assume that the limit set A (G) does not lie in a round k-sphere Sk C Sn, k < n - 1. Otherwise, if A(G) C Sk, we may assume that G C Mob(k) and then pass to a k-dimensional manifold. With this assumption made, int C (M) # 0. Hence the boundary 8 C (M) = H* / G, obtained as the quotient of the boundary of the convex hull C(A(G)) \ A(G) C ]fl11+1, is an (n - 1)-manifold. It is possible to introduce a distance function dd on H* induced by the hyperbolic metric d in j n+i Namely, for any two points p and q from a connected component of H*, dd(p, q) is defined to be the infimum of the hyperbolic lengths of rectifiable curves in H* connecting p and q. It converts H* and 8C(M) = H*/G into lengths spaces in the sense of Gromov, see § 2. The correctness of this definition is implied by the following fact: Lemma 6.93. In the above setting, the set of rectifiable curves on the boundary H* of the convex hull C(A(G)) = CG connecting two given points in a component of H. is not empty.

Proof. Let yo E int CG be a fixed point. Then Rn+t \ CG is homeomorphic to the product H. x (0, oo) where the first coordinate corresponds to the intersection point y fl H* of a geodesic ray y = [yo, oo) and H*, and the second coordinate equals the hyperbolic distance along y to H*. Let C C H* be a connected boundary component containing given points p and q. It corresponds to a component of IH[n+' \ CG C x (0, oo), whose any two points can be clearly connected by a rectifiable curve.

6. Uniformization

370

In order to find a rectifiable curve in H* connecting p and q, we push these points

along rays [yo, oo) inside C x (0, oo). So obtained points p', q' can be joined by a rectifiable curve which is projected by the retraction p : ]F3In+1 -+ CG onto arectifiable

curve in the boundary H. The last property (which completes the proof) is due to the fact that the retraction p does not increase hyperbolic distances, that is, for any X, y E Hn+1

d(P(x), P(y)) < d(x, y) .

(6.52)

Figure 116

To prove (6.52), we assume that p(x) ; p(y) and consider a hyperbolic geodesic

L containing these points and directed from p(x) to p(y). Then its segment I = [p(x), p(y)] C L belongs to CG due to the convexity of CG. Let n : H11+1 -+ L be the orthogonal projection in the hyperbolic space. Since p(x) is the closest to x point in CG (and hence in I) and, similarly, p(y) is the closest to y point in I, we have that

r(x) < p(x) < p(y) < ir(y), see Figure 116. This, together with the fact that the orthogonal projection r does not increase hyperbolic distances, completes the proof of (6.52) and hence of Lemma 6.93. Now, comparing the metric spaces (N, kN) and (Ne, dr), where kV is the canonical conformal metric on N and dd is the metric induced by the path metric on the component NN C H*, we get the following result known as the Sullivan theorem on convex hulls:

Theorem 6.94. Let N = S2/G, G C Mob(S2), 0 C Sn, be a non-elementary uniformizable conformal n-manifold whose fundamental group rr1(N) is finitely generated. In dimension n > 3, let N be in addition geometrically finite with n covered by strictly maximal balls B E Ss (G), and the limit set A (G) of the holonomy group G do not lie in an (n - 1)-sphere Sn-1 C Sn. Then there exists a K-quasiconformal homeomorphism f : (N, kN) -+ (Ne , dc) inducing a homotopy equivalence N ^-' Nc.

Furthermore, if n = 2 and 0 is simply connected, the constant K > 1 may be chosen universal, that is not depending on N.

The proof of this theorem is based on study of the boundary of the convex hull

Cr = C(A(G)) in ]}1n+1, similar to that in Theorem 6.84 (see Sullivan [3] and Epstein-Marden [1] in dimension n = 2 and Apanasov [29, 36] for n > 3). It seems the additional condition in dimension n > 3 on covering of S2 by strictly maximal

6.8. Canonical Riemannian metric on conformal manifolds

371

balls B E BS(G)} is superfluous, see Apanasov [29]. However another condition of geometrical finiteness in dimension n > 3 is important for our proof. It allows one to show that the set S2' C S2 (G) (defined as in (6.37)) is always a bijectivity subset

of the restriction p = p i c of the natural retraction p : W+1 -+ CG. Namely, in each hyperbolic stratum B' (in a maximal ball B C S2 (G)), the restriction pIBI is the rotation about the boundary (n - 1)-sphere aB by the right angle. To prove this fact (see Apanasov [30, Lemma 4.1]), one can use the following equivalent description of a hyperbolic stratum Bo in a maximal ball Bo E 2(G).

Let Bj, j E J = J(Bo), be all the maximal balls that intersect Bo. Then, for each such ball Bj, let soj and coj = cjo be (n - 1)-spheres in S' which contain the (n - 2)-sphere 8 Bo fl 8 B j and such that soj is orthogonal to So = 8 Bo, and coj divides the dihedral angle Bo f1 Bj into halves. We define exto (soj) and exto (coj) in the ball Bo as the components of the complements Bo \ soj and Bo \ coj, respectively, whose

closures contain the stratum B. Then we have (see Figure 117):

Ba =

I

I

exto(soj) C Bo =

jcJ(Bo)

I

I

exto(coj).

(6.53)

jEJ(Bo)

Figure 117

In particular, with the additional assumption that 2 (G) is covered by strictly max-

imal balls B E Ss (G), we have that 0' in (6.37) is the maximal bijectivity set of the retraction p. It implies that the boundary manifold H* = aCG C Ill[+1 is the union of totally geodesic hyperbolic strata p(B) which correspond to maximal balls B E B(S2). Among such strata we distinguish special singularities, which we call simple bending geodesics of H,k. More precisely, we call a geodesic t = (x, y) c H', with limit end points x, y E A (G) a bending geodesic if £ does not lie inside any totally geodesic k-dimensional stratum of the boundary surface H*, k > 2. We call

6. Uniformization

372

such a singular geodesic t = (x, y) a simple bending geodesic if its (limit) endpoints x and y belong to the boundary sphere a B of some strictly maximal ball B C 0 (G). Otherwise, we call £ a limit bending geodesic (one can show that it is in fact the limit of other bending geodesics on H,,). Using this notion, one can describe singularities of H,k as follows, see Apanasov [30, Lemma 4.4].

Lemma 6.95. Let G C Mob(n) be a Kleinian group whose limit set A(G) is not contained in an (n - 1)-sphere. Assume that G is finitely generated if n = 2 and is geometrically finite if n > 3. In the latter case, assume in addition that the discontinuity set Q (G) admits a strictly maximal ball covering. Then there exists a finite subfamily 2F = {B1, ... , Bm } C B'' (G) such that (a) G (int BI) fl G (int

0 for i 0 j, and m

92 (G) = U g(&); =1 gEG

(b) for each of finitely many (mod G) components S20 C 0 (G), there is a subset So C BF of joint maximal balls Bi such that the corresponding hyperbolic strata p (Be') in a component p (00) C H, are joined to each other alongJaces o f a finite subset W I ,..., Wk of their non-degenerate polyhedral angles; (c) for n = 2, the angles Wj have the magnitude r; and for n > 3, they have the combinatorial type of the angle

Xj = {(tx, xn) E ]I8r-1 x 118: x E Pj, t E R+};

(6.54)

where Pi C Sn-2 C Rn-1 is a spherical convex polyhedron, and the edge in the angle Wj corresponding to the edge ((0, xn) : xn E 1[8} C aXj is a simple bending geodesic if 8 Pj does not contain opposite (polar) points in the sphere

Sn-2;

(d) to every simple bending geodesic £ = (x, y) C H* (from the G-orbit of an edge of one of the angles Wj, j = 1, . . . , k), there corresponds a finite set £ (t) of maximal balls such that the axis 2' of the spherical polyhedral angle

B(i) = n{B : B E S(t))

(6.55)

is projected by the retraction p to the bending geodesic £ = p (t'). It should be mentioned that, due to the fact that the restriction p I B; of the retraction p

to a hyperbolic stratum BL C Bz E 2 (t) (see (6.55)) is a conformal homeomorphism, the polyhedral angle p-1(Wj) of the stratum Bj is congruent to Wj, has a geodesic edge £ 9 = stk, and is contained in a bigger spherical polyhedral angle Vj of

6.8. Canonical Riemannian metric on conformal manifolds

373

the same combinatorial type as that of Wj and whose edge is the axis Z' of B(t), see Figure 118.

Figure 118

The angle Vj is formed by the intersection of the balls bounded by the bisector spheres cik (instead of sik in the case of p-1(Wj )), k E J (i ). It is important to notice that the union of the polyhedral angles Vi,

U(.C') = U{Vi

:

Vi C Bi E S(t)},

(6.56)

forms a full neighborhood of the arc 2' C S2 (G), p(.f') = t, with the limit end points x, y E A (G). In dimension n = 3, it defines a splitting of a neighborhood U (t') of t' into non-degenerate dihedral angles Vi. This observation implies the existence of conical singularities of the boundary H,k of CG if n > 3:

Theorem 6.96. Let C C Mob(n), n > 3, be a Kleinian group which has at least three strictly maximal balls Bi C 0 (G) with two common limit points x and y, x, y E A(G) ni 8Bi, and with int(ni B1) 0. Then a neighborhood U(t) C H* of the simple bending geodesic £ = (x, y) C H,, is not isometrically developable in the hyperbolic n-space IHIn.

Proof. Due to Lemma 6.95, we have a clear necessary condition for an isometrical development of a neighborhood U(t) C H,, into HI to exist. Namely, the sum of spherical measures mii_1(Wj) of all polyhedral angles Wj on H. having the geodesic

374

6. Uniformization

f as the common edge must be the full (n - 1)-dimensional spherical measure. As we have seen, it is equivalent to the same condition on the polyhedral angles p-(Wj) of the hyperbolic strata BJ C Bj E 2(f). However, the polyhedral angles p-(Wj) correspond to the bigger polyhedral angles Vj C Bj in (6.56), that is to the polyhedral angles of the domains B C Bj defined in (6.53). The axis 2' = (x, y)c C 2 (G) is the common edge of the angles Vj which form its full neighborhood U(E') in (6.56).

So, ifl2(C)I>3then E mn-1(Wj) < mn-1(Sn-1) , and U(P) is not isometrically developable in ffi

We note that in dimension n = 2, we always have at most two maximal disks Bi C V with common limit points x, y Eli 8B; fl a D. That is why the boundary aC,a C 1H13 is always isometrically developable in the hyperbolic plane H2. It is easy to construct a Kleinian group G C Mob(n) which satisfies the conditions of Theorem 6.96 and whose discontinuity set 0 (G) is connected. More interesting is to find examples of such groups G with contractible components of S2 (G), especially

of quasi-Fuchsian groups G, whose limit set A(G) splits the discontinuity set into topological balls. We construct such groups as follows. 5. Quasi-Fuchsian groups with cone singularities of the convex hull. We start with balls D1, ... , Dk C S' in a general position, k > n > 3, such that their intersection

nt Di is a k-hedral angle D(x, y) with vertices x, y E li 8Di. Such a D(x, y) in II83, resembling a pea pod, is shown in Figure 119.

Figure 119

Let h E Mi b(n) be a loxodromic transformation which fixes both x and y and preserves D (x, y). Then the exterior Fh of the isometric spheres I (h) and I (h-1) is a fundamental polyhedron for the cyclic group (h), and its boundary spheres I (h) and I (h-1) are orthogonal to aD(x, y). Suppose that we have a finite family 4 = 3,y, k

'8xy = U (-Sij U 4Si) i,j=1

,

(6.57)

6.8. Canonical Riemannian metric on conformal manifolds

375

whose subfamilies -8i and -i j are as follows: (1) all spheres S E 4Si j are orthogonal to a Di fl a Dj;

(2) all spheres S E -Si are orthogonal to the (n - 1) -side of D(x, y) that lies in aDi; (3) any two spheres S and S' from different subfamilies -Si j and 8m1 (correspondingly, -Si and 8j) are disjoint;

(4) the dihedral angles bounded by spheres S, S' E -Sij U -Si have magnitudes 7r/m, m E 7G;

(5) the balls int S bounded by S E Sxy cover the boundary 8D(x, y) fl Fh.

Lemma 6.97. Given the angle D(x, y) = n1 Di C S", let Gxy C Mbb(n) be a group generated by the loxodromic element h and the reflections XS with respect to the spheres S from the family 3xy in (6.57), Gxy = (h, xs : S E S). Then the group Gxy is Kleinian, some its finite index subgroup is quasi-Fuchsian, and the convex hull boundary H* = aCG,,y has a cone singularity along its simple bending geodesic

,2=(x,y)CH*. Proof. The conditions (1)-(5) on the family Sxy in (6.57) together with Corollary 4.30 imply that Gxy is a Kleinian group. Arguments similar to those in Example 2.19 show that 0 (G) splits into two topological balls SZo and 521, which are invariant for a finite index subgroup Go C Gxy, that is Go is quasi-Fuchsian. Assuming that x, y are finite points in IR so that D (x, y) is bounded, let S20 be the component of S2 (Gxy) which lies in D (x, y). Then the second (unbounded) component 01 is covered by the G-orbit of maximal balls Bi = lR' \75i. The balls Bi are strictly maximal because each boundary sphere a Bi contains both limit points x, y, as well as their images Xs (x), Xs (y) where

S E -Sij U -Si C 4xy. Now, Theorem 6.96 completes the proof by showing that H* = a CG has a cone singularity along the geodesic Q = (x, y) C H* C H"+1 We illustrate the above construction in dimension n = 3.

Example 6.98 (Pea-pod quasi-Fuchsian group). Let x = 0, y = oo, and Di = {x E 1183

:

xi > 0}, i = 1, 2, 3, D(0, oo) = {x E 1[83

:

xi > 0, i = 1, 2, 3} ,

and %3000 be a family in (6.57) formed by the following spheres:

(1) S23 = S2 (a', r), S13 = S2(a2, r), S12 = S2(a3, r) are centered at points ai = 8 ei where ei are the unit coordinate vectors in R3, and the radius r and the number 8 are such that 28 sin(7r/8) = r ,/3-, that is:

r=8 (2--.,F2)/3.

(6.58)

(2) Sk = S2(6 (ei + ej)/,12-, r) are spheres of the radius r > 0 given in (6.58) and whose centers lie in the coordinate planes orthogonal to the k-th coordinate axis; here triples (i, j, k) are obtained from the triple (1, 2, 3) by permutations.

6. Uniformization

376

We also specify a homothety h generating a cyclic group (h),

=2(82-r2)/(232-r2-r 482-3r2)

h(x)=?x,

whose fundamental polyhedron Fh is bounded by two co-centric spheres I and I', I = S2(0, R), I' = S2(0, R'), with radii

-r+ 482-3r2

R

Since the distance between the centers of Si and Sj equals 8, the spheres Si and Sj are disjoint. Figure 120 shows the intersection of the constructed spheres with the plane {x E R3 : X3 = 0}, and their intersection with 8D(0, oo) is shown in Figure 121.

Figure 121

Figure 120

It is clear that all dihedral angles of the polyhedron

P =next S; next Sij nextInintI' ij equals 7r/3, and hence the conditions (l)-(5) in (6.57) hold. Therefore the group

G = Goo, = (h, xs : S E

in Lemma 6.97, generated by h and the reflections with respect to the spheres Si and Si j, i, j = 1, 2, 3, is a quasi-Fuchsian group in

S3. One of its discontinuity set components, 01 C S2 (G), is the G-orbit of the complement ll \ D(0, oo) which is the union of three strictly maximal balls Bi = {x E IIg3 : xi < 0}, i = 1, 2, 3. The corresponding hyperbolic strata B' C Bi are such that, for sufficiently large K > 0,

Bi'(K)=B1 f1 {x ER3 : x i <-K}={x ER3:xi <-K, xj > 0; i 0 j}.

Notes

377

In other words, for a sufficiently large i-th coordinate, BZ looks like the right dihedral angle with the following geodesic edge:

29={x ER3 : x i <-K, x j =0; j #i}. The retraction p : Q (G) -a H. = aCG maps all three edges £9, i = 1, 2, 3 onto an infinite ray in the simple bending geodesic f = (0, oo) C H* C 1H[4, whose neighborhood in H,, is the union of three right dihedral angles p(BI (K)). Obviously, the pleated manifold H. has a conical singularity along the geodesic f.

Remark 6.99. As we have seen, the stratification of the discontinuity set 0 (G) = U{B' : B' C B E .8(G) C S" by totally geodesic hyperbolic strata and the corresponding stratification of M = S2 (G)/ G by totally geodesic hyperbolic submanifolds define geodesic laminations of the boundary aCG of the convex hull and the boundary of the convex core QM) C H'+'/G, respectively. These laminations are projected to a lamination of the quotient manifold M = 2(G)/G. In particular, if a connected component Mo of Q (G) / G is homotopy equivalent to a complete hyperbolic manifold 1W'/ F, we obtain along this way a geodesic lamination of the hyperbolic manifold IEI'2/ F, perhaps with singular leaves of codimensions at least 2 (compare Lemma 6.97 and Example 6.98), which is related to a deformation of H'/ F to the manifold Mo C SZ (G) / G. In particular, in the case n = 2, we obtain usual geodesic laminations of hyperbolic surfaces all whose leaves are either closed or infinite geodesics, see § 4, Chapter 5, and which define deformations of H2/ F into Mo C S2 (G)/ G. We shall explain the situation with deformations of high-dimensional hyperbolic manifolds along their geodesic laminations in Chapter 7. In particular, the quasi-Fuchsian groups constructed in Example 6.98 can be obtained by deforming hyperbolic lattices r c Isom H13 in Isom 1H14.

Notes For original definitions (and applications to the uniformization) of length spaces, convexity and coarse hyperbolicity, see Gromov [1, 2], Aleksandrov, Berestovskij and Nikolaev [1] and Nikolaev [1]. A detailed account on the Geometrization Conjec-

ture 6.3 of Thurston [2] can be found in Scott [6]. The inquire in Section 3 of § 2 about splitting of 4-manifolds and their structures can be considered as a continuation of the study of Kahler structures on 4-manifolds by Wall [2, 3], involving homology 3-spheres, see also Chapter 1, § 7 and Apanasov [48]. The main reference for results about hyperbolic volumes of 3-manifolds is Thurston's lecture notes [1], see also Gromov [3] and Chapter 3, § 11. The proof of Theorem 6.49 for non-compact manifolds follows Wielenberg [3] and, for closed manifolds, Apanasov-Gutsul [1, 2]. For other proofs, see Kawauchi [1] and Vesnin [1]. Similar claim for volumes of non-compact hyperbolic 4-manifolds (Theorem 6.50) is due to

378

6. Uniformization

Ivansic [1, 2] where one can also find constructions of such manifolds whose volumes provide "half" of possible hyperbolic 4-volumes, see also Ratcliffe-Tschantz [1]. Theorem 6.45 on finiteness of hyperbolic structures on link complements in compact 4-manifolds is due to Ivangic [1, 3]. Also we follow this paper while describing topology of ends of finite volume hyperbolic 4-manifolds and their difference from those of hyperbolic 3-manifolds. The Conformal Uniformization Conjecture 6.25 was formulated in Apanasov [30]. The idea of conformal uniformization is originally due to N. Kuiper [1-3] and R. Kulkarni [1, 4]. Especially, the Kuiper's notion of boundary points of the development map provides the base for the present proof of the uniformization Theorem 6.60 which was originally proven by Kulkarni-Pinkall [1] for general geometric structures. Theorems 6.62 and 6.65 generalize a Gunning [1] result on projective structures on surfaces to the case of conformal manifolds. They were originally proven in KulkarniPinkall [1] and Kamishima [1], see also Gusevskii-Kapovich [1]. A progress in solution of the Conformal Uniformization Conjecture is presented in Section 3 of § 4 and §§ 5-7. Theorem 6.68 on connected sums is due to KulkarniPinkall [1]. Theorem 6.71 on torus sum of hyperbolic manifolds is due to Apanasov [30]. Our proof of Theorem 6.74 on Seifert manifolds (cf. Kapovich [1]) uses bounds in Corollary 6.77 on the Euler class of circle bundles with conformal structures, which sharpens bounds of Kapovich Lemma 6.75; for the best known bounds, see Kuiper [4, 5] and Luo [2]. The approach of § 7 to torus sum of conformal manifolds (one of which is Seifert) is due to Kapovich [1]. For a general account on geometric uniformization

of graph-manifolds, in particular for conformal and isometric uniformization, we refer reader to the original works by Buyalo-Kobel'skii [1, 2]. In Remark 6.81, we reduce the general problem of torus sum of two hyperbolic manifolds to a problem of deforming their hyperbolic structures into cone hyperbolic structures, see Chapter 7, §4.

Canonical smooth Riemannian metrics on conformal manifolds were independently constructed by Apanasov [39], Kulkarni-Pinkall [3] and Nayatani [1]. Our construction in § 8 presents this metric as a conformal analogue of the complex analytic Kobayashi metric and mainly follows to Apanasov [27, 29, 38, 39]. For other properties (and comparison) of such canonical metrics, see Maubon [1] and IzekiNayatani [1]. Theorem 6.94 is a generalization of the Sullivan [3] theorem on convex hulls. Its detailed proof can be found in Epstein-Marden [1] if dimension n = 2, and in Apanasov [29] if n > 3. Cone singularities of the convex hull in dimensions n > 3 (Theorem 6.96 and Example 6.98) were first discovered in Apanasov [27, 29]; see also Chapter 7, § 4.

Chapter 7

Theory of Deformations

§1. Deformations of geometric structures 1. Teichmaller spaces of marked structures. Let M be a smooth n-dimensional manifold (or orbifold) with a geometric structure modeled on some (X, G)-geometry, see Chapter 1. In other words, we have a pair (M, s) where s is an (X, G)-structure on M determined by a maximal (X, G)-atlas on M. Two (X, G)-structures (M, sl) and (M, s2) on M are said to be equivalent if there exists an (X, G)-diffeomorphism

f : (M, Si) - (M, s2) homotopic to the identity. An equivalence class of (X, G)structures on M is called a marked (X, G)-structure. We denote T(M) = T(M; X) the space of marked (X, G)-structures on M. The space T"(M; X) is covered by the space £(M) = A(M; X) of the development maps d: M -* X of (X, G)-structures, with a metrizable topology defined by the uniform convergence of the developments on compacts. So we may define a topology on 1(M) as the topology induced by £ (M)-topology. In particular, if our (X, G)-geometry is hyperbolic the above space 7(M) 7(M; IHI) becomes the Teichmtiller space of marked complete hyperbolic structures on M. In the case of conformal geometry, (X, G) = (S", Mob(n)), we denote the Teichmaller space 7 (M; S") of marked conformal structures on M by C (M). Here we are mainly interested in how one can describe a marked conformal or hyperbolic structure on a manifold M, that is what are parameters (moduli) of such structures.

2. Teichmaller space of Riemann surfaces. An excellent example of spaces of marked geometric structures is provided by the classical problem of moduli of Riemann surfaces S. Due to the Klein-Poincare uniformization theorem, each conformal structure on a closed surface S of genus g > 1 arises from a complete hyperbolic metric. It has been known since B. Riemann's work [1] that classes of conformal equivalence of such surfaces depend on 3g - 3 complex parameters which are called moduli. It had been already noticed in classical studies that the moduli problem, that is the problem of describing the space of parameters of conformal structures on S,

7. Theory of Deformations

380

becomes simpler if the confonnal equivalence is replaced by a weaker equivalence that fixes generators of the fundamental group 7c1 (S), i.e. makes marking of S. This leads to a branched covering of the space of Riemann surfaces, which can be described by a means of quasi-conformal mappings. This space of marked Riemannian surfaces was called the Teichmiiller space T (S), after 0. Teichmiiller who sketched a program of investigations in this direction. Thus, we consider a marked closed Riemann surface S of genus g > 1, i.e., a pair

(S, E) where g

{al,b1,...,ag,bg) C7r1(S),

fl[ai,bi]=1 i=1

is a singled out system of generators of the fundamental group it (S). Since two homeomorphisms S -* S' of Riemann surfaces are homotopic if and only if the induced isomorphisms 7r1 (S) --+ 7r1(S') coincide up to inner automorphisms of the fundamental groups, it follows that homeomorphisms of marked Riemann surfaces (S', E') are specified by their homotopy classes. (5, E) Furthermore, any two marked closed surfaces of the same genus are homeomorphic and even quasi-conformally equivalent. This follows from a topological equivalence of closed surfaces of the same genus and a Nielsen theorem which states that every automorphism of the fundamental group 7r1 (S) of a closed surface S can be realized geometrically, i.e., it can be induced by some topological automorphism of S, which is determined up to a homotopy, see Theorem 4.40. Then, having a homeomorphism f : S -+ S', one can construct a quasi-conformal homeomorphism homotopic to it. That construction may be extended to any surface S of finite topological type, see for details Krushkal' [1, §6, Ch. 1]. Now two homeomorphisms f1: S -+ S1 and f2: S -+ 52 are regarded as equivalent if f2 fl 1 is homotopic to a conformal homeomorphism h: S1 -3 S2. Passing to the

universal coverings of those surfaces by the unit disk U = B2(0, 1), we get that both f1 and f2 can be lifted to automorphisms f1 and f2 of the disk U such that the following two diagrams commute:

U fi U ni S

n'

f

Si

and extensions of the liftings f1 and f2 to 8U coincide on the boundary aU = S1. For a given surface S of genus > 1 with n punctures, the TeichmUller space T (5) of the surface S is formed by the set of equivalent classes [S] of marked Riemannian surfaces, or equivalently, by the set of equivalence classes [f ] of quasi-conformal

7.1. Deformations of geometric structures

381

homeomorphisms. There is the so-called TeichmUller metric p in the Teichmuller space T (S),

p([fil, [f21) =

inf

In

K(f2fi 1)

(7.1)

.fJE[.f1l1f2Elf2l

where K(f2 fi 1) is the quasi-conformal dilatation of the map f2 fi 1, see (1.3). For any other marked surface So E T (S), there exists an isometry of the spaces T (So) and T (S) given by the formula [f h] -+ [f ] where h is a homeomorphism, h: So -> S. Therefore, instead of the spaces T (S), one can speak of a space T (g, n) of equivalence classes of surfaces of the type (g, n). In this space, the isometries of the form [f h] -+ [f ] generate the so-called modular group Mod T (g, n) which acts discontinuously on T (g, n). Then the moduli space R(g, n) of Riemann surfaces of genus g withn punctures is obtained as the quotient, R (g, n) = T (g, n)/Mod T (g, n). Correct proofs of main theorems which give a clear description of the moduli space R (g, n) and Teichmiiller space T (g, n) belong to Ahlfors [8] and Bers [1]. Namely, the space T (g, n) is a cell of real dimension 6g - 6 + 2n, and one can introduce a unique complex analytic structure in T (g, n) compatible with the Teichmuller metric (7.1).

3. Variety of representations and stability. As we have seen in Chapter 6, the main tool for studying (X, G)-structures on M is the development map d: M -- X which induces the holonomy representation d*:7r1M G and thus defines the holonomy map

hol: T(M, X) -* Hom(n1(M), G)/G,

(7.2)

where the group G acts on the variety of representations Hom(nr1(M), G) by conjugation, and the image hol(M, s) of a structure (M, S) E T(M, X) is the equivalence class G d,k consisting of representations gd*g-1, g E G. However in general, the holonomy representations d,, do not uniquely determine (X, G)-structures on M. For example, two immersions of an annulus A = {x E R2: 1 < Ix I < 2} into the plane R2 one of which has a self-intersection, define Euclidean structures on A, both with trivial holonomy, but they are not equivalent in any reasonable sense. This shows a necessity to study local properties of both spaces r(M, X) and Hom(7r1(M), G) / G, where the latter space has the quotient topology induced by the algebraic convergence topology on the representation variety Hom(nl (M), G). In this topology, two representations of a group are close if they are close on generators of the group. Let IF be a finitely generated group and G a simple linear algebraic group defined over R whose real points form the group G. The group G acts algebraically on the variety Hom(F, G) by conjugation. Here Hom(F, G) is the set of complex points of an affine variety defined over R whose real points form Hom(F, G). Since G is simple and hence reductive, we have a quotient variety X(I', G) defined by the Gaction on Hom(F, G). The variety X (I', G) is called the character variety. Generally, the character variety X (F, G) is not isomorphic to the orbit space Hom(F, G)/G and is defined as follows.

7. Theory of Deformations

382

Let {yl, ... , y k } be a set of generators of r and { fl , ... , f') a set of generators of the algebra of invariant polynomials on Gk which are chosen so that they take real values on Gk. Then we have a map F: Hom (F, G)/G -+ cm (which is not necessarily injective) defined on representations p E Hom([', G) as follows: F(G.p)

= (f1 (P(Yi),...,P(Yk)),..., fm (P(Y1),...,P(Yk)))

(7.3)

Then the character variety X([', G) is the affine variety defined over R and corresponding to the ring of G-invariant polynomials on Hom(F, G). This is to say that X (F, G) is the image under F contained in an affine variety determined by relations among the generators {yt, ... , yk} and by relations among the invariant polynomials { fl, ... , fm }. The variety X (F, G) of real points of X (F, G) is the image under the map F of the classes of representations on which polynomials ft, ... fm take real values. Among representations p: [' -+ G, we consider stable representations, that is those

ones whose orbits G p are closed in Horn([', G) and whose stabilizers Z(p) in G are finite. A stable representation p is called good if its stabilizer Z(p) is the center ZG of the group G. Due to a result of Newstead [1, Prop. 3.8], the set S([') C Hom([', G) of all stable representations is Zariski open in Hom([', G), and the map F defined in (7.3) induces a homeomorphism from S([')/G onto an open subset of the character variety X(F, G). Moreover, characterizing stable representations p: IF -+ G as those representations whose images p (I') are not contained in any proper parabolic subgroup of G, one can show (in the case when stable real representations exist) that the actions of G on S(F) and of G on s(r) fl Hom([', G) are proper, see Johnson-Millson [1]. In what follows, we are interested in those groups r which are lattices in some geometries and hence are often rigid. However, the following observation still allows us to consider some non-trivial representation varieties of F. Let H be a classical simple algebraic group defined over 18 with real points H and po: r -+ H an embedding of a finitely presented group r into H such that poF is a lattice in H. In addition, we assume that the group H is represented in another algebraic group G defined over I[8 so that the representation H -+ G is defined over R and its image is not contained in a proper parabolic subgroup of G. Then the image under the composition

H

I-

(7.4)

of the representations po: F -* H C H and H -+ G is a discrete subgroup in G, and we can consider the variety of representations Hom(F, G). In particular, the above scheme leads to the variety of quasi-Fuchsian representations of a hyperbolic lattice F. Namely, let H = SO(n, 1) and G = SO(n + 1, 1) be linear algebraic groups of orientation-preserving isometries of the quadratic forms

-xo+xi +...xn

and

-xo+xi

correspondingly in Cn+1 and Cn+2, and the groups SO(n, 1) C SO(n + 1, 1) consist of their real points. In addition, let r c SO(n, 1) be a uniform hyperbolic lattice, that

7.1. Deformations of geometric structures

383

is SO(n, 1)/ F is compact. Then we have a discrete faithful representation in (7.4) which is the composition of inclusions r C SO(n, 1) and SO(n, 1) C SO(n + 1, 1). Abusing notations, we denote it again by po: I' c+ SO(n + 1, 1). All representations p: F -* SO(n + 1, 1) conjugate in SO(n -I- 1, 1) to the representation po are called Fuchsian. A representation p E Hom(F, SO(n + 1, 1)) is said to be quasi-Fuchsian if the action of the group pF on the n-sphere S" = 8]EII"+1 is quasi-conformally conjugate to the action of the Fuchsian group pol'. We denote by Rq f (F) the subspace

of all quasi-Fuchsian representations p E Hom(F, SO(n + 1, 1)) and by Tgf(I') = [i q f (F)/ SO(n + 1, 1) the Teichmiiller space of conjugacy classes of quasi-Fuchsian representations. In § §3-5, we shall give geometric conditions on a hyperbolic lattice r which guarantee non-triviality of the Teichmiiller space Tq f (I') (even for rigid lattices

F, i.e. for n > 3).

