Decorated Teichmuller Theory

The QGM Master Class Series Edited by Jørgen Ellegaard Andersen, Henning Haahr Andersen, Nigel Hitchin, Maxim Kontsevich, Robert Penner and Nicolai Reshetikhin The Centre for Quantum Geometry of Moduli Spaces (QGM) in Aarhus, Denmark, focuses on collaborative cutting-edge research and training in the quantum geometry of moduli spaces. This discipline lies at the interface between mathematics and theoretical physics combining ideas and techniques developed over the last decades for resolving the big challenge to provide solid mathematical foundations for a large class of quantum field theories. As part of its mission, QGM organizes a series of master classes each year continuing the tradition initiated by the former Center for the Topology and Quantization of Moduli Spaces. In these events, prominent scientists lecture on their research speciality starting from first principles. The courses are typically centered around various aspects of quantization and moduli spaces as well as other related subjects such as topological quantum field theory and quantum geometry and topology in more general contexts. This series contains lecture notes, textbooks and monographs arising from the master classes held at QGM. The guiding theme can be characterized as the study of geometrical aspects and mathematical foundations of quantum field theory and string theory.

Robert C. Penner

Decorated Teichmüller Theory

Author: Robert C. Penner Centre for Quantum Geometry of Moduli Spaces Aarhus University DK-8000C Aarhus Denmark Email: [email protected]

2010 Mathematics Subject Classification 30-02, 30F60, 32G15, 30F10, 30Fxx Key words: Teichmüller space, mapping class group, Riemann’s moduli space, quantum Teichmüller theory

ISBN 978-3-03719-075-3 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.

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Dedicated to my mother Beverly Preston Penner in memoriam

Prologue

This book is intended to be a self-contained and elementary presentation from first principles of a theory developed by the author and collaborators over the last 25 years. It has arisen from lecture notes for a master’s class on decorated Teichmüller theory taught at Aarhus University during August, 2006, under the aegis of the Center for the Topology and Quantization of Moduli Spaces, which has evolved into the Center for the Quantum Geometry of Moduli Spaces. It is an honor to present this inaugural volume in what is hoped will be a stimulating series based on master’s classes presented at the center. It is a great pleasure to thank my colleague, collaborator and friend Jørgen Ellegaard Andersen for organizing the master’s class and arranging my original visit to Aarhus, which is now my principal academic home. Let me also thank the other participants in the class, who made many valuable and sometimes critical comments, including Marcel Bökstedt, Bill Browder, Niels Gammalgaard, Magnus Lauridsen, especially Gregor Masbaum, Guillaume Theret, Rasmus Villemoes and Yannis Vlassopoulos. Further thanks are due to Alex Bene, Rinat Kashaev and Dylan Thurston for useful discussions. One goal of this monograph is to present global affine coordinates on decorated Teichmüller spaces as well as natural cell decompositions of these spaces, which are necessary for the quantization of Teichmüller spaces and for the simplest geometric occurrences of cluster algebras. However, there are many further applications and extensions of these basic ingredients discussed here as well, for instance, to algebraic number theory, harmonic analysis, profinite and pronilpotent versions of surfaces, the topology of Riemann’s moduli spaces, topological and conformal field theories and computational biology.

Foreword

One can argue that the general idea of a moduli space, viewed in its philosophical dimension, lies at the very heart of modern scientific thinking. A moduli space of whatever nature is an incarnation of the “manifold of possibilities”, be it the phase space of Solar System, the Hilbert space of quantum state vectors of a hydrogen atom, or, as in one of the papers of R. Penner and M. Waterman of 1993, the space of secondary structures of an RNA molecule. At the next stage of scientific reflection, “scientific laws” are introduced further constraining time evolution of a system in its phase space, or specifying a measure on it, its subsets and regions, say “equilibrium points” or “attractors”, that are more directly responsible for the explanation of observable phenomena. And as a final check and a hopeful triumph for a theory, observable phenomena and/or results of controlled experiments are seen to fit (or not to fit ...) their expected patterns in the relevant moduli space, the manifold of possibilities. In pure mathematics, “moduli” of the objects of a given type are associated with the image of a space of parameters on which such objects can depend. Historically early modern examples include Grassmannian spaces of linear subspaces, upper complex half-plane as parameter space for elliptic curves, and brilliant generalizations and innovations due to Riemann. In the second half of the XX century, such thinkers as Alexander Grothendieck and William Thurston contributed their very different visions to the development of this general idea. The book “Decorated Teichmüller Theory” by R. C. Penner is a beautifully presented survey of some of the most important work of the last two decades dedicated to the moduli of two-dimensional geometric objects, complex and/or Riemannian (metric) surfaces, compact or, more often, satisfying appropriate restrictions on boundaries (finite number of points, or a union of small horocycles etc.). One of the dominating characteristics of Penner’s approach is a rich and visually appealing representation of surfaces as embedded in three-dimensional Minkowski space and interacting there with piecewise linear structures such as triangulations, embedded graphs etc., described geometrically in terms of hyperbolic lengths, flatness and combinatorics. One can consider this construction as a descendant of classical Dirichlet–Voronoi methods in the theory of lattices. Such metric characteristics related to a single surface become then parameters on the moduli space of all such surfaces, and, as Penner’s body of research abundantly shows, they are so successfully constructed, that a host of known structures can be very efficiently described in their context. As examples, one can name the Weil–Petersson forms, the Thurston boundary of Teichmüller space, the action of the mapping class group on the decorated Teichmüller space, useful and beautiful cell decompositions, and many more. A version of Penner’s cell decomposition was used by Maxim Kontsevich

x in his spectacular proof of Ed Witten’s conjectures on the intersection numbers of moduli spaces. Returning to the general picture of “manifolds of possible” in science, one must now mention that hyperbolic surfaces embedded in a Minkowski space can be imagined as “world sheets” of quantum strings. This image motivated much interesting research and interaction between mathematicians and string theorists in the physical community inspired by Ed Witten. This interaction, to which Penner contributed a series of important insights and results, underlies one of the many aspects that will attract a student or a researcher to study this book. Yuri I. Manin

Preface

Teichmüller theory succeeds in describing and classifying geometric structures on surfaces. It was born in the work of O. Teichmüller using techniques of complex analysis [2], [65] and was transformed under W. Thurston’s influence [161], [42] using techniques of hyperbolic geometry. Decorated Teichmüller theory is an essentially combinatorial treatment of the Teichmüller theory of surfaces using techniques of hyperbolic geometry, where the surface is required to have punctures and/or boundary, and the punctures or boundaries often come equipped with a further “decoration” typically given by a real or positive real parameter which may be assigned to punctures, to boundary components, to distinguished points on boundary components, or to subsets of these sets. For example in the case of punctured surfaces, the decoration may describe a tuple of “horocycles”, one about each puncture. One studies in each case an appropriate so-called “decorated Teichmüller space”, which is typically a trivial bundle over the Teichmüller space with fiber RN or RN >0 , for some N  1, and the “mapping class group” action on Teichmüller space extends by permuting the parameters to an action on the decorated space itself. Thus, very little is lost in passing to the decorated space, but one must study punctured and/or bordered surfaces to get started. A main point of passing to the decorated space is that decorated Teichmüller space (more precisely, its quotient by the diagonal action of R>0 on decorations) admits a mapping class group-invariant “ideal simplicial decomposition”, by which we mean a decomposition into open simplices together with only certain of their boundary faces. Moreover, the cells in the decomposition are described sufficiently succinctly by an elaboration of graphs called “fatgraphs” so as to allow computations of invariants, for instance, presentations of the mapping class groups and calculations of cocycles. Our approach to the ideal simplicial decomposition assigns to each “decorated hyperbolic structure” a decomposition into polygons of the underlying surface, and it depends upon a convex hull construction in Minkowski space. The Poincaré dual of such a polygonal decomposition of the surface is a fatgraph, and one assigns a real number to each edge of the fatgraph called a “simplicial coordinate” using Minkowski geometry. Rather than the hyperbolic version of this ideal simplicial decomposition treated here, one may instead derive an analogous one in the setting of conformal (rather than hyperbolic) geometry relying on the foundational work of K. Strebel [153]. An explicit construction assigns to a fatgraph together with a tuple of positive real numbers, one number for each edge of the fatgraph, a “Jenkins–Strebel differential”, i.e., a meromorphic quadratic differential q whose horizontal trajectories foliate the underlying p surface-minus-fatgraph by simple closed curves with residues of q assigned at the punctures. Deep work of Strebel plus further results of Hubbard–Masur [66] shows

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that this assignment of conformal structure to a fatgraph-with-numbers establishes an isomorphism of Teichmüller space decorated by residues at the punctures with the natural space of all isotopy classes of fatgraphs-with-numbers embedded in the underlying surface. This decomposition agrees combinatorially with the hyperbolic one, but the two differ as point sets in decorated Teichmüller space. It is a basic distinction that the explicit constructions in the two theories “go in opposite directions” in the sense that the convex hull construction produces a fatgraphwith-numbers from a decorated hyperbolic structure, and the Strebel theory produces a conformal structure from a fatgraph-with-numbers1 . In particular, the inverse of the Strebel construction depends upon solving the “Beltrami equation” while the inverse of our construction amounts to the solution of an explicit family of “arithmetic problems”, one such system of integral algebraic equations for each trivalent graph. Whereas the solution to the Beltrami equation is highly transcendental, the solutions to our arithmetic problems are algebraic. It was D. Mumford who first observed the application of Strebel’s work to the combinatorics of Riemann’s moduli space as described in J. Harer’s landmark papers [57], [58], which gave the first substantial applications of the conformal version of the triangulation to the geometry of Riemann’s moduli space. At roughly the same time, the decorated Teichmüller theory gave the hyperbolic version described here by specializing to dimension two the general convex hull construction [41] of the author with D. Epstein for complete but non-compact finite-volume hyperbolic manifolds of any dimension. This is enough to describe the polygonal decomposition of the surface, but more work is required to show that the “putative cells are cells” in decorated Teichmüller theory. Subsequently, B. Bowditch and D. Epstein [22] gave a proof of the existence of the ideal simplicial decomposition of decorated Teichmüller space based on loci equidistant to specified horocycles, which coincides exactly with the convex hull construction here (as we show when we re-interpret the arithmetic problem geometrically). Another singular and fundamental aspect of decorated Teichmüller theory is that there are global affine coordinates on the decorated Teichmüller space called “lambda lengths” with remarkable properties. These are the ambient coordinates in which we formulate and solve the arithmetic problems and prove the existence of the ideal simplicial decomposition, which amounts to proving the unique solvability for lambda lengths from appropriate simplicial coordinates. These lambda length coordinates, which are essentially inner products in Minkowski space, are absolutely central to our treatment, and we unapologetically take a decidedly nineteenth century viewpoint and perform essentially all basic calculations in suitably normalized lambda lengths. We parenthetically mention that Wolpert [173] has recently shown that lambda lengths are convex along earthquake paths. 1 One may thus start with a decorated hyperbolic structure, apply our convex hull construction to produce a fatgraph-with-numbers, where the numbers are given by simplicial coordinates, and then misinterpret these numbers as Strebel coordinates so as to produce a map from decorated Teichmüller space to itself. This map is not the identity, but we conjecture that this map has bounded distortion in an appropriate sense.

Preface

xiii

These coordinates are not canonical in that they depend upon the choice of a suitable fatgraph in the surface just as coordinates on a vector space depend upon a choice of basis. However, the coordinate transformations corresponding to different choices of fatgraph faithfully describe the action of the mapping class group of the surface in coordinates and are calculable in terms of “Ptolemy transformations”, which play a role by now in a number of fields of mathematics, for example, in the study of “cluster algebras” [46], [45] and [47], and “quantum Teichmüller theory” [32], [78], [31], and [33]. Not only that, the “Weil–Petersson Kähler two form” [172], [173] admits a simple and compact expression2 in lambda lengths, which is also part-and-parcel of these other studies. Here, we shall compute and extend the basic WP Kähler two form and perform a few sample WP volume calculations partly as a paradigm for the general method of integration over moduli space. We shall also compute the Poincaré dual of the WP two form in Appendix B [133], which is primarily included because it illustrates further important general aspects of integration over Riemann’s moduli space. Another of the author’s papers [136], on the Gauss product of binary integral quadratic forms, is included as Appendix A because just as this volume itself begins essentially tabula rasa, so too this paper was intended to be a self-contained introduction to topics in algebraic number theory from first principles and hence may be useful for a similar audience. Furthermore, a joint paper [105] with Greg McShane is included as Appendix C because of the basic computations it describes on the asymptotics of lambda lengths during degeneration of the underlying surface. We are grateful to each of International Press, Springer Science and Business Media, and the Proceedings of Research Institute for Mathematical Sciences, Kyoto University, as well as to Greg McShane, for kind permission to include the appendices here. We have taken this opportunity to correct a few small calculational and other errors (which are explicitly noted in the text) and to present sometimes simpler and sometimes more detailed proofs than in the original papers. It is fair to say that results were not necessarily discovered in “correct” order temporally, so here we try to give a more systematic derivation of this theory from first principles. Variants and relative versions of the foregoing theory are discussed for “partially decorated surfaces” (where only certain of the punctures are decorated), for “bordered surfaces” (where the punctures are in effect required to lie in the boundary of the surface and all of them are decorated), 2 These two attributes of simple calculability, both for the action of the mapping class group and for the underlying symplectic geometry of the WP metric, distinguish lambda lengths among all known parametrizations of (decorated) Teichmüller spaces. So-called “Fricke coordinates” (i.e., entries of matrices in a Fuchsian group) transform explicitly under the action of the mapping class group with the WP two form unknown, and Fenchel–Nielsen coordinates, cf. Theorem 1.18 in Chapter 1, transform horribly under the mapping class group with the WP two form simply and beautifully expressed by Wolpert, cf. [172], [173]. In fact, our treatment of the WP two form is based on Wolpert’s formula Theorem 3.2 in Chapter 2, and if Fenchel–Nielsen coordinates are “length/twist” coordinates, then lambda lengths provide “length/length” coordinates on decorated Teichmüller space. Moreover, lambda lengths “tropicalize” to convenient coordinates on Thurston’s boundary as discussed in Section 5.4 of Chapter 5.

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and the general case (where both interior and boundary punctures are allowed and only certain of them are decorated). Most of the sections beyond the first two in Chapter 2 and all of Chapter 3 are independent, and all of them are optional. In fact after reading Chapter 1, a bee-line for the lambda length parameterization for punctured surfaces is directly to read the second section of Chapter 2, and a subsequent bee-line for the ideal simplicial decomposition of the decorated Teichmüller space of a punctured surface is to read the first four sections of Chapter 4. It may be useful to comment further here on various sections. In Chapter 2, Section 4 covers the parallel theory of undecorated “surfaces with holes” where boundary components can be deformed to punctures. This is useful for quantization including the Poisson structure inherited from the Weil–Petersson Kähler form, which is also discussed. Chapter 3 extends lambda lengths and associated structures from the setting of surfaces to the topological group of homeomorphisms of the circle suitably manifest as the space of all “tesselations of the Poincaré disk” and studies an associated infinitedimensional Lie algebra in Section 4. Section 3 treats a universal profinite object in Teichmüller theory, the “punctured hyperbolic solenoid”. The main applications of the theory to mapping class groups and moduli spaces are given in Chapter 5, and the final Chapter 6 covers further applications, where we have sometimes included particularly interesting or illustrative excerpts from more recent papers. There are clear extensions of aspects of the theory to possibly nonorientable two-dimensional orbifolds (for example, lambda lengths extend immediately to coordinates in the non-orientable case), but these have not yet been fully articulated. We have not strived for completeness in the bibliography, rather, we have cited papers and books whose bibliographies may be consulted for more complete references. Let us apologize here and now if the concomitant omissions from our listed references might cause offense. Let us also apologize for the quirk of notation that the surface F .G/ associated to a fatgraph G is sometimes taken to be a “skinny” surface with boundary and sometimes to be the punctured surface that arises by capping off each boundary component with a punctured disk, where the distinction will always be explicitly stated. Aarhus, Denmark November, 2011

Robert Penner

Contents Prologue

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Foreword

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Preface

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1 The basics 1 Cast of characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Mapping class group . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Decorated Teichmüller and Riemann moduli spaces . . . . . . . . . 1.3 Tinker-toys: fatgraphs and flips . . . . . . . . . . . . . . . . . . . 1.4 The hyperelliptic involution . . . . . . . . . . . . . . . . . . . . . 2 Three models for the hyperbolic plane . . . . . . . . . . . . . . . . . . 2.1 Upper half-plane and Poincaré disk . . . . . . . . . . . . . . . . . 2.2 Minkowski space . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Harmonic functions in Minkowski space . . . . . . . . . . . . . . 3 Farey tesselation and Gauss product . . . . . . . . . . . . . . . . . . . . 3.1 Farey tesselation and modular curve . . . . . . . . . . . . . . . . . 3.2 Gauss product and hypercycles . . . . . . . . . . . . . . . . . . . . 4 Basic definitions and formulas . . . . . . . . . . . . . . . . . . . . . . . 4.1 Pairs of horocycles and lambda lengths . . . . . . . . . . . . . . . 4.2 Triples of horocycles: h-lengths and equidistant points . . . . . . . 4.3 Quadruples of horocycles: Ptolemy equation and simplicial coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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47 47 47 51 55 56 62 67 68 69 73

Lambda lengths in infinite dimensions 1 Tesselations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Pinched lambda lengths . . . . . . . . . . . . . . . . . . . . . . . . . .