4. Holonomy theorem and Sullivan structural stability. As we have already observed, the holonomy representation alone cannot determine an (X, G)-structure on M. In particular, due to Maskit [8] and Goldman [2], there are different conformal structures on a hyperbolic n-manifold M, n > 2, with the same holonomy. One of them is uniformizable, and the other is non-uniformizable, with surjective development map d: M -+ S'n, d(M) = Sn. A crucial fact is that these different structures have different developments. This becomes clear because of the following holonomy theorem which is due to Lok [1], see also Thurston [1, §5.2] and Goldman [3].

Theorem 7.1. Let M be a closed (X, G)-manifold. Then the map

hol: 7(M, X) -+ Hom(7r1(M), G)/G ,

hol(s) = G d,, ,

is an open map which lifts to a local homeomorphism hol of the space of development

maps to the variety of representations n (M) -+ G,

hol: D(M, X) -+ Hom(rrl(M), G) . We notice that the map hol is not necessarily a local homeomorphism, especially in neighborhoods of structures with unstable holonomy representations. However,

for an (X, G)-structure (M, s) whose holonomy representation d,, is stable, there exist a neighborhood U = U (s) C T (M, X) and a neighborhood V = V (d,,) C Hom(.7rl(M), G)/G with finite stabilizer subgroups H1 and H2 of s and d* correspondingly (H1 C H2 in the natural sense) that specify finite coverings U and V of U and V, U_= U/Hl and V = V/H2, such that the map hol lifts to a local homeV. In particular, in neighborhoods of structures s with good omorphism hol: U holonomies d,,, the map hol itself is a local homeomorphism, see Johnson-Millson [1]. In the case of conformal structures, this fact can be seen as an application of the so-called stability theorem which was originally proved by D. Sullivan [9] for Kleinian groups on the plane. Here we recall that a convex-cocompact group G C Mob(n) is a geometrically finite discrete group which has no parabolic elements, see Theorem 4.16.

7. Theory of Deformations

384

Theorem 7.2. Let G C Mob(n) be a non-elementary convex-cocompact group, and {G,n} a sequence of discrete groups Gm C Mob(n) isomorphic to G which algebraically (on generators) converge to G. Then, for sufficiently large m, the groups Gm are convex co-compact, and their actions on the sphere Sn are quasi-conformally conjugate to the action of the group G. Proof. Since G is convex-cocompact, its limit set A(G) consists of conical limit points and is quasi-self similar due to Corollary 2.66. The quasi-self similarity means that, for

each x E A(G) and E > 0, an arbitrarily small piece A(G) fl B(x, e) of A(G) can be expanded to a standard size and then mapped into A (G) by a K-quasi-isometry where K is independent of both x and e. On the other hand, loxodromic elements of the groups Gm (whose fixed points are dense in A(Gm), see Corollary 2.13) approximate loxodromic elements of the group G. This fact together with the quasi-self similarity of A(G) implies that the limit sets A(Gm) converge to A(G) in the Hausdorff metric p (*, *) on the set of closed subsets in S. Here the distance p (X, Y) between two closed subsets X, Y C S' is defined as follows:

p(X, Y) = sup{s(x, Y), s(y, X) : x E X, y E Y}

(7.5)

where s is a spherical metric on S', see (1.10). It follows that any neighborhood U(A(G)) of A(G) in S' contains all but finitely many limit sets A(Gm). We may assume that all groups Gm and G act by hyperbolic isometries in the (n + 1)-ball Bn+1 bounded by the round sphere Sn, because Mob(n) = Isom H1+1 So we have Kleinian orbifolds

M = M(G) = [Hn+1 U S2(G)]/G

and Mn = M(Gm) = [lHIn+1 U Q(Gm)]/Gm

whose faithful holonomy representations dm*: nirbMm - G

Gm C Mob(n)

converge to the natural inclusion G C Mob(n) which is the holonomy representation d* of M. So we have a convergence of the development maps, lim dm = d, where :M &R'

dm: An

Bn+1

Let F C Bn+1 U 0 (G) be the d-image of a compact fundamental polyhedron for the co-compact action of G on the universal cover M, with transversally intersecting sides. Since A(Gm) converge to A(G) as Gm tends to G, the mappings dm are well defined in a neighborhood U (F) of F, and hence lim dm (F) = F. It follows that dm(F) fl A(Gm) = 0 for sufficiently large m, and hence dm(Mm) # Bn+1. In other words, the developments dm are coverings due to Theorem 6.62, and the groups Gm have compact fundamental polyhedra Fm C Bn+1 U S2 (Gm) for sufficiently large m > mo. It follows from Theorem 4.16 that the groups Gm are convexcocompact. Now a direct application of the isomorphism Theorem 4.41 provides canonical G-equivariant homeomorphisms cpm: A(G) -+ A(Gm) which are in fact quasi-symmetric, see Theorem 4.44.

7.1. Deformations of geometric structures

385

The orbifold M can be obtained by identification of equivalent sides of the fun-

damental polyhedron F by some elements 91, ... , gk E G. So, for a small open neighborhood U = U(F), M = U/($1, ... , gk). Taking sufficiently large m (so that Fm C U), we may obtain orbifolds Nm = U/(g1,,,,, ... , gk,m) where gi,,n E Gm correspond to g1 E G with respect to the holonomy representations d,,,*. Since the topological structure on M is determined by a finite number of relations among the generators g1, ... , gk of G on the neighborhood U, we have that the orbifolds N,,, are homeomorphic to M because we have exactly the same relations among generators , gk,m of Gm on U due to our conditions: G. = G, Gm -+ G. gl,m, Here we may clarify the above homeomorphism argument on the base of the Chernavskii's [1, 2] variational principal, by using Siebenmann's results on when a proper submersion is a bundle map. We note that such an approach to stability of deformations of geometric structures and the corresponding holonomy representations works well in more general situation of pinched negative curvatures and is due to Apanasov-Siebenmann [1]. We consider sets of hyperbolic isometries {51,,,, , ... , gk,m }, m > mp, as sufficiently

small perturbations of the set of (F-side-pairing) generators {g1, . . . , gk} of G such that each gj,,n maps a small open neighborhood Uj C U of j-th side sj C aF of the polyhedron F to a neighborhood U! C U of the image side s = gj (sj) C a F, so that we have a topological perturbation of F inside of U, with transversally intersecting sides. We denote I = {m > mp } U {oo} to be the compact set of our parameters, where oc corresponds to the generators of G. Now we may consider mappings {g1, ... , gk} which identify "sides" of the compact "polyhedron" in the product U x I such that,

in each m-level U x {m}, m E I, their actions are the same as those of the small perturbations {g1,,,,, ... , gk,m} of the generators {$1, ... , gk} of G. So we have (even for continuous parameters m E I) that the quotient (U x I)/{g1, ... , gk} is locally a Cartesian product, with compact fibers. Moreover, since all perturbed polyhedra are closed and lie in the (closed) complement to an open neighborhood of the limit set A(G) in the closed unit ball Bn+1, the map 7r: (U x I)/ {g1, ... , gk} --+ I is proper,

that is all preimages n-1(X) of compacta X C I are compact. So we can apply Ehresmann-Siebenmann results on deformations of homeomorphisms on stratified sets, which can be formulated as follows, see Siebenmann [1, §0 and Cor. 6.14]):

Lemma 7.3. Let I be compact and the submersion 7r: (U x I)/{g1, ... , gk} -+ I be proper. Suppose that its fiber 7r (x) for each x E I is compact, then 7r is a bundle map.

This implies that we have in fact a fiber bundle (U x I)/{g1, . . . , gk} -+ I over 1. On the other hand, since our parameters I lie in a closed interval, this bundle is trivial. So the quotient (U x I)/{g1, ... , gk} is homeomorphic to the product (F/{g1, ... , 9k}) x I, and all fibers Ni are homeomorphic to the limit fiber M. We claim that the orbifolds N,n are in fact conformally equivalent to M,,, (their interiors are hyperbolically isometric). Indeed, since {$1,m, ... , gk,m} are generators of the discrete group G,n, the image of the universal cover Nn, under the development

386

7. Theory of Deformations

map Sm: Nm -' Bn+1 is Bn+1 U S2 (G,,). So the development Sm is an equivariant covering, and we may pass to the quotient under the group Gm. It defines an isometric immersion int Nm 0r4 int Mm. However the spaces Nm and Mm have the same topological K(Gm, 1) type, which implies that the above immersion above is in fact a hyperbolic isometry, and Nm = Mm. Finally we have that the compact orbifolds M. are orbifold-homeomorphic to the compact orbifold M. Due to compactness, there exist a quasisymmetric (quasiconformal in the interior) homeomorphism fm: M -* Mm whose lift to the universal cover, fm: Bn+1 U Q (G) -+ Bn+1 U S2 (Gm), is a quasisymmetric (quasiconformal) G-equivariant homeomorphism conjugating the actions of our groups G and Gm = fm G f,n 1. Due to Theorem 4.45, the union f m = fm U(pm of fm and the canonical quasisymmetric homeomorphism cpm : A (G) --+ A (Gm) is a desired quasisymmet-

ric G-equivariant self-homeomorphism of the closed unit ball, f m: B"+l -* Bn+l that conjugates the actions of the groups G and Gm. Its restriction to S' = aBn+1 is a G-equivariant quasiconformal homeomorphism of the sphere Sn conjugating Mobius groups G and Gm. Now, let M be a conformal manifold/orbifold and 7"1(M) C T (M) its Teichmuller space of equivalence classes of marked conformal structures with faithful holonomy d,,: 7rl'rb (M) - Mob (n). Let also To (M) C 7(M) be a subset of (classes of) conformal structures on M whose development maps are non-surjective, i.e. 7o(M) is a subset of almost uniformizable conformal structures, see Theorem 6.63. On the base of the stability Theorem 7.2, one can describe the subspaces To (M) and 7"1(M) as follows.

Corollary 7.4. Let M be a closed manifold/orbifold with a uniformizable conformal structure. Then the subset

7"c (M) = (EM, c] E 91(M): d r rb (M) = G is convex cocompact, I ZG I< 00) (7.6)

is open in the Teichmiiller space 7'1(M), and 7o (M) is closed in 7" (M).

Proof. The first claim that 7, (M) is open in 7'1(M) immediately follows from Theorem 7.2. In order to prove the second claim, suppose that we have conformal structures (M, cm) E To (M) whose developments dm converge to the development d: M -+ Sn of a non-almost uniformizable structure (M, c) 7o(M), d(M) = Sn. Then, for

any x E Sn, there is a neighborhood U(x) such that U(x) C dm(M) for sufficiently large m > mx. This, together with compactness of Sn, implies that there is an integer mo = max{mxt, ... , mx.} such that dm(M) contains an open covering {U(xl), ... , U(xk)} of Sn. Thus dm (M) = Sn, which contradicts the non-surjectivity of dm .

One can apply the above result in the case of a closed hyperbolic manifold M = IHI'2/ r, r C Isom 1HIn. Namely, let 7, (M) be again the space of equivalence classes of

7.1. Deformations of geometric structures

387

marked conformal structures on M whose holonomy representations d*: F --+ Mob(n) are faithful and d* F are convex co-compact. Let also Tqf (M) be the subspace of T, (M) consisting of classes of quasi-Fuchsian structures. Such a quasi-Fuchsian structure

on M is characterized by the property that its development d: M = Hn -+ S" is the composition of the inclusion IIP = Bn(0, 1) C 1R' = S' and a F-equivariant quasiconformal map f : Sn -a S" so that the holonomy group d*1' C Mob(n) is a quasi-Fuchsian group, d* 1' = f r f -1.

Corollary 7.5. Let F C Isom TIP be a co-compact group and M = ff / 1'. Then the space T, (M) is open in the Teichmiiller space Tl (M) of marked conformal structures on M with faithful holonomy, Tq f (M) is a component of T', (M), and To(M) is closed in the Teichmiiller space T (M) of all marked conformal structures on M.

5. Tangent bundle to Teichmriller space and group cohomology. Let M be an (X, G)-manifold/orbifold with a fixed structure so E T(M, X). We define a local deformation of the (X, G)-structure so as a one-parameter fancily {st} c 9'(M, X) of (X, G)-structures. To describe such local deformations, we consider the corresponding family {At} of (X, G)-atlases At = {U,, qpi}iE, on M, each of which defines transition functions q = V (cpj)-1 which are restrictions to open subsets of X of elements of the group G. It implies that there are locally constant transition maps yt: X -± G of the space X to the group G, defined up to the composition with elements of G: yt(x) _ e E G if x E j(U/ f1 Uj) . (7.7)

Clearly, these transition maps satisfy the cocycle condition wherever it makes sense: '0if

tE(-s,s), s>0.

(7.8)

A local deformation {st} is said to be trivial if, up to (X, G)-diffeomorphisms, all structures st coincide with the original structure so.

In order to study local deformations of (X, G)-structures so E T (M, X) with stable holonomy representations po = d*, we reduce the problem to studying local deformations of the representations po E Hom(irl (M), G) in the representation variety Hom(7t1(M), G). Such deformations can be defined as follows. Let po E Hom(F, G) be an injective homomorphism of a given group r into a Lie group G with discrete image poF, and W C lt8"` a neighborhood of the origin. A deformation of the representation po is a mapping R: W -+ Hom(F, G) such that R(0) = po and R(t) = Pt, t E W, are injective homomorphisms pt: F -* G with discrete images pt F and, for all y E F, the mapping t H pt y is differentiable (or continuous) in W.

Two deformations R: W -- Hom(F, G) and R': W' -* Hom(F, G) are said to be equivalent if there are a diffeolnorphism f : W -* W', * (0) = 0, and a differentiable map 'Y: W -+ G such that R'(if (t)) = 41 (t)* R(t) where, for h E G, h* = Ad(h) denotes the induced inner automorphism of G, h* g = hgh-1. A deformation R is called trivial if it is equivalent to a constant deformation.

7. Theory of Deformations

388

To have type-preserving deformations of Mobius groups r C Mob(n), we may consider a special type of deformations, the so-called quasi-conformal deformations R of r, R: W --* Hom(r, Mob(n)), determined by families { ft}tEw of r-equivariant

quasi-conformal self-homeomorphisms ft: Sn -+ S, fo = id. Namely, the rule R(t) _ (ft)*, (ft). I' = ftrft 1, t E W, defines a quasi-conformal deformation of the inclusion po: F C Mob(n). The correctness of this definition is implied by the Liouville Theorem 1.6 and the following well-known fact about the pointwise convergence of quasiconformal mappings, see Gehring [1], Reshetnyak [1, 2], Vaisaila [1]:

Theorem 7.6. Let f : X --> YY be a sequence of K-quasiconformal homeomorphisms of metric spaces which pointwise converges to a homeomorphism f : X -> Y. Then the limit homeomorphism f is K-quasiconformal. Now we shall define infinitesimal deformations which form the tangent bundle of the representation variety Hom(l', G). This tangent bundle generally differs from the tangent space to a smooth space. Namely, for a real algebraic set X C RI and a point x E X, one can define the tangent cone T Cx of X at x as follows. Let a: (-e, s) -a X be a real analytic curve passing through x, a (0) = x, and a(t) = E0 o aktk its Taylor series about t = 0. Then the leading coefficient of the curve a at t = 0 is said to be

the first non-zero coefficient an, n > 0; ak = 0 for 0 < k < n. The set of leading coefficients of all such curves a, a(0) = x, forms the tangent cone TCx of X at x, which coincides with the tangent space to X if X is smooth at x. To realize this approach for varieties Hom(r, G), we start with recalling the group cohomology. Let V be a vector space and A: r -+ Aut V a representation. A 1-cocycle on the group r with coefficients in A (or a crossed homomorphism with values in V) is said to be a map c: IF -± V such that

c(8y) = c(8) -f- A(8) c(y)

,

for y, 8 E r

.

(7.9)

We denote by Z' (r, V) the space of all 1-cocycles on r with values in V. The subset B1(r, v) C Z1(r, V) of 1-coboundaries consists of those 1-cocycles c for which there are vectors v E V such that

c(y) = A(y) v - v,

for all y E r.

(7.10)

The first cohomology of the group r with coefficients in the representation A: r Aut V is now defined as

H1(r, V) = Z1(r, V)/B1(r, V).

(7.11)

In the same way, one defines the cohomology groups HP (r, v) for p > 2, see Eilenberg and MacLane [1].

7.1. Deformations of geometric structures

389

Now let p: I' -* G be a representation and R: (-e, s) -)- Hom(r, G) its local deformation, R(t) = pt, po = p. Differentiating the equality pt(8y) = pt (8) pt(y) at t = 0, we have for y, S E F: dt

Lo Pt(SY) = p(3) ((d I t=opt(Y) I + (dt It=opt(s)

p(Y) .

(7.12)

/the

In this equality (in the tangent space to G at point p(Sy)), the points on the righthand side denote left and right translations, respectively. As each vector in the Lie group G can be uniquely extended to a right-invariant vector field on G, one can follow

Weil [2] and define an element c(y) E g that is an extension of the tangent vector d Ir-o pt(y). This and (7.12) give: Tt

S,yEr,

(7.13)

which defines a crossed homomorphism c = CR: r --> g twisted by the adjoint endomorphism ad p: r - * Aut g of the representation p, see (7.9). It provides an embedding of the tangent cone to Hom(r, G) at p into Z1 (r, g). For this reason, we call Z1 (r, g) the space of infinitesimal deformations of the representation p. For a trivial deformation R of the representation p, R (t) = Ad gt p where gt E G

and R(t)y = gtp(y)gr 1, go = id, let v E g be a tangent vector to the curve (gt} at t = 0. Then, upon differentiation, we obtain that

c(y) =

I t_o(gtp(Y)gr

1) = v - adp(Y). v.

dt This and (7.10) implies that this 1-cocycle c tangent to a trivial deformation is a 1coboundary. Conversely, every 1-coboundary is tangent to a trivial deformation. It enables us to identify an infinitesimal deformation of a representation p = R (O), p: r -+ G, with the corresponding cohomology class in H1(r, g). In the case of conformal structures, we may assume that G = Mob(n) = Isom 1Hl't+1 and IF C Mob(n - 1) is a finitely generated discrete group conformally uniformizing a (hyperbolic) n-manifold M. So we can consider infinitesimal deformations corresponding to quasiconformal deformations of r, that is to families {ft } of r-equivariant quasiconformal automorphisms of the sphere Sn = IRn (or of the hyperbolic space). In the case of the hyperbolic space H n+1' the property of a quasiconformal homeHn+1 to be r-equivariant can be conveniently expressed in omorphism f : Hn+1 terms of distortion of tangent vectors, with respect to the hyperbolic metric in the tangent bundle THn+1, see Kourouniotis [1]. Namely, denoting the hyperbolic length of a vector u E TXHn+1 by 1j u IIx, we have the following criterium. Hn+1 a Lemma 7.7. Let r c Isom H1+1 be a discrete group and f : IH[n+1

quasiconformal homeomorphism, n > 2. Then f is r-equivariant, that is f r f -1 C Isom Hn+1 if and only if the following holds for ally E r, all vectors u E I[8n+1 and almost every x E ]H[n+1: (7.14)

390

7. Theory of Deformations

Proof. The necessity of (7.14) is obvious because each mapping f y f -'is a hyperbolic isometry provided f is r-equivariant. Conversely, assuming that (7.14) holds, wehave that II D(f yf -1) - v Il f y(x) = Ii v Il f(x) for all v c- ][8"+1 and a.e. x E IH[n+1. It implies that the quasiconformal dilatation K (f y f -1, x) equals 1 almost everywhere in Hn+t Thus, the homeomorphism f y f -'is a hyperbolic isometry due to Theorem 1.6.

Let X be a domain in W and F a 1-parameter family of a.e. differentiable home-

omorphisms of X into Rn, F: X x (-e, s) -* R", F(x, 0) = id. In addition, we assume that the map t i-+ F(x, t) = ft(x) is differentiable at t = 0 for almost all x E X, and define

d (7.15) oft(x) dt Then F(x, t) = x + ta(x) + o(t) and, if fi(x) is almost everywhere differentiable, o(t). Furthermore, the derivative of D1F(x, t) = E +

fi(x) =

I det D1F(x, t)1-2/n

. (D1F(x, t))T D1F(x, t)

with respect to t at t = 0 is twice the value of the operator

2

Here AT denotes the transpose of a matrix A, tr is the trace function, and E is the unit (n x n)-matrix. In mechanics, the operator SS is known as the deviator of the tensor of deformations, and the equation SS = 0 is called the Liouville condition for conformality of F(x, 0), see J. Serrin [1]. It was also considered by Reshetnyak [3] and Ahlfors [3, 5, 7] as a generalization of differentiation with respect to z. Its properties allow us to define quasiconformal vector fields as follows, see Ahifors [5].

A K-quasiconformal vectorfield in a domain X C R' is a continuous map : X JR of the W1,1oc(X)-class (i.e., the first derivatives Dolt of are locally integrable) such that the function x H ll Sg (x) II is of the L(X)-class, and the norm II Sd II < K almost everywhere. Such vector fields are integrable due to Reimann [1], Ahlfors [5], Semenov [1]:

Theorem 7.8. Let be a K-quasiconformal vector field in 1R with compact support. Then there exists a unique 1 parameter group of homeomorphisms Ft generated by the field , which satisfies the condition

Ft(x) = x +

f

t '(F.r(x))ds

0

and the homeomorphisms Ft are eK.I tI -quasiconformal maps.

We now assume that a family F (x, t) = ft (x) of quasiconformal self-homeomorphisms of the (n + 1)-ball Bn+1 = Hn+1 is equivariant with respect to a discrete group

7.1. Deformations of geometric structures

391

F C Mob(n) and defines a deformation R = {pt}, R: t H pt = (ft)*. The group Mob(n) acts in IH[n+1 as the isometry group, and one can therefore define its action on the set of continuous vector fields 17 in IH[n+1 as follows: (7.16) y - 17(x) = DY(Y-1(x)) ?7(y-1(x)). Then we take such a vector field 77 as the vector field given in (7.15). The F-

equivariantness of the family { ft} implies that the vector field c(y) defined as

c(y)(x) = '(x) - y .

(x)

for x E 1H[n+1, y E r ,

(7.17)

is compatible with the action of F in IHIn+1 and defines a class c: F

1)n+1 in the

,

cohomology group H1 (F, n+1). Here C)n+1 is the Lie algebra of fn+1 which is isomorphic to the Lie algebra mn of Mob (n), and hence the vector field (7.17) defines a class c in H1(r, mn). In fact, for all y E I' and X E IH["+1, we have:

c(y)(x) = fi(x) - y fi(x) = fi(x) - DY(Y-1(x)) (Y-1(x)) = D2F(x, 0) E + D1F(x, 0) Dy(y-1(x)) (-D2F(y-1(x), 0) E d

= dt

t_o(.ftYffi_1Y-1)(x) I

,

and therefore c(y) is an infinitesimal isometry of the hyperbolic space IH[n+1

In §§3-5, we shall provide conditions on hyperbolic (or, generally, conformal) manifolds M = IH[n/ F, n > 3, under which the cohomology groups H1 (r, ma), that is the spaces of infinitesimal deformations of the inclusions F C Mbb(n), are non-trivial.

Let (M, S) E 7(M, X) be a (X, G) -structure on a manifold M such that in some of its neighborhoods the holonomy map hol is a local homeomorphism, in particular a structure with a good holonomy. Then infinitesimal deformations of such an (X, G)structure (M, s) can be identified with those of its holonomy representation d*, i.e. with classes in H1 (P, g). One may expect however that not all infinitesimal deformations are integrable, that is not all of them correspond to deformations of representations

p E Hom(r, G)/G (or of structures s E 7 (M, X)). The first obstruction to that integration of infinitesimal deformations in H 1(I', g) is the second cohomology group H2(r,, g). To construct this obstruction, we can use the Kolmogorov-Alexander cup product, see Dold [1, §7.8]:

H1(r, g) ®H1(r, g)

H2(j,'9).

Namely, for any 1-cocycle c E Z1 (r, g) specifying an infinitesimal deformation for a curve {pt} C Hom(P, G), po = p, the cup-square of c defines the following 2-cocycle

c _ C E Z2(r,g):

(c - c)(y, S) _ [c(y), adp(y) C(8)];

Y, S E r,

(7.18)

where [*, *] denotes the Lie bracket in g. Then a necessary integrability condition of infinitesimal deformations can be formulated as follows, see Johnson-Millson [1, Pr. 2.1]:

7. Theory of Deformations

392

Theorem 7.9. Let {pt} C Hom(I', G)/G be a deformation of a good representation po and c E Z1 (P, g) its infinitesimal deformation. Then c '- c represents the zero element in the second cohomology H2(r, g). The integrability obstruction lying in H2(F, g) is the first of a sequence of such obstructions to the existence of a deformation {pt) C Hom(P, G) with a given tangent vector c r= Z 1(I', g). Constructions of them using the Massey products can be obtained from the general scheme of the Kodaira-Spencer deformation theory. We shall return to this in §3 in study of deformations of hyperbolic and conformal structures.

§2. Rigidity of hyperbolic structures Here we study deformations of hyperbolic structures, that is structures of Riemannian n-spaces M with complete metrics of constant negative sectional curvature -1. Since they are K(Jr, 1)-type spaces, one can identify them with isometric actions rorb(M) on the (universal covering) hyperbolic of their fundamental groups r'

n-space IV, M = Hn/ r, and study the corresponding representation variety Hom(r, Isom Hl)/ Isom Hn. 1. Spaces of finite hyperbolic volume. The first result on deformations of hyperbolic structures in dimension n > 3 follows from A. Weil's rigidity theorem [1]:

Theorem 7.10. Let G be a connected semisimple Lie group without compact components, whose Lie algebra g has no factors 5C(2, I[8), and P C G its uniform lattice.

Then the orbit G r of the inclusion F C G with respect to Ad G action is open in the variety of injective representations p E Hom(I', G). Due to this rigidity, there are no local deformations of compact hyperbolic spaces Hn/ P. Moreover, there is a stronger rigidity in a bigger class of finite volume locally symmetric rank one spaces, which is due to the remarkable theorem by G. Mostow [1, 2], G. Margulis [1] and G. Prasad [1]. We formulate it for the real hyperbolic case:

Theorem 7.11. Let I'1 and F2 be isomorphic lattices in the hyperbolic n-space H, n > 3, that is isomorphic discrete isometry groups with finite volume quotients Hn / I'!. Then they are conjugate in Isom Hn, and the original isomorphism is induced by this conjugation. The following geometric reformulation of this theorem implies that the Teichmiiller space T (M, Hn) of hyperbolic structures on a finite volume hyperbolic orbifold M consists of one point:

Theorem 7.12. Let M1 and M2 be complete hyperbolic n-orbifolds of finite volume, .jrb(M2) an isomorphism of their fundamental groups. n > 3, and rp: 7r' (M1) Then M1 and M2 are isometric, and the isometry is uniquely determined by the isomorphism cp, up to inner automorphisms of the fundamental groups. We shall examine the main stages of the proof of this rigidity theorem.

7.2. Rigidity of hyperbolic structures

393

First, we note that the isomorphism (p: n1 (M1) -* 7ri (M2) is induced by a homotopy equivalence M1 = M2 because both M1 and M2 are of K (7r, 1)-type. So there are

continuous mappings fl: M1 -> M2 and f2: M2 -a Mi such that fl f2 and f2 f, are homotopic to identity. Passing to the universal covering H1, one obtains equivariant liftings fi:]EII"

H.

The second step is that the equivariant homotopy equivalences it and f2 may be chosen pseudo-isometries of the hyperbolic n-space klrn, see (4.42). In the original proof of Mostow [1], they were especially taken as quasi-conformal maps. Then one can use Efremovich-Tikhomirova [1] Theorem 4.43 to extend the pseudo-isometries ff:1HI" -, En to homeomorphisms of the sphere at infinity, Si-1 = alEln. Due to Gehring [1, 2], the extension automorphisms of the sphere S'-' are quasiconformal homeomorphisms, provided n > 3. The third step uses the finite volume condition to prove the fact that there is a measurable conformal structure on the sphere Sn-1 = 3W invariant with respect to the action of a co-finite discrete group F C Isom Mn. This structure is unique up to a set of zero measure. In addition to the original proofs in Mostow [1, 2], Margulis [1] and Prasad [1], there are different proofs of this claim due to Gromov (see Thurston [1]), Sullivan [2] and Tukia [10, 12]. Later in Theorem 7.16, we shall give a detailed account on this claim in more general situation of infinite volume spaces. In the last fourth step, one proves that a quasiconformal automorphism of the sphere Sn-1 whose dilatation equals I almost everywhere is a MObius transformation A. It is given by a generalization (Gehring [1], Reshetnyak [1, 3]) of the Liouville Theorem 1.6; for its generalization to Carrot groups, see Capogna [1]. Thus, the groups G1, G2 C IsomH1 are conjugate by the transformation A E Mob(n - 1), G2 = AG1A-1.