88 88 95

3

Lambda lengths in finite dimensions 1 Cyclic and decorated polygons . . . . . 1.1 Decorated polygons . . . . . . . . 1.2 Cyclic polygons . . . . . . . . . . 2 Decorated Teichmüller spaces . . . . . . 2.1 Punctured surfaces . . . . . . . . . 2.2 Partially decorated surfaces . . . . 2.3 Bordered surfaces . . . . . . . . . 2.4 General case . . . . . . . . . . . . 3 Weil–Petersson Kähler two form . . . . 4 Shear coordinates on Teichmüller spaces

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3 Punctured solenoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4 Lie algebra of tesselations . . . . . . . . . . . . . . . . . . . . . . . . . 110 4

5

6

Decomposition of the decorated spaces 1 Convex hull construction . . . . . . . . . . . . 2 Arithmetic problem . . . . . . . . . . . . . . . 3 Putative cells are cells . . . . . . . . . . . . . . 4 First consequences . . . . . . . . . . . . . . . . 5 Partially decorated bordered surfaces . . . . . . 5.1 Partially decorated surfaces . . . . . . . . 5.2 Bordered surfaces . . . . . . . . . . . . . 5.3 General case . . . . . . . . . . . . . . . . 6 Fermionic formulation of the arithmetic problem 6.1 Arithmetic problem as fermionic integral . 6.2 Related super Lie algebras . . . . . . . . .

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124 124 130 140 150 155 155 160 164 165 165 169

Mapping class groupoids and moduli spaces 1 Ptolemy and mapping class groupoids . . . . 2 Integration over moduli spaces . . . . . . . . 3 Weil–Petersson volumes . . . . . . . . . . . . 4 Fatgraphs revisited . . . . . . . . . . . . . . . 4.1 Skinny surfaces associated to fatgraphs . 4.2 Fatgraphs and permutations . . . . . . . 4.3 Matrix models . . . . . . . . . . . . . . 4.4 Fatgraph tables for F21 and F31 . . . . . . 5 Compactifications . . . . . . . . . . . . . . . 5.1 No vanishing cycles . . . . . . . . . . . 5.2 Quotient arc complexes . . . . . . . . . 5.3 Low-dimensional quotient arc complexes 5.4 Laminations and tropicalization . . . . . 5.5 Deligne–Mumford compactification . . .

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174 174 181 184 193 194 198 202 207 213 213 218 223 231 235

Further applications 1 Universal Ptolemy and Thompson groups . . . 2 Nilpotent theory and three-manifold invariants 3 Open/closed strings, TFT and CFT . . . . . . 4 Computational biology . . . . . . . . . . . . 4.1 RNA . . . . . . . . . . . . . . . . . . . 4.2 Protein . . . . . . . . . . . . . . . . . .

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Epilogue

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Appendix A. Geometry of Gauss product

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Appendix B. Dual to the Kähler two form

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Appendix C. Stable curves and screens

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Bibliography

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List of Notation

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Index

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1 The basics 1 Cast of characters s Let F D Fg;r be a smooth oriented surface of genus g  0 with s  0 punctures and r  0 smooth boundary components. We shall usually require that 2g  2 C r C s > 0, i.e., F has negative Euler characteristic, and that r C s > 0, and we shall typically s 0 write Fgs D Fg;0 and Fg;r D Fg;r for notational simplicity.

1.1 Mapping class group Definition 1.1. Define the mapping class group MC.F / of F to be the group of all isotopy classes of orientation-preserving homeomorphisms of F , where the homeomorphisms and isotopies are required to fix each point of the boundary of F . Define the pure mapping class group PMC.F / < MC.F / to be the subgroup corresponding to homeomorphisms which furthermore fix each puncture. Really, this book is dedicated just to the study of these groups which play a central role in low-dimensional topology and geometry. See [19] for an extensive discussion of these and related groups. However, our analysis of these groups will take us far afield from the purely combinatorial group-theoretic considerations with which we begin in this section. There are especially simple and explicit generators for MC.F / going back to Max Dehn in the 1930s as follows. Definition 1.2. Suppose that c is a simple closed curve in F and define the (right) Dehn twist c W F ! F to be the homeomorphism supported on an annular neighborhood of c, where one cuts F along c, twists once to the right in the annular neighborhood, and then re-glues along c as indicated in Figure 1.1, where we illustrate the effect of c on a transverse arc. Notice that the isotopy class of c is independent of the choice of annular neighborhood and depends only on the isotopy class of c in F , and that the right-handed sense of c depends only upon the orientation of F and not upon any orientation of c. Furthermore, c is isotopic to the identity if and only if c is inessential, i.e., contractible to a point in F or to one of the punctures of F , and otherwise it is essential. Theorem 1.3 (Dehn, Lickorish, Humphreys, [67]). Adopt the notation of Figure 1.2 0 for the curves c0 ; : : : ; c2g in the surface Fg;1  Fg0 , where Fg0 arises by capping off

2

1 The basics

0 0 the boundary component of Fg;1 with a disk. Then the mapping class groups MC.Fg;1 / 0 and MC.Fg / are generated by the Dehn twists c0 ; : : : ; c2g .

c cut

F twist

c

reglue

Figure 1.1. Dehn twist homeomorphism.

b

c0 c5

c3

c2g1

c1 c2

c4

c2g

Figure 1.2. Dehn twist generators.

Let us remark that analogous generators are known for surfaces with s  1 and r  1 as well, and also for the pure mapping class groups (cf. [19]), but we will be satisfied here with this result. Furthermore, a different generating set will be derived for any surface with r C s ¤ 0 later (cf. Section 1 in Chapter 5). In fact, there are standard relations among these generators that were essentially already known to Dehn as follows, where we assume that all curves mentioned are simple closed curves. (Naturality) if f 2 MC.F / with f .c/ D d , then d D f c f 1 ; (Commutativity) if c and d are disjoint, then c d D d c ;

3

1 Cast of characters

(Braiding) if c and d intersect one another transversely in a single point, then c d c D d c d ; (Chain) if d1 , d2 are disjoint each meeting c transversely in a single point  1 , where b1 ; b2 are the as in Figure 1.3a, then .d1 c d2 /4 D b1 1 b2 boundary components of a neighborhood in F of c [ d1 [ d2 ; 0 (Lantern) if c12 , c23 , c13 is a configuration of curves in F0;4 pairwise intersecting one another transversely in pairs of points, where d1 , d2 , d3 , 0 d4 denote the boundary components of F0;4 as in Figure 1.3b, then c12 c23 c13 D d1 d2 d3 d4 ;

(Garside) in the notation of Figure 1.2, if we define the Garside word g D .c1 /.c2 c1 /.c3 c2 c1 / : : : .c2g : : : c2 c1 /; 0 then 4g is the Dehn twist along the boundary of Fg;1  Fg0 . In fact, this Garside relation is an algebraic consequence of the others.

d4

b1

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d1

c13

c

c12

d2 d2

d3 b2 a) Chain relation

c 23 b) Lantern relation

Figure 1.3. Chain and lantern relations.

The naturality and commutativity relations follow directly from the definitions. The remaining relations are actually kind of fun to verify, where one chooses a collection of arcs in each case that decomposes the surface in question into a disk, one confirms that the two sides of the asserted equation map each such arc to pairs of isotopic arcs, and one finally uses Alexander’s trick (that a homeomorphism of a disk which is the identity on the boundary is isotopic to the identity) to conclude that indeed the asserted relations hold in MC.F /. These verifications give the flavor of the combinatorial fun to be had with this description of the mapping class group. Let us say that a relation as above is non-separating in F if all of the curves in the relation are non-separating in the surface F . It required a long sequence of ideas from Dehn to Cerf to Hatcher/Thurston to Harer to Wajnryb to show that this list of easily verified relations in fact gives a complete presentation of MC.F /:

4

1 The basics

0 Theorem 1.4 (Wajnryb, [20], [165]). Consider the generators for MC.Fg;1 / described 0 in Theorem 1.3, omitting c0 for g D 1. Then a complete set of relations for MC.Fg;1 / is provided by commutativity and braiding among these generators plus the following relations: for g D 1, no additional relations; for g D 2, a single non-separating chain relation; for g  3, a single non-separating chain relation and a single nonseparating lantern relation. Furthermore, the kernel of the natural homomorphism 0 MC.Fg;1 / ! MC.Fg0 / is generated by the following relations: for g D 1, 41 D 1, and for g  2, 2g b 2 g D b , where b is the curve illustrated in Figure 1.2.

Corollary 1.5. For g  3, we have H1 .MC.Fg0 // D 0, i.e., the abelianization of the mapping class group is trivial. Proof. According to Theorem 1.3, the mapping class group of F D Fg0 is generated by Dehn twists along non-separating curves. If c and d are non-separating curves, then F  c is homeomorphic to F  d , and so c is conjugate in MC.F / to d by the naturality relation. It follows that the abelianization of MC.F / is a cyclic group. Since there is a non-separating lantern relation in F for g  3 and since the exponent sum of a lantern relation is one, it follows that this cyclic group is trivial. s Further analogous tricks with the lantern relation likewise prove that MC.Fg;r / also abelianizes to zero for g  3. This elementary argument belies the difficulty of the following important and fundamental open problem:

Problem 1.6. Calculate the homology or cohomology of the mapping class groups. 1.2 Decorated Teichmüller and Riemann moduli spaces. We shall develop tools towards addressing Problem 1.6 by considering actions of mapping class groups on appropriate spaces as follows. Supposing now that r D 0 for simplicity and that 2g  2 C s > 0, first roughly define the “Teichmüller space” T .F / of F D Fgs in any one of the following equivalent ways: T .F / D fcomplex structures on F g=isotopy D fconformal structures on F g=isotopy D fhyperbolic structures on F g=isotopy: Slightly more explicitly in the first two formulations, we consider complex or conformal structures on the fixed smooth manifold F modulo push-forward of structure by diffeomorphisms isotopic to the identity, where the equivalence of conformal and complex structures in this dimension is easily verified. See [1], [65] for a more complete discussion. Definition 1.7. In fact, it is the third formulation we shall develop here, where by a hyperbolic structure on F we mean a complete finite-area Riemannian metric of

1 Cast of characters

5

constant Gauss curvature 1 modulo push-forward by diffeomorphisms isotopic to the identity. The equivalence of conformal and hyperbolic structures is provided by the celebrated Uniformization Theorem of Koebe, Klein, Poincaré. See [1], [65] for more details. Let us now be precise with a careful definition of T .F / as a topological space in the hyperbolic setting. Definition 1.8. Define the Möbius group PSL2 .R/ to be the quotient of the group of two-by-two matrices of determinant one over the reals R, where the matrix     a b a b AD is identified with  A D : c d c d (The group PSL2 .C/ defined analogously over C is sometimes also called the Möbius group.) Much more will be said about this group in the next several sections, and see [12], [48] for further details. Definition 1.9. Consider the collection of all homomorphisms  W 1 .F / ! PSL2 .R/ from the fundamental group 1 .F / to PSL2 .R/, and define the sub-collection Homdfp .1 .F /; PSL2 .R// of all homomorphisms  so that the identity in PSL2 .R/ is not an accumulation point of the image of  ( is discrete),  is injective ( is faithful), and finally if  2 1 .F / is freely homotopic to a puncture of F , then the absolute value of the trace of ./ is 2 ( maps peripheral elements to parabolics). Such a subgroup of the Möbius group is called a (punctured) Fuchsian group. Choosing a basis for 1 .F /, this space of homomorphisms inherits a topology from that on the entries of the representing matrices; this topology is independent of the choice of basis since given two such choices, it is easy to see that each topology is finer than the other. Finally, PSL2 .R/ acts on this space of homomorphisms by conjugation, i.e., A 2 PSL2 .R/ acting on the representation  7! . / yields the representation  7! A. /A1 . Definition 1.10. Define the quotient space T .F / D Homdfp .1 .F /; PSL2 .R//=PSL2 .R/; to be the Teichmüller space of F . This may seem like a mouthful, but we shall develop tools to understand this space quite explicitly from geometric and combinatorial points of view. In fact, we shall see (cf. Theorems 2.5 and 4.2 in Chapter 2) that T .F / is homeomorphic to an open ball of dimension 6g  6 C 2s for F D Fgs with s  1 (this result holding also for s D 0). The

6

1 The basics

principal reason for introducing T .F / is that MC.F / acts on this ball, as is obvious from the rough definitions (using the fact that each isotopy class of homeomorphisms of a surface contains a diffeomorphism) since in each case we take the quotient by pushforward of structure by isotopy just as in the definition of the mapping class group. In our careful definition, MC.F / acts on 1 .F / by outer automorphisms, so the action on T .F / is well defined since we mod out by conjugation. Furthermore, we shall see that MC.F / acts on T .F / discretely but with fixed points of finite isotropy (i.e., the stabilizer in MC.F / of any point of T .F / is a finite subgroup), and these properties allow us to derive facts about MC.F / from this action. Arguably the other central character of our considerations beyond the mapping class group, we come to: Definition 1.11. The quotient M.F / D T .F /=MC.F / is Riemann’s moduli space. Closely related to Problem 1.6, we have the likewise fundamental open problem: Problem 1.12. Calculate the homology or cohomology groups of Riemann’s moduli spaces or their various compactifications [37], [15]. 0 There is a “stable” version [57] of the cohomology of the moduli spaces Fg;1 , and with the celebrated Madsen–Weiss–Tillman proof of Mumford’s Conjecture [104], [162], this stable algebra has been computed to be the free algebra generated by the Miller–Morita–Mumford or “MMM classes”, which are also sometimes called “tautological classes” [113], [117], [121].

Definition 1.13. It is an easy fact that a conformal structure on a punctured disk has a removable singularity and extends across the puncture. There are thus s C 1 sC1 s mappings i W M.Fg;r / ! M.Fg;r / gotten by what Lippa Bers amusingly called “forgetting the puncture”, for i D 1; : : : ; s C 1 depending upon which of the punctures we wish to forget.1 We may think of i as a bundle whose generic fibers are copies s sC1 of Fg;r , and as such, there is a fiberwise cotangent bundle over M.Fg;r / with Euler 2 sC1 nC2 sC1 class i 2 H .M.Fg;r /I Z/. The Gysin homomorphism H .M.Fg;r /I Z/ ! n s H .M.Fg;r /I Z/ arises by integration along the fibers, and the integrals Z nC1 s 2 H 2n .M.Fg;r /I Z/ n D i s Fg;r

are the MMM classes associated to the i th puncture, for n  0. If this definition is a bit much for you, do not despair; we give it here only for completeness and shall describe and compute these classes in various contexts later. 1 In the conformal setting, this map is just a triviality and is easily seen to be continuous, but hyperbolically, it is most mysterious and not much is known, i.e., finding the hyperbolic metric of the new conformal structure and thus the location of the forgotten puncture is an intractably difficult problem.

1 Cast of characters

7

Just as Teichmüller space T .F / is introduced to study Riemann’s moduli space M.F /, we next go one step further and define: Definition 1.14. For any surface F D Fgs with s  1, take the trivial Rs>0 -bundle over T .F / called the decorated Teichmüller space and denoted by Tz .F /. The action of MC.F / on T .F / lifts to an action on Tz .F / by permuting the positive real parameters assigned to the punctures. Thus, our basic cast of characters is summarized in the following diagram: MC.F /

å

MC.F /

å

Tz .F / #

T .F / # M .F /:

There are close cousins of the Teichmüller space of a surface F given any fixed Lie group G as follows: Definition 1.15. The moduli space of flat G connections is the space MG .F / D Hom.; G/=G; where  is the fundamental group of F and G acts on representations by conjugation, this time, with no restriction on the representing homomorphisms. This is a big swindle by us, of course, because the term already has its differential geometric meaning, and a series of great works by Atiyah, Bott, Narasimhan, Seshadri and others finally delivers this tinker-toy description of the moduli space. In fact, Teichmüller space is a component of the moduli space of flat PSL2 .R/ connections [53], and there are likewise comparable so-called Hitchin components [64] of other moduli spaces as well. Notice the weird terminology that Riemann’s moduli space is a quotient of Teichmüller space, which is a component of the moduli space of flat PSL2 .R/ connections, A still more tinker-toyish description of the moduli space of flat G connections is: Definition 1.16. Suppose that  is one-dimensional CW-complex and G is a Lie group. A G-graph connection on  is the assignment ge 2 G to each oriented edge e of  so that geN D ge1 if eN is the reverse orientation to e. Two such assignments ge ; he 2 G are 1 regarded as equivalent if there is kv 2 G, for each vertex v of  so that ge D kv he kw for each oriented edge e of  with initial point v and terminal point w. If F deformation retracts to   F , then a G-graph connection on  determines an isomorphism class of flat principal G bundles over F , and indeed, the moduli space of flat G connections on F is isomorphic to the space of G-graph connections on , cf. [36]. Finally, in order to develop some intuition about Teichmüller space, we shall briefly mention “Fenchel–Nielsen coordinates” [42], [171].

8

1 The basics

Definition 1.17. A pair of pants is a surface whose interior is homeomorphic to the thrice-punctured sphere, i.e., a sphere with several holes with geodesic boundary and several punctures, where the total number of holes plus punctures is three. A pants s decomposition of a surface F D Fg;r is a collection of disjointly embedded simple closed curves so that each complementary region in the surface is a pair of pants. One can prove2 that there is a unique hyperbolic structure on a pair of pants given only the hyperbolic lengths of its geodesic boundary components. The idea of Fenchel– Nielsen coordinates is to decompose F into pairs of pants, or rather, to glue pairs of pants together to produce F . When gluing pairs of pants along geodesic boundary components, there is evidently the remaining degree of freedom of twisting one component relative to the other before gluing. Thus, instructions to build F by gluing together pairs of pants include not only the lengths of curves in a pants decomposition but also a further “twisting parameter” one for each pants curve, with which one must struggle a bit to make precise by making certain natural conventions. Theorem 1.18 (Fenchel–Nielsen Theorem). For any pants decomposition P of a s surface F D Fg;r of negative Euler characteristic, the Teichmüller space T .F / is parametrized by the collection of all pairs .`c ; tc / 2 R>0  R, one pair for each pants curve c 2 P . This thus gives a nice intuitive description of the Teichmüller space of F : it is the space of all ways of gluing together and twisting pairs of pants in a pants decomposition of F . Furthermore, the action of a Dehn twist along one of the pants curves is just a linear map on Fenchel–Nielsen coordinates, but a supplementary set of Dehn twists that generate MC.F / has an intractably complicated action on these coordinates. 1.3 Tinker-toys: fatgraphs and flips. We next introduce the basic tools of combinatorial topology which we shall require for decorated Teichmüller theory: decompositions of a surface into collections of triangles, the trivalent “fatgraphs” dual to them, and the “flips” that act on both polygonal decompositions and on fatgraphs. For definiteness and simplicity, we shall continue to restrict our attention for the moment to surfaces F D Fgs without boundary. Definition 1.19. An ideal arc is (the isotopy class of) an embedded arc ˛ connecting s punctures in Fg; which is not homotopic to a point rel punctures. An ideal triangulation  of Fgs is a collection of ideal arcs so that each region complementary to  is a triangle with its vertices among the punctures. An ideal cell decomposition, sometimes abbreviated i.c.d., is a subset 0   of an ideal triangulation so that each complementary region to 0 is a polygon. 2 In fact, one produces hyperbolic structures with geodesic boundary on pairs of pants by doubling hyperbolic hexagons, whose interior angles are all right angles, along every other frontier edge. One shows that such “all right hexagons” are uniquely determined by every other frontier edge length.

1 Cast of characters

9

Examples of ideal triangulations of punctured surfaces are given on the top of Figure 1.4. If an ideal triangulation of Fgs contains e arcs and has t complementary triangles, then 3t D 2e, and the Euler characteristic of Fgs is given by 2  2g  s D t e D  13 e. Thus, there are e D 6g 6C3s ideal arcs in an ideal triangulation of Fgs , and there are t D 4g 4C2s complementary triangles. For instance in the surface F11 , each ideal triangulation contains exactly three distinct ideal cell decompositions since we may remove any one but no two arcs and preserve the condition that complementary regions are polygons. Likewise, for F04 , we may remove any arc or pair of arcs but only certain triples of arcs from the illustrated ideal triangulation to produce an ideal cell decomposition.

Figure 1.4. Examples of ideal triangulations and dual fatgraphs.

Definition 1.20. A graph is a one-dimensional CW-complex comprised of vertices and open edges. An edge of the first barycentric subdivision is called a half edge. The number of half edges incident on a vertex is its valence Dual to an ideal cell decomposition  of a surface F D Fgs , there is a graph G D G./ embedded in F which has one k-valent vertex for each polygon with k sides complementary to ; two vertices are joined by an edge whenever the corresponding polygons share an edge in their frontiers. Examples of these dual fatgraphs are given on the bottom of Figure 1.4. An orientation of the surface determines an orientation on the boundary of any polygon complementary to , i.e., a cyclic ordering on the half

10

1 The basics

edges incident on each vertex of G. Definition 1.21. A fatgraph or ribbon graph is a graph together with a family of cyclic orderings on the half edges about each vertex. The fatgraph G D G./ and its corresponding ideal cell decomposition  D .G/ are said to be dual. One can conveniently describe a fatgraph by drawing a planar projection of a graph in three-space, where the counter-clockwise ordering in the plane of projection determines the cyclic ordering about each vertex. For example, Figure 1.5 illustrates two different fattenings on an underlying graph with two trivalent vertices giving different fatgraphs. The figure also indicates how one takes a regular neighborhood of the vertex set in the plane of projection and attaches bands preserving orientations to produce a corresponding surface with boundary from a fatgraph3 . Capping off each boundary component with a punctured disk in the natural way produces a punctured surface F .G/ associated with a fatgraph G, where the two surfaces F11 and F03 correspond to the two fatgraphs in Figure 1.5 as indicated.

Fatgraph for F11

Fatgraph for F03

Figure 1.5. Two fattenings of a single graph.

Definition 1.22. Notice that the fatgraph G arises as a spine of F D F .G/, namely, a strong deformation retract of F , and its isotopy class in F is well defined. Definition 1.23. Notice that an arc e in an ideal triangulation  separates distinct triangles if and only if its dual edge in G has distinct endpoints, i.e., the dual edge is not a loop. In this case, e is one diagonal of an ideal quadrilateral complementary to .F  [/ [ e, and we may replace e by the other diagonal f of this quadrilateral to produce another ideal triangulation e D  [ ff g  feg of F as in Figure 1.6. 3 It is a fun game, sometimes called “Kirby’s game”, to traverse the boundary components directly on the fatgraph diagram.

11

1 Cast of characters

We say that e arises from  by a flip or a Whitehead move along e. Dually, the fatgraphs G./ and G.e / are likewise said to be related by a Whitehead move or flip, i.e., collapsing and expanding the edge of G./ dual to e in the natural way as in Figure 1.6.

e

d

f

e flip

c a

b G./

G.e /

Figure 1.6. Flips on triangulations and fatgraphs.

Fact 1.24 (Whitehead’s Classical Fact). Finite sequences of flips act transitively on the collection of ideal triangulations of a fixed surface. See [42] for the classical combinatorial proof of this. We shall give a different proof in Section 1 of Chapter 5 as the first in a hierarchy of such results. Indeed, we shall describe all relations among sequences of flips (thereby giving a presentation of the mapping class group), relations among relations, and so on. Summarizing these constructions and observations, we have: Theorem 1.25. For any fixed surface Fgs , there is a natural one-to-one correspondence between its ideal cell decompositions and its isotopy classes of fatgraph spines. Furthermore, the dual of a flip is a flip, and removing an edge, which is not a loop, from an ideal cell decomposition corresponds to collapsing the dual edge, which has distinct endpoints, in its dual fatgraph. The deeper structure of Tz .F / that is useful in studying M.F / and MC.F / comes in both algebraic and combinatorial guises, and ideal triangulations of F , or equivalently, trivalent fatgraphs embedded as spines of F correspondingly play two roles. The deeper algebraic structure derives from global affine coordinates on Tz .F /, one such positive real coordinate, called a “lambda length” (or “Penner coordinate” by some authors), for each arc in an ideal triangulation  of F , or equivalently, for each edge of its dual trivalent fatgraph G./. These basic lambda length coordinates give

12

1 The basics

another graphic description of Teichmüller space provided by gluing triangles along their edges. In effect, whereas the Fenchel–Nielsen coordinates are length/twist coordinates, lambda lengths are just lengths (specifically, distances between horocycles), and the action of MC.F / on T .F / is simple and beautiful with deep consequences. A flip has the effect of a so-called “Ptolemy transformation” f D .ac C bd /=e on these coordinates in the notation of Figure 1.6, where we identify an arc with its lambda length for convenience, and this leads for example to an explicit faithful algebraic representation of MC.F / as well as the direct connection with cluster varieties. Thus, ideal triangulations on the one hand play the role of a basis for global coordinates on the decorated space Tz .F /. The more involved combinatorial guise involves an MC.F /-invariant decomposition of Tz .F / as follows. Definition 1.26. An ideal simplicial decomposition is a CW-decomposition into cells each of which is identified with a closed simplex minus certain of its faces, where two such cells meet along a common face if at all. In fact, the collection of all ideal cell decompositions of F forms a partially ordered set under inclusion, and its geometric realization supporting the natural MC.F / action is equivariantly isomorphic to an ideal simplicial decomposition of Tz .F /=R>0 , where R>0 acts diagonally on decorations. Ideal triangulations thus also play the role of the names for the top-dimensional simplices in this decomposition of Tz .F /=R>0 , and ideal cell decompositions likewise play the role of the names for the higher-codimension cells. Moreover, flips correspond to crossing codimension-one faces of this decomposition. These ingredients of natural global coordinates and an invariant ideal simplicial decomposition descend to useful tools on moduli space itself as we shall see. 1.4 The hyperelliptic involution. Among this cast of characters for surfaces Fgs with s  1, there is one non-conformist, the surface F D F11 , where we shall modify the definitions of mapping class group and Teichmüller space given above. The issue is that the mapping class group contains a central element W F ! F of order 2, and it is convenient to re-define the mapping class group and Teichmüller space by taking the quotients by this involution. To describe the mapping , consider the surface F D F11 embedded in R3 as illustrated in Figure 1.7, where we have indicated a line ` passing through the puncture and meeting F in three further points. Definition 1.27. The hyperelliptic involution is described by rotation-by- about this line ` as indicated in the figure, which also illustrates two curves , , respectively the “meridian” and “longitude” of F , where and  meet at a fixed point of as also indicated in the figure. To prove the claims about , one first shows that Dehn twists on , act transitively on the set of all isotopy classes of essential arcs ˛ embedded in F with endpoints at

13

1 Cast of characters







`

Figure 1.7. The hyperelliptic involution.

the puncture. One proves this by induction on the number of times ˛ meets , finally 0 , which contains noting that F  ˛ is homeomorphic to the interior of the annulus F0;2 a unique essential embedded curve. It follows that Dehn twists on , together with generate the group of homeomorphisms of F modulo isotopy by using Alexander’s trick as before. Indeed, this is the rough paradigm for proving Theorem 1.3, which however involves many more cases and considerations. Since the isotopy classes of , are each invariant under , it follows from naturality of Dehn twists that is indeed central. Take the basepoint for the fundamental group of F as the point of intersection of  and and take the based loops given by specified orientations on , as generators for the fundamental group; the action of on Homdfp inverts each of these generators since reverses orientation on each of , . We therefore go back and re-define the mapping class group MC.F / and Teichmüller space T .F / for F D F11 taking the quotient by the hyperelliptic involution and shall use these modified definitions without further comment in the sequel. Among all orientable surfaces, there are precisely two others whose mapping class 0 groups admit analogous central involutions, namely, F1;0 , which supports the elliptic 0 involution, and F2;0 , which supports its own hyperelliptic involution. The definitions of these involutions are entirely analogous, given by rotation-by- about appropriate lines 0 0 respectively meeting F1;0 in four points and F2;0 in six points. The proofs that these involutions have the asserted properties follow the pattern above for F11 , and one again typically re-defines the mapping class group and Teichmüller space for these surfaces taking the quotient by these involutions. Indeed, we had already tacitly taken this quotient in stating several of the theorems in this section. Since we shall be concerned s in this volume with surfaces Fg;r with r C s  1, it is only F11 that is affected here by these considerations.

14

1 The basics

2 Three models for the hyperbolic plane The discovery of two-dimensional hyperbolic geometry in the nineteenth century independently by Klein, Poincaré and Lobachevsky solved a millennia-old problem on the consistency and independence of Euclid’s other axioms from his Fifth Axiom: given a line and a point not on that line, there is a unique line through the given point which is parallel to the given line, a well-known property of Euclidean geometry which fails in hyperbolic geometry. This section introduces three different models for the hyperbolic plane all of which are basic to our treatment of Teichmüller space. References for the material in this section are [12], [48], [128]. 2.1 Upper half-plane and Poincaré disk Definition 2.1. The first model is the upper half-plane U D f.x; y/ 2 R2 W y > 0g endowed with the Riemannian metric ds 2 D .dx 2 C dy 2 /=y 2 , which is in particular evidently conformal to the Euclidean upper half-plane, is complete, and which one computes to have constant Gauss curvature 1. We shall typically identify R2 with the complex plane C and hence U  Cpwith the complex numbers z D x C iy of positive imaginary part y > 0, where i D 1. One can check without difficulty that geodesics in U are given by circles perpendicular to the real axis together with the extreme case of rays with real endpoints parallel to the imaginary axis. Thus, one sees immediately that Euclid’s Fifth Axiom fails since there are in fact infinitely many lines parallel to a given line, i.e., disjoint from it, passing through a given point not on the given line, and it is an exhaustive exercise to check that Euclid’s remaining axioms do indeed hold in the upper half-plane. Let us emphasize that points of the extended real axis R [ f1g are not points in U, but rather are ideal points comprising a circle at infinity compactifying U to a closed ball x we shall denote U. The Möbius group PSL2 .R/ D SL2 .R/=.˙I / was already defined in the previous section, and an element of this group, which is called a Möbius transformation, acts on U by fractional linear transformation   az C b a b : W z 7! c d cz C d This action extends by the same formula to an action on the circle at infinity, where .a1 C b/=.c1 C d / is taken as a=c for instance. One directly checks that U is preserved by each Möbius transformation, that this is in fact an action by orientationpreserving isometries of U, and that PSL2 .R/ is indeed the full group of orientationpreserving isometries of U, where the last assertion follows from the easily verified fact that the Möbius group acts transitively on the tangent space to U. One reason that

2 Three models for the hyperbolic plane

15

the upper half-plane is useful for calculations in hyperbolic geometry is this simple expression for the action of isometries as fractional linear transformations. Lemma 2.2. The Möbius group PSL2 .R/ acts simply transitively on triples of ideal points in a fixed cyclic order in the circle at infinity. Proof. First suppose that u < v < w < 1 occur in this order and consider the transformation Au;v;w W z 7!

zu vw .v  w/z C .w  v/u D ; zw vu .v  u/z C .u  v/w

which evidently maps u 7! 0;

v 7! 1;

w 7! 1:

Since .v w/.uv/w .v u/.w v/u D .v w/.uv/.w u/ > 0 by hypothesis, it follows that Au;v;w 2 PSL2 .R/, i.e., the determinant is positive. The further cases where one of u, v, w might be 1 are handled similarly. It follows that the fractional 0 0 0 linear transformation A1 u0 ;v 0 ;w 0 Au;v;w maps u 7! u , v 7! v and w 7! w , so PSL2 .R/ indeed acts transitively. Furthermore, if the Möbius transformation B pointwise fixes some triple u, v, w, then   a b D Au;v;w BA1 u;v;w c d fixes each of 0, 1, 1, i.e., we have b D 0; d

aCb D 1; cCd

and

c D 0;

from which it follows that B D ˙I . Since the fixed points z D .azCb/=.czCd / of a Möbius transformation A D evidently satisfy the equation p 2cz D a  d ˙ .a C d /2  4 for c ¤ 0

a b  c d

(and for c D 0, we may conjugate by an appropriate Möbius transformation B noting that the fixed points of BAB 1 are the images of the fixed points of  Aunder B), direct calculation shows that there is the following trichotomy on A D ac db 2 PSL2 .R/: • if ja C d j < 2, then A is said to be elliptic, and there is a unique fixed point in U with A described as rotation about this fixed point; • if ja C d j D 2, then A is said to be parabolic, and there is a unique fixed point at infinity; • if ja C d j > 2, then A is said to be hyperbolic, and there is a pair of fixed points at infinity, which are the ideal points of a unique geodesic in U with A acting as translation along this geodesic. Furthermore, the translation length `A along this invariant geodesic is given by 2 cosh `2A D ja C d j.

16

1 The basics

The elliptic and hyperbolic cases are analogues of the familiar rotations and translations in Euclidean geometry, but the parabolic case is a novel aspect of hyperbolic geometry, upon which we shall comment further below. x at least three of which Definition 2.3. The cross ratio of four points w; x; y; z 2 U are distinct is given by Œw; x; y; z D

.w  x/.y  z/ x 2 U; .w  z/.y  x/

where if one of w, x, y, z is 1, then its common factors on the right-hand side cancel, and if x D y or w D z, then the cross ratio is taken as 1. Cross ratio is invariant under A 2 PSL2 .R/, i.e., Œw; x; y; z D ŒAw; Ax; Ay; Az ; x i.e., the and is a complete invariant of ordered four-tuples of distinct points in U, cross ratios of two four-tuples agree if and only if there is some Möbius transformation mapping one four-tuple to the other. In fact, w 7! Œw; x; y; z is evidently the value at w under the unique fractional linear transformation taking x 7! 0, y 7! 1, z 7! 1. x is real if and only if x, y, z, w lie on a common circle Furthermore, Œw; x; y; z 2 U x All these things can be routinely verified directly. or line in U. Given distinct ;  2 U, the hyperbolic distance between and  is given by logŒ ; 0 ; ; 0 , where 0 , 0 are the ideal endpoints of the geodesic connecting and  enumerated so that lies between 0 and . Finally, exchanging any two of w, x, y, z leaves  D Œw; x; y; z invariant, and the six possible remaining permutations of coordinates have cross ratios: ;

1  ;

 ; 1

1 ; 

1 ; 1

1 ; 

forming the group of anharmonic ratios. Definition 2.4. Our second model for the hyperbolic plane is the Poincaré disk D D fz 2 C W jzj D x 2 C y 2 < 1g endowed with the metric ds 2 D 4.dx 2 C dy 2 /=.1  jzj2 /2 , which is again a model conformal to the Euclidean metric this time on the open unit disk in C, and again is complete with constant Gauss curvature 1. Whereas the upper half-plane is useful for calculations, the Poincaré disk is useful for drawing pictures in part because the point 1 at infinity in U is seen to be “just 1 in C, the circle at infinity, which compactifies D another point” in the unit circle S1 to the closed unit disk in C.