2. Hyperbolic structures of infinite volume. Examining the above scheme of proof of the hyperbolic rigidity Theorem 7.11, one may notice that the main problem in extending this rigidity to a wider class of hyperbolic structures which may have infinite volume, is concerned with the third step of its proof. Namely, one needs the uniqueness of a conformal structure on the sphere alHkn compatible with the action of a discrete

group IF C Isom H. It was questioned by Mostow [2] (and Krushkal [3-5]) whether such uniqueness takes place in the case of those discrete groups whose limit sets A(F) coincide with the whole sphere M. In general, the uniqueness of a conformal structure at infinity and the rigidity of the corresponding hyperbolic structure fail, as the following construction shows. Theorem 7.13. In each dimension n, there are non-rigid discrete groups r c Isom kkn whose limit set A(r) is the whole sphere akiln.

Proof. We shall construct a discrete group r C Isom klrn, A(r) = aklln, together with

its non-trivial quasiconformal deformation {ct}, t E (-s, e), where Ot: lEn -* in are F-equivariant quasiconformal homeomorphisms. Let Q = {x E k8n-1 : 1xi 1 < 1/2} be a unit cube and {km} an increasing sequence of odd integers. We split up the cube Q into ki-1 equal subcubes Q1i and cast out

7. Theory of Deformations

394

the interior of the central one. Similarly, in the second step, we split up each of the remaining cubes Q1i into k2-1 equal subcubes Q2./ and cast out the interiors of the central ones. Suitably choosing the increasing integers km, e.g. km = 3m, and continuing the process of casting out, we finally obtain a limit continuum K C Q whose Lebesgue (n - 1)-measure mii_I (K) may equal any positive number in the interval (0, 1), see Hausdorff [1]. Let pl : lW -+ IR el be the orthogonal projection to the first coordinate axis III el,

and O1 C pi(Q\K) be the complement in [-1/2, 1/2] to the p1-projection of all casted out subcubes Qij, C Q. Then we define a map ft: Q x R -+ R1, t > 0, as follows:

lkt(xl)=

.ft(xi,...,xn)=(*t(X1),x2,...,x,),

f

X'

t(Y)dy,

(7.19)

0

where the measurable function Ot (y) is

(Pr(y)=

11, if ye01; it,

ifYE[-2

,2]\01.

It is clear that ft is a quasiconformal homeomorphism whose dilatation equals t if t > 1, and it is 1/t if 0 < t < 1. The map ft transforms Q x IR C RI to the set

ft (Q x R) = Qt, x IR = (x E r : Ix11 < ti/2, Ixi l < 1/2, 1 < i < n) , which can be obtained by stretching the set Q x R along the first coordinate axis IR el by a factor t1 > 0. This number t1 depends only on t and the sequence {km}, and can be equal to 1 only for t = 1. We notice that ft is conformal on the subset 0 x JR C Q x R. Namely, on each component of 0 x R, the map ft is a Euclidean translation by a vector r el whose length r depends on the component, see Figure 122.

d

a

f Y

Figure 122

To construct isomorphic discrete groups r and rt c Isom H", we consider a dense covering E of the set 0 by disjoint round n-balls B1, Bin R"-1 C 0, i E I, centered at

7.2. Rigidity of hyperbolic structures

395

points of the plane Rii-1 C 1R' . The map ft translates the family E = {Bi } to a family Et = {BI) of disjoint round balls which densely cover the image ft (A). Now we define

free discrete subgroups H and Ht in Isom Hn as the groups generated by reflections in boundary spheres { a B1 } and { a BI }, respectively. Taking free Abelian groups E, Et C IsomRn-1 whose generating translations identify the opposite faces of Q x R and Qt, x IR, correspondingly, we can use the Klein Combination Theorem 5.12 to form the desired discrete groups 1, and rt as the following free products,

r=E*H and rt=Et*Ht,

(7.20)

which act in the upper half-space R+ by hyperbolic isometries. Obviously, the limit sets A (r) and A (rt) coincide with the whole sphere ][8n-1 = ailW , and we can choose

fundamental polyhedra P(r), P(rt) c ]R+ as

P(r) _ (Q x ]I8+)\UintBi ,

P(rt) = (Qt, x 1R)\UintBf

.

(7.21)

Since ft (P (1')) = P(rt) and ft is r-equivariant on the boundary of the polyhedron P (F), we can equivariantly extend the map ft I P(r) to a r-equivariant quasiconformal homeomorphism 4?t: ]]8+ - R+ whose dilatation is the same as that of ft. Due to Theorem 4.43 and Gehring [1, 2], the homeomorphism It is a quasiconformal selfhomeomorphism of ]]8n. It clearly follows from the construction that rt = ctr4?r 1, and {cDt } is a quasiconformal deformation of the inclusion r C Isom H1. We claim that this is a non-trivial deformation. It preserves the subgroup of IF generated by reflections in the boundary spheres of the subfamily {B1}iEio, to C I, that consists of all balls that cover the central cube Quo. Hence, if this deformation

were trivial it would preserve the whole group IF, which is not true: rt # r for

t # 1. One can use another argument based on the action of our deformation on loxodromic subgroups of F. Namely, let y E r be a loxodromic element whose fixed points p and q are projected to different components of the set Al C R. Then the corresponding loxodromic element yt = I t y 4?I 1 of the group Pt has fixed points 1. It follows that the length of the hyperbolic Pt and qt, I Pt - qt 10 I P - q l if t translation yt is not equal to the length of the hyperbolic translation y, which is impossible for trivial deformations.

The above construction can be slightly modified to answer another question of L. Bers [5] and S. Krushkal [3-5]: Corollary 7.14. There are non-trivial quasiconformal deformations {DO} of Kleinian groups r C Mob (n), n > 1, whose restrictions to the discontinuity set 0 (r) C Sn are conformal. In particular, for n = 1, 2, there are Beltrami IF-differentials A(z)dzldz whose supports are contained in the limit set A(G) C C.

Proof. We can define desired Kleinian groups r c Mob(n) as subgroups of the discrete groups of isometries of the hyperbolic space Hn+1 constructed in Theorem 7.13, see (7.20). Namely, in the definition of those groups, we take a subfamily

7. Theory of Deformations

396

E0 C E = {Bi } obtained from E by excluding a finite subset of balls B11, ... , Bik, Bit fl (I[8" x {0}) C A. Then the resulting group r C Mob(n) - IsomHn+1 generated as in (7.20) by n Euclidean translations and by reflections in remaining spheres a Bj, Bj E E0, is a Kleinian group, and its discontinuity set S2 (r) C R is the r-orbit of the balls Bi, fl I[8n, j = 1, ... , k. Upon this adjustment, the construction in Theorem 7.13 gives us a quasiconformal deformation { l }t>o, 0o = id, of the group r where r-equivariant quasicon-

formal homeomorphisms '1 : ln+1 -- IEg+ 1 conjugate r to the Kleinian groups see (7.19) and (7.20). Furthermore, this deformation [(DO) is rt = cI non-trivial, and restrictions (Do I trl are conformal because of the definition (7.19) of the maps ft. In other words, the quasiconformality coefficients K(00, x) differ 1 for t ,-E 1 only at from 1 only on the limit set A(r). More precisely, K(4)0, x) points x from the F-orbit of the subset Kp C K that is projected to the Cantor set [-1/2, 1/2] \A1 in the first coordinate axis. In the case n = 2 or n = 1, that is for Kleinian groups on the extended complex plane C, it means that the complex characteristics µt (z) of our r-equivariant quasiconformal homeomorphisms cto IU are supported on the corresponding subset of the limit set A(r). F((D0)-1

Remark. We notice that the above constructions in Theorem 7.13 and Corollary 7.14 are based on the crucial fact that the subset of limit points A (r) fl P (r) on the boundary of the fundamental polyhedron P(r) in (7.21) has a positive measure in the sphere at infinity a1W1. We shall see below that this is a characteristic property of non-rigid hyperbolic groups and structures.

Due to the previous observations about non-rigid hyperbolic structures, we may restrict our attention to the class 3eng of hyperbolic n-orbifolds M = H11G without boundary whose holonomy groups G, ni rb (M) = G C Isom Hn, have fundamen-

tal polyhedra P (G) C HI with zero Lebesgue measure in the sphere at infinity, mn_1(P (G) fl 0lW1) = 0. Fixing a point 0 E M, we denote by M(r) = {x E M : d(0, x) < r} the r-neighborhood of this point in M with respect to the hyperbolic distance d, and D(r) C 1HI' a hyperbolic ball of radius r > 0. Then, due to Theorem 2.63, the class 3fng of hyperbolic orbifolds M can be characterized by any of the following equivalent conditions:

(1) limr.00 vol M(r)/ vol D(r) = 0; (2) the action of the holonomy group G on the sphere alHln is conservative; (3) the horospherical limit set Ah (G) has the full measure in a1HJ'. We shall show that the class 3fg consists of hyperbolic orbifolds that are rigid in the following sense.

Definition. A complete hyperbolic n-orbifold M = HI / G and its discrete holonomy group G C Isom ff3[n are called rigid in the sense of Mostow if any pseudo-isometry

between M and another complete hyperbolic n-orbifold M' = WIG' is homo-

7.2. Rigidity of hyperbolic structures

397

topic to an isometry, that is the induced isomorphism G -a G' is a conjugation in Isom ]EII" .

Theorem 7.15. Let M = Hn/G be a complete hyperbolic n-orbifold with the holonomy group G C Isom Hn, n > 3, which satisfies one of the following equivalent conditions; (1) the subset at infinity of a fundamental polyhedron P(G) C IH[n of G has zero measure in a1HI' ;

(2) limr,oo vol M(r)/ vol D(r) = 0; (3) the G-action on aIHIn is conservative; (4) the horospherical limit set Ah (G) has the full measure in a]Hv'. Then M and its holonomy group G are rigid in the sense of Mostow Before we go on with the proof, we notice that the condition n > 3 is essential. This is based on the fact that the extension of a quasi-conformal automorphism of a round 2-disc B2 to its boundary circle aB2 may be a non-absolutely continuous map of the circle, in contrast to the quasi-conformality of such extensions 8H' -+ a1HIn in higher

dimensions n > 3. In dimension n = 2, there are non-rigid groups G C Isom 112 satisfying all conditions of Theorem 7.15. Such groups were constructed in 19181921 by V. Golubev [1, pp. 322- 351, 401-403]. Namely, he constructed bounded automorphic functions on 1HI2 for each of the following classes of discrete groups: (1) for a group G C Isom1H[2 whose fundamental polygon P(G) C 1H12 has a zero measure subset at infinity a1HI2, of continuum cardinality; (2) for a group G C Isom 1H[2 whose fundamental polygon P (G) has a countable subset P(G) fl a1H[2.

Proof of Theorem 7.15. The result can be obtained by replacing the third step in the proof of the hyperbolic rigidity Theorem 7.11 by the following claim.

Theorem 7.16. Let G C Mob(n) be a discrete group, n > 2, and v a measurable conformal structure on the n-sphere S" invariant a.e. with respect to G. Then the structure v is the standard conformal structure of the sphere Sn, almost everywhere on the subset .7( C Sn with the conservative G-action.

Proof. This essentially follows Sullivan's proof of this fact in dimension n = 2, see [2], and is based on the observation that the conservative part d( C S2 of the G-action has no measurable a.e. G-invariant vector field. In dimension n > 3, comparing a conformal structure v on Sn with the standard one, one gets almost everywhere a field of ellipsoids, each of which is defined in the tangent space up to similarity. It defines tangent k-plane fields, 1 < k < n - 1, and reduces the proof to the following claim.

Theorem 7.17. Let G C Mob(n) be a discrete group, n > 2, and X C Sn a Ginvariant subset with the conservative G-action. Then d( has no G-invariant a.e. measurable tangent field of k -planes, 1 < k < n - 1. Proof. It splits up into two cases, the case of the infinite solid angle EgEG x9 n of the G-orbit of a ball in 1n+t, and the case of the finite solid angle, see Lemma 2.46.

398

7. Theory of Deformations

In the first case of the infinite solid angle, the proof follows from Theorem 2.47 which shows that the G-action on the set of hyperbolic geodesics is ergodic: any measurable G-invariant function cp on S' x S' is constant a.e. Indeed, if such a Ginvariant k-plane field existed, we would take the following G-invariant measurable function cp: S" x S' -+ R. Given points x, y E Sn, the value of the function O(x, y) measures the angle between the corresponding k-planes at x and y, up to parallel translation along the geodesic (x, y) C IH[n+1 The ergodicity then implies that (p is a constant a.e. on X, that is we have a continuous G-invariant k-plane field, which is impossible. In the second case, Y'gEG A-9-' < oo, we adjust the Sullivan's arguments with analytic functions on the plane C to the general situation. As in Chapter 2, §5, we

assume that each g E Mob(n) is decomposed into a composition of a hyperbolic translation with antipodal fixed points (poles of Sn) and an orthogonal transformation

in Sn. Then the linear derivative dg of a transformation g in the spherical metric varies in an interval [ 1 /A , A] with k > 1 and is constant on each of concentric spheres SB-1 C S" (interpolating between the fixed points of the hyperbolic translation). The

spheres SB-1 are parameterized by the angle 0, 0 < 0 < ir, where the value 0 = 0 corresponds to the fixed point at which dg = X. We now consider a sufficiently large K > 1 and, for all g E G having X > K2, we define a spherical concentric annulus Ag which is the union of the spheres Sa-t with

parameter 0 from the interval irK/;. < 0 < 7r/K. The complement S"\Ag is the union of two components: the smaller one Eg in which 0 < 0 < it K/.X, and the larger one C. in which nK < 0 < it. We define a continuous piecewise approximation Dg of the linear derivative dg by putting X

Dg(x) _

XOT

Kz,

1'f xE E g, if x E Ag n Se-1

(7.22)

if X E Cg.

Let us rewrite this with respect to the Euclidean _metric in R. To do that, we consider y E Mob(IW'), y (x) = ,lx, and pass from kn = Rn U {oo} to the unit sphere Sn by using the stereographic projection (1.9). Then the spherical metric in Rn is given by ds = Idxl/(1 + Ix12), and thus the linear derivative of y is dy(x) _ A(l + Ixl2)/(1 + I;,xl2). Therefore the domain Eg C IIln is the ball {x E Rn : JxJ K/A} where dy(x) varies in the interval [1,,(1 + K2/A.)/(1 + K2)

,

A] C A[1/(l + K2)

,

1].

(7.23)

The domain Cy is {x E I[8n : 1/K < lxi}. Here dy(x) varies in the interval

[1/;,, A(1+1/K2)/(1 + A2K2)] C 1/X. [1, 1 + K2]

.

(7.24)

7.2. Rigidity of hyperbolic structures

399

This shows that, in the annulus Ay = {x E Rn : K/?. < Ixl < 1/K}, the ratio dy (x)/Dy (x) = dy (x) ), Ix 12 varies in the interval Ipxl2

X(1 + 1/K2) XIx12

[ 1+IAx2'

IAx12

1

] C

1

' I I+1/K2+K2 1

]

.

(7.25)

Passing to the stratification of the sphere S'\{x, y} with two punctured points by (n - 1) -spheres SB-1 parameterized by the angle 0, 0 <_ 0 <_ rr, we see from (7.23)(7.25) that, for all elements g E G with ),g > K2, the ratio dg(x)/Dg(x) is uniformly bounded in Eg U Cg and is arbitrarily close to 1 in the annulus Ag for sufficiently large K. It implies that there are constants bK and CK that depend only on K and such that

lim cK=1,

K-+oo

volE5 <

K diam Ag <

bn, volg(Cg)
,

diam g(Ag) <

2K .

(7.26)

The last bound in (7.26) follows from the fact that, up to a factor close to 1 for sufficiently large K, the size of the spheres g(SB-1), S' -1 C Ag, is the same as that of the spheres S8'-1 with 01 = n2(A0)-1. Since rrK/X < 0 < n/K in the annulus Ag, it implies that the angle 01 lies in the interval [n K/A, 7r/K] whose size determines the constant CK.

Now let X C Sn be the set where the group G acts conservatively. This means that, for any X C X of positive measure mn (X) > 0, there exist infinitely many elements g E G such that mn (X fl g(X)) > 0. For a concentric annulus A C Sn, let the modules mod(A) denote the logarithm of the ratio of radii of its boundary spheres. Then the above bounds imply:

Lemma 7.18. Let G C Mob(n) be a discrete group with EG 'Xg n < oo and the conservative part X C Sn. Then, for a given subset X C X and constants A, 8 > 0, there exist a point x, an element g E G and a concentric annulus A C Sn such that

(1) xEXflA,g(x)EX; (2) diameters of A and g (A) are less than 8; (3) moduli of A and g(A) are arbitrarily close to A; (4) the linear derivative dg (x) is constant on each of concentric (n -1)-spheres in A, and its variation in A is arbitrarily close to exp(2A).

Proof. Let K > 1 be so large that, in notations of (7.26), 2cK/K < S. Since G is discrete, we can cast finitely many of its elements out to make Ag/K2 > e° for the remaining g E G. Additionally casting out finitely many elements of G, we can make the sum EgEG, vol(Eg U g(Cg)) over the remaining elements g E G to be less than Mn (X) because of (7.26) and the fact that EG )'9_' < oo. Then, removing the infinite

union UgEG,(Eg U g(Cg)) from X, we get a set X' C X with mn(X') > 0. Since

7. Theory of Deformations

400

X' is a subset of the G-conservative part X, there is one of the remaining elements,

g E G', such that mn (g-1(X') fl X') > 0. The previous construction implies that 9-1 (XI) fl X' C Ag because any x E g-t (X') fl X' does not belong to Eg, and g(x) ¢ g(Cg). Since dg(x) is arbitrarily close to Dg(x) in Ag for sufficiently large K, we have from (7.25) that dg (x) varies in Ag in the interval [K2/,lg, .kg/K2]. Due to .lglK2 > exp(A), this interval contains an interval of the length exp(2A) consisting of values dg (x), x E Ag. Let such an interval define the annulus A in question, which then satisfies the conditions (1) and (4). The annulus A satisfies the condition (2) because of (7.26) and A C Ag. To prove the remaining claim (3), we notice that dg(x) behaves in Ag and thus in A as 7r2(.lgO2)-1, because of (7.22) and (7.25). At the same time, In dg (x) varies in A by 2A, which implies that the modulus of A is close to A = (In 022/02)/2 =1n(92/01). As the annulus g(A) is bounded by spheres whose sizes are defined by parameters r91 e2A and rB2 with some r > 0, the modulus

of g(A) approximately is A = 2A - ln(02/01) = ln(rOle24/rB2). This completes the proof of the lemma.

Now we pass from the sphere Sn to the space Rn. Here let there be given a set B and a subset A C B with a volume proportion rl, rl = mn (A)/mn (B), and let B' = g(B) for g E Mob(n). Then the set A' = g(A) is contained in B' in the volume proportion at least rl', mn(g(A)) '

=

mn (g(B))

(7.27)

where d is the maximum ratio of the volume distortion at various points of B. In fact, Rn, that property is true for any absolutely Lebesgue continuous mapping g: IR' which is implied by the following observation.

By using an affine scaling, we may assume that mn(B) = 1 and the volume distortion varies in B in the interval [1, d]. Then the worst case occurs when the volume is preserved in all points of A and is disturbed d times in B\A. In this case, we obviously have

mn(g(A)) mn(g(B))

q

- rl + d(1 - rl) >- 1 - d(1 - tl)

,

which proves (7.27). We now define a sector box of shape A as a set VA obtained by intersecting an annulus AA and a circular solid cone Co, VA = AA fl Co, where AA is bounded by spheres centered at a point po E 1Rn with radii ro and roe°, po is the vertex of Co, and the ruling of the cone Co forms an angle A /2 with the direction vo E Rn of its axis. Lemma 7.19. Let X C 11 have a positive measure, and let rl and A be given positive numbers. Then there exist a subset X' C X of positive measure and S > 0 such that each sector box VA of shape A and diameter less than 8 containing a point of X' also contains at least the proportion rl of X, m, (VA fl X)/mn(VA) > i.

7.2. Rigidity of hyperbolic structures

401

Proof. Since the class of sector boxes of shape A can be obtained by similarity transformations from one of these boxes, it is possible to use such sector boxes instead of round balls in the Lebesgue theorem, cf. E. Stein [1, Ch. 1 §2]. This theorem states that almost all points of a set of positive measure are density points of the set. Therefore, for almost all x E X, there exists a largest positive Sx > 0 such that the proportion of X in boxes containing x and of diameter at most Sx is at least ri. Then the function x i-+ SX is a positive measurable function which, on a subset X' C X of positive measure, has to be greater than some number S > 0. This proves the claim. Proof of Theorem 7.17. Now we continue the proof by using the above results. Let us assume that the conservative part X C R' has atangent k-plane field E, 1 < k < n-1, which is G-invariant a.e. We may find a subset X C .7C of positive measure on which the k-plane field E is almost parallel, up to a small angle A, that is it varies on E only in an interval of inclination of length A /2. Due to Lemma 7.19, for a fixed number

n, 0 < j < 1, there exist a Si > 0 and a subset X' C X of positive measure such that each sector box Vo of shape A, of diameter less than 81, and containing a point xo from X' contains at least the proportion n of the set X. Let xo be a density point of X', and 7r: Rn -+ Sn be the stereographic projection with respect to that point, that is the map in (1.9) where the origin is at the point xo. Let B' = B(xo, 32) be a ball centered at xo whose radius 82 is sufficiently small so that the distortion of the stereographic projection Tr on that ball 2B' = B(xo, 232) is as close to 1 as we need for the following. Now, for given A, S = min{S 1, 82}, and a set Y = X' n B' C X of positive measure

(considered on Sn), we apply Lemma 7.18 to find a point x, an element g E G and a concentric annulus A (put back on II8n and centered at the center of the isometric sphere I (g)) that satisfy the conditions (1)-(4) in Lemma 7.18. In the annulus Q we consider sector boxes Vo of shape A which contain the point x and are centered at g-l(oo). Due to (7.26), diameters of VA and g(VA) are less than S < 81. Thus, our construction implies that Vo and g(VV) both contain q-portions of the set X, because x E VA C A, g(x) E g(VA) C g(A), and x and g(x) both belong to X'. Due to (4) in Lemma 7.18, the variation of dg (x) in VA is close to exp(2A), and hence (7.27) implies that the proportion of the subset of points y E Vo such that g(y) E X is as follows: (7.28) mn(V° n g-1(X)) > z' = 1 -e 2A(l - rl). Mn (VA)

This and (7.27) imply that

mn(Vt n x n g-1 (x))

> 1-[(1-n)+(1-n')I>ri-e2°(1-q),

mn(V,)

that is the set x n g-1 (X) gives at least an ho-portion of the sector box Vo, no

r - e2' (1 - 77).

=

Now let us consider all vectors connecting the center point defining the sector box VA to points of X n g-1 (X). Clearly, the variation of angles between these vectors is at

7. Theory of Deformations

402

least Ono. Indeed, in the worst case when the set Vo fl x fl g-1 (X) is the intersection Vo fl V in which V is a solid round cone with the same center as VA and a ruling angle t, we have:

f

A/2

r/2

smn-2 cpdtp > io ,

fSinn-2 rpdc

.

0

This implies that t > 0 no11(n-1) > t1 rJo, which can be applied to the .s-parallel G-invariant field E of k-planes. For two parallel vectors v and w at points of the isometric sphere I (g), g E G, whose radius vectors have an angle a, 0 < a < n, their images g(v) and g(w) (in the tangent space), or equivalently, the images I(v) and I(w) under the reflection I with respect to the sphere I (g), have the angle 2a, see Figure 123. I

Figure 123

Since the center g-1 (cc) of the isometric sphere I (g) varies (dense) in a positive measure set X, we may assume that our vector field is not radial with respect to that center. This extends the above observation to almost parallel vector fields. Therefore, if the angles between k-planes in the field E varied in an interval [-a, -a + 0], where

a, 0 < a < 0, is some normalization, the angles between their images in the field g(E) would vary in the interval [20rjo - a, 20r70 - a + 0]. However, since X is a subset of the conservative part X of the G-action, points of the latter interval must have the distance to the origin at most A. Clearly, this is impossible for the numbers no sufficiently close to 1.

7.2. Rigidity of hyperbolic structures

403

This contradiction completes the proof of Theorem 7.17, and hence of Theorems 7.15 and 7.16.

3. Conformal rigidity of manifolds with boundary. We can apply the above rigidity Theorem 7.15 to answer the following natural question posed by L. Bers [5] and S. Krushkal' [3-5]:

Let M be a conformal n-orbifold with boundary, n > 3, whose conformal struc-

ture in int M is induced by a complete hyperbolic metric, and f : M -+ M' be a quasiconformal map onto a similar conformal orbifold M' with boundary whose restriction to the boundary, 8M a M', is conformal. Is there a hyperbolic isometry F: int M -* int M' that induces the same isomorphism F* = f* of the fundamental groups, Trirb(M) -->. 7r orb (M,) ?

In Corollary 7.14, we have already shown that in order to have a positive answer to this question, one needs to assure that the holonomy group G of M has a zero measure set of limit points at the boundary of its fundamental polyhedron. We shall show that this condition is also sufficient.

Theorem 7.20. Let M and M' be conformal n-orbifolds with boundary whose structures on their interiors are induced by complete hyperbolic metrics pM and pM,, and let f : M -> M' be a quasiconformal map which is conformal on the boundary. If, for some (any) positive number s, the s-neighborhood CE C M of the convex core C(M) C M is as follows: lim.

yol{x E CE : pM (O, x) < r)

r *ooo vol{x E IHIn

: d(0, x) < r}

0,

(7.29)

then there exists a hyperbolic isometry F: int M -* int M' inducing the isomorphism orb 7r,orb (11'1), F* = f*. *:7r1 (M) .fi

Proof. Due to our assumptions, we may choose developments d: M -* Sn and Sn so that d(M), d(M') C Rn+, the holonomy groups G = d*Tr1(M) and d': M' G' = d* r1(M') are Kleinian, and M = [R+ U S2(G)]/G, M' = [R+ U S2(G')]/G' are the discontinuity sets of G and G. Then the map where 0 (G), 0 (G') C l18n-1

f lifts to an equivariant quasiconformal homeomorphism f : lR' + -+ Rn+ which is conformal on the discontinuity set, cZ (G) -a 92 (G'). We claim that the condition (7.29) is equivalent to the condition on the limit subset P* of the boundary a P of a fundamental polyhedron P = P (G) C Rn+ to have zero

(n - 1)-measure:

mn-1(P*) = 0 ,

P*

= P fl A(G) .

(7.30)

If the set P* has positive measure, almost all its points x E P* are density points, and we have lim r-+O

mn-1(P* n Bn-I (x, r)) mn_1(Bn-1(x, r))

= 1,

x E P* a.e.

(7.31)

7. Theory of Deformations

404

We recall that the convex core C(M) C M is the quotient of the convex hull, [C (A (G)) \A (G)] / G, that is C (M) can be obtained by identifying G-equivalent points

of the boundary of PP = P fl C(A (G)), see Theorem 3.19. Let xo E Pc be a lift of a point 0 E C(M) in (7.29). Then mn_1(P*) equals the solid angle under which the set P* is seen from the point xo, and (7.31) implies that the limit in (7.29) tends to mn_1(P*). It shows that P* is in fact a zero measure set. Conversely, let us assume that mn_1(P*) = 0. Let Sn-1(xo, r) be the hyperbolic sphere centered at the point xo E Pc of a radius r > 0, and Ur,6 its intersection with an e-neighborhood U,(P,) C P of the truncated polyhedron PP = P fl C(A(G)). Then the set Ur,E is seen from the point xo under a solid angle w(r). Due to convexity of PP and U6(Pa) and (7.30), we have that the solid angle co(r) decreases and tends to zero as the radius r increases to oc. Let an-1 (r) be the hyperbolic (n - 1)-measure of Sn-1(xo, r). Then we can write: vol{x E CE : pM(0, x) < r} vol{x E IH[n : d(xo, x) < r}

_

vol{x E U,(P,) : d(xo, x) < r} vo1Bn(xo, r)

(7-32)

f 0an-1(r)w(r)dr/ J 0 an-1(r) dr. Due to limr, w (r) = 0, for any 8 > 0 there is a small ra > 0 such that w (r) < 8 for all r > ra. It implies that the left-hand side in (7.32) is bounded by r3

j an-1(r)w(r) dr 01

for an-1(r) dr

8

_r an-(r) dr

+ for an-1(r) dr

(7.33)

The first term in (7.33) tends to zero as r increases, and the second term is bounded by an arbitrarily small 8 > 0. This shows that the limit in (7.29) equals zero, that is the conditions (7.29) and (7.30) are equivalent. To complete the proof, we can apply the fact given by Corollary 2.61 that the dissipative part of the G-action on Rn-1 is precisely the orbit G(P fl Rn-1). It shows that the group G acts conservatively on the limit set A (G) C Rn-1 of positive measure

only in the case when mn_1(P*) = 0. In this case, the restriction of the map f to Rn-1 is a 1-quasiconformal homeomorphism of Rn-1 and hence, due to Theorem 1.6, it is a Mobius transformation F conjugating the actions of the groups G and G'. Due to Corollary 7.14, the equivalent conditions (7.29) and (7.30) are essential for the conformal rigidity of deformations. However, they are not necessary conditions for that rigidity in dimensions n > 4. It is shown by the following construction.

Example 7.21 (A rigid manifold M = Hn/ G with rapid growth of volume). Let S C Rn-1 be a Peano surface of positive measure mn_1(S) > 0, which splits up Rn-1 into two domains. Considering an infinite family E of disjoint round balls {Bi } which

densely cover the surface S, we generate a discrete group G = GE C Mob(n -1)

7.2. Rigidity of hyperbolic structures

405

whose generators are reflections in the boundary spheres 8 Bi, see Theorem 5.12. To

construct such a family E, we can follow the construction of the covering E of a Peano curve on the plane, see Abikoff [1], to make the group G = GE to have two G-invariant components, S2o U S21 = S2 (G). Since 0, the constructed group G has a positive (n - 1)-measure subset P* = P (G) n A (G) of limit boundary points of the fundamental polyhedron P(G) C W , mn_t (P*) > 0. We claim that the group G C Mob(n - 1) is conformally rigid, that is, for any quasiconformal G-equivariant homeomorphism f : l[8"+ -+ Y'-+ which is conformal

on 0 (G), the group G' = f Gf -1 can be obtained by an inner automorphism of Mob(n - 1). Indeed, due to the Liouville Theorem 1.6 and the condition that n > 4,

the conformal maps fo = f I S20 and fl = f 121 are restrictions of some Mobius transformations y1, Y2 E Mob(n - 1). Since both components 01 and 02 are Ginvariant, fo = fl on the set of fixed points of elements g E G. Due to Corollary 2.13, this set is dense in A(G), and thus Y1IA(G) = Y2I A(G) It follows that the map yi U Y2 = y is a Mobius transformation conjugating G and G'. Finally we notice that the above argument shows conformal rigidity of any functional group G C Mob(n), n > 3, having an invariant component of the discontinuity set, cf. Krushkal' [4].

4. Weak rigidity of hyperbolic structures of infinite volume. Let r be a torsion free, finitely generated group. An n-dimensional hyperbolic structure on the group F is determined by a pair (N, gyp) where N is an oriented hyperbolic n-manifold and

(p: r -* nl(N) is an isomorphism. Two pairs (N1, cpl) and (N2, cp2) determine the same hyperbolic structure on F if there is an orientation preserving isometry I: N1 --> N2 such that the isomorphism X02 and the composition I*cpl differ by an inner automorphism of F. We denote the set of all (classes of) n-dimensional hyperbolic

structures on I, by Je' (r). Clearly, this set is naturally identified with the variety of conjugacy classes of faithful representations p: r -* SO°(n, 1) with discrete images, which is a subvariety of Hom(r, G)/G with G = SO°(n, 1) - Isom+En, see §1.3 and Corollary 7.4. This yields a topology on . '1(F), the topology of algebraic convergence of representations where two representations are close if they are close on generators. As an example, we describe the space je2(F) of hyperbolic 2-structures on the fundamental group r = Trl(Sg) of a closed surface Sg of genus g > 1. It turns out that .3e2(F) is the Teichmtiller space T (I').

Teichmiiller space of a surface group. Due to the Klein-Poincare uniformization theorem, we may assume that the surface group r = rrl (Sg) is a torsion free Fuchsian group IF C M6b(2) which preserves the upper half-plane R+ = (z E C : Im z > 01.

As in §1.1, we need to consider quasi-conformal r-equivariant homeomorphisms f : C -* C, that is homeomorphisms whose quasiconformality characteristics µ(z) define 17-invariant Beltrami differentials µ(z)dz/dz,

A(y(z))y'(z)/Y'(z) = µ(z)

for y E r, z E C.

(7.34)

406

7. Theory of Deformations

We can use Beltrami differentials to identify the Teichmiiller space T (F) = 3e2(P)

with the Teichmiiller space T(g) of marked Riemann surfaces of genus g > 1. Namely, let MS'(C, F) be the complex Banach space of functions µ E such that µ(z)dzldz is a Beltrami P-differential, and µ(z) = µ(z), z E C, with the norm Ilµll = 111111L., and let Mi (cC, r) = {µ E MS(cC, r) : Ilµll < 1} be its unit ball. Let Q (I') be the group of F-equivariant quasiconformal automorphisms of the upper half plane 1f8+ with fixed points 0, 1 and oo, that is it consists of automorphisms f µ with complex characteristics µ E M i ((C,1'). The group Q(F) contains the normal

subgroup Q0 (l') a Q(F) consisting of those f µ E Q(F) that pointwise fix the real axis 118 = 81[8+. Then the space T(F) of equivalence classes of faithful discrete representations p: F --* Isom 1H[2 consists of classes of quasiconformal deformations,

T(r) = Q(r)/Qo(r)

(7.35)

Each P-equivariant homeomorphism f µ E Q (r) is projected to a quasiconformal homeomorphism of the surface S9 = R2 /r. On the other hand, the equivalence in (7.35) with respect to the subgroup Qo(r) corresponds to the homotopy equivalence of homeomorphisms of the surface Sg, that is to marking of Ss. It implies that the Teichmiiller space T (r) = 3e2 (F) can be identified with the Teichmiiller space T (g) of marked Riemann surfaces Sg. The same rule as in (7.1) defines the Teichmuller metric p ([ f µl ], [fµ2]) on T (F), which turns T (r) into a complete connected metric space that can be embedded as an open domain in (C3g-3

There is another approach to the Teichmiiller space T(F) which can be identified with the space of quasi-Fuchsian deformations of the group r. Namely, instead of the space MS (C, r), one can consider the space

M(C, r) = {µ E L. (C) : µ(z) = 0 for z E R2 , µ(z)dz/dz is r-invariant} , and its unit ball Ml (C, F)= {µ E M(C, IF): II µ II < 11. Then each µ E Ml (C, F) defines a unique quasiconformal homeomorphism fµ: C -* C with complex characteristic µ(z), which fixes points 0, 1, and oo and is conformal in the lower half-plane R2 C C. It is clear that two such homeomorphisms f1,1 and fµ2 coincide on the real axis R C C if and only if the corresponding homeomorphisms f Al and f F12 coincide on R. Here Ai E Mi (C, F), i = 1, 2, are obtained from the µi by extending them from 1112+ to 1[82 by symmetry:

µi (z) =

µi (z) ,

I µi (z-),

Im z > 0 Im z < 0 .

This implies a one-to-one correspondence between the restrictions fµ I R and the classes

[f 4], that is the points of T (r), see (7.35). In fact, for z E R+ and i = 1, 2, we have

7.2. Rigidity of hyperbolic structures

407

f '! (z) = hi f, (z) where hl is a conformal map of the domain f,', (R+) onto R. So if [f Al ] = [f 411 then the function

h(w) =

11f h2h 1

1(w) ,

l .fµ2 fµ,1 (w) ,

w E f1,1(R2F)

w E fµ1(82)

extends to a r-equivariant homeomorphism of C that is conformal in C\fµ1(IR). Thus h is a Mobius transformation which, due to the normalization, is in fact the identity.

Therefore, we can identify classes [f w] E T (I') in (7.35) with classes [f,,] of I'-equivariant quasiconformal homeomorphisms f j': C -* C up to Mobius transformations, that is with M&(2)-orbits of quasi-Fuchsian groups:

T(r) - {Gµ : Gu = furfµ 1: µ E M(C, F)}/M6b(2)

.

(7.36)

Returning to the general space 3en (F), we immediately see from the hyperbolic rigidity Theorem 7.11, that the space 3en (r) degenerates to a single point if n > 3 and r is the fundamental group of a hyperbolic n-manifold of finite volume. It suggests looking at unbounded sequences in general spaces 3C" (I') in order to understand whether the corresponding hyperbolic structures on the group I' degenerate. In particular, for a surface group r, 3e2 (F) C 3en(r) for all n > 2, and such degeneration may take the place, so that 3en (I') may be noncompact. However there are fundamental groups I' of hyperbolic n-manifolds of infinite volume which have a weak hyperbolic rigidity in the sense that the spaces 3en (r) are compact. The first rigidity result of this sort (for the fundamental group of a compact acylindrical 3-manifold) is due to W. Thurston [4] (for its conformal version, see Canary [1]):

Theorem 7.22. Let r be the finitely generated fundamental group of a hyperbolic 3-manifold whose variety 3e3 (r) is noncompact. Then r has a decomposition of one of the following types: (1) IF A *c B is a nontrivial free product with a cyclic amalgam C.

(2) r = A*c is an HNN-extension of A with cyclic C.

In the proof of this result, the fact that r is a 3-manifold group is important but the dimension of the hyperbolic structures is not essential. Namely, utilizing tools of algebraic geometry, Morgan-Shalen [1, 2] and Morgan [2] have built upon ideas of Culler-Shalen [1] and generalized the above theorem:

Theorem 7.23. Let r = nl (M) be the finitely generated fundamental group of a hyperbolic 3-manifold. Then, for any n > 3, the space 3en (r) is compact if and only if r does not have a decomposition of one of the following types: (1) r = A *c B with a cyclic subgroup C of infinite index in A and B;

(2) r = A*c with C cyclic. Notice that it readily follows from the above theorems that 3en (1r1 (M)) is compact if and only if 3e3 (7r1(M)) is compact. It implies:

408

7. Theory of Deformations

Corollary 7.24. Let M be a finite volume hyperbolic 3-manifold with the fundamental

group r = 7rl (M). Then 3f n (r) is compact for all n > 3. There is another class of groups r for which similar results hold. We state them in the case of no parabolic subgroups, see Morgan [3]:

Theorem 7.25. Let M be a closed oriented k-manifold, k >_ 3, with contractible universal covering k and r = ni (M). Then, for any n, the space 3Cn (r) is compact provided that r has no decomposition of one of the following types: (I) r = A *c B with virtually Abelian subgroup C, C A, C B; (II) r = A*c with virtually Abelian C. Corollary 7.26. Let M be a closed hyperbolic k-manifold, k > 3, with the fundamental group F = Ttl (M). Then 3en (F) is compact for any n.

The proofs Morgan-Shalen [1, 2] and Morgan [2] of these results are beyond the scope of this book. They use algebraic geometry tools and are based on studying actions of the fundamental groups n1 (M) of hyperbolic k-manifolds on ]1-trees, that

is on countable, increasing unions of metric trees. In fact, the space of actions of F on I8-trees is an analogue of the character variety of representations of r into SO(n, 1), and the ideal points of the obtained compactification of can be also identified with actions of r on R-trees. Then it can be shown that there are no points in the boundary X" (F) \ Xn (r), which implies that Rn (F) is compact. A different geometric proof, which utilizes the notion of convergence of compact metric spaces developed by M. Gromov [4], can be found in Bestvina [1].

§3. Quasi-Fuchsian structures: bendings Due to the hyperbolic rigidity of a finite volume hyperbolic n-manifold M = Hn/ F, n > 3, F C Isom Hn, one has that the Teichmi ller space 7 (M, IHI') of hyperbolic structures on M degenerates into a point. What can we say about conformal structures on M? In particular, what are the structures obtained by quasiconformal deformations of the conformal structure (M, co) induced by the hyperbolic metric? The holonomy representations p: itl (M) -* Mob(n) of the latter structures are compositions fpo of the (Fuchsian) holonomy representation po of the structure

(M, co), po: ni(M) = F C Mob(n - 1) -+ Mob(n), and the conjugations by Fequivariant quasiconformal homeomorphisms f : Sn -+ Sn. Such structures, their holonomy representations, and the holonomy groups pF are called quasi-Fuchsian. And, as we have shown in the previous section in dimension n = 2, the space of equivalence classes of quasi-Fuchsian groups on the plane is isomorphic to the Teichmiiller

space T(F) = 3e2(F). However, in dimensions n > 3, the varieties 7 (F) and 3Cn (F) are different. In fact, despite the degeneration of 3.' (F) to a point, the Teichmiiller space of quasiFuchsian structures Tgf(M) C 7 (M, Mob(n)) is often non-degenerate. The aim of

409

7.3. Quasi-Fuchsian structures: bendings

this and the next two sections is to present different types of non-trivial quasi-Fuchsian deformations.

1. Bendings along totally geodesic hypersurfaces. We start with a historically first construction (Apanasov [12]) of a smooth curve 1: (0, 1) - Tgf(r) which defines a smooth nontrivial quasiconformal deformation of a co-compact discrete group r c Isom I3[3. Due to (7.14) and (7.15), it simultaneously implies nontriviality of the tangent bundle to Tqf (r), i.e. nontriviality of the group cohomology H1 (r, m3) ; 0 with coefficients in the Lie algebra m3 of the Mobius group Mob(3), which answers a question in Borel-Wallach [1, p. 221-224]. Example 7.27 (A smooth curve in Tq f (r)). Let r C Isom IEi<3 be a co-compact group acting in a round 3-ball B C R3 as follows. We fix a unit cube Q = {x E R3 : IX, 1 <

1/2, i = 1, 2, 3) and draw spheres Si, 1 < i < 8, centered at its vertices with radii /3-/3. The spheres Si and Sj are disjoint if their centers are not endpoints of an edge of Q. Otherwise, the exterior dihedral angle bounded by Si and Sj is n/3. We add six disjoint spheres Si, 9 < i < 14: S9

= S2(rel, p),

S12 =

S2

(-re2, P),

S10 = S2(-rel, p) , S13 =

S2

(re3, P),

Sn1

=

S2

(re2, P)

,

S14 = S2(-re3, P)

,

where r = 5/6, p =

10/6 and vectors ei form the orthonormal basis in R3, so that the new spheres are orthogonal to the previous ones. All so constructed 14 spheres are orthogonal to the boundary sphere So = 8B of the ball B = B3 (0, 15/6). Now let r be the index two subgroup of orientation preserving elements of the group generated by reflections 1i with respect to the spheres Si, 1 < i < 14. Due to the angle conditions and Corollary 2.39 (Theorem 4.23), the group r is a discrete group co-compactly acting in B3 = H3 by hyperbolic isometries.

To deform the group r into a smooth family It = f Fft-1 C M6b(3) of quasiFuchsian groups in S3, we can deform the family E _ {Si } into new families Et = {Sill whose spheres have the same magnitudes of the dihedral intersection angles. For example, we may replace the last sphere S14 centered at - 5 e3 by the sphere S14 centered at -tea and such that it orthogonally intersects the same spheres Si, i # 14, as the sphere S14 does. This condition implies that the radius of S14 equals t2 -- t H- 5/12, and the parameter t may vary in an interval (to, t1), to _ 5(1 + 11 + 12

516)

< t < t1 = 5(1 +

5/6)

.

(7.37)

For better understanding of the corresponding deformation of the hyperbolic orbifold B3/ r, it is worth mentioning that the deformed sphere S14 canbe obtained from S14 by rotating it in 1811 about a hyperbolic 2-plane which orthogonally intersects the sphere So C R3 at infinity of H4 = R along a circle.

7. Theory of Deformations

410

Let Ft C Mob+(3) be the groups generated by such families Et of spheres S. we see Applying Corollary 2.39 to their hyperbolic isometry actions in 1I8+ that they are Kleinian groups, and their fundamental polyhedra in S3 may be chosen as follows:

Ft = Pt U.I1(Pt),

Pt = n (IR3\intSit)

.

1
Since Ft splits into bounded and unbounded components, Fo and F1, the discontinuity

set Sgt = S2(rt) C S3 consists of two rt-invariant topological balls S20 ' = rt(FF) and ct = rt(Fl) whose common boundary is the limit set A(Ft), see Lemma 2.22 and arguments in Example 2.19. Let us distinguish everything related to the group r = r5/6 by discarding the corresponding parameter t, t = 5/6. We can define r-equivariant quasiconformal homeomorphisms q)t: S3 -- S3 by defining such r-equivariant maps on the fundamental polyhedron F, separately on each of its components:

f : Fo __> F4

,

f l : Fl -

F1, ,

fi5'6 =id .

The above homeomorphisms fit are obtained by uniform compressions and stretching of FO and F1 along a family of disjoint circles which are orthogonal to all spheres Si41

to < t < t1. These circles are trajectories of the mentioned rotations in R+ about a fixed 2-sphere orthogonal to aR+ = ][83. Clearly, the quasiconformal dilatations of the maps fo and fl are bounded and tend to 1 as t goes to 5/6. We can F-equivariantly 00 ' and c21 -3 extend the maps fo and fl to quasiconformal homeomorphisms S2o S4 which, together with the canonical homeomorphism A(r) --* A(rt) induced by the isomorphisms r = rt (Theorem 4.41), define a r-equivariant quasiconformal homeomorphisms fit: S3 S3, 05/6 = id, rt = (Ptr(Dr 1. This defines a curve (D: (to, t1) y 3"y f (r), c (t) = (ct),k. Due to the above argument with rotating in IR4, the curve (D is smooth (real analytic).

The deformation 4) of the inclusion r c M6b(2) C M6b(3) is nontrivial. In fact, it is the identity on the subgroup of r generated by the first 13 generators yi, 1 < i < 13, excluding the generator Y14 corresponding to S14. However, fi ¢ id on the whole r because OtY14071 ; Y14 for all parameters t E (to, t1), t # 5/6. The method of deformations used in the above construction works as well in the general situation of geometrically finite groups r C Isom H" provided some geometric condition on r, see Apanasov [24, 36]. The matter can be greatly clarified by Thurston's idea to describe such class of deformations of hyperbolic surfaces by bendings of these surfaces along geodesics, see Mickey Mouse Example 3.20. Below, this idea is realized in arbitrary dimensions. Let r c Isom IH1 be a discrete group, not necessarily a lattice. It conformally

acts in the round balls Bxi(0, 1) C B"+1(0, 1) regarded as Poincare models of the hyperbolic spaces H" C Hn+1 with the distance d. The main condition on r is that

7.3. Quasi-Fuchsian structures: bendings

411

it has a bending hyperplane P C H. This means that there is a hyperplane P C IHI" such that

inf{d(y(P), P) : y E r\I'P} = dp > 0,

(7.38)

where we additionally assume that the stabilizer Fp = {y E I' : y(P) = P} of P in r is a proper subgroup, Fp r. By conjugating r, we may assume that the origin

lies in one of the components of B"\r(P) where ' = r(P) is the infinite set of distinct 17-translates of P. For each (n - 1)-plane R E Y, let Ro C ][fln+1 be an n-plane which orthogonally

intersects W along R, and let Ro denote the closure in IE1[n+1 = Bn+l of the half space bounded by Ro and disjoint from 0. We define an order relation in the set P by setting for planes R, Q E Y :

Ro CQO.

(7.39)

It follows from (7.38) that, for any R, Q E J", either the half spaces Ro and Qo are disjoint or one of them contains the other. Thus the relation (7.39) is a partial ordering such that q-1 is order-preserving. To do in Y. We can define a bijection q: N -* that, we consider the sequence of closed balls

Bi = Bn+1(0, 1 - 1 / i) C B'i+1(0, 1) ;

iEN,

i >2,

each of which may intersect at most finitely many planes Ro, R E ?, because of discreteness of F. Hence we can define the bijection q by using the finite subsets

{REJ :RonB;¢0},i>2. Now we specify numbers a and such that 0 < < n/2 and 0 < a < n -2 and C -+ C, see Figure 124, which bends define a bending homeomorphism = the real axis 118 C C at 0 by the angle a: z

z, exp(ia)

(z) -

z exp(ia(1 - (argz - )/(n z exp(ia(1 + (argz + )/(n -

Figure 124

if argzl ? n if argil _< if < argz < n if - n < argz <

-.

7. Theory of Deformations

412

For negative a, 2 - n < a < 0, we set 0a, (z) =

Clearly,

is quasiconformal, and it is conformal in a c-cone neighborhood of the real axis R because its linear distortion is given by

1

if IargzI if I arg z l <

(n - 2

if

1

K(O«,f, z) _

r-

- n < argz < -c .

Identifying the 2-plane Ren + Ren+1 with the complex plane C = Ren + i Ren+1, we define bending homeomorphisms in II8n+1 by extending the homeomorphisms ¢a, 5 from this 2-plane to Il8n+1 as follows:

/+= a

(xl,...,xn+1) H (xl,...,xn-1,Oa,5(xn+ixn+1)), (x1, ... , Xn+l) H (XI, ... , xn-1, J . Oa, ' J (xn + ixn+l)) ,

where j denotes the involution j : C -+ C, j (z) = z. Clearly, the homeomorphisms i/r+ preserve the unit ball Bn+1 and the half space V++1 = {x E IlBn+l : xl > 0}. Furthermore, for all g E Mob(B') = Isom Hn that preserve Bn-1, we have:

get = i because 1/r

* ±g i?Fg

if gIBn-1 E Mob+(Bn-1)

(7.40)

if gIBf_1 E Mob_(Bn-1) ,

IIRn_1 = id.

Now we can define an elementary bending *R: jn+1 11n+1 associated with any bending hyperplane R E P, which bends the plane IH[n inside W+1 along R C W and

pointwise fixes the component of Bn\R (a half space in IE3In\R = Bn\R) containing 0. Namely, let gR E Isom+ HI act in Bn+1 = Hn+1 so that gR(Bn-1)

= R,

gR ({x E Bn+1 : xn > 0}>

= Ro

.

(7.41)

We set 1/rR = gR 1/'+ 9R- which does not depend on the choice of gR due to (7.40). Thus we have a sequence of elementary bendings *j: H1+1 , 11n+1 enumerated with

respect to the bijection q: N -- P. We shall show that the compositions of elementary bendings 1/r11Jr2 ... 1/ri converge to a F-equivariant homeomorphism. To do that, we need to control two types of support

domains of the elementary bendings. Namely, for a quasiconformal homeomorphism f of a domain D C R1, we define

sp(f) = {x E D : f (x) Ox} ,

gsp(f) = {x E D : K (f, x) # l } .

7.3. Quasi-Fuchsian structures: bendings

413

Lemma 7.28. Let F C Isom IH[n be a discrete group acting in the balls Bn C Bn+1 with a bending hyperplane P C IH1 as in (7.38), and let a, E R be as follows:

0<

0 < a < Tr - 2 and cosh(dp /2) > 1/ sin

< .rr/2 ,

.

(7.42)

Then the elementary bendings *i, i > 1, have the following properties: (i) gsp(Vfi) fl qsp(*j) = O for i ,-b j. (ii) For i 0 j, either sp(1/ri) and sp(1/rj) are disjoint, or one of these domains contains the other. In the latter case, if sp(1Jri) C sp(1(rj) then i > j. (iii) If i > j then gsp(1lrj) n sp(1/ri) = 0.

(iv) limi,,o diam(sp('i)) = 0. (v) For every y E IF, there is a bijection N -a N which maps i to j so that, for the planes R = q (i) and Q = q (j), we have that y (R) = Q and IID

i(x) . ull*i(x) = IID(ijy)(x) ullify(x)

(7.43)

for all u E Rn+1 and almost every x E IHI1+1 at which 11r= and 1lrj both are differentiable.

Proof. Let R, Q E 5' be the planes corresponding to i and j, that is R = q(i) and Q = q(j) where the bijection q: N -* ./" induces the order (7.39). We have their elementary bendings 1/ri and 1/rj defined in (7.41), t/r; = gR 1lr+gR 1 and 1/rj = gQlJr+gQ1 for some gR, gQ E Isom+ H. Due to conformality of gR and gQ, gsp(1//'i) = gR(gsp(i'+)),

gsp(Ij) = gQ(gsp( '+)), sp('j) = gsp(1Gj) U Q .

sp(fi) = gsp(ii) U Rp ,

(7.44)

From the construction of 1/r±, we see that gsp(1/r±) does not intersect any of the hyperplanes H0 C 1H[n+1 that are orthogonal to Hn and located at the distance more than a = arccosh(1/ sin ) from the plane ]H[n-1 C if2 along which *'f bends lIP in 1H[n+l Indeed, the subdomain of the half-space R+ 1 = {x E I[81+1 : x1 > 01 where the bending z/c differs from a conformal map is the following domain:

gsp(i/i) = {x E l[8n+ : Ixn/xn-1I > tan } . 1

Since the intersection IHIn fl Ho is a hemisphere orthogonal to the boundary plane : x1 = 0}, the condition d(Ho,1H[n-1) > a implies that

8118+ = {x E I

(xi -I- xn)/xn_1 < tan2 i;

for x E H0,

and hence gsp(i/r+) fl Ho = 0. This fact and (7.44) show that gsp(1/ri) and gsp(1/rj) do not intersect any of the hyperplanes in Hn+1 orthogonal to lHln and distant more than a > 0 from R and Q,

7. Theory of Deformations

414

respectively. Using the condition in (7.42) that dp > 2a, we see from d(R, Q) > dp that there is a hyperplane Ho C Hn+1 orthogonal to H' such that gsp(*i) and gsp(*j) lie in different components of HI+1\Ho. This implies the claim (i). The proof of the claims (ii) and (iii) proceeds similarly, on the base of (7.44) and the definition of the order (7.39). To prove the claim (iv) that diameters of sp(r(ri) C Bn+1(0, 1) tend to 0, we use the fact that at most finitely many planes R E P intersect the hyperbolic ball D (0, a) C Bn+1 (0, 1). This is due to discreteness of r. Therefore, for sufficiently large i, we may assume that d(0, Ro) > a. From the above argument, we also have that sp(*i) is disjoint from any hyperplane in Hn+1 orthogonal to lH[n and distant more than a from Ro . Let Ho be such a hyperplane orthogonal to the perpendicular from

0 to Ro at the distance a from Ro, and let r be the Euclidean radius of the sphere supporting Ho. Then, putting d(0, Ro) = s, we have:

r2+1=(r+tanhs2tl 2

2

r=1/sinh(s- t),

and hence diam(sp(*i)) < 2/ sinh(s - t). Now, for a given s > 0, we can find a sufficiently large is such that for all i > is

the distance s = d(0, Ro) is as big as s, e > 2/ sinh(s - t). It implies that, for all i > is, diam(sp(*i)) < s, i.e. (iv) holds. The proof of the last claim (v) follows from the chain rule, together with (7.40) and the fact that HD

(x) . uII,,-(x) = IID* (x) ull*+(x),

for all u E liin+l and those points x E Bn+1 where *+ and ,- are both differentiable. This completes the proof of the Lemma.

Theorem 7.29. Let r C Isom HI be a discrete group with a bending hyperplane P C IEiln as in (7.38) and real numbers a and as in (7.42). Then the map

fa, _ urn 1//'11//'2... *i : W+1 -, i-). oo

Hn+1

(7.45)

is a r-equivariant quasiconformal homeomorphism.

Instead of proving this theorem we shall prove a more general statement which immediately implies Theorem 7.29. Theorem 7.30. Let r c Mob(B"+1) be a discrete group and {1/ri } a sequence of Kquasiconformal homeomorphisms of B1+1 satisfying the conditions (i)-(v) of Lemma 7.28. Then the sequence of homeomorphisms fi = 1/x1 ... *i converges uniformly on Bn+1 to a K-quasiconformal homeomorphism f = lim fi which is r-equivariant. Proof. Given s > 0, let is be chosen due to (iv) so that diam(sp(r/ri )) < s for all i > is, and let l > m > i5. We claim that the homeomorphisms fm 1 fi = */rm+1 ...1//T are

7.3. Quasi-Fuchsian structures: bendings

415

uniformly s-close to the identity, and hence the sequence { fi } uniformly converges to the limit f : Bn+1 -k Bn+1 which is K-quasiconformal due to Theorem 1.5. To show that property of fn 1 fi, we look at its behavior on sp(fm 1 fl) which is the union sp(1/r,n+1) U . . . U sp(1/ri). For X E sp(fm 1 fi), let j = j (x) be the minimal

integer i E N such that i > m + 1 and x E sp(1/ri). It follows that if x E sp(1/ri) for

in + 1 < i < 1 then sp(*i) n sp(irj) 0 0 and, due to (ii), sp(1/ri) c sp(1Jrj). Since i'i (sp(1/ri)) = sp(1/ri) for any homeomorphism 1/ri, it follows that f;' fl(x) E sp(1Jrj),

where j = j (x) > iE. Therefore sup, Ifm 1 f1(x) - xl < diam(sp(i/rj)) < e, and hence f = lim fi is a K-quasiconformal homeomorphism. To prove that the limit f is F-equivariant, it is enough to show (due to Lemma 7.7) that, for all y E I' and points x E B1+1 where f and f y are both differentiable, (7.46) II D(fy)(x) - ullfy(x) = IIDf(x) ulif(x) . First, let x V Ui gsp(i/ri). Then (v) implies that, for y E r and every i, there is

some j such that q (j) = y (q (i )) and (7.43) holds. This shows that y (x) 0 gsp(1Jrj ). As i runs through N, the number j corresponding to i also takes all values in N, and

hence y(x) V gsp(i/rj) for all j E N. Therefore, f is conformal at x and y(x), and (7.46) holds.

Now let us assume that x E gsp(1/rj) for some j. Due to the condition (iii), gsp(1/rj) n sp(iri) = 0 for all i > j, and hence f (x) = fj(x) = 1/r1 ... *j (x). For x E sp(irj) and for any 1 < k < j, we have due to (ii) that 1/k ... l/rj(x) E sp(lfrl) if and only if 1/rk_1 ...1/rj(x) E sp(1/ri). By induction, we have then that, for any l < j, x E sp(*I) if and only if 1/ri ... ifrj (x) E sp(1/ri).

Now, for any i < j, let i' = min{k E N : i + 1 < k < j, sp(ljrj) C sp(i/rk)}. Then, for any 1, i + 1 < l < i', we see that 1/ri 1/rj(x) ¢ sp(1/ri), and hence t+1 ...*j (x) = 1/ir ...1/rj (x). This shows that *j +I ...1/rj (x) E sp(1/ri,) which, together with (iii), implies that *i+.1 ...1/rj (x) ¢ gsp(1/ri ). Since this is true for any

i < j, we have: (7.47)

IIDf(x).ullf(x)=IID*j(x).uII*;(x) Due to (v), for y e F there is k E N such that II Di/i (x) - uII,/rr(x) = II D( 'kY)(x) . ull*kY(x),

u E IIgn+1

(7.48)

This implies that K(1(rj, x) = K(1/rk, y (x)), and hence y (x) E gsp(>Jrk). By the above argument, f y (x) = 1/r1 ... 1/rk y (x) and

llD(fY)(x) uIIfy(x) = IID(hkY)(x)

uII*ky(x)

(7.49)

Now (7.47)-(7.49) imply that the homeomorphism f satisfies (7.46) and thus is Fequivariant. This completes the proof of both Theorems 7.30 and 7.29. We notice that the F-equivariant quasiconformal map f,,,C:Hn+1 --+ Hn+1 constructed in (7.45) does not depend on the choice of the bijection q : N -+ i with order preserving q-1 and induces the isomorphism of r onto a quasi-Fuchsian group Fa,

fa*: r a l"', = fa,Crfa

1

C Isomfflln+l

416

7. Theory of Deformations

which does not depend on the choice of in (7.42). The family f f,,.1 of such representations defines a smooth quasiconformal deformation P of the inclusion r c Isom H' C Isom H+ 1, called a bending deformation,

fi: (-0, 0) -* Hom(r, Isom Hn+1),

(a) = fa* ,

(7.50)

where the number 0 is such that

0 < 0 < ri = 2 arccos(l/ cosh(dp/2)) < tr . The smoothness of the bending deformation follows from its algebraic description below, which also shows that the deformation in Example 7.27 is in fact a bending. Let R C H'+1 be an (n - 1)-plane and AR (a) E Isom IH1+1 a unique hyperbolic isometry which is the identity on R and is isometrically conjugate to an orthogonal transformation in Bn+1 (0, 1) whose matrix is as follows:

E

0

0

0

cosa - sina

0

sina

cos a

For a bending hyperplane P C ]HIW of a discrete group r C Isom H, let U1 and U2 be the components of IH[n\r(P) adjacent to P and such that U2 C PO +, and 171 and

r2 be the stabilizer subgroups of U1 and U2 in r. Theorem 7.31. The bending deformation fi in (7.50) of a discrete group r c Isom+ 1HI

associated to a bending plane P C H' can be expressed as follows: (I) If no element of r maps U1 to U2 then r = r1 *rP F2 is a free product with amalgamated stabilizer subgroup r p, and

(a)(Y) =

v

ify E rl ,

AP(a)YAp'(a),

ifY E r2.