2 Three models for the hyperbolic plane

17

There are explicit mutually inverse isometries, called the Cayley transforms, between U and D as follows: zi 1Cw U ! D; z 7! ; D ! U; w 7! i ; zCi 1w x to the points 1, i , C1 2 S 1 . which map the respective points 0, 1, 1 at infinity in U 1 The action of the Möbius group on D is given by conjugating the action of fractional linear transformations on U by the Cayley transform. 2.2 Minkowski space Definition 2.5. Our third model for the hyperbolic plane is useful both for figures and for calculations, as we shall see. It is described as a subset of Minkowski three-space, which is defined to be R3 endowed with the indefinite pairing h  ;  i W R3  R3 ! R;

h.x; y; z/; .x 0 ; y 0 ; z 0 /i 7! xx 0 C yy 0  zz 0 ;

where x, y, z denote the usual coordinates on R3 . Definition 2.6. There are several characteristic subspaces of Minkowski three-space, which are illustrated in Figure 2.1, as follows: H

LC H

Figure 2.1. Minkowski three-space.

the upper sheet of the hyperboloid H D fu D .x; y; z/ 2 R3 W hu; ui D 1 and z > 0gI

18

1 The basics

the open positive light-cone LC D fu D .x; y; z/ 2 R3 W hu; ui D 0 and z > 0gI and the hyperboloid of one sheet H D fu D .x; y; z/ 2 R3 W hu; ui D C1g: Notice that if u; v 2 LC , then hu; vi  0 with equality if and only if u is a scalar multiple of v, and if u 2 LC ; v 2 H, then hu; vi < 0, where both inequalities follow from the Cauchy–Schwarz inequality. We shall say that an affine plane … in R3 is elliptic, parabolic, or hyperbolic, respectively, if it is so in the sense of the Greeks, i.e., the corresponding conic section … \ .LC [ LC / has the corresponding attribute. One checks without difficulty that a plane … D fu 2 R3 W hu; vi D g with Minkowski normal v, for some 2 R, is elliptic, parabolic, or hyperbolic, respectively, if and only if hv; vi is negative, vanishes, or is positive. One furthermore easily proves: Lemma 2.7. The Minkowski pairing h  ;  i restricts to a bilinear pairing on … that is definite, degenerate, or indefinite, respectively, precisely when … is elliptic, parabolic, or hyperbolic. In particular, the upper sheet H of the hyperboloid has all tangent planes elliptic and so inherits an honest Riemannian metric from the indefinite pairing on Minkowski space. The upper sheet H together with this induced metric forms our third model for the hyperbolic plane. 1 Identify the closed unit disk D [ S1 with the horizontal disk at height zero in Minkowski space. An isometry between H and D is given by central projection N from the point .0; 0; 1/, i.e., the line segment with endpoints .0; 0; 1/ and v 2 H meets the unit disk D at the point v; N that is, x W H ! D;

.x; y; z/ 7! .x; y; z/ D

1 .x; y/; 1Cz

establishes an isometry between these two models, where the inverse is given by D ! H;

.x; y/ 7!

1 .2x; 2y; 1 C x 2 C y 2 /: 1  x2  y2

Furthermore, this projection extends to a natural mapping 1 x W LC ! S1 ;

1 .x; y; z/ 7! .x; y; z/ D p .x; y; 0/; x2 C y2

2 Three models for the hyperbolic plane

19

1 where the fiber over a point of S1 is a corresponding ray in LC . Finally composing this central projection with the Cayley transform, explicit formulas that will be useful in subsequent calculations give isometries:

H ! U; and U ! H;

x C iy 7!

.x; y; z/ 7!

i y ; zx

1 2 .x C y 2  1; 2x; x 2 C y 2 C 1/: 2y

The action of PSL2 .R/ on H can be computed by conjugating its action on U by these isometries, and one finds that PSL2 .R/ is exactly the group of real three-bythree matrices of determinant one which preserve the Minkowski pairing as well as preserving the upper sheet H, i.e., PSL2 .R/ is isomorphic to the component SOC .2; 1/ of the identity in SO.2; 1/, which is of index four, preserving not only the orientation but also the upper sheet of the two-sheeted hyperboloid. Notice that elements of SOC .2; 1/ have determinant one and hence are volume-preserving. More explicitly, we may identify R3 with the collection B2 .R/ of all real symmetric bilinear forms on two indeterminates via   zx y 3 ; R ! B2 .R/; .x; y; z/ 7! y zCx and then the action of A 2 PSL2 .R/ is simply given by A W B2 .R/ ! B2 .R/;

B 7! At BA;

where At denotes the transpose of A. Thus, the natural action of the Möbius group by change of basis for bilinear forms coincides with its action as hyperbolic isometries. (This theme will be further developed in the next section.) Since an isometry of Minkowski space preserving an affine plane must preserve its normal vectors, we conclude: Lemma 2.8. If a Möbius transformation A leaves invariant an affine plane which is elliptic, parabolic, or hyperbolic, respectively, then A itself must also be elliptic, parabolic, or hyperbolic. Geodesics in H are intersections with planes through the origin whose Minkowski normal lies in H as one can check directly, and several standard formulas relating Minkowski inner products and hyperbolic lengths and angles can then also be directly derived: Lemma 2.9. If u; v 2 H, then hu; vi2 D cosh2 ı; where ı is the hyperbolic distance between u and v.

20

1 The basics

If u 2 H and v 2 H , then hu; vi2 D sinh2 ı; where ı denotes the hyperbolic distance from u to the geodesic determined by v. If u; v 2 H , then ´ cosh2 ı if the corresponding geodesics are disjoint; 2 hu; vi D cos2  if the corresponding geodesics intersect; where in the first case, ı is the infimum of hyperbolic distances between points on the two geodesics, and in the second case,  is their angle of intersection. We finally come to a definition that is basic for our considerations and gives a geometric interpretation to the points of LC . A “horocycle” in U is either a Euclidean circle tangent to the real axis or a horizontal Euclidean line parallel to the real axis. Applying the Cayley transform, a horocycle in D is thus a Euclidean circle tangent 1 to S1 . This definition leaves something to be desired since we relied on the underlying Euclidean metric of the hyperbolic Riemannian manifold U, so here is a better and more invariant definition: Definition 2.10. Choose a point p in the hyperbolic plane and a tangent direction v at p, and consider a family of hyperbolic circles whose radius and center diverge in such a controlled manner as to pass through p with tangent direction v; such a sequence of hyperbolic circles has a well-defined limit, defined to be a horocycle. Thus, a horocycle 1 , and the point of tangency is called the in D is indeed a Euclidean circle tangent to S1 center of the horocycle, with a similar remark and definition for U. Of course, an analogous family of Euclidean circles in the Euclidean plane limits to a Euclidean line, so the existence of horocycles is a truly new phenomenon in hyperbolic geometry. One can check that yet another invariant definition is that a horocycle is a smooth curve in the hyperbolic plane of constant geodesic curvature one. A direct calculation using the formulas given earlier allows the description of horocycles in Minkowski space, and one finds: Lemma 2.11. The assignment LC ! fhorocycles in Hg;

p u 7! h.u/ D fv 2 H W hu; vi D 1= 2g;

establishes an isomorphism between points of LC and the collection of all horocycles 1 N in H. Furthermore, the center of the corresponding horocycle h.u/ in D is uN 2 S1 , 1 N p , where u D .x; y; z/. and the Euclidean radius of h.u/ in D is 1Cz

2

21

2 Three models for the hyperbolic plane

p The funny choice of constant 1= 2 in Lemma 2.11 will be explained later; any negative constant would do just as well here. Indeed, given u 2 LC , the plane fv 2 R3 W hu; vi D 0g is the tangent plane to LC at u, and any affine translate fv 2 R3 W hu; vi D t g, for t < 0, intersects H in a horocycle. Let us finally re-consider the action of A 2 SOC .2; 1/ PSL2 .R/ on Minkowski space now armed with this notion of horocycle. As a unimodular linear map acting on R3 , A has at most three non-zero eigenvalues, and there are various cases: • A is hyperbolic if it has an eigenvalue  with jj ¤ 1, so  is real and positive with corresponding simple eigenvector (ray) contained in LC ; there is one other eigenvector contained in LC with eigenvalue 1 , and these two eigenvectors correspond to the ideal points at infinity of the invariant geodesic; there is a third eigenvector on H with eigenvalue 1 which corresponds to the invariant geodesic. • A is parabolic if there is a unique eigenvector contained in LC with eigenvalue 1 and no eigenvector on H; by Lemma 2.11, there is a corresponding foliation of H by horocycles which is leafwise invariant under A. • A is elliptic if all its eigenvalues lie on the unit circle with a unique eigenvector meeting H corresponding to the unique fixed point of A in H. 2.3 Harmonic functions in Minkowski space. There is another basic formula from [51] to give at this point, which will not be required in the sequel and is included just because it is perhaps interesting (towards developing a “Poincaré series” type expression for harmonic functions on a punctured surface): Lemma 2.12. For any q in Minkowski three-space, the function fq .p/ D

hq  p; qi ; 1Cz

where p D .x; y; z/ 2 H;

satisfies the differential equation of the conformal factor fq2  log fq D hq; qi.1 C hq; qi/; where  denotes the Laplacian. Proof. Adopting the usual polar coordinates rei on D, the mapping D ! H takes the form 1 .2r cos ; 2r sin ; 1 C r 2 / rei 7! 1  r2 in standard Euclidean coordinates on Minkowski space, so in the basis 1

u D 2 2 .1; 0; 1/;

1

v D 2 2 .1; 0; 1/;

1

w D 2 2 .0; 1; 1/;

the image p 2 H is expressed as 1 C r 2 C 2r.sin   cos / 1 C r 2 C 2r.sin  C cos  / 2r sin  uC vp w: p p 2.1  r 2 / 2.1  r 2 / 2.1  r 2 /

22

1 The basics

Setting q D ˛u C ˇv C  w, we find that p p fq .p/ D r 2 Œ˛ C ˇ C 2 C 2 2.˛ˇ C ˛ C ˇ / =2 2 p C rŒ4 sin  C 2.˛  ˇ/ cos  =2 2 p p C Œ˛ C ˇ C 2  2 2.˛ˇ C ˛ C ˇ / =2 2 D Ar 2 C Br C C; where A and C are independent of , so @2 fq D 2A; @r 2

@ 2 fq D Br; @ 2

@fq D 2Ar C B; @r

and

p @fq D rŒ2 cos  C .ˇ  ˛/ sin  = 2: @ We finally calculate that fq2  log fq D ŒAr 2 C Br C C Œ2A C  Œ2Ar C B 2 

1 Œ2 2r

2ArCB r



B r

cos  C .ˇ  ˛/ sin 

D 4AŒAr C Br C C  Œ4A2 r 2 C B 2 C 4ABr 2

 Œ2 2 cos2  C D 4AC 

1 Œ.ˇ 2

.ˇ ˛/2 sin2 2 2 2

 C 2.ˇ  ˛/ cos  sin 

 ˛/ C 4

D 4.˛ˇ C ˛ C ˇ /2 C 2.˛ˇ C ˛ C ˇ / D hq; qi.1 C hq; qi/ as was claimed. Corollary 2.13. If q 2 .H [ H/ [ .LC [ LC /, then log fq .p/ is a harmonic function of p 2 H. Thus if q1 , q2 both lie in this subspace, then log

hq1  p; q1 i hq2  p; q2 i

is harmonic in p, and in particular, 1 hp; qi  1 log hp; qi C 1 2 is Green’s function on H with pole q 2 H.

23

3 Farey tesselation and Gauss product

3 Farey tesselation and Gauss product Before pressing forward with geometry and Teichmüller theory, we cannot resist digressing here to briefly discuss topics from algebraic number theory. The first part of this section is explicated in [48], and Appendix A [136] treats the latter part of this section in much greater detail with complete proofs and with a discussion of the background number theory from first principles. See [28], [175] for a detailed treatment of quadratic forms. 3.1 Farey tesselation and modular curve. Let us begin in the upper half-plane model U and consider the horocycle hn of Euclidean diameter 1 centered at n 2 Z  R, for each n 2 Z. Thus, hn is tangent to hn˙1 and is disjoint from the remaining horocycles. We also add the horocycle h1 centered at infinity given by the horizontal line at height one, which is tangent to each hn . Two consecutive horocycles hn , hnC1 determine a triangular region bounded by the interval Œn; n C 1  R together with the horocyclic segments connecting the centers of the horocycles to the point of tangency of hn and hnC1 . There is a unique horocycle contained in such a triangular region which is tangent to hn and hnC1 as well as tangent to the real axis, and we let hnC 1 denote this horocycle, which is evidently tangent to the 2

real axis at the half-integer point n C 12 and of Euclidean diameter 14 . We may continue recursively in this manner, adding new horocycles tangent to the real axis and tangent to pairs of consecutive tangent horocycles in order to produce a family of horocycles H in U. See Figure 3.1. h1

h1

h 1

h1 2

2

1

h1

h0

 12

0

1 2

1

Figure 3.1. Horocyclic packing of U.

Lemma 3.1. There is a unique horocycle in H centered at each extended rational point x D Q [ f1g  C, x and the horocycle centered at p 2 Q has Euclidean diameter Q q p 1 , where is written in reduced form except with n 2 Z written as n1 and 1 D ˙ 10 . 2 q q

24

1 The basics

x are tangent to Furthermore, the horocycles in H centered at distinct points pq ; rs 2 Q one another if and only if ps  qr D ˙1, and in this case, the horocycle in H tangent x to these two horocycles is centered at pCr 2 Q. qCs

It is not hard to prove this lemma inductively starting with the second sentence. In fact, this result was discovered by the mineralogist J. Farey and solved the longstanding problem of giving a one-to-one enumeration of the rational numbers. After Farey published his empirical findings, Cauchy essentially immediately supplied the inductive proofs. Definition 3.2. Now define the Farey tesselation to be the collection of hyperbolic geodesics in U that connect centers of tangent horocyles in H ; see Figure 3.2. Thus,

 11

 23  12  13

 01 D

0 1

1 1 2 1 4 3 5 2

3 2 3 5 3 4

1 1

Figure 3.2. Farey tesselation of U.

the Farey tesselation of U is a countable collection of geodesics that decompose U into regions called ideal triangles, i.e., regions bounded by three disjoint geodesics pairwise sharing ideal points at infinity. Definition 3.3. Define the Farey tesselation  of D to be the image of the Farey tesselation of U under the Cayley transform as illustrated in Figure 3.3, where  is regarded as a set of geodesics. The Möbius group PSL2 .R/ contains the discrete group PSL2 .Z/ consisting of all two-by-two integral matrices of determinant one again modulo the equivalence relation identifying the matrix A with its negative A. This subgroup (or sometimes its deprojectivization or two-fold cover SL2 .Z/) is called the (classical) modular group and plays a basic role in number theory, as we shall partly explain in this section. It is also among the simplest of mapping class groups, namely, the mapping class group of F11 (where we have taken the quotient by the hyperelliptic involution).

25

3 Farey tesselation and Gauss product

1 3

1 2

2 3

1 1

3 2

2 1

3 1

1 2i 1C2i 1 3i 1C3i 1 4i 1C4i

4 1

1 4

Ci

2 3i 2C3i

3 2i 3C2i 2 i 2Ci

UT T C2

TU

U C2

T

U 0 1

0 1

1 0

1 0

1

1 U

1 4

1 3

1 2 2 3

1 1

2 3 1 2

3 1

4 1

1 4i 1C4i 1 3i 1C3i

1

U

1 2i 1C2i 2 3i 2C3i

labeling of vertices

1

C1

T

1

U

1

T

2

U T

3 i 3Ci 4 i 4Ci

1

i

T

2 1

2 i 3 2i 2Ci 3C2i

4 i 4Ci 3 i 3Ci

labeling of edges

Figure 3.3. Farey tesselation of D.