(II) If an element yo e r maps U1 to U2 then r = (r1, yo) is an HNN-extension of the subgroup r1 by the element yo, and

(a)(Y) =

Y,

171

AP(a)yo,

ifY = yo.

Proof. We note that any element of r is uniquely determined by its restriction to U1 or to U2 because r c Isom+ IF. Now the proof follows from the fact that

faIU, =id

and

faIU2 =Ap(a)IU2.

o

From such an algebraic form of the bending deformation $, we notice that it is the identity on the subgroup r1 C r. It implies that the deformation f is trivial if and only if it is the identity on r\r1, too. Geometrically, this condition can be reformulated as follows.

7.3. Quasi-Fuchsian structures: bendings

417

Theorem 7.32. Let $ be the bending deformation of a non-elementary discrete group F C Isom IHIn along a bending plane P C Hn. Then ,6 is a nontrivial deformation if and only if the plane P intersects the interior of the convex hull Cr = C(A(P)). Now we shall extend the definition of bending deformations to the case of bending along several bending planes which are not F-equivalent. As before, let P be a countable P-invariant set of disjoint bending hyperplanes in 1HIn which splits into k equivalence classes .l i, i = 1, ... , k, which we identify with vectors ei of the standard basis of lRk. We order planes in 3P as in (7.39) and enumerate them by using a bijection q: N -+ 3) such that q-1 is order preserving. The condition (7.38) for bending planes in 3n can be rewritten as

di=inf{d(P,Q):P,QE3,P¢Q}>O, i=1,...,k, and we may assume that d,, = inf{d(P, Q) : P, Q E .P, P # Q} > 0 where di > d,,p. Similarly, instead of angles a and in (7.42), we define P-invariant functions a: J" -> (-3r, rr) and : 3 -* (0,.7r/2) which have constant values ai and on P-orbits Pi and satisfy the conditions described below. Let W C Rk be a neighborhood of the origin whose closure is contained in U = {u E IIBk

: -7r < ui < n, cos(ui/2) > 1/cosh(di/2)},

and let y E U be a point with positive coordinates yi > 0 such that the interior of the convex hull C(y) C U of the points (±yl, . . . , ±yk) contains W. Then the ,, = i. It follows that I'-invariant function is defined by yi = n I

sin i = cos(yi/2) > 1/ cosh(di/2)

.

Thus, for a function a: J" -* (-jr, jr) such that (al, ... , ak) E W, we have that la(R)I

REd

.

This allows us to define elementary bendings ii as 'Ni = VIR = gR* (R),S(R)gR1

where gR E Isom+ IHIn is as in (7.41). Then Lemma 7.28 shows that the sequence { *11 satisfies the conditions of Theorem 7.29 and defines a family of quasiconformal f'-equivariant bending homeomorphisms f,,,: Hn+1 -* lH[n+1 w E W, and hence the bending deformation ,8: W -* Hom(P, Isom1Hln+1) , P (w) = (fw),k

.

(7.51)

To give conditions when the bending deformation ,8 really depends on k parameters,

let us say that a set X C Sn of at least two points spans an m-sphere S'n C Sn if S'n is the sphere of lowest dimension which contains X.

7. Theory of Deformations

418

Theorem 7.33. Let $ be the bending deformation in (7.51) of a discrete group r c Isom IH[n along the family P of bending planes such that, for each P E Y, the deformation Pi along J` i = F(P) is not trivial and A(F) fl aP spans aP. Then

p o f: W -> Hom(F, Isom Hn+1)/ Isom W+1 is an embedding, where p is the natural quotient projection

p: Hom (r, Isom r+1) -). Hom(r, Isom IH[n+1)/ Isom Hn+1 Proof. It is enough to prove that, for any two non-trivial bendings along Yi and Yj, 0, a2 0 0 in their domains of definition, the say flu and f2, and parameters a1 existence of an isometry g E Isom IHIn+1 such that

,8i(a1)(Y) = g (f2(a2)(Y)) g-1 ,

YEF,

(7.52)

implies that al = a2 and R1 = JP2. Let Pi (ai) _ (fi )* where fi : Hn+1 --, IIIIn+1, i = 1, 2, are the corresponding bending homeomorphisms. Then (7.52) implies that fi 1 gf2 commutes with elements of r, and hence it pointwise fixes the limit set A(r):

gf2(x) = fi (x) ,

x E A(r).

(7.53)

i = 1, 2, we have that rp, 0 r, and hence A(r)\aPi ; 0. For each Pi E This, together with the condition that A (r) fl a Pi spans a Pi, shows that A (r) spans the sphere Sn-1 = 81H[n. We claim that this fact and (7.53) imply that g = f1 f;-1 in Hn

Indeed, since the restrictions fi I a (Y pi) are conformal for ally E r, the maps fi I A(F)

determine the maps fi Ir(api)* On the other hand, if U is a component of Hn\r(P1) and R, Q E 9'i he in 8 U, then fi conformally maps U into the sphere spanned by the set fi (Q U R). Therefore fi I H,, are defined by the maps fi IA fry, i = 1, 2. Due to (7.53), the latter two maps differ by a conformal map g. Thus fl f2 1 = g E Isom H1. In particular, g maps totally geodesic pieces of f2 (Hn) to totally geodesic pieces of

fl (H'), which implies that the angles a1 and a2 are equal. Since f 1gf2(r(P2)) _ r(P1), we have from (7.53) that r(P2) = r(P1), i.e. 511 = .P2. We notice that bending planes P C HI for a discrete group r c Isom HI correspond to embedded totally geodesic suborbifolds P/ rp c IHI'/ r. This and Theorem 7.33 immediately imply:

Corollary 7.34. Let M be a hyperbolic n-orbifold of finite volume which contains k disjoint totally geodesic connected (n - 1)-dimensional suborbifolds. Then the bending deformation of M along these geodesic suborbifolds defines an embedding

7.3. Quasi-Fuchsian structures: bendings

419

Bk y 7'"(M, S') of a k-ball into the Teichmuller space of marked conformal (quasiFuchsian) structures on M. We note that if S is an embedded totally geodesic hypersurface of finite (n - 1)volume As in a hyperbolic n-manifold M, then parameters of bending deformation,

-+ 7,(M, Sn),

RS:

fS(a) = (fa)* ,

vary in the interval whose length depends only on As. It follows from the fact that the supremum q of bending parameters is as in (7.50), and from the Basmajian [1, 2] tubular neighborhood theorem for such hypersurfaces:

11 = 2 arccos(l/ cosh(ds/2)) ,

dS = (Vn-1)-1(AS) ,

where Vk(r) is the volume of the hyperbolic k-ball of radius r. Another remark is that, for a closed Riemann surface M = Sg of genus g > 1, the dimension k in Corollary 7.34 equals 3g - 3, the half of dim T (g).

2. Infinitesimal bendings and singularities of Teichmuller space 9'(M). Here we shall describe the infinitesimal deformations, that is the elements of H 1(F', F)n+1), corresponding to bending deformations. We start with calculating the vector fields in Hn+1 corresponding to the family of bending homeomorphisms fa in (7.45) which were defined by elementary bendings Tfi« : Hn+1 -_+ Ifl[1+1 and 0a: C -a C. Denoting the complex conjugation by j, j (z) = z, we have from the definition of 0a: d da a=0

a (Z) -' da

ZEC. a=0

this defines the following vector fields:

For the bending homeomorphisms

*,+ a=0

d

(x)

*a (x) ,

da a=0

* (x) _ (0,...,0,-xn+1,xn),

x E Bn+1(0, 1) = Hn+1

a=0

Then, for any bending hyperplane R C ]H[n, R E Y, and for the associated bending homeomorphism *R,d = gR*+ 9R 1 with gR E Isom+ Hn defined in (7.41), we can use the action of Isom ]IIn+1 on vector fields in Hn+1 given in (7.16) to define the following vector field: d 1/JR,a(x) = gR 9(x) , da a=0

x E Hn+1

This enables us to consider the vector field in (7.15) for the quasiconformal F'equivariant map fa: lRln+1 , Hn+1 as follows: si(x)=

da a=0

fa(x) =

9R .9(x). Re?

7. Theory of Deformations

420

As we have seen in (7.16) and (7.17), the vector field defines an infinitesimal deformation c E H1(r, b,+ 1) tangent to the bending deformation B: (-rl, rj) --+ Hom(r, IsomIEI"+1), f (a) = (f«)*, where 4n+1 is the Lie algebra of IsomHn+1 (isomorphic to the Lie algebra mn of Mob(n)). Namely, for y E r, the 1-cocycle c E H1(r, Cln+1) corresponds to the vector field (cf. (7.17)):

c(Y)(x) =

d

((f«)*(Y)(x)) _ fi(x)

Y

[gR 'O(X) - (Ygy-1(R) '0W]

E [gR ' OW - (YgR) ' 8(x)] REP

RE.1

If y (y-1(R)+) = R+ then 9R-1 Ygy-1(R) preserves the orientation of BI-1 (0, 1), and

we have that ygy-1(R) 0(x) = gR 8(x). Otherwise, gRIYgy-1(R) reverses the orientation of Bn-1 (0, 1), and we have due to (7.40) that (Ygy-1(R))

' 8(x) = gR

da «-0

gR .8(x)-(Ygy-1(R)).8(x) =gR'X(x) This completes the proof of the following fact.

Theorem 7.35. Let I' C Isom H" be a finitely generated discrete group with a bending hyperplane P C fflln. Then the infinitesimal deformation for the bending ,8 along P is presented by the crossed homomorphism cp E H1(r, C)n+1),

E REr'(P)

cP(Y)(x)= R9

gR'X(x),

yEr.

y(y-1(R)p)

It is possible to use infinitesimal bending deformations to study whether the TeichmUller space T (M, Sn) of a conformal (in particular, hyperbolic) n-manifold M and

the space Jk(r) of hyperbolic structures on Il (M) = r are smooth.

If this were the case (as in dimension n = 2), the sum c = c1 + c2 of two infinitesimal deformations Cl, c2 E H1 (IF, Ejn+1) tangent to two smooth curves in the Teichmtiller space would be integrable, that is c would be tangent to another smooth

curve in Hom(r, IsomHEE'+1). It is natural to take 1-cocycles cl and c2 defined by bending deformations P1 and ,B2 each of which bends the manifold M along the corresponding totally geodesic hypersurface Si C M, which we assume to exist and to have non-empty intersection, S1 fl S2 # 0. However, in dimensions n > 4, one can find such closed hyperbolic (arithmetic) n-manifolds M with intersecting totally geodesic bending submanifolds S1 and S2, S1 fl s2 # 0, that the sum c = c1 + c2 E H1(r,1jn+1) of their infinitesimal bending deformations is not integrable, see Johnson-Millson [1]. The obstruction for its integration is a non-zero element c - C E H2(r, lln+1), see Theorem 7.9.

7.3. Quasi-Fuchsian structures: bendings

421

It is still an open problem whether the discovered singularities of the varieties of representations Hom(rri (M), Isom H'+1) or Teichmiiller spaces T (M, S'), n >_ 4, are topological, that is the varieties are not manifolds near the inclusion of the group r. Another open problem is the existence of singularities in the space of conformal structures on a hyperbolic 3-manifold M (or of the variety of discrete representations Hom(7r1(M), Isom H4)). In fact, the first obstruction for integrability of infinitesimal deformations which was used in the case n > 4 (on the base of Theorem 7.9) vanishes in the case of n = 3 because it belongs to the subspace H2(r, 173) of H2(F, 44), and H2(F,133) = H1(r, 43) = 0 for a lattice F C Isom H3. Moreover, there are some evidences for a conjecture that, for every cocompact discrete group F C Isom H3, the variety of representations Hom(r, Isom H4) is smooth at the inclusion r c Isom H4. We shall prove this in the case of reflection groups F, where simultaneous bendings along intersecting totally geodesic suborbifolds are possible. 3. Deformations of reflections groups. The first examples of conformally rigid

co-finite hyperbolic groups F C Isom Hn, n > 3, were discrete groups generated by reflections in the sides of hyperbolic simplices An C 1H0, see Apanasov [11]. Here we consider the simplest case of uniform hyperbolic lattices F C Isom H3 generated by reflections in sides of a compact polyhedron P C H3 of the simple type, see Chapter 3, §9, that is a polyhedron each whose vertex belongs to precisely 3 edges. Such type

of polyhedra is generic because almost all Dirichlet polyhedra Dy(F), y E H3, are simple, see J¢rgensen-Marden [2] and Lemma 2.34. Theorem 7.36. Let F C H3 be a compact simple polyhedron with k sides such that the group FF C Isom H3 generated by reflections in sides of F is discrete and F is its fundamental polyhedron. Then, near the inclusion po: FF C Isom H4, the variety Hom(FF, Isom H4) of discrete representations is a smooth manifold of dimension

k-4.

Before we go on with the proof, we need some facts about geometry of round (n - 1) -spheres in Sn, see Luo [1]. Let Chn be the set of such spheres with a natural topology. It is a smooth (n + 1)-manifold, and Mob(n) acts on Cyn. Obviously, instead of round spheres s E CSn, we may consider totally geodesic hyperplanes H C Hn+l,

aH = S. Lemma 7.37. For any (n + 1) hyperplanes H1, ... , Hn+t C Hn+l, one of the following holds:

(i) the closures Hi have a common point p E fi Hi, p E 8H,+1; (ii) all Hi, 1 < i < n + 1, intersect at a common point in H"+'; (iii) there is a hyperplane Ho C Hn+1 orthogonal to all hyperplanes Hi. In the spherical situation, this assertion is equivalent to the following.

Lemma 7.37'. For any (n + 1) round hyperspheres S1,

,

Sn+1 C Sn, one of the

7. Theory of Deformations

422

following holds: (i) (Euclidean) There is a common pointx E flSi. (ii) (Spherical) The spheres Si are totally geodesic in some metric of constant positive curvature on Sn; (iii) (Hyperbolic) There is a hypersphere So C Sn orthogonal to all spheres Si.

Proof. Let S' be a round sphere in lI8n+1 and Mob(n) acts on Sn as a subgroup of PGL(n +2, R), see Chapter 1. Then every sphere S E CSn is the intersection of Sn with some affine hyperplane P C ]I8n+1, and we denote Sn fl P = S(P). Let P,k E l[8pn+1

be the polar of S(P) with respect to Sn, that is the point whose tangent cone to Sn touches Sn at S(P) = S fl P, and let P be the closure of P in R1P"+1. Then it follows from the orthogonality property in the projective Klein model of the hyperbolic space (in the ball Bn+1, aBn+l = Sn, see Chapter 1, §3.2) that two spheres S(P) and S(Q) in CSn are orthogonal if and only if P. E Q (equivalently, Q,k (=- P). Now, for given spheres S(Pl),... , S(Pn+l) E CSn, let P be the extended hyperplane in inn+l which contains the polars Pi*, 1 < i < n + 1. Then we have the following trichotomy: 1. The intersection P fl S3 is empty. Then the polar P, lies in the ball Bn+1 aBn+l = Sn, in the intersection n Pi. Applying Mob(n) C PGL(n + 2), we may assume that P. is the center of Sn. Hence S(Pi) are "great spheres" in Sn, i.e. totally geodesic in the spherical geometry, and (ii) holds. 2. The intersection P fl Sn is a point. Then X E Pi, 1 < n + 1, and (i) holds.

3. The intersection P fl Sn = S(P) is a sphere. Clearly, it is orthogonal to all

S(Pi), 1
Corollary 7.38. Let Sl, ... , Sn+l E CSn be hyperspheres with no common orthogonal (n - 2)-sphere. Then the unique sphere So E CSn that is orthogonal to all Si, 1 < i < n + 1, smoothly depends on (Si, ... , Sn+1) E 'Sr. o Proof of Theorem 7.36. Let a (S1, S2) be the magnitude of the external dihedral angle bounded by round intersecting spheres S1, S2 C 1i83. Considering the Poincare model of iiii in the unit ball B3 C 1183, we have a collection {SO, ... , So } C 63 of round k

spheres containing the sides of the fundamental polyhedron F C H3. Due to the Poincare-Aleksandrov Theorem 4.23, the reflection group 17F C Isom iiii is discrete and F is its fundamental polyhedron if and only if all dihedral angles a(S?, S9) of F are integer parts of r. Thus, rF has the following presentation, see Theorem 2.37:

(YiYj)n`J=1)

where yi are reflections in the sides of F (in spheres Si), nij = Jr/a(S?, S9) E Z if

Si fl Sj 0 0 and i 0 j, and nii = 1.

7.3. Quasi-Fuchsian structures: bendings

423

For any other configuration C5 = (Sl, ... , Sk) E C53 of Spheres in S3 with the same dihedral angles a(S1, Sj) = a(S?, Si), the group r C Mob(3) generated by reflections in the spheres Si, i = 1, ... , k, is isomorphic to 1'F. Therefore the problem of deforming the representation po: I'F C Mob(3) is equivalent to the problem of deforming the configuration 6o = (SO, ..., Sk) E (5k so that the deformation preserves the dihedral angles between the neighboring spheres. Enumerating the edges of the polyhedron F and the corresponding intersection circles SI fl Sj, we have a map: A3: C53/Mob+(3)--> R1,

A3([6]) = a = (al ... , a1)

E ]I1

where al, .. . , al are the dihedral angles corresponding to the l intersecting circles

Si flSj,am=a(Si,Sj), 1 <m
A2([S]) = a = (al, .. , a1) E ][81,

where am = a(Sr, Sj), 1 < m < 1, and the initial configuration 6o is mapped to CIO = A20o]) whose coordinates are angles am = a(S?, Si). Since F C ]H[3 is compact and rF is hyperbolically rigid due to Theorem 7.10, we have that H1(rF, [33) = 0 and the map A2 is an immersion at the point [C5o]. Moreover, in a neighborhood of the point [C5o], the dimension of C52/ Mob+(2) equals 1 = dim ]R1, i.e. the number of edges of F, and hence the map A2 is also a submersion

in a neighborhood of [80]. Indeed, if v is the number of vertices of F, then due to the simplicity of F we have that 3v = 21. On the other hand, using the Euler characteristic of 8 F -_ S2, we see that v - l + k = 2. It implies that l = 3k - 6 which is the dimension of C5z/ Mob+(2). Due to the fact that A2 is a submersion near [6o], the map A3 is also a submersion

near [6o]. Therefore, in a neighborhood of [8o], the variety Hom(rF, Isomf4) coincides with A31(as) and hence is a smooth manifold. Its dimension is equal to

dim(C53/Mob+(3))-1=4k-10-1=k-4. Now, we obtain immediately:

Corollary 7.39. Let r'F C Isom 1H13 be a discrete group generated by reflections in the sides of its fundamental polyhedron F C IH[3. Then rF is conformally rigid, Tq f(I'F) = {a point}, if and only if F is a simplex.

We remark that the above condition k = 4 is equivalent to saying that the orbifold IH13/ rF is sufficiently large, i.e. it contains an incompressible suborbifold, see Thurston [1, Ch. 13].

424

7. Theory of Deformations

Example 7.40 (Simultaneous bending along intersecting geodesic surfaces). Here we shall construct a co-compact discrete group r' = TF C Isom iEll3 generated by reflections in sides of a hyperbolic cube F C E3 and such that: (1) Hom(F, IsomlHI4) is a smooth 2-manifold near po: F C IsomH4. (2) There are two bending curves ,Bl: (-s, e) -> Hom(F, Isom THI4) passing through po and corresponding to two intersecting totally geodesic suborbifolds S1, S2 C HI / F, Si n s2 0. (3) The sum of the infinitesimal bending deformations along S1, S2 C iH[3 / F, c = c1+ c2 E H 1(F, C14), is integrable; in other words, there is a deformation fi of po tangent to c, which simultaneously bends IH13 / T along two intersecting geodesic suborbifolds S1 and S2.

Figure 125. A hyperbolic cube F

As the hyperbolic cube F C H3, we consider a convex polyhedron shown in Figure 125 whose edges labeled by integers i have dihedral angles equal 7r/i. The existence of such an F follows from Andreev's Theorem 2.41. Let us show that M = E 3/ F has exactly two totally geodesic orbifolds which intersect along a geodesic covered by the "vertical axis" of F. First of all there are only three incompressible suborbifolds M; C M. The boundaries of their supports IM1 I are shown in Figure 125

as curves St, i = 1, 2, 3. The "horizontal" orbifold M3, 8IM31= 83, cannot be geodesic. Otherwise, the orthogonality of the corresponding plane M3 to a F would imply that the upper half F3 of F\M has such angles (see Figure 126) that the corresponding (hyperbolic) orbifold has an incompressible Euclidean rectangle orbifold, which is impossible. Let us consider the suborbifold M2 C M. We claim that it is homotopic to a totally geodesic one. To show that we take a polyhedron F2 as shown in Figure 127. It exists

7.4. Quasi-Fuchsian structures: cone deformations

425

due to Theorem 2.41. We see that its front face is a rectangle symmetric with respect to its axis 1. Hence we can rotate FZ by the angle 7r about the axis 1, and the union of F2 and its rotation image is in fact isometric to F. It follows from the equalities of angles and Theorem 2.41.

Figure 126

Figure 127

The same argument implies that the orbifold M1 C M is totally geodesic. The application of Theorem 7.29 completes the construction.

§4. Quasi-Fuchsian structures: cone deformations Here we describe quasi-Fuchsian structures on hyperbolic n-manifolds M obtained by (non-bending) cone deformations. We start with the first examples of such deformations (Apanasov [35, 41]) which show a relation of such deformations to cone hyperbolic structures on the original complete hyperbolic manifolds M. The main idea is an equivariant deforming of the universal covering M embedded in the hyperbolic (n + 1)-space Hn+1 as a hyperplane H C 11n+1, to one of the hypersurfaces H« C jp1n+l, a E (-s, e), such that Ha have convex hyperbolic structures with cone singularities along geodesic subspaces of co-dimension at least two. These singular geodesic subspaces correspond to totally geodesic submanifolds in M.

Example 7.41 (Deforming pea-pod groups). Let r c Isom H3 be a hyperbolic lattice whose fundamental polyhedron F C ]HL3 has all dihedral angles equal to 7r/3 and is bounded by eight 2-planes whose circles at infinity a1Ell3 are shown in Figure 128. Assuming the Poincare half-space model 1R

H3, we define generators of r

426

7. Theory of Deformations

as a hyperbolic translation x F-a kx, .l E 1[8.+, which identifies the opposite sides of F corresponding to concentric circles in Figure 128, and reflections in the other sides

of F.

Figure 128

It is easy to see that the group F has three bending planes P1, P2, P3 C 1H13 whose boundaries at infinity correspond to the following lines: {x E ]I82

V 3 X21, {x E II82 : x1 = -,-./3: x1 = 01, {x E II82 : x1 = -/3,V 3 - X21. (7.54)

Let po: F C Isom 10 be a natural inclusion and a E (ao, 2n/3] a parameter with a lower bound ao, 0 < ao < n/2. We shall define a quasiconformal deformation of po, a H Pa, Pen/3 = po, induced by F-equivariant quasiconformal homeomorphisms f« 14 --- IH14, pa = (fa)*po This deformation deforms the hyperbolic structure on H3/ r along three totally geodesic surfaces with boundary, Sp = P2/Gp,, i = 1, 2, 3, corresponding to (7.54), and along their common boundary geodesic l C H3 / F which is covered by (x E 1R+ : xl = x2 = 0). Namely, it deforms the hyperbolic structure on lH[3/ F into conformal structures which correspond to singular hyperbolic structures on H3/ r whose singular set consists of the union of the geodesic surfaces U Sl\l (where the singularity looks like a bending singularity) and the closed geodesic l with a cone singularity of positive curvature. To construct the representations pa, a E (ao, 2n/3], we consider quasi-Fuchsian

pea-pod groups ra = pa F C Isom H4 defined as in Example 6.98. Namely, let Da C R3 be a regular trihedral angle with unit edge vectors vl (a) = (1, 0, 0), v2(a) and v3 (a) whose dot products are v; (a) vj (a) = cos a, i j. Note that a is the value of the side angle of Da at vertex 0 and, for a = 27r/3, the boundary of D2n13 is the plane {x E 1183 : X3 = 0}. We define a fundamental polyhedron F« C II83 for

7.4. Quasi-Fuchsian structures: cone deformations

427

ra as the polyhedron bounded by the following eight spheres whose intersection with the angle Da is indicated in Figure 121: S2

Sk (a) =

8a

vi (a) + vj (a) 2(1 + cos a)

Sij(a) = S2 (8avk(a), ra)

ra

>

1l

la = S2 (0, Ra) ,

/ ,

(7.55)

Ic, = S2(0, R'a)

Here S2 (x, r) denotes the 2-sphere centered at x E R3 with a radius r > 0, the integers i, j, k are obtained by a permutation of 1,2,3, and the numbers 8a, ra, Ra and Ra are determined (modulo a positive factor) by the equalities 26,, sin

a 4

= J ra

,

Ra =

With a normalization Sa =

ra +

4S'2 -

3r) /2

,

Ra = (82

- ra)/Ra

this implies

a a Ra = J cos 4 - sin 4 , a a RU = cos 4 -{- sin 4 .

a

ra = sin 4 , (7.56)

A direct calculation shows that the dihedral angles of the polyhedra Fa bounded by spheres (7.55) do not depend on a, ao < a < 27r/3, and all equal it/3. Here the limit value ao, ao < i , corresponds to the case when the spheres Si j (ao) and Sik (ao), j # k, are tangent. Now we can define discrete groups ra c Mob(3) isomorphic to r as the groups generated by hyperbolic translations x i-+ X ,,x, Xa E R+, which transfer Ia to la, and by reflections in the other spheres in (7.55). Obviously, the representations pa: r -* ra can be induced by homeomorphisms fa: H4 -+ JUl4 obtained as requivariant extensions of quasiconformal homeomorphisms of fundamental polyhedra in ]H14, F = F2,r/3 -+ Fa, which carry sides on hyperplanes bounded by spheres in (7.55) with a = 2n/3 to the corresponding sides with parameter a E (ao, 21r/3), see Apanasov [41].

It now follows from (7.55) and (7.56) that the constructed deformation {pa} C Hom(r, Isom H4) is a real-analytic curve passing through po: F C Isomlll4. Moreover, as we have seen in Example 6.98, the groups ra = par are quasi-Fuchsian groups, and the image Ha = fa(1E13) is a pleated hypersurface in iiii with cone singularities, described in Lemma 6.97. Namely, the (vertical) hyperbolic geodesic £ = (0, oo) C R3 C ]R is mapped to a singular geodesic fa (C) = £ C H2 whose neighborhood U(t) is the union of three 3-dimensional dihedral angles with magnitudes a, a < 2rr/3. Hence U(l) C Ha has a cone singularity along fa(e). Such a cone U (t) whose defect along its singular geodesic f equals (2n - 3a) > 0 can be developed into a totally geodesic 3-plane by means of only a quasi-isometry stretching its 3-dimensional dihedral angles with magnitudes a to dihedral angles with

428

7. Theory of Deformations

magnitudes 27r/3. This shows that its distortion should be at least 2rr/3a > 1. On the other hand, in the case of bending deformations, we have already seen that if the image of a totally geodesic plane is a pleated surface in H4, it can be isometrically developed into a totally geodesic plane. This implies that there are neighborhoods of the constructed representations pa E Hom(F, Isom JH[4) which do not contain bending

representations of r associated with the bending planes P1, P2 and P3.

The basic idea of the above cone deformation can be seen as follows. For a closed geodesic l C H3/ F = M which may be a common boundary of totally geodesic surfaces S1, ... , Sk C H3 / I', we need to deform (equivariantly) the universal covering M = H3 into a pleated embedded surface Ha C H4 so that neighborhoods of any geodesic Z in the lift .n-1(l) are deformed into cones like U(Z) C Ha, and singularities of these cones have positive curvature along the geodesic images of

Z C n-1(1). The hyperbolic structure on I3/ r projected from such a pleated 3surface Ha C H4 is called a polyhedral hyperbolic cone structure. It induces a conformal (quasi-Fuchsian) structure on 1U13/ ' as the structure projected from the topological 3-ball in aH4 bounded by a Ha. It is an open problem, for which closed geodesics l in a hyperbolic 3-orbifold M, such cone deformations of a given hyperbolic structure on M exist (see also HodgsonKerckhoff [1] and Kojima [4]). It is possible to use the Andreev-Rivin Theorems 2.41-2.43 in order to answer this question affirmatively in a special case of geodesics which are common boundaries of at least three embedded geodesic surfaces Si C M with common boundary and with the property that there are isometries ri: M -+ M each of which pointwise fixes the corresponding surface Si C M. The following example provides a polyhedral cone deformation of such a symmetric closed hyperbolic 3-manifold. Example 7.42 (Singular-cone defonnation of a closed manifold). Let M be a closed (hyperbolic) 3-manifold constructed as follows. We take two tetrahedra A and A' with sides A, B, C, D and A', B', C', D', respectively, and glue their sides D and D' together. Then the remaining sides of the obtained polyhedron P are pairwise glued as shown in the spherical development of a P in Figure 129, so that the arrows on the 00

Figure 129

7.4. Quasi-Fuchsian structures: cone deformations

429

edges match. The resulting C W-complex has one edge and one vertex, and the link of the vertex is a surface of genus 2. Removing a neighborhood of the vertex, we make the complex into a manifold N with boundary. Then the desired closed manifold M is the double of the manifold N along its boundary.

The manifold M is hyperbolic. To see that, we can reconstruct the manifold N by gluing regular truncated hyperbolic simplices, see Figure 130. We get such a truncated hyperbolic simplex by expanding a regular hyperbolic simplex in the projective Klein model of IHI3 so that its vertices vi first touch the sphere at infinity 0H3 and then, continuing the expansion, go beyond aH3. Let Ht c 1HI3 be the halfspaces containing the center of B(0, 1) = IHI3 and bounded by hyperbolic planes Hi that are orthogonal to geodesics in 1H13 emanating from the vertices vi lying beyond OIH[3. Then the intersection of the simplex with li H: is a regular geodesic truncated I

hyperbolic simplex whose dihedral angles are all equal a, 0 < a < it/3. Here a = n/3 corresponds to the regular ideal hyperbolic simplex, see Figure 131.