Lemma 3.4. The modular group leaves invariant the Farey tesselation  , mapping geodesics in  to geodesics in  and complementary ideal triangles to complementary ideal triangles, and any orientation-preserving homeomorphism of the circle leaving invariant  in this manner lies in the modular group. Furthermore, the modular group acts simply transitively on the oriented edges of  . A generating set is given by any pair of       0 1 1 1 1 0 SD ; T D ; U D ; 1 0 0 1 1 1 where T 1 D SUS and U 1 D S T S ; a presentation in the generators S, T is given by S 2 D 1 D .S T /3 , so the modular group is abstractly the free product Z=2Z Z=3Z. A fundamental domain for the action of the modular group on U is given by fx C iy 2 U W x 2 C y 2 > 1 and jxj < 12 g. In particular and in other words, one can construct the Farey tesselation  by taking the orbit of the ideal triangle in D with vertices 01 , 10 , 11 under the group of hyperbolic isometries generated by reflections in its sides. The orientation-preserving subgroup of this group of reflections is the modular group PSL2 .Z/. We might analogously define a tesselation by starting with any ideal triangle in D, and there is then an element of the Möbius group PSL2 .R/ mapping  to this alternate tesselation according to Lemma 2.2. We choose the special triangle with vertices 01 , 10 , 11 , or any image of it under PSL2 .Z/, in order to guarantee a kind of “rational structure” of  , which is at once at the heart of elementary number theory and is also manifest in our lambda length coordinates on T .F11 /, cf. Example 4.5 of Chapter 4. For example, an especially beautiful combinatorial fact is that the continued fraction

26

1 The basics

expansion of a rational number pq can be read off from the sequence of right or left turns in the Farey tesselation  connecting the point i to pq . Let us just illustrate with 5 an example, the rational 13 . Starting from i one makes the following sequence of 5 : RRLRL. Reading off the turns right (R) or left (L) in triangles of  to arrive at 13 number of consecutive turns 2(R), 1(L), 1(R), 1(L), we calculate the continued fraction expansion: 1 5 D : 13 1 2C 1 1C 1 1C 1C1 The proof of this general fact (which follows from Lemmas 3.1 and 3.4 and the definition of continued fractions) is left to the reader. Insofar as the modular group acts simply transitively on the oriented edges of  according to Lemma 3.4, we may identify these two sets. If A 2 PSL2 .Z/, then we let A act on the right on the points at infinity (following Gauss’ convention in fact), and hence on oriented geodesics, and shall thus identify an oriented edge of  with the image under the right action of A of the distinguished oriented edge from 0 to 1. This gives a labeling of the edges  by elements of PSL2 .Z/ as illustrated on the right in Figure 3.3. Furthermore, reversing the orientation on an edge labeled A corresponds to re-labeling SA. Another number-theoretic aspect of the current discussion involves so-called elliptic curves, namely, discrete subgroups ƒ of C of rank 2, also called lattices. The quotient of C by ƒ is a flat torus together with a distinguished point corresponding to 0 2 C, and up to conformal equivalence of this torus, we may take ƒ to be generated by the unit 1 2 C and a complex number  of positive imaginary part, i.e., the analogue of Teichmüller space (modulo the elliptic involution) for elliptic curves is precisely the upper half-plane U. The mapping class group (modulo the elliptic involution) of this torus-with-distinguished-point is precisely the modular group PSL2 .Z/ acting by fractional linear transformation on U. Definition 3.5. The analogue of Riemann’s moduli space for lattices is the modular curve U=PSL2 .Z/. A fundamental domain for the action of PSL2 .Z/ on U is illustrated on the left of Figure 3.4 and the modular curve itself on the right. It is not quite a manifold, rather, it is an orbifold, namely, neighborhoods of its point are modeled on quotients of Euclidean space by a finite group action, in this case, the finite groups being the trivial group, Z=2Z, and Z=3Z, where the orders of the non-trivial groups are indicated near the corresponding points in the figure. In fact, we may remove the distinguished point of the elliptic curve to produce a once-punctured torus, whose Teichmüller space is U, whose mapping class group is

27

3 Farey tesselation and Gauss product

1

1 2

0

1 2

1

2 3 Figure 3.4. The modular curve.

PSL2 .Z/, and whose moduli space is the modular curve as we shall see in Section 4 of Chapter 4. The modular curve is the unique moduli space of a two-dimensional orbifold which is itself a two-dimensional orbifold. We shall depend vitally on the Farey tesselation  and its invariance under the modular group PSL2 .Z/ in parts of the sequel. 3.2 Gauss product and hypercycles. Our principal topic for this section is a commutative product of suitable binary quadratic forms, to which we finally turn our attention, and since this material is developed from first principles in Appendix A, our discussion here will just serve to introduce the main ideas (as well as include a basic remark we neglected to mention explicitly in the published paper). Recall the space B2 .R/ of symmetric real bilinear pairings discussed in the previous section, where corresponding to the point .x; y; z/ in Minkowski three-space, we have considered the pairing   zx y 2 B2 .R/ y zCx with its corresponding quadratic form .z x/ 2 C2y C.z Cx/2 , where ,  denote indeterminates. We shall now restrict our attention to the subspace B2 .Z/ corresponding to integral binary forms and shall let the integral quadratic form a 2 C b  C c2 , where a; b; c 2 Z, be denoted simply by Œa; b; c . We say that Œa; b; c is primitive if a, b, c have no common divisors, i.e., gcdfa; b; cg D 1, and define the discriminant of Œa; b; c to be D D b 2  4ac. In particular, b is even if and only if D is equivalent to zero modulo four, and b is odd if and only if D is equivalent to one modulo four. The quadratic

28

1 The basics

form Œa; b; c is said to be definite form or imaginary form if D < 0, and it is said to be indefinite or real if D > 0; otherwise if D D 0, it is said to be degenerate. The modular group acts on integral quadratic forms by change of basis, as in the previous section, and leaves invariant both primitivity of the form as well as the discriminant as one easily confirms. We let ŒŒa; b; c denote the corresponding orbit and consider the set G .D/ D fŒŒa; b; c W Œa; b; c is primitive of discriminant Dg: Gauss showed that for each discriminant D, the set G .D/ has the structure of a finite abelian group, and we shall formulate a version of this group law presently. It is an important open problem to explicitly calculate the orders of these groups, the so-called “class numbers”, cf. [28]. Definition 3.6. We say that two classes of integral quadratic forms Œf1 , Œf2 are unitable if their respective discriminants D1 , D2 satisfy Dj D tj2 d for some tj 2 Z, i.e., if D1 D2 is the square of some integer, and in this case, we may define tj0 D tj =gcdft1 ; t2 g. Two unitable forms f1 ; f2 are said to be concordant if there are respective representatives of their PSL2 .Z/-orbits of the form Œa1 ; t10 b; t10 a2 c 2

and Œa2 ; t20 b; t20 a1 c : 2

Given concordant forms f1 , f2 , we define the (Gauss) product of their respective classes Œf1 , Œf2 to be Œf1 Œf2 D ŒŒa1 a2 ; b; c : Theorem 3.7. Given two unitable classes, there exist concordant representatives, and the class ` of the product is well defined. Fixing a square-free discriminant d and setting .d / D t1 G .t 2 d /, the product gives .d / the structure of an abelian semigroup, and the restriction of the product gives G .t 2 d / < .d / the structure of a finite abelian group, for each t  1. As we shall see, the elementary yet crucial geometric point about concordant forms is: Lemma 3.8. Suppose that Œaj ; bj ; cj , for j D 1; 2, are primitive forms of respective discriminants D1 , D2 , where b1 b2  0. Then the two forms are concordant if and only if the following two conditions hold: • D1 b22 D D2 b12 , and thus Dj D tj2 d for some tj 2 Z, and we define bj D btj0 for i D 1; 2; • 4a1 a2 divides b 2  d.gcdft1 ; t2 g/2 . Turning attention now to the case of definite quadratic forms (which we should remark is the “easy” case of quadratic forms, where our understanding is much more complete), the key geometric point is that integral points .x; y; z/ of Minkowski threespace which lie inside of LC (i.e., have positive z-coordinate and negative Minkowski

3 Farey tesselation and Gauss product

29

length) at once correspond to definite integral quadratic forms Œz  x; 2y; z C x as above and to suitable points of H, namely, the ray from the origin in Minkowski three-space through the integral point .x; y; z/ inside LC meets H in the point 1=.z 2  x 2  y 2 /.x; y; z/. The PSL2 .Z/-orbit ŒŒz  x; 2y; z C x thus corresponds to a point of the modular curve, so the Gauss product defines an abelian group structure on appropriate subsets of the modular curve. We ask (and partly answer) whether this product can be understood geometrically in these terms. The critical property of concordant forms is that the first condition in Lemma 3.8 is projectively invariant, i.e., it is defined on projective classes of definite quadratic forms and hence determines some condition on points of H. By definition, a definite quadratic form Œa; b; c has exactly two roots of az 2 Cbz C c D 0, and exactly one of these roots r r D r b p !Œa;b;c D  D Ci Ci 2 fz D x C iy W x; y 2 2 Q and y > 0g; 2 2a 4a q s called the primitive root, lies in U. (One can identify elliptic curves with quadratic forms by identifying the primitive root with the invariant  of the elliptic curve discussed before, and then this locus of primitive roots of definite integral quadratic forms corresponds to the collection of elliptic curves that admit “complex multiplication”, cf. [29], [175].) Lemma 3.9. Given an integral point .x; y; z/ of Minkowski three-space inside LC with corresponding point v 2 H, the Cayley transform in U of the central projection vN q 2D p agrees with the primitive root !Œzx;2y;zCx . Furthermore, given a point ! D q Ci rs , the primitive form proportional to Œq 2 s; 2pqs; p 2 s C q 2 r has ! as its primitive root. The proof is an elementary but somewhat involved calculation left to the reader (using the formulas in Section 2.2); it is the first part of this lemma that we neglected to explicitly mention in Appendix A. We are led to consider the level sets of D=a2 ; D=c 2 and especially D=b 2 , for if two primitive definite forms lie on a common level set of one of these functions, then they q are unitable. Setting ! D that • D=a2 D ˛ 2 ()

r s

Ci

D

˛2 4

r s

D u C iv 2 U, we compute from Lemma 3.9

() v D ˛2 ; 2

D ˇ 2 pq 2 () v D ˙ˇu; q q 2 () pq 2 rs C rs D 2 1 () u2 C .v   1 /2 D  2 ,

• D=b 2 D ˇ 2 () • D=c 2 D  2

p q

r s

where ˛; ˇ;  > 0. These respective loci in U are thus horizontal lines, rays from the origin, and circles tangent to R at zero as illustrated in Figure 3.5. The first and last cases are the by-now familiar horocycles centered at zero and infinity, and the second case is new to us and bears further discussion.

30

1 The basics D b2

D ˇ 2 ; b > 0

bD0

D b2

D ˇ 2 ; b < 0

D a2

D ˛12

D c2

D 22

D a2

D ˛22

D c2

D 12

Figure 3.5. Horocycles and hypercycles in U.

Definition 3.10. A ı-hypercycle to a geodesic in U is a component of the locus of points at distance ı  0 from the geodesic, and there are thus two ı-hypercycles for each ı > 0 while the 0-hypercycle is just the geodesic itself. Put another way, one can check that a hypercycle is a locus of constant geodesic curvature between zero and one, and one can thus think of hypercycles as interpolating between geodesics (of curvature zero) and horocycles (of curvature one). In particular, hypercycles to the imaginary ray in U are precisely Euclidean rays from the origin, i.e., the second case above corresponds to hypercycles. Projections to the modular curve of horocycles centered at infinity or of hypercycles to the imaginary ray will be called simply horocycles and hypercycles in the modular curve as illustrated respectively on the right and left in Figure 3.6. Corollary 3.11. Two classes of definite primitive quadratic forms are unitable if and only if they lie on a common hypercycle in the modular curve. Furthermore, if the classes lie on a common horocycle in the modular curve, then they are unitable as well. It is worth emphasizing that two primitive definite quadratic forms on a common hypercycle may not be concordant for that hypercycle, i.e., the second condition in Lemma 3.8 does not follow from the first. We say that a definite form f translates to another form f 0 if there is some uppertriangular element of the modular group sending f to f 0 . Here is our promised geometric description of the Gauss product for definite forms: Theorem 3.12. Suppose that f1 , f2 are concordant primitive definite forms with corresponding hypercycle h. Then the product Œf1 Œf2 is represented by the point f 2 h closest to the origin with the property that whenever f1 and f2 translate, possibly by different elements of the modular group, to concordant forms on a common hypercycle h0 , then f also translates to h0 . It would be most interesting to formulate a corresponding geometric description of the Gauss product for indefinite forms. In this case, the natural geometric invariant of

4 Basic definitions and formulas

31

Figure 3.6. Hypercycle and horocycles in U=PSL2 .R/.

an indefinite form Œa; b; c is the hyperbolic geodesic connecting the two real roots of az 2 C bz C c D 0.

4 Basic definitions and formulas This section is principally based on [128] but also includes calculations from [107], [134], [51]. Throughout, we shall conveniently normalize using the following result. Lemma 4.1. Given three distinct rays r1 ; r2 ; r3  LC from the origin, there are unique uj 2 rj so that huj ; uk i D 1 for j ¤ k, j; k D 1; 2; 3. Proof. Choose any vj 2 rj for j D 1; 2; 3. We seek ˛j 2 R>0 so that h˛j vj ; ˛k vk i D 1 for j ¤ k, i.e., ˛j ˛k D hvj ; vk i1 , and there is a unique positive solution, namely, s hvk ; v` i ˛j D for fj; k; `g D f1; 2; 3g: hvj ; vk ihvj ; v` i Corollary 4.2. The group SOC .2; 1/ acts simply transitively on positively oriented triples of distinct rays in LC . This is the version in Minkowski space of the familiar 3-effectiveness of the action of the Möbius group on triples of positively oriented points in the circle at infinity in Lemma 2.2. In particular, it follows that any Möbius transformation fixing distinct rays

32

1 The basics

in the light-cone is necessarily the identity. We remark that we shall use Corollary 4.2 without apology to normalize subsequent calculations and directly compute various quantities with a decidedly nineteenth-century ethos. We shall sequentially analyze pairs, triples and quadruples of horocycles, and to this end, introduce a convenient basis for Minkowski three-space as follows. Definition 4.3. Define the standard light-cone basis 1 u D p .1; 0; 1/; 2

1 v D p .1; 0; 1/; 2

wD

p

2.0; 1; 1/;

so that u; v; w 2 LC , hu; vi D hu; wi D hv; wi D 1, and uN 7! 0;

vN 7! 1;

w x 7! 1

under the Cayley transform. The standard basis vectors in R3 are expressed in the light-cone basis as vu .1; 0; 0/ D p ; 2

.0; 1; 0/ D

uCvw ; p 2

vCu .0; 0; 1/ D p : 2

4.1 Pairs of horocycles and lambda lengths. Here is the definition of our basic coordinates on decorated surfaces: Definition 4.4. Given a pair of horocycles h1 , h2 , say in D, with distinct centers, consider the geodesic  in D connecting their centers. Let ı denote the signed hyperbolic distance along  between the points h1 \  and h2 \ , where the sign of ı is taken to be positive if and only if h1 and h2 are disjoint. See Figure 4.1. Define the lambda length of h1 , h2 to be p .h1 ; h2 / D

eı:

h1 h1

ı

ı 

 h2

h2

Figure 4.1. Lambda length.

4 Basic definitions and formulas

33

These are our basic invariants, and essentially all of our calculations will be performed using them. (In our papers before p 2006 and in particular in Appendix p B, ı the lambda length was defined to be 2e with the “funny” constant 1= 2 in Lemma 2.11 replaced by 1.) Put another way, a decoration on a hyperbolic geodesic is the specification of a pair of horocycles, one centered at each of its ideal points, and the lambda length is an invariant of a decorated geodesic. Thus though a hyperbolic geodesic has infinite length, a decoration gives a way to truncate it and thereby define a sensible finite length ı, essentially the lambda length. These invariants are so fundamental to our treatment that we shall often simply identify a decorated geodesic with its lambda length when no confusion may arise. Lemma 4.5. Suppose that u1 ; u2 2 LC do not lie on a common ray in LC and let h.u1 /; h.u2 / be the horocycles corresponding to these points via affine duality in Lemma 2.11. Then the lambda length is given by p .h.u1 /; h.u2 // D hu1 ; u2 i: The identification of horocycles with points of LC via affine duality in Lemma 2.11 is therefore also geometrically natural in this sense. Proof. We may normalize by Corollary 4.2 using the standard light-cone basis so that u1 D t1 u and u2 D t2 v for some t1 ; t2 2 R>0 . We seek the points j 2 h.uj /, for j D 1; 2, on the geodesic connecting the centers of h.u1 /, h.u2 /, i.e., we seek j D xj u C yj v so that 1 t1 y1 D h1 ; t1 ui D  p D h2 ; t2 vi D t2 x2 2 and 1 ; 2 2 H, i.e., 2xj yj D hj ; j i D 1

for j D 1; 2:

Solving these equations, we find 1 t1 1 D p u C p v; 2 2t1

1 t2 2 D p u C p v: 2t2 2

Thus from Lemma 2.9, we have cosh2 ı D h1 ; 2 i2 D

1 2

2 .t1 t2 C .t1 t2 /1 / ;

so hu1 ; u2 i D ht1 u; t2 vi D t1 t2 gives e˙ı D hu1 ; u2 i: Since ı ! 1 as t1 ! 1 or t2 ! 1 by the last part of Lemma 2.11, we must take the plus sign, completing the proof.

34

1 The basics

Armed with this result, direct calculation in the upper half-plane yields: Corollary 4.6. Given horocycles hj in U with distinct centers xj 2 R of respective Euclidean diameters j , for j D 1; 2, the lambda length is given by jx1  x2 j .h1 ; h2 / D p I 1 2 likewise, given the hororcycle h centered at infinity of height H , the lambda length is given by s .h; h1 / D

H : 1

  Furthermore, if A D ac db is a Möbius transformation with Ah1 centered at Ax1 2 R, then Ah1 has Euclidean diameter 1 =.cx1 C d /2 ; likewise, if Ax1 D 1, then Ah1 2 has height 1 1 =c . We conclude that Euclidean diameters of horocycles in the upper half-plane scale by derivatives under the action of Möbius transformations. An interesting side-note that follows from Corollary 4.6 is that the lambda length of any pair of horocycles in the Farey tesselation  decorated as in the previous section is actually an integer.4 Lemma 4.7. Suppose that u1 2 LC with corresponding horocycle h.u1 /. If u2 2 H, then 1 hu1 ; u2 i D  p e ı ; 2 where ı is the signed hyperbolic distance from u2 to h.u1 / taken with a positive sign if and only if u2 lies outside h.u1 /. If u3 2 H with corresponding geodesic 3 which is not asymptotic to the center of h.u1 /, then let u4 be the orthogonal projection of the center of h.u1 / to 3 . Then hu1 ; u3 i2 D

1 2ı e ; 2

where ı is the signed hyperbolic distance from u4 to 3 taken with a positive sign if and only if 3 and h.u1 / are disjoint. 4

At the risk of proving beyond any doubt that we have too much spare time, let us remark that there is a musical instrument, the “hormonica”, based on this observation as follows. Begin with the Farey tesselation  as the untuned instrument regarded as drawn before you on the computer screen. Perform a sequence of flips by serially selecting edges to produce another tesselation  of the Poincaré disk likewise displayed on the computer screen. Choose some basic frequency, say middle C, to represent unity, so that any natural number may be interpreted as a multiple of this frequency. In this way, each edge of  with its integral lambda length can be “plucked” to produce a corresponding tone. Moreover, each triangle complementary to  can be “tapped” to produce a triple of tones or chord. Attributes such as duration or timbre could be introduced as further aspects of the tuning process. Maybe this is crazy, but it could be fun. On the other hand, it is difficult to probe the combinatorics of moduli spaces visually, and the analogous hormonicas based upon tesselations of a fixed surface could provide an auditory tool towards this end.

35

4 Basic definitions and formulas

Proof. Let u, v, w be the standard light-cone basis. For the first part by Corollary 4.2, we may assume that u1 is a multiple of u and the geodesic  connecting the center of h.u1 / to u2 is asymptotic to u and v. Applying a hyperbolic isometry fixing , we may furthermore assume that u1 D t u for some t > 0 and u2 D p1 .u C v/, so 2 p hu1 ; u2 i D  pt , i.e., t D  2hu1 ; u2 i. The point 2

 D h.u1 / \  D xu C yv 2 H; for some x, y, satisfies  p1 D hu1 ; i D ty, so y D p1 and x D pt since  2 H. 2 2t 2 According to Lemma 2.9, the hyperbolic distance ı between  and u2 is given by cosh2 ı D hu2 ; i2 D

˝

p1 .u 2

˛2

C v/; p1 .t u C 1t v/ 2

D

1 2

2 .t C 1t / ;

p whence e˙ı D t D  2hu1 ; u2 i, and the positive sign follows from the definition of signed distance and the asymptotics of Lemma 2.11. For the second part using the earlier notation, we may assume that u1 D t u, for some t > 0 and u3 D ˙ p1 .v  u/, so 2

˝ ˛ hu1 ; u3 i D t u; ˙ p1 .v  u/ D  pt : 2

2

As in the previous calculation,  D h.u1 / \  D h; u3 i D

˝

p1 .t u 2

pt u 2

C

p1 , 2t

so

 ˛  C 1t v/; ˙ p1 .v  u/ D  12 1t  t : 2

By Lemma 2.9, h; u3 i2 D sinh2 ı, so t D e˙ı . The positive sign again follows from the definition of signed distance and the asymptotics of Lemma 2.11. 4.2 Triples of horocycles: h-lengths and equidistant points. The lambda length is thus an invariant of a pair of horocycles or of a decorated geodesic, and we turn our attention next to triples of horocycles. Just as in Lemma 4.1, given a triple of positive numbers a, b, e, there are unique positive multiples ˛u, ˇv,  w of the standard lightcone basis realizing h˛u; ˇvi D e 2 ;

h˛u;  wi D a2 ;

hˇv;  wi D b 2 ;

namely, ˛D

ae ; b

ˇD

be ; a

D

ab : e

Corollary 4.8. Ordered triples of lambda lengths give a parametrization of Möbiusorbits of ordered triples of horocycles with distinct centers.

36

1 The basics

Lemma 4.9. Given a triple of horocycles h1 , h2 , h3 with distinct centers, let j D .hk ; h` / denote the lambda lengths and let j denote the geodesic connecting the centers of hk , h` for fj; k; `g D f1; 2; 3g. Then the hyperbolic length of the horocyclic  segment in hj between k and ` is given by kj` . See Figure 4.2. 1 0 2

h1

0

2 0 1  2

1

h2

h0

2 0 1

Figure 4.2. h-length.

Proof. Again, we use Corollary 4.2 and the earlier calculation to arrange that the three u, be v, ab w. We seek points in LC corresponding to the triple of horocycles are ae b a e the point  D xu C yv 2 H so that D be E be be 1 h; vi D  x;  p D ; v D a a a 2 i.e., x D

p1 a . 2 be

Now using 1 D h; i, we find y D

be uC p v D Dp 2be 2a a



1 , 2x

so

a be a be  C ; 0; C 2be 2a 2be 2a



y 1 as a vector in R3 . Using the explicit mapping H ! U given by .x; y; z/ 7! . zx ; zx /, be we compute the imaginary part of the image of  to be a ; since u, v, w respectively map to 0; 1; 1, the hyperbolic segment lying in h. be v/ maps to the horizontal segment of a be a Euclidean length one at height a , which has hyperbolic length be using the expression for the hyperbolic metric in U. The other formulas follow by symmetry.

(This corrects Proposition 2.8 in [128].) Definition 4.10. A decoration on an ideal triangle is a triple of horocycles, one centered at each ideal point of the triangle. Define a sector to be an end of an ideal triangle, so associated with a decorated ideal triangle, each sector has a corresponding horocyclic arc, whose length is called the h-length of the sector.

4 Basic definitions and formulas

37

We have just shown that the h-length of a sector is the opposite lambda length divided by the product of adjacent lambda p lengths, and it was to guarantee this formula that we took the funny constant 1= 2 in Lemma 2.11. It follows from Lemma 4.9 that the product of the h-lengths of two sectors in a decorated ideal triangle is the reciprocal of the square of the lambda length of the decorated geodesic connecting the sectors, i.e., in the notation of Figure 4.2, we have j k D 12 . k ` j ` `

Lemma 4.11. Given u1 ; u2 ; u3 2 LC no two of which lie on a common ray in LC , define j D .h.uk /; h.u` // for fj; k; `g D f1; 2; 3g. The affine plane containing u1 ; u2 ; u3 is elliptic if and only if the strict triangle inequalities hold among 1 , 2 , 3 , it is parabolic if and only if some triangle equality holds, and it is hyperbolic if and only if some weak triangle inequality fails. Proof. The tangent space to the affine plane is spanned by v1 D u1  u3 and v2 D u2  u3 , and we compute that hvj ; vj i D 22k for fj; kg D f1; 2g while hv1 ; v2 i D 21 C 22  23 . The determinant of the Minkowski pairing restricted to the affine plane is thus hv1 ; v1 ihv2 ; v2 i  hv1 ; v2 i2 D 421 22  .21 C 22  23 /2 D 41  42  43 C 221 22 C 221 23 C 222 23 D .1 C 2 C 3 /.1 C 2  3 /.1 C 3  2 /.2 C 3  1 /; and the result then follows from Lemma 2.7 since at most one factor in the last expression can be non-positive for 1 ; 2 ; 3 > 0. Lemma 4.12. Given u1 ; u2 ; u3 2 LC , define j D .h.uk /; h.u` // for fj; k; `g D f1; 2; 3g, and let h1 , h2 , h3 denote the corresponding horocycles which we assume have pairwise distinct centers. Then there is a point equidistant to h1 , h2 , h3 if and only if 1 , 2 , 3 satisfy the strict triangle inequalities, and in this case, the equidistant point  is unique. Furthermore in this case, let ˛j denote the h-length of the sector corresponding to hj and let j be the geodesic connecting the centers of hk , h` , for fj; k; `g D f1; 2; 3g. Then the geodesic connecting  to the center of hj is at signed distance 12 .˛1 C ˛2 C ˛3 /  ˛k from k along hj , where the sign is positive if and only if  lies on the same side of k as the center of hk , for fj; k; `g D f1; 2; 3g. See Figures 4.3a and 4.3b. Proof. As usual, we shall compute in the standard light-cone basis u, v, w and arrange u, be v, ab w. that the three points in LC corresponding to the triple of horocycles are ae b a e By the first part of Lemma 4.7, we seek  D xu C yv C zw 2 H so that h; ae ui D h; be vi D h; ab wi: b a e

38

1 The basics h1

h1

˛2 ˛1 ˛3 2

˛1 C˛2 ˛3 2

3



˛1 C˛3 ˛2 2

2 ˛2

˛3 h3

˛2 C˛3 ˛1 2

h2

3

2

1

˛2



˛1

˛1

˛3

1

h3

h2

a) Equidistant point inside

b) Equidistant point outside

Figure 4.3. Equidistant points to horocycles.

Define A D y C z, B D x C z, C D x C y, so A ae D B be D C ab by equidistance. b a e We must solve the equation 2 D A.B C C  A/ C B.A C C  B/ C C.A C B  C /; where A D

b2 C e2

2

and B D ae2 C . Thus,  2 2 2 2 2 2 D C 2 be2 ae2 C 1  be2 C ae2 be2 C 1 

a2 e2



C

 b2 e2

C

a2 e2

1



C2 Œ2a2 b 2 C 2a2 e 2 C 2b 2 e 2  a4  b 4  e 4 e4 C2 D 4 .a C b C e/.a C b  e/.a C e  b/.b C e  a/; e so there is indeed a solution if and only if a, b, e satisfy all three possible strict triangle inequalities. Furthermore in this case setting D

K D .a C b C e/.a C b  e/.a C e  b/.b C e  a/; r

we find

2 2 2 2 .b ; a ; e / K and must take the positive sign to guarantee a positive z-coordinate in Minkowski three-space. Thus, the unique equidistant point is given by r 1 D Œ.a2 C e 2  b 2 /u C .b 2 C e 2  a2 /v C .a2 C b 2  e 2 /w : 2K .A; B; C / D ˙

4 Basic definitions and formulas

39

Let us find a Minkowski normal n for the hyperbolic plane through the origin determining the geodesic  passing through  and u. To this end, hn; ui D 0 gives n D xu C yv  yw, for some x; y, and hn; i D 0 further gives p 0 D  2K hn; i D .b 2 C e 2  a2 /.x  y/ C .a2 C b 2  e 2 /.x C y/ D 2b 2 x C 2.a2  e 2 /y: A normal to the plane containing  is thus given by n D .e 2  a2 /u C b 2 v  b 2 w: Next, we seek a point r D xu C yv C zw 2 LC so that the ideal points of  in D 1 are r; N uN 2 S1 , that is, 0 D hn; ri D y.e 2  a2  b 2 / C z.e 2 C b 2  a2 /; so r 2 LC gives 0 D xz.b 2 C e 2  a2 / C z 2 .b 2 C e 2  a2 / C xz .a2 C b 2  e 2 /: Thus, xz D

a2 b 2 e 2 2 z , 2b 2

and it follows that such an r 2 LC is given by

r D .a2 C b 2  e 2 /.a2  b 2  e 2 / u C 2b 2 .b 2 C e 2  a2 / v C 2b 2 .a2 C b 2  e 2 /w: We may finally compute the length along the horocycle h. ae u/ between  \h. ae u/ b b and the geodesic asymptotic to the rays of u, v using Lemma 4.9 and find this length to be v r u   u h be v; ri 1 a b e .a2 C b 2  e 2 /2 a t ˙  ae D ˙ ; C  D ˙ 4a2 b 2 c 2 2 be ae ab h b u; rih ae u; be vi b a where the plus sign finally follows by comparison with the expression above for  and the definition of the signed distance. The other formulas follow by symmetry. It is interesting that the fact that every triangle inscribes in a circle in Euclidean space is a condition equivalent to Euclid’s Fifth Postulate, i.e., for any triple of distinct points there is a unique equidistant point. This condition thus fails in hyperbolic space, but according to the lemma, there is nevertheless an equidistant point to a triple of horocycles under the articulated condition on lambda lengths. 4.3 Quadruples of horocycles: Ptolemy equation and simplicial coordinates. We finally turn our attention to quadruples of horocycles with distinct centers, or put another way, decorated ideal quadrilaterals, that is, ideal quadrilaterals with a horocycle centered at each ideal point. There is a basic calculation from which several results are derived as follows:

40

1 The basics

Calculation 4.13 (Basic Calculation). Consider multiples u0 D ae u; v 0 D be v; w 0 D b a ab w of the standard light-cone basis u, v, w as before. Given two further positive real e numbers c, d , we claim that there is a unique point  D xu C yv C zw 2 LC so that h; u0 i D d 2 , h; v 0 i D c 2 , and ; w lie on opposite sides of plane through the origin containing u0 and v 0 . Indeed, we have the equations ae ae h; ui D  .y C z/; b b be be 2 0 c D h; v i D h; vi D  .x C z/; a a

d 2 D h; u0 i D

which give y D we find 0D

bd 2 ae

z, x D 

ac 2 be

z. Now using that  2 LC , i.e., 0 D xy Cxz Cyz, 









abc 2 d 2 c2d 2 bd 2 ac 2 ac 2 bd 2 C z2  z C C z zC  z z D 2  z2: 2 be ae be ae e abe

Thus, z D ˙ cd , and we must take the minus sign to have  on the correct side of the e plane through the origin containing u0 and v 0 . It follows that the unique solution is given by d cd c .ac C bd /u C .ac C bd /v  w D eb ea e completing the Basic Calculation and proving the claim, which is formalized in the next lemma. Lemma 4.14. Given u1 ; u2 2 LC and real numbers 1 , 2 , 3 satisfying hu1 ; u2 i D 23 , there is a unique point u3 on either side of the plane through the origin containing u1 , u2 so that hu2 ; u3 i D 21 and hu1 ; u3 i D 22 . Furthermore, the ratio 1 =2 uniquely determines the ray in LC containing u3 . Proof. The first part follows from the uniqueness of solution in the Basic Calculation, and the second part follows from homogeneity of the solution. Corollary 4.15. Ordered five-tuples of positive lambda lengths give a complete invariant of Möbius-orbits of ordered quadruples of horocycles with distinct centers. Proof. Given an ideal quadrilateral, choose a diagonal decomposing it into a pair of adjacent ideal triangles. Lambda lengths of the frontier edges of the quadrilateral together with the lambda length of the chosen diagonal uniquely determine the Möbiusorbit of a decoration on the ideal quadrilateral by the Basic Calculation 4.13. Several other important results which are illustrated in Figure 4.4 also follow from the Basic Calculation. 1 Corollary 4.16. Suppose u1 ; u2 ; u3 ; u4 2 LC , where uN 1 ; uN 2 ; uN 3 ; uN 4 2 S1 are distinct p and occur in this counter-clockwise cyclic order, and let j k D .uj ; uk / D huj ; uk i, for j; k D 1; 2; 3; 4, denote the lambda lengths. Then:

41

4 Basic definitions and formulas

a) Ptolemy equation: 13 24 D 12 34 C 14 23 . b) Cross ratio: The conformal map sending uN 1 7! 0, uN 2 7! 1, and uN 3 7! 1 also 14 sends uN 4 7!  23 ; in other words, the cosine of the angle at the point of 12 34 intersection from the geodesic asymptotic to uN 2 , uN 4 to the geodesic asymptotic 12 34 14 23 to uN 1 , uN 3 is given by 12 . 34 C14 23 c) Shear coordinate: Letting  be the geodesic with ideal points uN 1 , uN 3 and dropping perpendiculars from uN 2 and uN 4 to  , the signed distance between the points 14 of intersection with  is given by log 23 , where the sign is positive if and only 12 34 if these points lie to the right of one another along  . d) Opposite distance: If ı denotes the hyperbolic distance between the geodesic spanned by u1 , u4 and the geodesic spanned by u2 , u3 , then  12 34 2 : cosh2 ı D 1 C 2 14 23  bd ac

e

c

d

d acCbd e

1 0

0 1

c

a

a

b

b

a) Ptolemy equation

log bd ac

b) Cross ratio

c c

d

d

ı

b a

a b

c) Shear coordinate

d) Opposite distance

Figure 4.4. Formulas for quadruples of horocycles.