Figure 130

Figure 131

Since all 12 edges of two truncated hyperbolic simplices are equivalent, the cone angle at the singular set of N is 12a. It meets the totally geodesic boundary aN at two points. The boundary a N is formed by triangular faces of the truncated simplices and

is a non-singular hyperbolic surface if and only if a = n/6. Therefore, doubling N along the boundary a N provides a hyperbolic cone structure (M, cu,) on M, with cone

singularity along an embedded circle Sl y l C M. For a = n/6, this hyperbolic cone structure is a complete hyperbolic structure on M, and for 0 < a < 7r/6, the singular closed geodesic l C M has positive curvature. Finally, we note that doubling sides A, B, C and D of the hyperbolic truncated simplex provides four totally geodesic surfaces SA, SB, SC and SD in M with common boundary geodesic 1. The universal covering of (M, ca) can be taken to be a pleated 3-cell Ha C H4 consisting of totally geodesic truncated simplices and whose singular

7. Theory of Deformations

430

set is the union of the disjoint geodesics £ with cone angle 12a. Here 12a < 2n if 0 < a < n/6, and the geodesics £ are the boundaries of totally geodesic pieces of Ha which universally cover the surfaces SA, SB, Sc and SD.

Let Ira: Ha -> (M, ca) be the universal projection which induce the holonomy

representation pa: Jrl (M) -+ Fa C Isom 1H4 with discrete image ra, Ha/ ra = (M, ca). For a = 7r/6,17,/6 = r c Isom H3 C Isom R4 preserves 3-plane 1H13 C H4 which universally covers the hyperbolic manifold M =H3/ F. Therefore the family of (quasi-Fuchsian) representations {pa } defines a smooth curve in Hom(F, Isom H4) and a smooth deformation in the Teichmiiller space 3"(M, S3) of marked conformal structures on M. Each of these structures is obtained as the quotient Ba/ ra N M where a quasiconformal ball Ba C S3 = dlH[4 is bounded by the sphere aHa -_ S2 at infinity.

In the above constructions of cone defomations of hyperbolic structures, we had hyperbolic (polyhedral) cone structures which locally (in neighborhoods of singluar geodesics) looked like the union of totally geodesic three-dimensional dihedral angles bounded by geodesic half-surfaces with common geodesic (singular) boundary. The next construction provides a cone deformation of a closed hyperbolic 3-manifold M along an isolated singular geodesic which is not a common boundary of geodesic surfaces in M. This cone deformation is determined by a smooth family of cone hyperbolic structures (M, ca) on M =1HP / F such that, in a neighborhood of a singular geodesic l C M, (M, ca) looks like a round 3-cone having a positive curvature along its singular geodesic. In other words, the universal covering Ha of (M, ca) is embedded in lHl4 as a pleated 3-cell whose disjoint singular geodesics have neighborhoods in the form of round 3-dimensional cones. Another interesting feature of the obtained quasiFuchsian groups Ft C Isom ll]I is that the boundary of the convex hull C (A(Ft )) C H4 has a non-discrete set of bending geodesics. We define such "smooth-cone" deformations on special hyperbolic 3-manfilds M = H31 F, vol M < oc, which are similar to the Jorgensen [3] manifold fibered

over S1. Like the Jorgensen group r,j C Isom 1EI3, our groups r = Fo ' Ho are semidirect products of geometrically infinite normal subgroups FO and subgroups H0

of similarities with presentations H0 = (a, b : akbm = 1) for some k, m c Z. Example 7.43 (Smooth-cone deformation of a closed hyperbolic manifold). To define

a desired group F = Fo x Ho C Isom lEl(3, we fix an arbitrary number X > 1 and consider a family E0 of hyperbolic planes in 1l8+ determined by boundary circles si = S1(zt, ri) C C =air+\{oo} centered at zt E C with radii ri > 0 as follows:

(1) ri=1,r2=A, r3=A3; (2) arcs of the circles s; bound a triangle having all angles equal to n/4. For the chosen three circles, let zo be the most distant from zt intersection point of two circles: {Z E C: Iz2 - Z1/1Z1 - ZI = .k} (1 {Z E (C : IZ3 - ZI/IZ1 - ZI = A3)

.

7.4. Quasi-Fuchsian structures: cone deformations

431

Let also denote by a > 0 and fi > 0 the angles between the vectors (Z2 - zo) and (z3 - zo) and the vector (zi - zo), respectively, see Figure 132.

Figure 132

Up to a Euclidean translation, we may assume that zo = 0 and zi - zo are the radius-

vectors of the points zi, i = 1, 2, 3. Then we define loxodromic transformations ht(z) = Xe`az and h2(z) = JA3e-'Pz such that hl(st) = s2 and h2(sl) = S3. It follows from the definition of the angles a = a (A) and P = P (A) that lim a (A) lim $ (),) = 0 as A -* 1 and, for sufficiently large values of A, the sum a (X.) + P (A) becomes larger than 'r/2. This shows that there exist an integer m and a corresponding value A = Ji(m) > 1 such that, in addition to (1) and (2), the following holds:

(3) 2a(A) + $(),) = 27r/m; (4) Ho = (ht, h2) is an elementary discrete group with the presentation:

Ho=(hi,h2:hlmh2m=1).

7. Theory of Deformations

432

We note that, assuming zo = 0 and due to the law of cosines, the numbers m E N,

= .l(m) > 1, a = a(,L(m)) and Iz1I obtained in the above construction can be shown to satisfy the following equations: A2+

6+ A6

/+1=

;3,/2-

+),4,/2-

1Z,12(l

- 2A cos(a) +,k2) cos(27r

1 = Iz112(1- 2A3

M

- 3a) +,X 6)

cos(2rC

+ ;' 3 = Iz112(a,2 - 2A4

M

- 2a) + A6)

(7.57) .

Now consider the circle h1(s3) = h2(s2) and denote by s4 the circle orthogonal to it and to the circles s2 and S3. Similarly, denote by s5 the circle disjoint from s4 and orthogonal to the circles Si, s2 and s3, see Figure 132. Then we define the family E0 of 2-spheres in 1R3 (planes in lFll3 =1R+) as the Ho-orbit of the 2-spheres Si = S2 (z, , rl ) orthogonally intersecting the boundary plane C of 13 along the circles si, 1 < i < 5.

The spheres from Eo bound a conical (infinite sided) spherical polyhedron whose dihedral angles equal either n/2 or n14,

P=fl{exth(S1):hEHo,i=1,4,5},

p

C 1[8

(7.58)

and whose component P+ C ]R+ is a convex hyperbolic polyhedron. Obviously, the group Fo c Isom1H13 c Mob(3) generated by reflections in the spheres from the family Eo is discrete, P+ is its fundamental polyhedron, and the group Ho acts on Fo by automorphisms because hFoh-1 = Fo for any h E Ho. The last claim follows from the fact that h(P) = P for any h E Ho. This shows that the semidirect product Fo x Ho = ro C Isom 1H13 C Mob(3) is a discrete group. Moreover, the group ro is a geometrically finite (co-compact) group because Il8+/ J,0 = P+/ I'o is compact. So there is a finite index subgroup in F0 which is the fundamental group of a closed hyperbolic manifold M whose closed geodesic l C M is covered by the axis £ = (0, oo) C R+ of the loxodromic subgroup Ho.

Now we shall construct a smooth-cone deformation of ro (and of the manifold M along the geodesic 1) by defining a smooth family {rt} of quasi-Fuchsian groups Ft C Mob(3), t E (-s, s), where the value s > 0 will be defined below.

Namely, just as for ro, consider for any t E (-s, s) and for any number A > 1 three spheres Si (t) = S2 (xt , rt ), i = 1, 2, 3, whose centers x1 E cC x R = 1R3 and radii rt > 0 are as follows: 'X3; xt = (zt , t), xt = (Zt , ),t), xt = (Zt , Xt );

(lt) rt = 1+ rt =

rt =

(2t) each sphere Si (t) intersects two other spheres at the angles 7r/4. Let x° = (z°, 0) be an analogue of the point zo, that is the point from the following intersection that is most distant from the point xt :

(cCx{0})n{xER3:Ixt-x1/Ixt -xI=.,}n{xE1183:Ixt -xI/Ixt-xI=J3}.

7.4. Quasi-Fuchsian structures: cone deformations

433

Clearly, the point xo smoothly depends on t and limbo xfl = zo. Let at = a (t, X) > 0 and , t = (t, )) > 0 be the angles between the orthogonal projections of the vectors (xt - x°), (xr - x?) and (xt - xO), (xt - x?), respectively, onto the plane C x {0} C II83. Making a translation x i-* (x - xO), if necessary, we may assume that xO = 0 and define rotations in 1183 with the orthogonal matrices sin at

0

At = - sin at cosat

0

0

1

cos at

0

and

Br =

cos t -sin 0t sin Pt 0

cos Ot 0

0 0 1

So we have loxodromic transformation his, h2t E Mob+(118+), hit(x) = .At x, h2t(x) = ),3 Bt x, such that h1t(Si(t)) = S2(t) and h2t(Sl(t)) = S3(t). Moreover, for the integer m determined while defining the group Ho, one can find a number ).t = X(m, t) > 1 such that: (3r) 3a(t,.L(m, t)) + $(t, ))(m, t)) = 2n/m; (4t) the loxodromic group Ht = (hlt, h2t) is an elementary discrete group with presentation

Ht = (hit, h2t : h3'nh2m = 1) .

(7.59)

As in (7.57), for x° = 0, the numbers At = X (m, t), at = a(t, A(m, t)) and Izr I are solutions of the following system of equations obtained by applying the law of cosines to the above geometric construction:

xt + t + 1 = Izr I (1 6 +,X

cos(at) + 2) + t2(xt _

1)2

27r

1 - 2,1r cos(- - 3ar) + X6 +

+ 1 = Izi 12 /

1)2

m

C

t -t cos(- - 2at) +

x6 +),4t V/2 + a,2t=lztl1 2 11\'X2 t

2X4

27-

m

.16) t

+

t-

Here we notice that, due to the smoothness of this system with respect to the parameter t, the spheres Si (t) and the group Ht in (7.59) depend smoothly on t and tend to the spheres Si = Si (0) and to the group Ho defined from (7.57) as t tends to zero. Now we define two disjoint spheres S4(t) and S5(t) such that:

1. S4 (t) is orthogonal to the spheres S2(t), S3(t) and htt(S3(t)) = h2t(S2(t)), 2. S5 (t) is orthogonal to the spheres Si (t), S2 (t) and S3 (t);

3. S4(t) and S5(t) are disjoint from the spheres h(S1(t)) for h E Ht and

i=1,2,3. We remark that the number e > 0 bounding the parameter t, -s < t < s, is determined by the last condition. Let us consider a (deformed) family Et = (h(Si (t)) : h E Ht , i = 1, 4, 5} whose spheres bound a cone-shaped spherical polyhedron Pt C 1183 with dihedral angles 7r/2

434

7. Theory of Deformations

and 7r/4 (as for P = Po). We define a discrete group Ft C Mob(3) generated by reflections in sides of Pt (in spheres of Et). This group is preserved by conjugations by elements h of the group Ht, hFth-1 = Ft. It implies that the semidirect product Ft = Ft x Ht is a discrete group isomorphic to the group F0. Moreover, by extending the spheres of the families Eo and Et to the 3-planes in the hyperbolic space IH[4 = 18+ which bound convex hyperbolic polyhedra p4 and

P4, we can construct a ro-equivariant quasiconformal homeomorphism ft:1

that induces the isomorphism Fo - Ft, (ft)*(ro) = rt = ftrofr 1. This homeomorphism ft maps a neighborhood U(2) C 1E13 of the axis t of the loxodromic 11

subgroup Ho to a pleated round-cone-shaped 3-surface Ut c ft(H3) = Ht C 1E14 whose singular geodesic ft (t) has positive curvature. This can be shown by applying the argument of Lemma 6.97 to the boundary of the convex hull C(A(Ft)) C 1E14, because the discontinuity set o (rt) of the quasi-Fuchsian group rt contains a solid round cone Ct with the axis {x E 183 : xl = x2 = 0} whose boundary has a dense subset of straight rays (0, oo) containing limit points of the group F.

§5. Bendings along surfaces with boundaries Here, in addition to bending and cone deformations of conformal structures on M, we define one more type of deformations which bend a given conformal structure along collections of hypersurfaces with common boundary and, in particular, along pleated hypersurfaces in M. Although our construction properly works for general conformal manifolds, we consider it for simplicity in the case of hyperbolic structures.

Bending along hinge-collections of geodesic hypercells. Let r c Isom 1H[1, n > 3, be a discrete group, not necessarily a lattice, which conformally acts in the unit balls Bn(0, 1) C Bn+1(0, 1) considered as the hyperbolic spaces Hn C 1i
position means that the union of any two surfaces Si and SJ is not totally geodesic near the boundary S, and no surface Si has a self-intersection along 8. Otherwise, if Si U Sj

is totally geodesic near S = Si fl Sj, we need to have k > 5. In the second exclusive case of self-intersections, we need to have k > 4 + m where m is the number of such self-intersections along S.

7.5. Bendings along surfaces with boundaries

435

Passing to the universal covering rc: M = H' -> M, we obtain an infinite family E of hyperbolic geodesic hypercells in ]HI" which universally cover the hingecollection {S1, ... , Sk} C M. This family E is the F-orbit of a k-element hinge-

collection of geodesic hypercells P1, ..., Pk C ll", ni P, = H = S =

E3[n-2,

rr(Pi) = Si, so it has the following properties: (1) Any two geodesic cells P, Q E E either intersect along their common boundary (n - 2)-plane H = P fl Q projected by 7r to the geodesic boundary S = ni Si, or have a uniform positive distance between them, d (P, Q) > do > 0. In the first case, H is the common boundary of a k-element hinge-collection

EH={R1,...,Rk}=g({P1,...,Pk)),gEF. (2) Let Fi = ni rb (Si) be the stabilizer in F of the hyperbolic plane containing the geodesic hypercell Pi C EH. Then, similarly to (7.38),

inf{d(Pi,g(Pi)):gEF\Fi,10.

(7.60)

Starting with such a family E of k-element hinge-collections EH = {P1, ... , Pk}, we can define elementary hinge homeomorphisms (PH: 1BIn+1 --> Hn+1 This homeomorphism cpH bends the hyperbolic n-space along the half-planes P1, ... , Pk that contain the geodesic hypercells Pi, ... , Pk E EH so that cpH is conformal in dihedral neighborhoods of components of H' \{ P1, ... , Pk }, and there are hyperbolic rotations

hi in Eli's+1 pointwise fixing the (n - 2)-plane H and such that cpHFicpH1 = hiFih! 1

,

1 < i < k.

(7.61)

We define coH in the same way as the elementary bendings *R in (7.40), (7.41) and Figure 124. The only difference in the present construction is that, instead of dihedral neighborhoods of half-spaces H' \R, we consider dihedral neighborhoods U 3 of components Pig of IEEn\{P1, ... , Pk} bounded by half-planes Pi and Pj. Each such Ui j is the intersection of dihedral neighborhoods of the corresponding half-spaces

Pit, P C l l bounded by planes that contain Pi and P, Pii = Pi+ fl Pt, see Figure 133. Namely, we define (PH as a quasiconformal map in H1+1 which preserves all hyperbolic 3-planes 1H13 orthogonal to the (n - 2)-plane H as well as all hyperbolic 2-spheres in these 3-planes 113 centered at points x E H fl 1113. Then qOH can be determined by its restriction to such a round sphere S2 C H3. So we define this restriction cPH I S2 as a quasiconformal automorphism of the sphere S2 which is a spherical isometry in each of the domains S2 n Uij. In other words, 'PH I S2 deforms a spherical k-gon on the sphere S2 (whose vertices are intersection points p, fl S2) so that the sides of this k-gon are rigid while the angles vary. It easily follows from our conditions on the number k that such homeomorphisms cPH I S2 (and hence (PH) exist and depend on as many parameters as the number of parameters of isometric deformations of a spherical k-gon on the 2-sphere S2, which have hinges at its vertices.

436

7. Theory of Deformations

In particular, for a generic k-element hinge-collection EH, there are precisely (k - 3) such parameters for an elementary hinge-homeomorphism (PH.

Figure 133

In Figure 134, we schematically show the behavior of such a homeomorphism qOH on the boundary aln in the case n = 3.

Figure 134

We remark that cOH simultaneously bends the n-plane Hn in Hn+l along halfplanes Pi containing geodesic hypercells P; E EH such that the bending angles

al = at (t1, ... , tk_3), i = 1, . . . , k, are functions of parameters determining (PH.

7.5. Bendings along surfaces with boundaries

437

Hence the bending angles ai cannot be chosen arbitrarily. That is why, even in the case when the union Pi U Pj is totally geodesic, we do not bend Hn along the plane Pi U P. It is quite possible that cei (t1, ... , tk-3) 0 ca3 (t1,

... , tk_3); in particular, this is the case if k = 4. Thus we have to bend Hn along P, and Pj separately. Now, enumerating the orbit r(H) of (n - 2)-planes in HI compatible with the enumeration of bending half-planes Pi containing Pi E E (cf. (7.39)), we can use elementary homeomorphisms (pi = coH; to define r-equivariant quasiconformalhomeomorphisms

f = ft: Hn+1 --, Hn+1,

f = jliM *00

(floV2o...Wj)

,

.f*: r ->

frf-t ,

(7.62)

which depend on (k - 3) -dimensional parameter t = (t1, ... , tk-3). Hence we have a (k - 3)-dimensional deformation in Hom(F, Isom Hn+1), which we call a bending of F along a hinge-collection of geodesic hypercells. Correspondingly, we have a bending of the conformal structure of M = HI/ IF along a hinge-collection of totally geodesic hypersurfaces with boundary. Clearly, a bending along a totally geodesic plane (surface) defined in §3 is a particular case of this deformation. A direct application of Theorem 7.32 gives us a condition for nontriviality of the deformation in (7.62):

Lemma 7.44. Let $: Wk-3 --> Hom(r, IsomHn+1) be a bending deformation in (7.62) of a non-elementary discrete group r C Isom Hn along a hinge collection {P1, ... , Pk} C Hn of geodesic hypercells such that the bending angles ai (t), t E Wk-3, 1 < i < k, are not constant. Then fi is nontrivial if and only if the union Ui Pi intersects the interior of the convex hull Cr = C(A(r)). 0

We can also apply the algebraic description of bendings along planes (Theorem 7.31), which involves amalgamated free product and HNN-decomposition of the group r, to obtain an algebraic description of bendings along hinge-collections of hypercells with boundary. This shows that the deformation in (7.62) smoothly depends on its (k - 3) -dimensional parameter t. Finally, the above observation, Lemma 7.44 and Theorem 7.33 imply: Theorem 7.45. Let 18 be a bending deformation in (7.62) of a discrete group r C Isom Hn along a generic hinge-collection {Pt, ..., Pk} C Hn of geodesic hypercells with boundary such that the corresponding bending angles ai (t) are not constant, int Cr fl (Ui Pi) 0 0, and the limit subsets A (r)f18 Pi span (n-2)-spheres containing 80o Pi C 8Hn, 1 < i < k. Then p o ,B: Wk-3 --+ Hom(r, Isom lHln+1)/ Isom IH[n+1

is an embedding of an open domain projection

Wk-3 C

1Rk-3

where p is the natural quotient

p: Hom(r, Isom Hn+1) -* Hom(r, Isom H+')/ Isom Hn+1

0

438

7. Theory of Deformations

Corollary 7.46. Let M be a hyperbolic n-manifold with a generic hinge-collection Es = {S1, ... , Sk} C M of totally geodesic hypersurfaces with a common boundary component 8 and finite area. Then the bending deformation of M along ES defines an embedding Bk-3 y Tqf (M, Sn) of a (k - 3) -ball into the Teichmiiller space of marked quasi-Fuchsian structures on M. 2. Bendings of 3-manifolds along arbitrarily many half-surfaces. Here we present a construction of closed hyperbolic 3-orbifolds Mi which have hinge-collections E1 =

IS,, ..., Sk, } C Mi with arbitrarily many half-surfaces, lim ki = oo as i -* oo. Although the closed manifolds constructed in Theorem 6.49 have this feature, we shall give a simpler example.

Example 7.47. To construct such hyperbolic 3-orbifolds Mi, it is enough to find a family of hyperbolic lattices Fi, i E N, i > 3, with the following properties. Each group 1'i has i bending half-planes in R3

lH[3 passing through the common geodesic

H = {x E R3 : xl = x2 = 0}. The dihedral angles bounded by these planes have

Figure 135

magnitudes that are multiples of ai = n/ i. For this purpose, for a given i E N, we shall construct a discrete group Gi C Isom IH[3 associated with a dihedral angle in IH13 of magnitude 27r/ i with the edge H. The group Gi is generated by a hyperbolic

translation go: x H hix, )i > 1, and by reflections with respect to spheres aj, 1 < i < 5, shown in Figure 135, where go(o'o) = oo.

7.5. Bendings along surfaces with boundaries

439

Here the dihedral angles bounded by spheres Qi and uj, 1 < i, j < 4, equal 7r/3. All other dihedral angles equal 7r/2. The existence of such a hyperbolic polyhedron Fi C 1H13 follows from the Andreev Theorem 2.41 (for a direct construction, see Apanasov-Tetenov [5]). So it defines a number Xi > 1 so that the transformation go translates the side oo c a Fi to the side Q6. In particular, Figure 135 shows the

orthogonal projection to aH3 of such a polyhedron Fi with i = 6, ai = rr/6 and = (8 + 33)/4. Now we construct the desired lattice ri as a discrete group generated by i block groups: hi=(U1GiU-i

*

:j=0,1,...,i-1)

Uj Gi U-i

Gi-l.i

* Uj+1 Gi U-(j+t) Gi+i,i+2 Gi.j+i

where U E 0(3) is a rotation about H by the angle 27r/i and the amalgamated U- j

are conjugate to the group Go,1 generated by go and reflections in the sides v3 and 0_0_4. For i = 6, the obtained hinge-collection EH consists of 6 half-planes, and hence the bending P along EH depends on 3 parameters.

subgroups Gj,j+1 = Ui Go, I

Figure 136

In Figure 136, these three parameters are shown as some angles which determine a "hinge"-deformation of 2-plane 1[82, with hinges along given rays and with rigid complements of these rays.

440

7. Theory of Deformations

The result of this deformation is a 6-hedral angle shown in Figure 136. It corresponds to a deformation of a spherical 6-gon with rigid sides shown in Figure 137.

Figure 137

3. Bending along pleated surfaces. It is possible to apply the above bending along hinge-collections of hypersurfaces Si C M to define a bending deformation of a hyperbolic n-manifold M along a pleated hypersurface S C M with compact (n - 2)dimensional singularities. It is important because some hyperbolic manifolds have no totally geodesic surfaces but do have pleated ones, see Adams-Reid [1]. Such a bending along a pleated surface S C M preserves S and bends by an angle each geodesic transversally intersecting S. Namely, let the pleated surface S be the union of totally geodesic surfaces with boundary, S = S1 U .. U such that S,j = S, fl S1 is either empty or a geodesic (n - 2)-dimensional common boundary of adjacent pieces, and these boundaries are disjoint. Additionally, let us assume that there are generic hinge-collections of totally geodesic hypersurfaces associated to each singularity component S,j c S and such that the adjacent to each 8(/ surfaces Si, Sj C S bound a dihedral angle in the complement in M to each of these hinge-collections. Then we construct a bending fi of M along all these hinge-collections with the boundaries 8,j. Due to the definition of such a bending, each geodesic surface in the hinge-collections (and in particular, the original ones Si) is preserved; the magnitudes of dihedral angles bounded by every two surfaces (and hence, angles between Si and Sj) are preserved, as well. Furthermore, each geodesic transversally intersecting Si C S is bent by some angle a, . Therefore, the obtained bending fi is in particular a bending along the given pleated hypersurface S C M. Especially, the new conformal structures on M obtained by the bending 8 along S have such holonomy representations d*: 7r1(M) -+ Isom H1+1 that the subgroup d*ir1(S) of the holonomy group d,,7r1(M) does not depend on parameters of the bending ,B.

7.6. Global properties of deformation spaces

441

§6. Global properties of deformation spaces In the previous sections, we studied quasi-Fuchsian conformal structures (M, c) on a hyperbolic n-manifold M, (M, c) E Tq f(M) C T (M, S"), which correspond to quasi-Fuchsian deformations in Tq f (r) of the inclusion nl (M) r c Isom IHIIn+1. Here the Fuchsian group r preserves a round ball B" C Sn = )H'+1 and conformally acts in this ball B" as a discrete group of isometries of the hyperbolic n-space 1H[n, H" / r = M. Due to the Sullivan structural stability Theorem 7.2 (see Corollaries 7.4 and 7.5), the spaces 9-q f (M) and Tq f (r) are open connected components of the Teich-

maller space T1(M) c T (M; Sn) of marked conformal uniformizable structures on M and, correspondingly, of the variety Hom(r, Isom Hn+1)/ Isom IHIn+1 = Hom(r, O(n + 1, 1))/ O(n + 1, 1)

.

Here we shall describe some global properties of the Teichmaller space Tl (M, Sn)

and the varieties .R1(r, O(m + 1, 1)) / O(m + 1, 1) with m > n. Here M is a closed hyperbolic n-orbifold with nirb(M) = F C O(n, 1) C O(n + 1, 1), and the variety `J21 (r, O(m + 1, 1)) consists of all faithful representations p: r -+ O(m + 1, 1) with discrete images pr. We shall pay special attention to 3-dimensional orbifolds M.

1. The boundary of Teichmaller space. We notice that in general 71 (M, Sn) is a proper subspace of T (M, Sn). In particular, there are examples of conformal structures

on a closed 3-manifold M whose holonomy groups d*nl (M) = G C M6b(3) are (non-classical) Schottky groups in S3 and hence are not isomorphic to Jr1(M), see Gusevskii [9]. Another important observation is that, for m > n,

71(M) = Tl (M, S") C J21(r, O(m + 1,1))/ O(m + 1, 1) - R'"+1(r)

.

(7.63)

This fact and the weak hyperbolic rigidity (see Corollary 7.26) provide a natural compactification of Teichmaller spaces:

Corollary 7.48. Let M be a closed hyperbolic n-orbifold, n > 3. Then the Teichmiiller space 71(M) of marked uniformizable conformal structures on M with faithful holonomies has a natural compactification T1(M) C 3fn+l (F) all whose points cor-

respond to discrete faithful representations p: r = rr'rb(M) -+ Mob(n). This result allows us to define the boundary of Teichmaller spaces 71(M) and their

quasi-Fuchsian components Tq f (M) as aT1(M) = T1(M)\T1(M) and a7q f(M) _ Tqf (M)\Tq f (M). The latter boundary consists of structures on M whose holonomies

are limits of conjugations fl r ft 1 of the inclusion r c Isom IHIIn C Mob(n) by equivariant quasiconformal homeomorphisms fl: Sn -+ Sn.

In dimension n = 2, due to basic results by Bers [3], Maskit [2] and Abikoff [2], the boundary points of the Teichmaller space T (g) of closed Riemann surfaces of genus g > 1 correspond to the so-called B-groups. Each such a Kleinian group G C M6b(2) has an invariant simply connected component S2o of the discontinuity

7. Theory of Deformations

442

set but is not quasi-Fuchsian. There are B-groups of three types: non-degenerate (i.e. geometrically finite), degenerate (see §3.3), and partially degenerate. Groups from the last class can be obtained from degenerate and non-degenerate B-groups by Maskit combinations. In dimensions n > 3, almost nothing is known about geometrically infinite groups on the boundary aT (M) of the Teichmiiller space of conformal structures on a closed hyperbolic n-manifold M. However, even non-degenerate boundary groups acquire new properties. We shall describe some such groups which have the so-called accidental parabolic elements (limits of sequences of loxodromic elements).

Theorem 7.49. The boundary aTq f (M) of the quasi-Fuchsian component of the Teichmiiller space of a closed hyperbolic n-orbifold M = Hn/ r with at least two disjoint totally geodesic (n - 1)-suborbifolds, n > 3, has points whose holonomy groups G = d*r C Mob(n) are non-degenerate B-groups with discontinuity set S2 (G) split into two G-invariant components, S2 (G) = 0o U 521, and which belong to one of the following two (realizable) classes:

(I) 0o is a quasiconformal n-ball, while the second component is not-contrac-

tible: 7r1(01) / 0 for i = 1, ... , n -2; (II) both components Qo and S21 are non-contractible: 7t; (SZj) # O for j = 0, 1

andi = 1,...,n-2. Proof. First we construct the desired boundary groups G, E aTq f(r), i = 1, 2, in dimension n = 3 where we use the closed hyperbolic orbifold M = H3/ r from Example 7.27. Namely, we choose the groups G 1 and G2 to be the groups r (ti, 5/6) and

r(t1, to) from the family {r (t, r)} of groups r(t, r) = r c M6b(3) defined by parameters t and t from the interval (7.37). Namely, the parameters t and r define spheres

Si (t, r), 1 < i < 14, obtained from the corresponding spheres Si = Si (5/6, 5/6) which determine the group r in Example 7.37. Specifically, Si (t, r) = Si for i 0 13, 14. The last two spheres S13 = S2 (re3, p) and S14 = S2 (-re3, p) with radius p = 10/6 and r = 5/6 are replaced by the spheres S13(t, T) and S14(t, r) centered

at points tea = (0, 0, t) and -re3 with radii t2 - -t+ 5/12 and

t2 - -r+ 5/12,

respectively. The radii are such thatthe spheres S13 (t, r) and S14 (t, r) have orthogonal

intersections with other spheres Si, i 0 13, 14. As we have seen in Example 7.37, for parameters t and t in the interval (to, ti) in (7.37), there are equivariant quasiconformal homeomorphisms fits: S3 -> S3 such that r(t, t) = 4>t,rr(Dr 1. However, the limit Kleinian groups

GI = r(t1, r) = t_+t, lira r(t, r)

and

G2 = F(t1, to) = lim r(t, t)

t-t,

r-+ro

are not quasi-Fuchsian.

Namely their fundamental polyhedra F11,,, Ft,,ro C S3 have two components and Ft ro. The polyhedron F,r T

each: bounded FO,r and FO TO and unbounded Ftl

7.6. Global properties of deformation spaces

443

is contractible (as are all non-limit polyhedra F°7), and the corresponding component Qo C 0 (G1) is a quasiconformal ball,

00 = U g(FP,z) _'h,T0(B3(0, -/-15/6)). gEGI

However, the three other polyhedra have nontrivial fundamental groups:

nl(Fl,T) = nl(F'1,to) _ Jr1(Ft,,,)) = 7G4.