42

1 The basics

Proof. Adopting the notation of the Basic Calculation 4.13 with 12 D a, 23 D b, 34 D c, 14 D d , 13 D e, and 24 D f , we find D

f 2 D  ;

i ab E ab h c d .ac C bd /2 ; w D .ac C bd / C .ac C bd / D e e be ea e2

proving part a). For part b) again in the notation of the Basic Calculation, write  as a vector in R3 as c d cd .ac C bd /u C .ac C bd /v  w eb ea e      p cd 1 d p cd  c c 1 d D p  ae C .ac C bd /; 2 ; p .ac C bd /  2 e be e 2 ae be 2 r

D

and apply the transformation H ! U where .x; y; z/ 7! .y C i /=.z  x/ to find the real part p  2 cd bd bd D D ; p cd p ce bd C ac C bd ac  2 e C 2 be .ac C bd / from which directly follows the asserted formula for the cosine. For part c), a Minkowski normal to the plane determining the geodesic asymptotic to u; v is evidently u C v  w, so by the last part of Lemma 2.9, if another such Minkowski normal n D xu C yv C zw corresponds to a perpendicular geodesic, then 0 D hn; u C v  wi D 2z, and furthermore, n 2 H gives 1 D hn; ni D 2xy. Thus, 1 a perpendicular geodesic has corresponding unit normal of the form n D xu  2x v. In particular, the unit normal nw of the perpendicular geodesic asymptotic to w thus 1 1 v; wi D 2x  x, and so nw D p1 u  p1 v. Likewise has 0 D hnw ; wi D hxu  2x 2 2 again in the notation of the Basic Calculation, the unit normal n of the perpendicular geodesic asymptotic to  thus has 0 D hn ; i D 1 c d cd E D xu  v; .ac C bd /u C .ac C bd /v  w ae e  

2x be d 1 c cd cd D x C .ac C bd /  .ac C bd /  ae e 2x be e

2

 bd 1 ac 2 D x C ; ae 2x be ac , and furthermore comparing with the expression in part b) for the so x D ˙ p1 bd 2 cross ratio and by definition of the vector u, we see that we must take the positive sign. ac Thus, we find that n D p1 bd u  p1 bd v. ac 2

2

4 Basic definitions and formulas

43

Again applying the last part of Lemma 2.9, the desired distance ı between the perpendiculars is given by D 1 1 bd E2 1 1 ac cosh2 ı D hnw ; n i2 D p u  p v; p u p v 2 2 2 bd 2 ac

  1 bd ac 2 D : C 2 ac bd Thus, we find ı D ˙ log bd , and the positive sign follows again from the expression ac for cross ratios in part b) and the definition of x. Finally for part d) in the notation of the Basic Calculation, the Minkowski normal in H to the geodesic spanned by v; w is p1 .u  v  w/, and the Minkowski normal 2

2 H to the geodesic spanned by u;  satisfies h ; ui D 0 D h ; i and is easily then computed to be

   1 ac 1C2 uCvw : p bd 2 The result follows upon taking the Minkowski products of these normals according to Lemma 2.9. Notice the similarity of Corollary 4.16a with Ptolemy’s classical theorem that a Euclidean quadrilateral inscribes in a circle if and only if the product of diagonal lengths is the sum of products of opposite lengths. Indeed, a more conceptual proof of Corollary 4.16a depends upon the observation that the putative formula is independent under scaling each of the points u1 , u2 , u3 , u4 separately, so we may scale these four points so as to lie at a common height in Minkowski three-space, say at height one. Since the induced metric is twice the usual Euclidean metric on this horizontal plane, and since the intersection with the light-cone is a round circle in the induced metric, Ptolemy’s classical result implies our ideal hyperbolic version. On the other hand, a Euclidean quadrilateral that inscribes in a circle is characterized by four degrees of freedom while the hyperbolic version admits five degrees of freedom, so the Euclidean interpretation only explains a restricted case of the general Ptolemy equation corresponding to coplanarity in Minkowski space. The expression for the cosine in Corollary 4.16b is likewise independent under scaling, so the ideal hyperbolic version again follows from the Euclidean version, which is itself tantamount to the Euclidean Law of Cosines. For our next formula, let us observe that the Möbius group action on Minkowski space is through (Euclidean) volume-preserving linear mappings SOC .2; 1/. Thus, the volume of the tetrahedron spanned by four points in LC is a well-defined invariant of the SOC .2; 1/-orbit such a four-tuple, and our next result calculates this invariant, which will play a fundamental role in the sequel. 1 Corollary 4.17. Suppose u1 ; u2 ; u3 ; u4 2 LC , where uN 1 ; uN 2 ; uN 3 ; uN 4 2 S1 are distinct and occur in this counter-clockwise cyclic order, and let j k D .uj ; uk / D

44

1 The basics

p huj ; uk i, for j; k D 1; 2; 3; 4, denote the lambda lengths. Then the signed volume of the tetrahedron determined by these four points is given by

2  p 12 C 223  213 234 C 214  213 ; C 2 2 12 23 34 14 12 23 13 34 14 13 where the volume is positive if and only if the Euclidean segment connecting u1 ; u3 lies below the Euclidean segment connecting u2 , u4 . Proof. As usual, we normalize by putting the points into standard position and rely on the Basic Calculation 4.13. Thus, we have the four points in LC given in the usual coordinates on R3 as: ae 1 ae uD p .1; 0; 1/; b 2 b be 1 be vDp .1; 0; 1/; a 2 a p ab ab w D 2 .0; 1; 1/; e e      1 2cd 2cd c c d d Dp .ac C bd /; .ac C bd /  ;  ; C e ae be e 2 ae be and subtracting ab w from the others, we therefore have the three corresponding Eue clidean displacement vectors 



1 .bd  ac/.bd C ac/ 2.ab C cd / .ac C bd /2 2.ab C cd / ; ; ;  p abe e abe e 2   1 ae 2ab ae 2ab ; ; ;  p  b e b e 2   1 be 2ab be 2ab : ; ;  p e a e 2 a Now take the triple scalar product of these three vectors in this order and pull out the obvious common factors to get: ˇ ˇ p ˇ.bd  ac/.bd C ac/ ab.ab C cd / .ac C bd /2  2ab.ab C cd /ˇ ˇ ˇ 2 ˇ 2 2 2 2 ˇ: ae ab ae  2ab ˇ ˇ a2 b 2 e 3 ˇ 2 2 2 2 ˇ be a b be  2a b Adding twice the second column plus the first column to the third column, we find: ˇ ˇ p ˇ.bd  ac/.bd C ac/ ab.ab C cd / 2bd.ac C bd /ˇ ˇ ˇ 2 ˇ 2 2 ˇ ae ab 0 ˇ ˇ a2 b 2 e 3 ˇ 2 2 2 ˇ a b 2be be ˇ 2 2 ˇ p ˇb d  a2 c 2 ab C cd bd.ac C bd /ˇ ˇ 2 2 ˇˇ ˇ; b 0 ae 2 D ˇ ˇ 3 abe ˇ 2 2 ˇ be a be

4 Basic definitions and formulas

45

and now directly taking the determinant by expanding along the third column gives p 2 2 2 Œae .ab C cd / C b.b 2 d 2  a2 c 2 /  a2 d.ac C bd /  b 2 d.ac C bd / ae p 2 2 2 Œe .ab C cd /  bac 2  ad.ac C bd /  b 2 cd D e

2  p a C b2  e2 c 2 C d 2  e2 D 2 2abcd C abe cde as was claimed. 2 C2 2

23 13 It is worth emphasizing that the expression 12 that occurs in our volume 12 23 13 calculation in Corollary 4.17 had already arisen in Lemma 4.12 in our calculation of equidistant points, and it is furthermore worth emphasizing that this same expression

212 C 223  213 12 23 13 D C  12 23 13 23 13 12 13 12 23 is actually linear in the h-lengths by Lemma 4.9. We are aware of no a priori geometric or other reason for these “happy coincidences”, and we shall exploit them in subsequent discussions.



d

c

 ˇ

e

a

ı "

˛ b

Figure 4.5. Simplicial coordinate.

Definition 4.18. Suppose that e is a diagonal of a decorated quadrilateral with boundary edges a, b, c, d as illustrated in Figure 4.5, and identify an edge with its lambda length for convenience. Define the simplicial coordinate of e to be the quantity a2 C b 2  e 2 c 2 C d 2  e2 C D ˛ C ˇ  " C  C ı  ; abe cde where the Greek letters indicate the h-lengths of nearby sectors illustrated in Figure 4.5.

46

1 The basics

There is actually another simple expression for the volume or simplicial coordinate: Corollary 4.19. In the notation of Corollary 4.17, the signed volume is also expressed as p 2 2Œ24 .12 14 C 23 34 /  13 .12 23 C 14 34 / Proof. This is an algebraic consequence of Corollaries 4.16a and 4.17 since a2 C b 2  e 2 c 2 C d 2  e2 1 C D Œcd.a2 C b 2  e 2 / C ab.c 2 C d 2  e 2 / abe cde abcde 1 D Œ.ac C bd /.ad C bc/  e 2 .ab C cd / abcde 1 D Œef .ad C bc/  e 2 .ab C cd / abcde 1 D Œf .ad C bc/  e.ab C cd / abcd is the lambda length in the usual notation of the Basic Calculation, where f D acCbd e ab of the horocycles corresponding to  and e w according to Ptolemy’s equation.

2 Lambda lengths in finite dimensions

There are various finite-dimensional manifestations of the basic lambda length coordinates. These range from cyclic polygons, to decorated hyperbolic structures on punctured surfaces, to hyperbolic structures on bordered surfaces. All of these incarnations of lambda lengths are treated in this chapter together with the corresponding mapping class group action and invariant Weil–Petersson Kähler two form or Poisson structure.

1 Cyclic and decorated polygons This section is based on [128] and treats decorated and cyclic polygons, which provide a reasonable arena in which to foreshadow much of the subsequent theory as we do here. We study “cyclic Euclidean polygons”, i.e., polygons in the Euclidean plane which inscribe in a circle, corresponding to coplanar vertices in the light-cone lying in an elliptic plane, and more generally “decorated ideal polygons”, i.e., ideal polygons in the hyperbolic plane together with a specification of distinguished horocycles, one centered at each vertex, corresponding to general configurations of vertices in the lightcone. 1.1 Decorated polygons Definition 1.1. Let Q denote an arbitrary abstract oriented polygon with n sides, called an n-gon, for n  3, with vertex set Q0 and one of its edges distinguished. Define a decorated hyperbolic structure on Q to be a mapping W Q0 ! LC which is order1 agree with those of  in the preserving, i.e., the separation properties of ./ N 2 S1 0 1 boundary of Q , where ./ N denotes the image of ./ under the projection LC ! S1 in Section 2 of Chapter 1. The space of all such mappings with the compact-open topology (i.e., a sub-basis for this Hausdorff topology is given by those functions that map a specified compact set in the domain to a specified open set in the range) is denoted by Tz 0 .Q/, and a Möbius transformation  2 PSL2 .R/ SOC .2; 1/ acts on

2 Tz 0 .Q/ by 7!  ı with quotient space Tz .Q/ D Tz 0 .Q/=PSL2 .R/; the decorated Teichmüller space of the abstract polygon Q.

48

2 Lambda lengths in finite dimensions

Definition 1.2. The convex hull of a set in a metric space is the smallest convex subspace containing the set. An ideal hyperbolic n-gon, for n  3, is the hyperbolic convex 1 hull R of the geodesics determined by a collection of n distinct points in the circle S1 at infinity, where one boundary edge of R is distinguished. A decoration on R is the further specification of a collection of n horocycles, one centered at each ideal vertex of R. Thus if Q is an abstract n-gon, then Tz .Q/ is the space of Möbius-orbits of all decorated ideal hyperbolic n-gons. Choose a triangulation  of the abstract polygon Q, i.e.,  is the set of sides of Q plus a chosen collection of chords of Q, disjoint except perhaps for endpoints, with each component of Q  [ a triangle with vertices in Q0 . Proposition 1.3. Fix any triangulation  of a polygon Q with n  3 sides. Then the mapping which assigns to each arc in  the lambda length of the horocycles centered at its endpoints gives a real-analytic homeomorphism Tz .Q/ ! R >0 : Proof. The mapping in the theorem is well defined since lambda lengths are by definition invariant under the diagonal action of the Möbius group SOC .2; 1/ on LC  LC . We must produce a two-sided inverse to this map and let  W  ! R>0 be any function. To this end, choose a triangle t complementary to , and regard the standard lightcone basis of the last section as fixed a priori. There are unique points lying in the rays from the origin containing the standard light-cone basis elements and realizing as lambda lengths the values that  takes on the arcs in the frontier of t in their clockwise cyclic order, which comes from the orientation of Q, by Lemma 4.8 of Chapter 1. Letting t 0 D fx; y; zg denote the vertices of t , we have thus defined respective lifts

.x/; .y/; .z/ 2 LC . Consider a triangle t 0 with vertices x; y; w 2 Q0 whose interior is complementary to . There is a unique point .w/ 2 LC lying on the opposite side of the plane through the origin containing .x/; .y/ from .z/ likewise realizing as lambda lengths the values of  on the edges with endpoints x, w and y, w according to Lemma 4.14 of Chapter 1. By construction, this definition of on w, x, y, z satisfies the requirement of being order-preserving in the sense of Definition 1.1. Continue in this way lifting all the neighboring triangles of t, then their neighboring triangles, and so on, to construct a mapping W Q0 ! LC . This function is well defined up to post-composition with a Möbius transformation by the uniqueness statements in Lemmas 4.8 and 4.14 of Chapter 1, by the Möbius invariance of lambda lengths, and by Corollary 4.2 of Chapter 1. Furthermore, such a representative is uniquely determined by fixing any triangle complementary to  and any three basis rays in LC to contain the images of its vertices as in Lemma 4.8 with the corresponding induction again based on Lemma 4.14 of Chapter 1.

49

1 Cyclic and decorated polygons

This gives a continuous two-sided inverse to the real-analytic mapping in the proposition, as required. Indeed, the formulas of the Basic Calculation 4.13 of Chapter 1 in the previous section show that the inverse mapping is also real-analytic. Recall from Section 1.3 of Chapter 1 that a flip along an edge e interior to Q in a triangulation  of Q is the triangulation that results after removing e from  to produce a unique complementary quadrilateral and replacing e by the other diagonal f of this quadrilateral as illustrated again for convenience in Figure 1.1. We also depict the dual trivalent fatgraph embedded in the polygon as well as the effect of a flip on it and shall find it useful to associate the lambda length of an ideal arc instead with the corresponding edge in its dual fatgraph. More generally, any decomposition of a polygon into sub-polygons, determines its dual fatgraph which has one k-valent vertex for each sub k-gon. d

d

f D

e

acCbd e

flip a

c

c

a

b

b

Figure 1.1. Flips on a polygon and its dual fatgraph.