(7.64)

This follows from the fact that each of the spheres S13 (ti , r), S13 (t1, To) and S14(tl, ro) has four tangent points with the other spheres Si, 9 <_ i <_ 12, and the tangent points lie outside the spheres Si, 1 < i < 8. Thus, the tangent points are parabolic fixed points of some elements in the groups G1 and G2. Lemma 2.22 implies that the domains G 1(Fti ), G2 (F0, , re), and G2 (F i to) are invariant (non-simply connected) components of the discontinuity sets 0 (G1) and 0 (G2), respectively. Finally we notice that G1- and G2-orbits of the (accidental) parabolic fixed points are dense in A (Gj) = 8 S2i (GJ ), j = 1, 2, i = 0, 1, hence the homotopy groups in the claim are infinitely generated.

Remark 7.50. Obviously, the above proof works as well in the general case of a lattice r C Isom HI with two bending hyperplanes - compare the construction of such bendings in §3 (Theorem 7.29). Here the boundary groups with desired properties correspond to (discrete) groups with accidental parabolic subgroups of rank one.

2. Connected components of Teichmuller spaces. As we have seen, the Teichmiiller space T(g) of marked Riemann surfaces of genus g > 1 is a cell R6g-6 and hence is connected. In contrast to this, here we shall show that Teichmiiller spaces 71 (M) of conformal structures on (closed) hyperbolic 3-manifolds M may have more than one connected component. It is implied by the following claim.

Theorem 7.51. There are (uniform) hyperbolic lattices r C Isom H3 whose Teichmuller spaces 7'1(F) are not connected. Moreover, the complements 7'1(F)\7q f(F) are not empty.

Proof. As the desired uniform hyperbolic lattices F C Isom 1H13, we take the groups constructed in the proof of Theorem 5.44 and Lemma 5.45. Assuming that such a hyperbolic lattice r acts in a round ball B3 C R3 (co-compactly, by Mobius transformations), and due to the proof of Lemma 5.45 we have that there exists a quasiconformal R3, see Figure 84. This embedding f conjugates equivariant embedding f : B3

the group r so that G = f If -1 C Mob(3) is a convex-cocompact Kleinian group whose discontinuity set 52 (G) has two invariant components, 7 (G) = 00 U 521. Here 520 = f (B3) is a quasiconformal 3-ball, but 521 is non-simply connected. In fact, the

boundary 8S2o = as21 = A(G) is an embedded 2-sphere, f (8B3) = A(G), which has a dense subset of points where it is wildly knotted.

7. Theory of Deformations

444

We claim that the conformal structure (M, cG) on M =1Ei[3 / P induced by the nat-

ural projection lrG: 520 -+ S2o/G - M lies in the complement Ti(M)\7gf(M). Otherwise the structure (M, cG) would be the limit of quasi-Fuchsian structures (M, ci) E Tq f (M). However, the corresponding holonomy groups Fl E T"q f(F) and the group G are convex cocompact. Hence, due to the structural stability Theorem 7.2, sufficiently close groups G and F1 would be conjugate by a quasiconformal homeomorphism fi : S3 _+ S3, fi (A (]Pi)) = A (G). This is impossible because A (G) is a wildly knotted 2-sphere in S3.

Remark 7.52. First, we notice that, as it follows from the construction of the group G C Mob (3) above, the non-connectedness of Teichmuller spaces 7"1(M) and T1 (F) is based on the existence of a sufficiently large number of disjoint totally geodesic surfaces in M = IH13/ F. However, it is unknown how many connected components there are in Teichmiiller spaces 7'1(M) and what invariants may distinguish them. One of the possible approaches might be using such invariants of conformal structures on M as the r1-invariant and the Chem-Simons functional which are locally constant in T1 (M), see Atiyah-Patodi-Singer [1], Chern-Simons [1] and Apanasov [39]. Secondly, as it was recently discovered in dimension two by Anderson-Canary [1], the deformation space of a geometrically finite Kleinian group G C PSL2(C) may also have many connected components. Namely, the (Teichmiiller) space of geometrically finite faithful representations of G into PSL2(C) modulo conjugacy has several components but its closure in the entire representation variety modulo conjugacy is connected. This phenomenon has some different reasons in algebraic convergence (see Remark after Theorem 5.27) and in topology of compact hyperbolizable 3-manifolds M = (1613 U 0 (G))/G with nonempty incompressible boundary, greatly clarified by Anderson-Canary-McCullough [1], see also Canary-McCullough [1]. Now, instead of closed hyperbolic 3-orbifolds M =1H13 / IF and Teichmuller spaces T1 (M) and 71 (F), we may consider conformal manifolds fibered over M and associated varieties of representations of their holonomy groups. It is possible to interpret these varieties of representations as the spaces of uniformizable m-dimensional con-

formal structures on the fundamental group r C Isom 1FF, n > 2, of the base M of such a fibration. Namely, a conformal m-dimensional structure on the group P is determined by a pair {N, rp} where N is a conformal m-manifold with a non-surjective development

d: N -+ S'n of its universal covering space N onto d(N) = 0 C S'n and cp is a monomorphism, gyp: P -* nl (N), corresponding to the short exact sequence:

0-

r ( ) -+ 7r1(N)- F-> 0.

(7.65)

Due to the uniformization Theorems 6.62 and 6.65, we may assume here that the

development map d and the natural projection 1r: 0 -+ S2/G = N are covering maps which factor the universal projection N -+ N. We say that two pairs {No, (po} and {N1, pl} determine the same conformal structure on F if there is an orientation

7.6. Global properties of deformation spaces

445

preserving conformal homeomorphism f : No _* Ni such that and Cpl differ (up to the isotropy subgroup Z(po) of the inclusion po: I' C Mob(m)) by an inner automorphism of r. The set 7m (r) of equivalence classes [N, gyp] is called the space of conformal m-dimensional structures on a given group r. As an example, we consider the fundamental group r = nl (Sg) of a hyperbolic surface Sg of genus g > 1. Then we the space 73 (r) of conformal 3-structures on r is determined by Seifert fibrations N over the surface Sg (for fibrations with nontrivial Euler numbers, see Gromov-Lawson-Thurston [1] and Corollary 6.77). As it follows from Theorem 6.74, such conformal 3-manifolds N have either H2 x l[8- or SL2 Rgeometries. We refer the reader to Ohshika [1] for a description of Teichmiiller spaces of such Seifert fibered 3-manifolds. Here we only notice that the Teichmiiller spaces

73(F) have many connected components. This fact is based on the existence of Seifert fibrations with non-zero Euler classes (Corollary 6.77) and on the topology of r-actions in S3. To explain how it is related to that topology, we present a simple example which has inspired constructions of conformal structures on total spaces of S1-bundles over surfaces Sg, see Corollary 6.77.

Example 7.53 (Groups with limit circles S1 y S3). For simplicity, let r c Isom 8312 be a discrete group generated by reflections in the sides of an ideal polygon in H2 of finite area. Representing the hyperbolic plane 1112 as the Poincare model in the round disk B2 in the plane R2 = R2 X {O} C R3, we have a round circle S' C R3 covered by sequentially tangent closed round 3-balls bi with disjoint interiors, bi n bj = 0 for I i - j I ; 1 mod k, 1 < i < k, whose boundary spheres are orthogonal to S1, see Figure 138. So we have a natural inclusion po: r c M6b(3) as the group generated

by reflections in the spheres 8bi, 1 < i < k, and A(F) = S1. The group r is not "good"in the sense of § 1.3 since its centralizer Z(r) C M6b(3) is infinite. However, taking its quasiconformal deformation in 3f3(r) or 7 4(r), we easily obtain a good representation with trivial centralizer.

Figure 138

Assuming that the number k is sufficiently large, we can construct faithful dis-

446

7. Theory of Deformations

crete representations p: F - * Mob (3) which lie in components of the variety r3 (F) different from that containing the inclusion po. To do that, we take a nontrivial polygonal knot K C R3 whose k vertices are tangent

points of closed round 3-balls Bi, 1 < i < k, with disjoint interiors, Bi fl Bj _ 0 for Ii - j 10 1 mod k. Then we define a desired representation p: r -k Mob(3) so that pI' = G is a discrete group generated by reflections in the boundary spheres aBi,

1
Figure 139

Obviously, the inclusion po: F c Mob(3) of the group r and the constructed representation pK: r -+ M6b(3), pK(F) = G, lie in different components of the variety

7.6. Global properties of deformation spaces

447

r3(1') because of the structural stability Theorem 7.2. Here the existence of parabolic elements in r and G is not essential because the isomorphism I' -a G is type preserving. Remark 7.54. One can extend the above construction to the case of cocompact groups r C Isom IH[3 and non-necessarily knotted embeddings S1 S3, see Apanasov [36, p. 421-423], Kuiper [4, 5] and Chapter 6, §6. In all these cases, we also have many connected components of the variety T3 (I'), their number being a function of F.

Hyperbolic lattices and 2-knots in S4. Now we consider the above problem of connectedness of the variety 3em+1(r), m > 4, of conjugacy classes of faithful discrete representations p: I, --+ Mob (m) of a uniform lattice r c Isom]Hl3 in the hyperbolic 3-space. As we observed, it contains the space 7' (r) of m-dimensional uniformizable conformal structures on IF which satisfy the exact sequence (7.65). Obviously, the Teichmiiller space T1 (r) is a subspace of Tm (r). Nevertheless, the fact obtained in Theorem 7.51 that the variety T1(r) is not connected does not imply non-connectedness of the bigger variety 7' (r). The topological obstruction for connectedness of 71 (F) (nontrivial knotting of the limit 2-sphere A (G) C S3, see the proof of Theorem 7.51) is not an obstruction in dimensions m > 4. In fact, the nerve of that knotting S2 _+ S3 is 1-dimensional, and hence the knotted in S3 2-sphere A(G) is unknotted in Sm, m > 4. However, we shall show that in general the varieties 74(r) C 3e5(r) are nonconnected. To do that we shall establish a link between components of these varieties and 2-dimensional knots in the 4-sphere S4 which carry conservative dynamics of conformal actions of a given uniform hyperbolic lattice r c Isom H3. We start with recalling on 2-knots in S4.

Knotted 2-spheres in the 4-sphere S4. A two-knot K C S4 is an embedding K: S2 y S4 of the oriented 2-sphere into the oriented 4-sphere. Two 2-knots KI and K2 in S4 are called equivalent (or of the same type) if there exists an orientation preserving homeomorphism f : S4 _+ S4 such that f K1 = K2. Obviously, it is an equivalence relation, and we call the equivalence class [K] of a 2-knot K C S4 the knot type of K. Those 2-knots that are equivalent to the natural inclusion S2 C S4 are called trivial or unknotted. The simplest examples of nontrivial 2-knots in S4 can be obtained by using the so-called suspensions and spins of classical knots in S3, see Artin [1], Rolfsen [1, §3J]. Namely, for a classical nontrivial 1-knot k C S3, let points a and b lie in the disjoint components S+ and S4 of S4\S3. Then the join K of the knot k with {a} U {b} is called the suspension of the 1-knot k, see Figure 140. It holds that nl (S4\K) - nl (S3\k), so the obtained 2-knot K is nontrivial. To construct a spun 2-knot K C S4, we consider a classical knot k in the halfspace II8+ _ {x E 1184 : X4 = 0, x3 > 0}, such that the boundary 2-plane R2 = {x E Il84

: X3 = x4 = 0} intersects the knot k along a single are a. Then spinning the

complementary arc P = cl(k\a) about the plane R2 sweeps out a 2-sphere K C ]R4.

7. Theory of Deformations

448

Obviously, K n 111 consists of a simple loop kt. Providing an orientation on kt which

is coherent to that of the are 0 C k1, we see that kl is representing the 1-knot that is obtained as the connected sum k#(-k) C R3, see Figure 141.

Figure 141. Spun 2-knot

Figure 140. Suspension 2-knot

The 2-sphere K C S4 is oriented so that the orientation of the 2-disk K fl S4 is coherent to that of kt. So obtained 2-knot K: S2 y S4 is called the spun 2-knot of a given 1-knot k C S3, see Figure 141. Since every loop in S4\K can be continuously deformed in l4 \P, we have that 7rt

(S4 \K)

= zrl ('\P) = n1(R3\k) .

We note also that every spun 2-knot K C S4 is a ribbon 2-knot. Such 2-knots generalize classical ribbon knots in S3, see Rolfsen [1] and Burde-Zieschang [1]. They can be obtained as follows, see Suzuki [1] and Figure 142.

S° 2

Ff JAR, -1

L __U

---2 St

11

r

lJ

Sm2

J Figure 142. A ribbon 2-knot with in fusion

7.6. Global properties of deformation spaces

449

Let So U U Sm C Il84 be a trivial 2-link with (m + 1) components (which are trivial non-linked 2-knots) and fl, ... , fm are appropriate embeddings of 3-balls, fi : [0, 1] x B2 y R4, which make m fusions of the 2-link. Each of the embeddings

fi is such that fi ([0, 1] x B2) fl (So u

U Sm) = fi ({0, 11 x B2)

has an orientation coherent with that of the 2-link, and the disks fi ({0) x B2) and fi ({ 11 x B2) are contained in different components of the link. Then the connected sum of the spheres So, ..., Sm and the sphere fi (3([O, 1] x B2)) represented by the homological sum

(SoU...USm)+fi(a([0,1]xB2))=SoU...US,-t is a trivial 2-link with (m - 1) components. Continuing this process of fusions on the link, we finally obtain a 2-knot which is called a ribbon 2-knot with m fusions, see Figure 142.

Similarly, for n > 4, one can define (n - 2)-knots K: Sr-2 ' Sn in the nsphere Sn as well as consider the above classes of (n - 2)-knots, in particular, ribbon

(n - 2)-knots. Now we can describe a connection between varieties of discrete representations of

hyperbolic lattices r C Isom

Ifln-1

and (n - 2)-knots in Sn.

Theorem 7.55. For a given nontrivial ribbon (n - 2)-knot K C Sn, n > 3, there exists a discrete faithful representation p: I' --* Mob(n) of a uniform hyperbolic lattice r C Isom Ip[n-1 such that the Kleinian group G = pI' acts ergodically on the everywhere wild (n - 2) -knot K,, = A (G) C Sn obtained as an infinite compounding of the knot K.

Proof. Basic construction. The proof of the above theorem is based on the "blockbuilding method" (see Chapter 5, §5.4) and geometrically controlled PL-approximations of smooth ribbon (n -2) -knots K C Sn. Namely, we may assume that the (n -2)Sn in the spheres So,._ , Sm C S' and the embeddings fi : Bn-1 [0, l ] x Bn-2 definition of a given ribbon (n - 2) -knot K C Sn are taken in the conformal category. That means that all spheres involved are round spheres, and each image fi (Bn-1) is contained in the union of finitely many round (n -1)-balls Bj in Sn,1 < j < ji, such that the boundary spheres of any two adjacent balls intersect each other along a round (n - 3)-sphere, see Figure 143. In other words, the (n - 1)-dimensional ribbon fi (Bn-1) can be obtained from a flat ribbon in Rn-1, which is the union of round balls, by sequential bendings along (n - 2)-planes. We do that by using the bending construction in §3. Let us assume in addition that in each round (n -1)-ball Bj in the construction (either a ball from one of the ribbons fi (Bn-1) or one of the balls bounded by the spheres

7. Theory of Deformations

450

Figure 143

Sk, 0 < k < m) there is a discrete action of a hyperbolic group Gj C Isom }JJ 1 = Mob(Bj). Up to isotopy of the (n - 2)-knot K and the family E of (n - 1)-balls Bj, we may assume that the groups G j have bending hyperbolic (n - 2)-planes whose boundaries at infinity 8Bj are the intersection spheres Sj = 8Bj fl 8Bj+1 for the adjacent balls Bj and Bj+1, and that the stabilizers of Sj in Gj and Gj+1 coincide. We denote such stabilizers by F j = G j fl G j+i . This property guarantees that the amalgamated free product G = ...

* Gj * Gj+1

r;-,

rj

*

rj+i

... C Mob(n)

(7.66)

is a Kleinian group isomorphic to a uniform hyperbolic lattice r C Isom 1HIn+1 As the result of this geometric construction, we have that our ribbon (n - 2) -knot K C Sn is represented as the union KO of m -I-1 disjoint round (n - 1)-spheres with 2m deleted disjoint round (n - 1)-balls on them, 2m

(SOU.US,)\UB7-1;

B7-1CSk

1
i=1

and disjoint (n - 2)-dimensional cylinders corresponding to the ribbons f1, ... , f,, of the knot K. These disjoint cylinders are the unions of spherical (n - 2)-dimensional annuli with disjoint interiors which lie on the boundary spheres 8Bj of the round (n - 1)-balls Bj in the construction. Due to the choice of the block-groups Gj C Mob(Bj) C Mob(n), the knot KO - K lies in the interior of the complement of a fundamental polyhedron F C S" of the product group G in (7.66). Thus Sn\F is a regular neighborhood of KO. Furthermore, sequentially using bending deformations described in §3, we can construct a uniform hyperbolic lattice r c Isom IHIn+1 which

7.6. Global properties of deformation spaces

451

has the same amalgamated free product structure as the group G in (7.66) (and hence, it is isomorphic to G) and conformally acts in one of the (n - 1)-dimensional balls BJ. We do that in a way similar to the proof of Lemma 5.45. Now a direct application of the isomorphism Theorem 4.41 shows that the limit set A (G) of the group G - F C Isom IHI' 1 is an (n - 2)-knot in S". We claim that it is the desired everywhere wild knot K00 C S' obtained by the infinite compounding of the knot Ko = K, compare Figure 139:

K00 = .

#K#K#K#

(7.67)

.

Before we go on with the proof, we shall show how this construction works in the case of 2-knots. We illustrate it by a simplest ribbon 2-knot K C 3184 obtained by one fusion from two unlinked 2-spheres So and Sl, see Figure 142. This knot K is in fact the spun 2-knot of the classical trefoil knot k C R3, see Figure 141.

We can take the knot K C S4 to be a PL-knot shown in Figure 144, that is as a ribbon 2-knot obtained by one fusion of two unlinked boundaries of 3-dimensional cubes Qo and Q1 in 3-planes I[83 x {0}, 3183 x {54} C 1L84, respectively.

To have a better geometric control on the blocks that we are going to use in the construction, we shall use (instead of the spheres So and S1 in the definition of a ribbon knot) the cubes 8 Qi with edges parallel to the coordinate axes in 3184. As the R4 we shall also use the union of 3-dimensional (smaller) cubes ribbon fl: B3

Qj, 2 < j < m, in R4 as it is indicated in Figure 144.

R

Figure 144. Spun 2-knot of the trefoil as PL-ribbon knot

452

7. Theory of Deformations

Here the edges of the equal small cubes Qj, j > 2, are parallel to the coordinate axes in 1R4. Their size shall be later determined by the size of the first two bigger cubes Qo = Q1. The plus signs in Figure 144 show the character of intersections in our 3-dimensional projection of K of the boundaries 8 Qo and 8 Q 1 with the boundary

of the 3-dimensional tube which is the union U21jIm Q j of small cubes. Later we shall also explain our choice of parallel 3-planes 1[83 x {-27}, 1183 x {0}, 1R3 x {54} and 1183 x {81 } in R4 which contain some of the cubes Q j and are orthogonally joined by the tubes that are the boundaries of the union of the remaining small cubes. Now we define discrete block-groups G j associated with the cubes Q3. Although Gj are isomorphic to hyperbolic isometry groups in H3, it is more convenient to use quasi-Fuchsian deformations of these hyperbolic groups (bendings as in Example 7.27) so that the obtained groups G j match the cubes Q j in the following sense. Assuming K = K0, we cover the 2-knot K C Uo< j<m 8 Q j by a family E = {bji } of closed round 4-balls bji whose boundary spheres 8bji are orthogonal to K. Namely, in the first step, we take 4-balls b j centered at the vertices of the cubes Q, 0 < j < m,

whose radii rji are equal to each other if either j = 0, 1 or 2 < j < m. One more condition on these radii rji is that bji n bkl ,-E 0 only if the centers of the different balls bji and bkl are the ends of a common 1-edge of one of the cubes. In the latter case, the magnitude of the exterior dihedral angle bounded by the spheres abji and 8bki should

equal rr/3. These 4-balls bji do not cover the entire knot K. On each square 2-side X c a Qj n K, we have uncovered 4-gon bounded by circular arcs. We cover such a 4-gon by five additional 4-balls bji centered at X and whose boundary spheres abji intersect (orthogonally) only those previously constructed balls that are centered at the

vertices of X. Among these five new balls, the first four sequentially intersect each other with the external dihedral angles 7r/3. The fifth ball is centered at the center of X and (orthogonally) intersects only the last new four balls, see Figure 81. After that, we still have uncovered those two 2-sides X0 and X1 of the big cubes Qo and Q1 that are (orthogonally) joined by the tube U2<j<m Qj. Here we assume that Qo n Q2 and Q1 n Qm are small squares centered at the centers of Xo and X1, respectively. Furthermore, we choose the size of the cubes Q j so that Q j, j > 2, are unit cubes and the cubes Qo and Q 1 have the size which matches the covering family {bji }. In fact, the boundary spheres of the four additional balls centered at int(Xo) orthogonally intersect the corresponding spheres centered at the four vertices of the small cube Q2, see Figure 145. This completes the construction of the family E _ {bji } of 4-balls

that cover the knot K. The union of these balls, Ui j int bji = N(K), is a regular neighborhood of the PL-ribbon knot K. Notice that we can take the size of cubes Q3' j > 2, arbitrarily smaller than that of the cubes Qo and Q 1. To do that, we repeat the above process of covering the sides Xo and X1 by balls bji where, instead of the vertices of X0 (and X1), we take the centers of the four new small balls. Then each of the annuli in Xo\Q2 and Xl\Qm will be covered by (4 + 8k) additional balls bji (for sufficiently large integer k > 0) instead of the above four additional balls corresponding to k = 0. This allows us to take the

7.6. Global properties of deformation spaces

453

I[84 as thin as we need. ribbon fl : B3 We define a discrete block-group GJ associated with a cube Ql, 0 < j < m, as the group generated by reflections with respect to all spheres a bpi , that is, withrespect to all spheres 8bbl that intersect the cube Qj. Obviously, Gj is discrete because all spherical

dihedral angles with edges abbt fl abjl are either 7r/3 or it/2, see Theorem 4.29 and Corollary 2.39. Furthermore, Gj preserves each of (coordinate) 3-planes R3 C IlS4 that contain the cube Qj . In such a 3-plane 93, the group Gj can be deformed by bendings to a Fuchsian group acting in a 3-ball B 3 C W. That is why we can consider the groups Gj as discrete subgroups in Isom1H13, Gj - G C Isomffl[3.

Q

Figure 145

For any two adjacent cubes Qj and QJ+1, the groups Gj and Gj+1 have a common

subgroup rj = Gi n Gj+1 which is generated by four reflections with respect to the spheres centered at the vertices of the square Qj fl Qj+1. So we can apply the Maskit combination Theorem 5.17 and obtain a Kleinian group G C MSb(4) as the free amalgamated product in (7.66). For the group G, we can take the complement of a regular neighborhood N (K) of the knot K to be a fundamental polyhedron P C S4:

P = R4\N(K) ,

N(K) = U int bit. i,i

(7.68)

454

7. Theory of Deformations

We remark that, for each amalgamated free product Gj * Gj+l, we can use a r; bending deformation along the hyperbolic 2-plane Hj whose boundary circle alj is the limit circle of the amalgama subgroup F3. As a result, we get a new hyperbolic isometry group G C Isom H3 which is isomorphic to Gj P G j+l . Applying this I

process m times, we obtain a cocompact discrete group F C Isom 1H13 isomorphic to the group G. In dimension n = 4, there is another (non-algorithmical) way to get such a unique hyperbolic lattice I, by using the Andreev-Rivin classification of hyperbolic compact polyhedra in 1H13. Namely, the boundary a P of the polyhedron in (7.68) has the combinatorial type of S2 x S1 where the 2-sphere S2 is decomposed into the union of spherical polygons. In fact, a P is the union of 3-sides each of which is the annulus on a sphere abji, i.e. each 3-side is the product of a spherical 2-dimensional polygon Dji and the circle S1. The dihedral angles between such 3-sides are determined by the corresponding 3-dimensional dihedral angles bounded by 2-spheres abji n i3 in the corresponding 3-planes I[83 C 1184, so they are either r/3 or.7r/2, and the AndreevRivin conditions apply, see Theorems 2.41 and 2.43. It follows that the combinatorial

type of the 4-polyhedron P determines the combinatorial type of a 3-dimensional compact hyperbolic polyhedron P C 11113, with the same magnitudes of dihedral angles as those for P. Thus the group F C Isom IlI3 generated by reflections in sides of P is a uniform hyperbolic lattice isomorphic to the group G.

Remark 7.56. The above observation that we can take the ribbon fl: B3 -* 1l "arbitrarily thin" makes it possible to apply the above block-groups Gj C Mob(4), Gi =_ G C Isom 1183, to represent an arbitrary ribbon 2-knot K C S4 as the knot which he on the boundary of the union of 3-cubes similar the above cubes Qj. To finish the proof of Theorem 7.55, we need to show that the knot KOO = A (G) in (7.67) is an everywhere wild (n - 2) -knot if the original knot K C S" is nontrivial. Let 6 = G(Ui j abji) be the G-orbit of the boundary spheres abji of the balls in the covering E of the PL-knot K. We can use the word norm I g I of elements g E G with respect to generators of G, which are reflections in sides of the fundamental polyhedron P = S"\N(K) in (7.68), to define a partial ordering on Ch. Namely, for two spheres S1, S2 E 6, we say S2 S1 if int S2 C int S1. It allows us to enumerate the set 66 by a bijection q: N - * 6 so that it is compatible with the ordering of (5, that is the map q-1 preserves this partial order. Then we have a nested sequence of compacts in SZ (G),

(7.69)

where the elements gi E G are the reflections with respect to i -th spheres Si E (5 each

of which contains a side of the (i - 1)-th polyhedron Pi-1. The complement S"\Pi of each of the compacta Pi in (7.69) is a regular open neighborhood of an (n - 2) -knot Ki which is obtained from the knot K by sequential

7.6. Global properties of deformation spaces

455

connected sums:

Ko = K, K1 = Ko#Ko, ..., Ki = Ki-1#Ki-1, ....

(7.70)

Since the limit set A(G) = S"\S2 (G) is homeomorphic to the limit set A(1') = Sr-2 (Theorem 4.41), A (G) is anembedded (n - 2)-sphere in S" . We denote A (G) _

K,,, and claim that it is an everywhere wild (n - 2)-knot in S". Obviously, K. = n i N (Ki) where, for any i, N (Ki) = S" \ Pt is a regular neighborhood of the knot Ki in (7.70). Due to (7.69), the nested sequence {N(K1)} is decreasing to its intersection, Koo.

Due to the Alexander duality (Spanier [1]) applied to 0(G) = S"\K., we have that H1 (Q (G); Z) = H' 2(S,-2; Z) = Z. Thus we can consider an infinite cyclic covering space S2 of Q (G). Now we are concerned with the integral homology H, (Q) = H,, (Q; Z) with A-module structure where A denotes the ring of finite Laurent polynomials with integer coefficients. Namely, choosing a generator -r: S2 -+ S2

of the deck transformation group of the cyclic covering 0 -* 0 (G), we define the product of an element p(t) = ci ti e A with an element a E Hj(S2) as S

E Hj (S2).

Here t*: Hj (2) -+ HJ (S2) is the homology isomorphism induced by -r. Thus it defines the A-module H,k(S2) which is known as the Alexander invariant of the knot

Kw C S'. As a shorthand description of the first homology Hi (S2) of the infinite cyclic covering of the knot K,, complement 0 (G), one can also use its Alexander polynomial AKA (t), see Rolfsen [1, Ch. 71. Lemma 7.57. Let SZ and Pk, k > 0, be infinite cyclic coverings of 0 (G) and Pk in (7.69). Then we have a nested sequence

Proof. The nested sequence in (7.69) defines a sequence of monomorphisms of the fundamental groups as follows:

ir1(Po)-37r,(Pl) ±...-+ 7L1(Pk)-* Furthermore, we have the following commutative diagram:

...7r1(Pk-1)

7r1(Pk)

`

j

...

ik

1

1

...H1(Pk-1)

`

ik*

>

Hi(Pk) `

1

...

`> Hi(2(G))

7. Theory of Deformations

456

Here the vertical maps correspond to the Abelinization. So the lemma follows. Lemma 7.58. Hi 6S2; Z) =A 0.

-

Proof. Due to Lemma 7.57, we have excisive triads defined by X = P; and Y = SZ\ Pk. So we have a Mayer-Vietoris exact sequence as follows:

H2( X n Y )

H2 (X) ED H2 (Y)

Hl ( X n Y )

H l (X) ® Hl (Y)

H2 (X U Y) >

Hl (X U Y)

> ...

where XnY=aPkandXUY=S2. For each k > 0, the boundary a Pk is homeornorphic to the product Sr-2 X S1. It Sn_2 x R, and hence H1 (a Pk) = 0. Therefore, the map

implies that a Pk

H1(Pk) ® H1(S2\Pk) -.) H1(S2) is injective. On the other hand, H1(Pk) is not trivial due to the initial condition that the (n - 2)-

knot K C Sn is nontrivial. This completes the proof.

To finish the proof of Theorem 7.55, we observe that the (n - 2)-knot K. = A(G) C Sn is nontrivial due to nontriviality of its Alexander invariant H*(S2), which follows from Lemma 7.58. On the other hand, K. is invariant for a non-elementary Kleinian group G C MSb(n), and hence it is a wild knot due to Kulkarni [5]. The latter fact also follows from Lemma 7.58 and the additivity of the Alexander invariant H,k (S2) with respect to connected sum (7.70) of knots, see Levine [1]. Obviously, any point z E & = A (G) that is the attractive fixed point of a loxodromic element g E G is a wild point of the knot Koo. The proof is completed by the fact (Corollary 2.13) that such loxodromic fixed points are dense in the limit set A(G) = Kw.

Due to the isomorphism Theorem 4.43, we have an additional metric property of

the knot K.: Corollary 7.59. For a given nontrivial ribbon (n - 2)-knot K C Sn, n > 3, there is a quasisymmetric embedding f : Sn-2 Sn whose image is an everywhere wild (n - 2)-knot K = f (Sn-2), infinitely compounded from K. Theorem 7.60. Let IF C Isom IH[n-1 be any uniform lattice from Theorem 7.55, and n > 4. Then the varieties (F) and M n+1 (F) of conformal and hyperbolic structures on F (respectively, the classes of discrete faithful representations) are not connected. Proof. The result immediately follows from Theorem 7.55 and the structural stability Theorem 7.2 because both groups r C Isom H n-1 and G C Mob(n) constructed in the proof of Theorem 7.55 are convex cocompact and isomorphic.

Notes

457

We conclude this section by pointing out a significant difference between the Dehn surgery on high-dimensional knots and the Dehn surgery on classical 1-knots in S3. This affects the properties of fiber bundles over hyperbolic n-manifolds, n >_ 3, and makes conformal structures on a hyperbolic lattice r more rigid. As we have seen in Chapter 6, §6, the existence of conformal structures on nontrivial circle and 2-plane bundles over a closed hyperbolic surface Sg is related to a conformal

realization of Dehn surgery on a classical knot S1 y S3. However, an analogue of Dehn surgery for high-dimensional knots Si-2 _a S' is very rigid. This is related to the fact, for the first time observed by H. Gluck [1] in dimension n = 4 (see also Browder [1], Lashof-Shaneson [1] and Swamp [1]), that two homeomorphisms of the

boundary 8N(K) ~ Stt-2 x S1 of a regular neighborhood N(K) c Sn-2 x B2 of a (n -2) -knot K C Sn, n > 4, are pseudo-isotopic if and only if they are homotopic. The group of pseudo-isotopy classes of homeomorphisms of Sn-2 x S1 is thus isomorphic to Z2 X Z2 X Z2. Here the first two factors correspond to orientation-reversals of Sn-2 and S1 respectively, and the third is generated by the following homeomorphism Sn-2 X S1 _* 5n-2 x S1, 27:

71(x, B) _ (T (0)(x), 0) ;

x E Sn-2,

0 E S1

,

(7.71)

where z(0) is the rotation of the sphere Sn-2 about its polar Sn-4 through the angle 0. Therefore, in contrast to the classical 1-knots in S3, each (n - 2)-knot K C Sn, n > 4, has the only one nontrivial Dehn surgery which is determined by the homeomorphism (7.71). This makes fiber bundles over closed hyperbolic (n -1)-manifolds M, whose fibers are either 2-planes or circles and which have either hyperbolic structures (of infinite volume) or conformal structures, respectively, more rigid than the analo-

gous fibrations over surfaces Sg, see Chapter 6, §6 and Gromov-Lawson-Thurston [1]; for such non-trivial bundles over 3-dimensional M, see Apanasov [43] and later Belegradek [1]. In particular it implies, see Apanasov [43]:

Theorem 7.61. For a given closed hyperbolic n-manifold M, n > 3, there are at most two non-equivalent circle (or 2 -plane) bundles over M allowing uniformizable conformal structures (or a complete hyperbolic metric, respectively). This finiteness of bundles over hyperbolic manifolds perfectly matches the finiteness of hyperbolic structures on codimension-2 complements in a compact m-manifolds, m > 4, due to Ivansic [1, 3], see Proposition 6.44 and Theorem 6.45.

Notes For Teichmuller spaces of Riemann surfaces and their generalizations, we refer the reader to remarkable surveys by L. Bers [4-6] and to a source of examples and exercises by S. Krushkal, B. Apanasov and N. Gusevskii [3]. The most close to our exposition

458

7. Theory of Deformations

is the real analytic approach to Teichmiiller spaces of Riemann surfaces in Abikoff [3]. For algebraic approach to moduli spaces, see Newstead [1] and D. Johnson and J. Millson [1]. Partially, we follow the latter work which has a detailed account on stable representations, the tangent space to the variety of representations and the holonomy Theorem 7.1 which is originally due to W. Lok [1], see also Thurston [1] and Goldman [3]. The structural stability Theorem 7.2 was essentially proved by D. Sullivan [9] who considered the case of Kleinian groups on the plane. Our proof extends the structural stability to the general case of Kleinian groups in the n-sphere. It implies clear proofs of Corollaries 7.4 and 7.5 on subspaces of Teichmiiller spaces of conformal structures; for quasi-Fuchsian subspace, see also Johnson-Millson [1]. For another approach to perturbations and stability of geometric structures with negative curvature, see Apanasov-Siebenmann [1]. Theorem 7.13, which describes a class of non-rigid hyperbolic groups, is due to Apanasov [7]. Due to D. Sullivan's rigidity Theorem 7.15, the complement of this class consists of rigid groups. Our proof of Theorem 7.15 essentially follows to ideas of the original proof in the 2-dimensional case by Sullivan [2]. The construction of conformally non-rigid groups in Corollary 7.14 is due to Apanasov [ 11 ]. Theorem 7.20

is due to Apanasov [16] and Sullivan [2]. Its rigidity sufficient condition is not a necessary condition in higher dimensions due to Example 7.21 by Apanasov [16, 19]. The construction of a nontrivial curve in the space of quasi-Fuchsian groups in the n-sphere in Example 7.27 was the first result (due to Apanasov [11, 12], see also Apanasov-Tetenov [1] and Tetenov [1]) on Teichmiiller spaces of hyperbolic nmanifolds. It was generalized, due to the Thurston [ 1] idea of bending of surfaces, in order to estimate a lower bound for the corresponding Teichmtiller spaces. These (equivalent) estimates were independently obtained by Apanasov [24], Johnson-Millson [1] and Kourouniotis [1], see also Sullivan [4] and Lafontaine [1]. Here we describe these

bendings deformations in Theorems 7.30-7.35 following to Kourouniotis [1]. An application of bendings showing that the Teichmiiller spaces of hyperbolic n-manifolds, n > 4, may have singularities (Theorem 7.9) is due to Johnson-Millson [1]. For 3-manifolds, it is conjectured that Teichmiiller spaces are smooth, which is true for reflection groups due to Kapovich's Theorem 7.36 and Example 7.40. Its Corollary 7.39 shows that simplex reflection groups are conformally rigid. This observation is originally due to Apanasov [11], see also Apanasov [44]. §4 and §5 present another types of quasi-Fuchsian defomations of a hyperbolic manifold different from bendings. The cone deformations (polyhedral in Example 7.41 and smooth in Example 7.43) were originally introduced by Apanasov [35, 41]. Later, another construction of a polyhedral cone deformation (Example 7.42) was presented by Tan [1]. The last class of deformations in §5, the bendings along hypersurfaces with boundary and pleated surfaces (Theorems 7.44-7.47) was defined by Apanasov-Tetenov [4, 5]. An example of such deformations was independently given by Tan [1]. For general relation between singular polyhedral structures and conformal structures on a hyperbolic manifold, see Luo [3].

Notes

459

The compactification of the Teichmuller space of a hyperbolic manifold in Corollary 7.48 (Apanasov [42]) is essentially due to the weak rigidity of hyperbolic structures (Theorems 7.22-7.26) discovered by Thurston [4], Morgan-Shalen [1, 2] and Morgan [2, 3]. Theorem 7.49 is due to Apanasov [11, 12]. Non-connectedness of the Teichmuller space of a closed hyperbolic manifold (Theorem 7.51) was discovered by Apanasov [37]. Example 7.53 is due to Apanasov [10, 11]; see also Maskit [12] and Gromov-Lawson-Thurston [1]. Its construction is related to nonconnectedness of Teichmuller space of conformal structures on Seifert 3-manifolds, see Ohshika [1] Gromov-Lawson-Thurston [1]; for generalizations, see Apanasov [36, p. 419-423] and Kuiper [4,5]. The link between varieties of representations of hyperbolic 3-lattices and knotted 2-spheres in the 4-sphere (Theorem 7.55) was discovered by Apanasov [43-45]. The presented proof follows Apanasov [43, 46]. Another approach to that has been presented four years later by Belegradek [1]. Corollary 7.59 and Theorem 7.60 (on nonconnectedness of the varieties of conformal and heperbolic structures on a closed hyperbolic 3-manifold) are due to Apanasov [43-46]. Uniqueness of non-trivial bundles over a closed hyperbolic n-manifold in Theorem 7.61 is due to Apanasov [43].

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Corrected Index Boris N. Apanasov, Conformal Geometry of Discrete Groups and Manifolds Walter de Gruyter, Berlin New York 2000

Abstract polyhedron, 182 accidental parabolic element, 442 action of a group conservative, 87, 97, 397 discontinuous, 41, 54 discrete, 41 dissipative, 87, 94 ergodic, 86, 95, 449 free, 54 on geodesics, 88, 95 on set of horoballs, 121 on vector fields, 419 recurrent, 87 affine structure, 3 Ahlfors finiteness theorem, 155, 277, 282 0-measure problem, 164, 255 Aleksandrov complex, 179 theorem on space tessellation, 180 generalization, 186 Alexander duality theorem, 208 invariant of knot, 455 polynomial, 455 sphere theorem, 267, 220 algebraic convergence of representations, 111,248 algebraic surfaces, 139

Andreev classification of crystallographic groups, 82 hyperbolic polyhedron theorems, 82 Andreev-Thurston rigidity of circular coverings, 268 approximation point, 91, 161 arithmetic group, 130, 133, 136 arithmeticity conditions, 133 Armstrong theorem, 55 aspherical manifold, 219 asymptotic geodesics, 16 atlas of manifold, 1 atlas of orbifold, 34 atom, 60 atoroidal manifold, 294 automorphic function, 383 automatic structure on group, 204 averaging operator, 88, 89

Beardon-Maskit condition, 161 theorem, 157 Beltrami differential, 405 bending, 408 deformation, 409, 416 along hinge collection, 434, 438 along pleated surface, 440 along submanifold, 409, 418 with boundary, 434, 438

514

geodesic, 360 homeomorphism, 272, 411 hyperplane, 411 hypersurface, 418 infinitesimal deformation, 419, 420 line of pleated surface, 360 Betti number, 298 B-group, 60, 239, 338, 442 Bieberbach decomposition, 72, 103 theorem, 46, 105 block-building, 262, 269, 277 block groups, 262 blowing hyperbolic orbifolds, 243 blow-up of 4-manifolds, 301 Bonahon compactification of hyperbolic manifolds, 251 condition, 246, 249 theorem on hyperbolic ends, 249 Borromean rings, 35, 150, 242 Busemann function, 16 Cannon theorem, 196 Canonical hyperbolic stratification, 356 Canonical splitting of 3-manifolds, 295 Cantor set, 291 Cartan-Hadamard theorem, 293 Cayley graph, 192 embedding, 195 geodesics, 192 quasigeodesics, 203 tree, 193 chain of hyperbolic spaces, 363 character variety, 381 charts in atlas, I changes, 1 folding, 34 local, 1 chordal metric, 8 circle bundle, 312, 457 Euler number, 220, 348 over surface, 220, 312, 343 over 2-torus, 30 total space, 220, 343 C-matrix, 78 codimension-k complement, 312

Index

coefficient of quasiconformality, 5 cohomology of group, 388 commensurability condition, 143 commensurable groups, 133, 140, 176 commutator mapping, 104 compact core of manifold, 221, 246, 258 theorem, 221 compactification of manifold, 244, 251 compact-open topology, 41 complement of knot/link, 222, 303 complete (X, G)-manifold, 24 completion of group, 192 complexity of manifold, 321 complex projective space, 299 component of discontinuity set, 60 invariant, 60 stabilizer, 60 cone deformation, 425, 428, 430 cone-like singularity, 84, 373 cone point, 84 cone structure, 307 cone 3-sphere, 307 conformal chain, 370 cone structure, 307 dynamical system, 98 homeomorphism, 5, 306 manifold, 6 metric, 355, 363 pseudo-distance, 364 rigidity, 403, 423 round ball, 363 sphere, 4, 307 structure, 7, 324 almost uniformizable, 326, 329 canonical metric, 365 connected sum, 333 integrable, 7 marked, 380 measurable, 393 non-uniformizable, 326, 329 on torus bundle, 325 torus sum, 350 uniformizable, 306, 324 with faithful holonomy, 386

Index

uniformization, 329 of connected sum, 333 of Seifert manifold, 342 conformally flat manifold, 7 congruence subgroup, 141 conical limit point, 91 conjugacy problem, 51, 202 connected sum of manifolds, 221 convergence group, 40 discrete, 41 elementary, 44 non-elementary, 49 of groups algebraic, 111, 248 geometric, 310 of manifolds, 310 of representations, 111, 248 of transformations, 107, 414 convex core, 168 hull, 123, 127 boundary, 124, 370

cone singularities, 373 extremal points, 360 minimal retract, 124 natural retraction, 124 set, 123 coset decomposition, 230, 235 covering of 3-sphere, 241 Coxeter diagram, 79 group, 80 crossed homomorphism, 388 for bending, 420 crystallographic reflection group, 78 Andreev classification, 82 cup product, 298, 391 curvature, 2 cusp end, 166 standard, 167 neighborhood, 160 point, 160, 161 cylinder theorem, 218 Defining relation, 76

515

deformation bending, 409, 416 cone, 425, 428, 430 equivalence, 389 infinitesimal, 389 integrability obstruction, 391 trivial, 387, 437 of homeomorphisms, 385 Siebenmann theorem, 385 Dehn algorithm, 202 lemma, 218 reduced word, 202 surgery, 223, 313 along figure-eight knot, 314 along Whitehead link, 313 generalized, 223 parameters, 223 density point, 94, 96 desplicing operation, 304 developing map (development), 23, 326 boundary point, 326 deviator of deformation tensor, 390 dilatation of quasi-conformal mapping, 5 Dirichlet polyhedron, 70 discontinuity set, 42, 60 disk sum of manifolds, 222, 227 distortion, linear, 6, 14 distortion of Mobius transformation, 14, 16

Donaldson polynomials, 301 double density of loxodromic points, 50 double of manifold, 237, 335 dynamical system, 98 Edge cycle, 76 Efremovich-Tikhomirova extension, 201 Eilenberg-MacLane cohomology, 388 end of manifold, 167 equivalence, 167 geometrically finite, 247 geometrically infinite, 247 geometrically tame, 247 isolated, 167 parabolic cusp, 166 simply degenerate, 249, 252

516

standard cusp, 167 topologically tame, 248 Enriques surface, 139 Euclidean isometry, 9 Euclidean similarity, 9 Euler number, 220 Euler characteristic, 224 exceptional fiber of a Seifert space, 219 excisive triad, 205 expanding cover, 98 extended n-plane, 7

Farrell-Hsiang theorem, 311 Fenchel-Nielsen theorem, 200 Fibonacci group, 144 manifold, 146 fibered solid torus, 219 fiber of fibration, 1, 219 figure-eight knot complement, 18 filling of a space, 179 incomplete, 181 locally finite, 186 locally similar, 182 non-regular, 180 projection, 179 regular, 180 rigidity, 190 finiteness theorems for Kleinian groups, 155, 170 finite-sided polyhedron, 63 flat manifold, 27, 311 folding map, 34 Fomenko-Matveev-Weeks manifold, 323 four-dimensional geometries, 31 four-manifold decomposition, 301 Fox-Artin sphere, 275 wild arc, 267 free amalgamated product, 225 Freedman theorem, 300 free product of groups, 225 fundamental domain, 62 local finiteness, 63 fundamental group of orbifold, 35 fundamental polyhedron, 63 Dirichlet, 70

Index

Ford, 68 generic, 71 isometric, 67, 70 Poincare-Aleksandrov theorem, 186, 190 pseudo-Dirichlet, 74, 164 side complexity, 138 with convex ends, 172

g-manifold, 3 Gabai rigidity theorem, 293 Garland-Raghunathan rigidity theorem, 190 Garnett point, 96 Gauss-Bonnet theorem, 315 Gauss map, 83 genus of knot, 446 geodesic, 2 geodesic submanifold, 141 geometrical finiteness, 153, 157, 166, 170, 175, 204 geometric isomorphism of groups, 199, 200 geometry, 3

S2 x R, 28 IH[2 x R, 28, 330 SL2I[8, 28, 305

conformal, 4 Euclidean, 27 four-dimensional, 31 homogeneous, 25 hyperbolic, 6, 11, 12, 13, 14 nilpotent, 29 solvable, 30, 32 spherical, 4, 26 three-dimensional, 26 Gieseking manifold, 17 Goldman-Kamishima uniformizaton theorem, 325 Golubev non-rigid groups, 397 Gram matrix, 78 graph homology sphere, 305 graph manifold, 321 Green function, 88 bounded, 88 Green measure, 88 Gromov concept of uniformization, 292

517

Index

hyperbolic group, 204 invariant, 315 norm, 315 theorem on volumes, 315 Gromov Piatetskii-Shapiro non-arithmetic groups, 140 group Abelian, 44 algebraic, 382 arithmetic, 133 co-compact, 98, 165 co-finite, 78 conformally rigid, 423 convex co-compact, 384 crystallographic, 78 classification, 82 degenerate, 59 discontinuous, 41, 54 discrete, 41 elementary, 44 Fibonacci, 144 free, 193, 240 free Abelian, 27, 193, 308 Fuchsian, 56 first kind, 56 second kind, 56 function, 60, 238, 251 generated by reflections, 78 geometrically finite, 153 geometrically infinite, 155, 168 geometrically tame, 247 Heisenberg, 29 Kleinian, 41 Lanner, 80 Lorentz, 12 Mobius, 9, 20 linear representation, 21 modular, 381 nilpotent, 29, 32, 102 non-arithmetic 140, 150 non-degenerate B-group, 328 non-rigid, 393 Picard, 42 properly discontinuous, 326 quasi-arithmetic, 133 quasi-Fuchsian, 56, 238, 324, 408 reflection, 77

rigid, 396 Schottky, 52, 57, 291 classical, 58 non-classical, 58 solvable, 30, 32 surface, 241 unimodular, 26 universal, 242 virtually Abelian, 102, 325 discrete, 102, 325 nilpotent, 102, 325 solvable, 102, 325 web, 58 with limit circle, 445 Haken manifold, 39, 219 half-space model of hyperbolic space, 14 handlebody, 228 harmonic function, 88, 91 Hausdorff dimension, 165 of the limit set, 165 measure, 165 metric, 121 h-cobordism, 257, 299, 302 with boundary, 257 Heegard diagram, 146 Heisenberg group, 29 Hermite-Mahler compactness criterion, 131 hinge collection, 434 hinge homeomorphism, 435 HNN-extension, 234 holomorphic chain, 363 holonomy homomorphism, 23, 381 representation, 23, 381 theorem, 383 homeomorphism conformal, 5, 306 elliptic, 44, 47, 52 G-compatible, 191, 389 loxodromic, 44, 47, 52 parabolic, 44, 47, 52 periodic, 47 quasiconformal, 5, 271, 380, 388 quasisymmetric, 201, 275

518

homogeneous coordinates, 5 homology cobordism, 257 nontrivial 261 with boundary, 257 homology 3-sphere, 301, 303 Brieskorn, 305 graph, 305 hyperbolic, 303 Poincare, 304 Seifert, 304 classification, 304 homotopy equivalence, 62 Hopf manifold, 28, 295 horoball, 16 precisely invariant, 117 radius, 122 horocyclic disk, 233 horosphere, 16 critical, 96 horospherical limit point, 95 hyperbolic arithmetics, 130 cobordism, 261 non-trivial, 261 distance, 11-13 geodesic, 13 geometry, 11 isometry, 12 manifold, 18, 308, 310 metric, 12 plane, 12 (Lobachevsky) space, 11-15

ball model, 13 half-space model, 14 hemisphere model, 13 hyperboloid model, 12 projective model, 12 pseudo-Euclidean model, 11 stratification, 356 structure, 17 complete, 181 geometrically finite, 153, 157, 166, 170, 175, 204 incomplete, 181 on group, 405 simplex, 17 complex parameters, 185

Index

ideal, 17, 315 of maximal volume, 315 regular, 17 volume, 149,310 function, 149, 311, 316, 320 growth, 97, 396, 404 minimal, 323 set, 311, 316

Ideal polyhedron decomposition, 129 ideal simplex, 17 decomposition, 129 identity condition, 23 incompressible tori, 294 index of subgroup, 22 infinitesimal deformation, 389 non-integrable, 391, 420 injectivity radius, 107 interbreeding construction, 142 intersection form, 298 direct sum, 299 rank, 298 signature, 298 type, 288 intersection index, 250 intersection subgroup, 277, 287 inversion, 8 involution topological, 52 irrational, parabolic group, 113, 157 parabolic point, 158 screw translation, 157 isometric sphere, 9 generalized, 70 Ivansic finiteness theorem, 315

Jacobian, 69 Jacobi matrix, 5, 6

Jaco-Shalen-Johannson splitting theorem, 295 Jones polynomial, 306 Jordan Lemma, 105 Jorgensen group, 430

hyperbolic volume theorem, 311 Julia set, 98 surface, 354

519

Index

Klein bottle, 4 combination theorem, 225 geometry, 3 Klein-Poincare uniformization theorem, 290 Kneser theorem, 218 knot complement, 222, 310, 322 fattening, 267 genus, 446 ribbon, 448 satellite, 309 tones, 309 trivial, 447 2-dimensional, 447 wild, 267 universal, 222 Kobayashi holomorphic chain, 363 metric, 363 Koebe theorem on cuts, 291 uniformization principle, 291 Kolmogorov-Alexander cup product, 391 Krein-Milman theorem, 360 K3-surface, 140 Kuiper uniformization theorems, 324 Kulkarni-Pinkall uniformization theorems, 327, 328 Kummer surface, 140, 301

Lanner group, 80 simplex, 80 lamination, 249, 361 ending, 250 geodesic, 249, 377 lattice, 131, 382 Lebesgue cover number, 99 density point, 94, 96 length function, 2, 292 length space, 292 convex, 2, 292 lens space, 4 Lie subgroups of Mobius group, 19, 20

limit point, 42 closest orbit, 95 conical, 91 Garnett, 96 horospherical, 95 set, 42, 86 as the group completion, 197 horospherical, 95 quasi-self similar, 98 wildly knotted, 275, 445, 449 link, 303 complement, 222, 310, 311 Seifert, 303 simple, 303 standard meridian, 303 standard parallel, 303 of vertex, 83 spherical polar, 83 universal, 222 Liouville conformality condition, 390 theorem, 9 Lobachevsky function, 149 (hyperbolic) space, 11-15 locally extended residual finiteness, 280 Lok holonomy theorem, 383 loop theorem, 218 Lorentz transformation, 12 Luzin theorem, 96 Manifold, 1 affine, 3 arithmetic, 142 aspherical, 219 atoroidal, 294 closed, 1 conformal, 6 conformally flat, 7 elliptic, 4 Euclidean, 27, 311 Fibonacci, 146 flat, 27, 311 geometrically tame, 245, 247 graph, 321 Haken, 219 Heisenberg, 30

520

Hopf, 295, 324 hyperbolic, 4, 101, 310 of finite volume, 310, 392 of minimal volume, 323 rigid, 392, 396 hyperbolically conformal, 365, 366 irreducible, 218 Kleinian, 217, 256 double, 237 end, 239 of B-group 239 of free group, 240 of function group, 238 of quasi-Fuchsian group, 238 of surface group, 241 non-elementary, 359 non-smoothable, 300 piecewise linear, 2 Poincare, 220 prime, 218 Riemannian, 2 rigid, 396 Seifert, 28, 219 virtually trivial, 28 similarity, 324 smooth, 1, 302 solvable, 31 sufficiently large, 219 topological, I topologically tame, 107, 248 universal, 38 Whitehead, 53 with boundary, 1 Margulis constant, 106 cusp, 168 lemma, 101 manifold decomposition, 101 super-rigidity, 130 tube, 168 marked surface, 380 Maskit combination theorems, 230, 235 matrix representation of Mobius group, 21 Matveev complexity of manifolds, 321 maximal ball, 356 covering, 356 marked, 363

Index

strictly, 359 Mayer-Vietoris sequence, 205, 206, 211 McMullen theorem, 309 Mickey Mouse example, 125, 410 Milnor fiber, 302 Minkowski convex hull, 127 space, 11, 121 Mobius transformation, 9 elliptic, 19 hyperbolic, 20 irrational parabolic, 72 loxodromic, 19 parabolic, 19 purely parabolic, 20 rotational part, 102 modular group, 381 moduli problem, 379 Montesinos knot, 305 Morgan-Shalen-Thurston weak rigidity, 407 Mostow rigidity theorem, 392

Nielsen convex hull, 123 realization problem, 199 Nikulin finiteness theorems, 137 polyhedron side complexity, 138 Nil-geometry, 29 non-arithmetic group, 140 Novikov conjecture, 262 nucleus, 302 Orbifold, 34 bad, 38 billiard table, 35 Borromean rings, 35 conformal, 55, 306 covering, 37 fundamental group, 38 geometric, 39 good, 38 Haken, 39 Kleinian, 217 singular set, 35 universal covering, 38

521

Index

orbit space, 219 Parabolic combination, 339 cusp, 160 end, 163 r-neighborhood, 160 fixed point, 93, 160 group of irrational type, 72 transformation, 19 perfect set, 43 P-harmonic function, 91 Picard group, 42 pleated surface, 247 PL-manifold, 2 Poincare conjecture, 243 duality, 211, 258 homology sphere, 304 polyhedron theorem, 77, 188 point of approximation, 91 Poisson formula, 88 measure, 88, 94 polyhedron, 63 abstract, 182 combinatorial type, 82, 182 hyperbolic, 63 ideal, 85 side, 63 simple, 82 precisely invariant domain, 117 projective space, 5 projective structure, 326 proper discontinuity set, 326 proper limit set, 326 proper map, 186 pseudo-Dirichlet plane, 74 pseudo-distance, 364 pseudo-isometry, 201 pseudo-manifold, 55, 224 punctured disk sum, 233 punctured solid cylinder, 176 punctured solid torus, 176

Quadratic form admissible, 139 automorphism group, 130

reflective, 139 2-reflective, 139 quasi-conformal compactness property, 6 deformation of a group, 387 group, 41, 51 mapping, 5 vector field, 390 quasi-conformality coefficient, 5 quasi-translation, 53 quotient space, 54 non-separable, 54 Reflection, 8 reflection group, 77 arithmetic, 133 complex arithmetic, 136 crystallographic, 77, 85 deformation, 421 finiteness theorem, 139 Riemannian manifold, 2, 355, 365 rigidity of space fillings, 190 ring of integers, 133 Rivin polyhedron theorems, 84 Rokhlin theorem, 300 rotational part, 102, 177

Sample polyhedron, 179 Saveliev invariant, 305 Schottky group, 52, 57, 291 classical, 58 non-classical, 58 Scott compact core theorem, 221, 276, 284 sectional curvature, 2 Seifert fibered space conjecture, 296 link, 304 manifold, 27, 219 conformal gluing, 351, 352 orbit space, 219 projection, 219 Seifert-Van Kampen theorem, 218 Selberg lemma, 22 shortest curve, 2 Siebenmann theorem, 385 similarity, 9 manifold, 324 simple link, 303

522

simple polyhedron, 82 singular set of orbifold, 35 local group, 35 SL2 R-geometry, 28 Smith conjecture, 52 Sol-geometry, 30, 32 special spine, 321 vertex, 321 spun 2-knot, 448 Sperner condition, 53 sphere theorem, 218 spherical metric, 8, 87 spine, 321 almost special, 321 splice of two links, 303 splitting of elliptic surfaces, 302 of 4-manifold, 297, 301 of K3-surface, 301 theorem, 294, 304 of 3-manifold, 295 (S2 x R)-geometry, 28 stable group representation, 382 good, 382 Stallings theorem, 221 standard cusp end, 167 standard longitude, 303 standard meridian, 303 Steiner problem, 85 stereographic projection, 7 Sullivan convex hull theorem, 370 cusp finiteness theorem, 156, 277, 286 ergodic theorems, 88, 95 microscope, 98 rigidity theorem, 397 stability theorem, 384 suspension of 1-knot, 447 Tangent fiber bundle, I Thurston geometrization, 293 hyperbolization theorem, 308 non-rigidity theorem, 190 structural conjecture, 293, 307, 309 structural theorem for knot complements, 310

Index

Teichmuller metric, 381 space, 379 boundary, 441 connected components, 443, 456 of Fuchsian group, 405 of Riemann surfaces, 379 singularities, 391, 420 surface group, 405 tangent bundle, 387 tessellation of hyperbolic manifold, 129 tessellation of space, 180 locally finite, 65 regular, 180 rigidity, 190 Penrose, 180 Tetenov finiteness theorem on invariant components, 205, 207 group with k invariant component, 204 hyperbolic cobordism theorem, 258 thick submanifold, 101 thin submanifold, 101, 111, 114 three-dimensional geometries, 26 Tietze transformations, 271 Todd-Coxeter algorithm, 204 topology compact-open, 41 of algebraic convergence, 381 of geometric convergence, 310 on MSbius group, 41 Zariski, 382 topological dilation, 53 topological translation, 53 torus, 3 affine, 3

bundle over circle, 31, 325 Euclidean (flat), 3 knot, 309 shell, 4 sum of manifolds, 350 theorem, 296 tree, n-dimensional, 60 truncated icosahedron, 265 Tukia compactness principle, 166 isomorphism theorems, 200, 201

523

Index

theorem on limit set, 165 Turk's head knot, 149 two-reflection, 139 type-preserving isomorphism, 200 Uniformization conformal, 306, 324, 350 conjecture, 306 of connected sum, 333 of flat connected sum, 335 of Seifert manifolds, 342 criteria, 325, 329 geometrical, 293, 309 topological, of 3-manifolds, 242 uniformizable (X, G)-structure, 324 unimodular form, 300 universal group, 242 universal link, 222, 337 Variety of representations, 381 non-connected, 443, 456 quotient, 381 tangent cone, 387 Vietoris topology, 121 Vinberg arithmeticity conditions, 133 finiteness theorem, 138 Wang theorem, 310 weak hyperbolic rigidity, 407 web group, 58 Weil rigidity theorem, 392

Whitehead group, 262 link, 182 complement, 182 double, 335 3-manifold, 53 wild limit knot, 447, 449 wild limit sphere, 275 wildly quasi-symmetric embedding, 275 word metric, 192 natural form, 226, 234 problem, 202 Dehn-Cannon solution, 203 reduced,203

(X, G)atlas, 23 development, 23 geometry, 3 manifold, 3 complete, 24 orbifold, 35, 38 structure, 35, 379 almost uniformizable, 326, 328 local deformation, 387 marked, 379 moduli, 379 stability, 384, 385 uniformizable, 324