Lemma 1.4. Finite sequences of flips act transitively on the set of all ideal triangulations of a finite-sided polygon. Proof. Proceed by induction on the number n  4 of sides of the polygon Q. For the basis step n D 4, the result follows directly from the fact that there are exactly two distinct triangulations of Q which are related by a flip. For the induction step, suppose that ; 0 are two triangulations of Q, where we assume here and below that arcs in  meet 0 a minimal number of times in their proper isotopy classes. The result follows easily from the inductive hypothesis under the further assumption that , 0 share a common interior edge. This observation is the basis step for a second induction on ˚ min # e \ [0 ; e2

where the count both here and below is only over points of transverse intersection in the interior of Q. For the induction step of this second induction, suppose that e 2  minimizes and that #fe \ [0 g ¤ 0. Consider the triangle t complementary to 0 containing among

50

2 Lambda lengths in finite dimensions

its vertices a specified endpoint of e, and let e 0 2 0 denote the edge of t opposite this endpoint. Perform a flip along e 0 in 0 to produce the triangulation 00 of Q. By construction, we have ˚ ˚ # e \ [00 D # e \ [0  1; and the result therefore follows by induction. Thus, we think of the choice of triangulation  of Q as a choice of global chart on z T .Q/, with the transition functions given by compositions of Ptolemy transformations along interior edges. This is a kind of easy paradigm for the action of a general mapping class group on its decorated Teichmüller space as we shall see. We mention all of this here because lambda lengths on (n C 3)-gons give precisely the model for cluster algebras of type An . Likewise, the parallel theory for once-punctured polygons gives the model for cluster algebras of type Dn . Furthermore, there is a closed invariant two form on the decorated Teichmüller space Tz .Q/ of a polygon Q as follows. Given an ideal triangulation  of Q, define the two form of Tz .Q/ by X d log a ^ d log b C d log b ^ d log c C d log c ^ d log a; ! D 2 where the sum is over all triangles complementary to  whose consecutive edges in the clockwise ordering have respective lambda lengths a, b, c. Proposition 1.5. If , 0 are two ideal triangulations of Q, then ! D !0 as two forms on Tz .Q/, i.e., this two form is well defined independent of ideal triangulation. Proof. In light of Lemma 1.4, it suffices to prove the result in case , 0 differ by a single flip along an ideal arc e 2 . To this end in the notation above, adopt the convenient convention that we identify the lambda length of an arc with the arc itself as usual, and set xQ D d log x D dx , for x D a; : : : ; f , writing the wedge product x simply as a product. In particular, by the Ptolemy relation ef D ac C bd , we have   1 eQ C fQ D ac.aQ C c/ Q C bd.bQ C dQ / : ac C bd We may compute the relevant contribution to !0 to be fQcQ C cQ bQ C bQ fQ C fQaQ C aQ dQ C dQ fQ D aQ dQ C cQ bQ C fQ.aQ C cQ  bQ  dQ / D aQ dQ C cQ bQ  e. Q aQ C cQ  bQ  dQ / ac bd  .aQ C c/. Q bQ C dQ / C .bQ C dQ /.aQ C c/ Q ac C bd ac C bd D aQ dQ C cQ bQ  e. Q aQ C cQ  bQ  dQ / C .bQ C dQ /.aQ C c/ Q Q Q Q Q D eQ b C b aQ C aQ eQ C eQ d C d cQ C cQ e; Q

1 Cyclic and decorated polygons

51

which therefore agrees with the corresponding contribution to ! . Thus, the two form is well defined independent of the choice of triangulation by Lemma 1.4 and is the analogue of the Weil–Petersson two form in this context as we shall prove in Section 3. The considerations so far serially recur in our later discussions: for decorated surfaces in the next section, the polygon Q will be taken as a fundamental domain for the action of a Fuchsian group; for circle homeomorphisms in Section 1 of Chapter 3, we take a weak limit as the number of sides goes to infinity to study tesselations of D; and for the solenoid in Section 3 of Chapter 3, we consider suitably quasi-periodic tesselations. Fix a triangulation  of the polygon Q and some assignment of lambda lengths to the arcs in . For any arc in  interior to Q, we may define a corresponding simplicial coordinate using the formula in Definition 4.18 of Chapter 1. Proposition 1.6. A decorated hyperbolic structure on the polygon Q determined by

W Q0 ! LC has .Q0 / coplanar in Minkowski space if and only if all simplicial coordinates vanish for some, and hence any, triangulation of Q. Proof. This follows directly from Lemma 4.17 of Chapter 1. There is a natural baby version of the “convex hull construction” we shall subsequently study already here in this case of decorated hyperbolic structures on polygons. Namely, given W Q0 ! LC , we may take the convex hull of the set .Q0 / in the underlying vector space structure of Minkowski three-space to determine a convex body. Since SOC .2; 1/ acts linearly, the orbit class of this body is actually well defined. We might take the underlying combinatorial class of this convex body as an invariant of the SOC .2; 1/-orbit of and decompose Tz .Q/ into subsets with common combinatorial invariant. In analogy to the upcoming ideal triangulation of the decorated Teichmüller space of a punctured surface, one might hope to show that these decomposition elements are cells, but we do not know how to prove this. We shall in Theorems 5.15 and 5.19 of Chapter 4 give a related construction that indeed gives a natural ideal triangulation of Tz .Q/ when we study bordered surfaces in Section 5 of Chapter 4, but the construction of the combinatorial invariant is different from what we have just discussed. In effect, we double the n-gon along its boundary arcs to produce an n-times punctured sphere supporting the natural involution swapping the two copies of the n-gon and rely on the decorated hyperbolic structures on the punctured sphere that are invariant under this involution. 1.2 Cyclic polygons. We turn finally to the other principal topic of this section: polygons that inscribe in a circle. Definition 1.7. A cyclic Euclidean polygon P is a closed finite-sided convex polygon P in the plane R2 whose vertices inscribe in a circle C  R2 . We say that P is r-cyclic

52

2 Lambda lengths in finite dimensions

if C has radius r. There are two essential cases: either the center C lies in P and P is an on-center cyclic polygon, or it does not and P is an off-center cyclic polygon. An off-center polygon has a unique longest edge b1 , which separates P from the center C , and in any case, the Euclidean edge-lengths b1 ; b2 ; : : : ; bn (which we shall identify with the edges themselves in the rest of this section) of course satisfy the strict generalized triangle inequalities: X bi ; for any i D 1; 2; : : : ; n. bj < i¤j

The connection with lambda lengths is basic and easy to explain: Suppose that uk 2 LC , for k D 1; 2; : : : ; n, is a collection of coplanar points in LC in Minkowski space R3 lying in an elliptic plane … 2 R3 , i.e., … \ LC is an elliptic conic section. As noted before, there is then an element of SOC .2; 1/ PSL2 .R/ taking … to a horizontal plane …h in R3 at height z D h, and this Möbius transformation is unique up to post-composition with elliptics fixing .0; 0; 1/ 2 R3 . A small further calculation shows that the Minkowski pairing restricts on any horizontal plane …h to twice the Euclidean metric induced on …h by the standard Euclidean metric on R3 …h (as was mentioned before), and the intersection LC \ …h is a round circle in this induced structure. Thus, the coplanar collection of points fuk gn1  LC \ …h determines a cyclic Euclidean polygon p P in the plane provided the points lie in an elliptic plane, and the lambda length hui ; uj i, for i ¤ j , agrees with twice the Euclidean length of the corresponding chord or side of P . It is natural to ask: How can we characterize or parametrize decorated hyperbolic structures on a polygon that lie in an elliptic plane? We answer this question completely here by studying cyclic Euclidean polygons: Theorem 1.8. Suppose that .b1 ; b2 ; : : : ; bn /, for n  3, is a tuple of positive numbers satisfying the strict triangle inequalities X bi ; for j D 1; 2; : : : ; n. bj < i¤j

Then there exists a unique r > 0 and an Euclidean r-cyclic polygon P , unique up to orientation-preserving isometry of the plane, whose Euclidean edge-lengths in order starting from some vertex of P are the given tuple .b1 ; b2 ; : : : ; bn /. One can think of this as a generalization of the side-side-side property of Euclidean triangles. One might also think of the uniqueness part of this as a version of Cauchy’s thesis on flexibility of convex surfaces, where of course planar polygons are not in general determined by their edge-lengths, but it follows from this theorem that cyclic polygons are so determined. The existence part of this theorem further gives the cyclically ordered tuple .b1 ; b2 ; : : : ; bn / as a complete modulus for isometry classes of cyclic Euclidean polygons.

1 Cyclic and decorated polygons

53

More to the point for applications using the discussion before on elliptic planes, the cyclically ordered tuple of lambda lengths of the edges of P gives a complete modulus for Möbius-orbits of configurations of finitely many points in LC constrained to lie in elliptic planes. For completeness, we shall repeat the proof of Theorem 1.8 from [128] and define p †.a; b; c/ D .a C b C c/.a C b  c/.a C c  b/.b C c  a/=abc for any triple a, b, c satisfying the usual strict triangle inequalities. Lemma 1.9. Suppose that P is a triangle with edges cyclically ordered a, b, c with some choice of first edge a. Then P is 1-cyclic if and only if †.a; b; c/ D 1. Furthermore, P is off-center with first edge long if and only if a2 > b 2 C c 2 . Proof. Heron’s formula relates the semi-perimeter s D 12 .a C b C c/ and the area A of P by p A D s.s  a/.s  b/.s  c/; and A is easily seen to also be given by A D abc=4r, where r is the radius of the circle circumscribing the triangle with consecutive edge-lengths a, b, c. Equate these two expressions for A and solve for † D 1=r to prove the first part. The second part follows from the Euclidean Law of Cosines. Proof of Theorem 1.8. Suppose without loss that b1  bi for all i D 1; 2; : : : ; n. The idea of the proof is to inscribe line segments in R2 of the consecutive prescribed lengths in a circle Cr of some large radius r  b1 =2. We may then vary r and apply the Mean Value Theorem to prove existence. In fact, if r > b1 =2, then there are two ways to inscribe the first arc of length b1 in Cr so that the arc has .r; 0/ 2 Cr  R2 as an endpoint, where the two possibilities correspond to the on- and off-center cases. We shall proceed more analytically and let ˇi .r/ D sin1 .bi =2r/;

for i D 1; 2; : : : ; n,

be half the angle subtended by a chord of length bi in Cr . Each ˇi is a strictly monotone decreasing function of r, and we define

.r/ D   ˇ1 .r/;

.r/ D ˇ1 .r/:

There is thus an r-cyclic n-gon realizing the tuple .bi /n1 as edge-lengths if and only if !.r/ D

n X

ˇi .r/ 2 f .r/; .r/g;

iD2

where !.r/ D .r/ gives an on-center and !.r/ D .r/ gives an off-center polygon. In fact, ˇ D .r/ and ˇ D .r/ are the respective upper and lower sheets of r D 1 b csc ˇ. 2 1

54

2 Lambda lengths in finite dimensions

To prove the existence, notice that !.r/ (and also .r/ and .r/) is a strictly monotone decreasing (increasing and decreasing, respectively) function of r, and limr!1 .r/=!.r/ D b1 =

n X

bi < 1;

iD2

limr!1 .r/=!.r/ D 1: On the other hand for r D b1 =2, either !.b1 =2/ 

 D .b1 =2/ D .b1 =2/; 2

which shows that !.r0 / D .r0 / for some unique r0  b1 =2, or perhaps !.b1 =2/ <

 ; 2

which shows that !.r1 / D .r1 / for at least one r1  b1 =2. This completes the proof of existence. For uniqueness, we first claim that .r/  !.r/ is a strictly monotone decreasing function whenever it is non-negative. To this end, compute tan ˇi dˇi D ; dr r whence

n 1 X d tan ˇi  tan ˇ1 : ..r/  !.r// D dr r iD2

However, an easy induction proves that if 2 C    C m  1  i D 1; 2; : : : ; m  3, then

 2

for i > 0 with

tan 2 C    C tan m < tan 1 ; from which the claim follows. A final application of the Mean Value Theorem shows that !.r/ equals one of

.r/; .r/ for exactly one value of r, as desired. Corollary 1.10. Fix a tuple .1 ; 2 ; : : : ; n / of putative boundary lambda lengths of inequalities j < P an abstract polygon that satisfy the generalized strict triangle C k¤j k , for each j D 1; 2; : : : ; n. Then there is a unique SO .2; 1/-orbit of decorated ideal hyperbolic n-gons realizing these numbers as boundary lambda lengths whose corresponding vertices in LC lie in an elliptic plane. Furthermore for any triangulation  of Q, we may compute lambda lengths of arcs in  and hence corresponding simplicial coordinates on arcs in  interior to the polygon according to Definition 4.18 of Chapter 1, and all of these simplicial coordinates necessarily vanish.

2 Decorated Teichmüller spaces

55

Proof. The first part follows from Corollary 1.10 and our earlier discussion noting that the plane containing .Q0 / is automatically elliptic by Lemma 4.11 of Chapter 1. The second part follows from Proposition 1.6. In light of Corollary 1.10, we may think of fixing the tuple bE of side lengths of a cyclic n-gon starting with some chosen edge, which thus determines the length (the lambda length of a decorated hyperbolic polygon whose vertices lie in an elliptic plane, E of the chord or twice the Euclidean length according to the previous discussion) j;k .b/ connecting the jth and kth vertices, for distinct j; k D 1; : : : n. For example, consider a quadrilateral as before with diagonals e and f , where e separates boundary edges a, b from c, d , and let E and F denote the simplicial coordinates of edges e and f respectively. By Proposition 1.6, the corresponding four points in LC are coplanar if and only if E D 0, or equivalently F D 0. Moreover, Ptolemy’s equation Corollary 4.16a and the formula in Definition 4.18 of Chapter 1 for simplicial coordinates give s .ac C bd /.ad C bc/ ; eD .ab C cd / s .ac C bd /.ab C cd / f D : .ad C bc/ Even for pentagons, we do not know a closed-form expression for the lengths of chords in terms of side lengths, but one can further manipulate the analogous formulas to find that the chord lengths are a certain real root of polynomials of degree ten. In fact, we have come to think of these functions j;k as a kind of “primitive” of our theory. E for various j; k on the Furthermore, the problem raised here of calculating j;k .b/ edges of a triangulation of a cyclic polygon in terms of the boundary lengths bE is the simplest (relative) version of the basic “arithmetic problem” in decorated Teichmüller theory which we shall discuss at length in Section 2 of Chapter 4. In effect, the general arithmetic problem of solving for lambda lengths from simplicial coordinates specializes here to the case of vanishing simplicial coordinates interior to a polygon. In both cases, one finds concrete solvable – but not explicitly solvable – systems of algebraic equations on lambda lengths. We have long hoped that this might lead to a fruitful Galois-theoretic interpretation of our theory but so far without satisfaction.

2 Decorated Teichmüller spaces We shall give in this section based on [128], [131], [139] parametrizations of several versions of decorated Teichmüller spaces, and we begin with the basic version discussed in Chapter 1 for a punctured surface F D Fgs with s  1.

56

2 Lambda lengths in finite dimensions

2.1 Punctured surfaces. A point of the Teichmüller space T .F / is a conjugacy class of discrete and faithful representations  W 1 .F / ! PSL2 .R/ of the fundamental group into the Möbius group so that peripheral elements map to parabolics as we have discussed in Section 1.2 of Chapter 1. The subgroup  D .1 .F // < PSL2 .R/ acts by isometries on the hyperbolic plane, say in the Poincaré disk model D, and there is thus an induced hyperbolic structure on the surface F D D= . Equivalently in the Minkowski model, we identify  with a subgroup (of the same name)  < SOC .2; 1/, and the hyperbolic structure on the surface is given by F D H= . A point of the decorated Teichmüller space Tz .F / D T .F /  Rs>0 of F D Fgs includes also the further specification of one positive real number to each puncture, and the geometric interpretation of these further parameters is as follows. 1 Definition 2.1. Suppose that x 2 S1 is the fixed point of some parabolic transformation in  acting on D. A horocycle in the hyperbolic plane centered at x projects into the surface F D D= to a closed curve in F , and we refer to such a curve as a horocycle in F , which is possibly immersed but not embedded; we say that the horocycle is centered at the puncture of F corresponding to x.

Since a horocycle in F is closed, it has a well-defined finite hyperbolic length in the hyperbolic structure on F , and it is this length that we shall identify with the additional real parameter in the fiber of Tz .F / assigned to the corresponding puncture of F . Definition 2.2. A decorated hyperbolic structure on F D Fgs is the specification of a conjugacy class of Fuchsian groups  as above together with the additional data of an s-tuple of horocycles, one about each puncture. Taking hyperbolic lengths of horocycles as coordinates on the fibers Rs>0 of the decorated bundle Tz .F / ! T .F /, this gives a geometric interpretation to the points of Tz .F /. Let us emphasize that though a horocycle is necessarily a closed curve, it is not necessarily a simple curve in F though a “short enough” horocycle (depending upon the group ) is always a simple curve separating the puncture from the rest of the surface: Lemma 2.3. If a collection of distinct horocycles centered at punctures of a hyperbolic surface each has hyperbolic length less than one, then the horocycles are disjointly embedded. Proof. We shall first show that each unit horocycle is embedded. To this end, given a Fuchsian group  < PSL2 .R/ with primitive parabolic A 2 , we may conjugate    in PSL2 .R/ so that A W z 7! z C 1, i.e., A D 10 11 . Choose any B D ac db 2  so that B is not a power of A.

57

2 Decorated Teichmüller spaces

We claim that then jcj  1. To see this, define further elements of  by B1 D B and BkC1 D Bk A Bk1    1 1 dk ak bk D 0 1 ck dk ck   1  ak ck ak2 D ; ck2 1 C a k ck

bk ak



k1

for k  1. If jcj < 1, then ck D c 2 tends to zero as k increases, which implies that also ak ; bk ; dk tend to one. Thus, Bk tends to A, which would violate discreteness of , so indeed, it must be that jcj  1. Now, B.1/ D ac , and the unit horocycle H centered at infinity lies at height one. We claim that B maps H to a horocycle of Euclidean diameter at most c 2 . The y imaginary part of Bz is given by jczCd , where z D x C iy, so the Euclidean diameter j2 of B.H / is given by ˇ ˇ a.x C i / C b

diam B.H / D supx2R ˇˇ

c.x C i / C d ˇ ˇ ˇ ˇ 1 1 ˇ D supx2R ˇˇ ˇ d c xCi C c D

ˇ



a ˇˇ cˇ

1 1 jcj2

as required. Finally, we show that unit horocycles are disjointly embedded. We may again conjugate so that a primitive parabolic A 2  maps z ! z C 1 and let H denote the horocycle at height one in the upper half-plane U. If another hororcycle H 0 in U meets H , then it must have diameter at least 1, from which it follows that A.H 0 / \ H 0 ¤ ;. Thus, no horocycle H 0 in U whose projection to the surface is embedded can meet H . As a point of notation, we shall typically regard  2 T .F / suppressing the fact z 2 Tz .F / denote the that  is only defined up to conjugacy, and we shall likewise let  specification of underlying group  together with a decoration. z 2 T .F /, Definition 2.4. To any ideal arc ˛ connecting punctures in F and any  z in the natural way as the lambda we may define the associated lambda length .˛I / length of any -geodesic representative for ˛ in the hyperbolic plane with decoration z induced from . In other words, straighten ˛ to a -geodesic in F , truncate using the horocycles from the decoration, let ı denote the signed hyperbolic length of this truncated geodesic,

58

2 Lambda lengths in finite dimensions

p z D e ı . In still other words by Lemma 4.5 of Chapter 1, a lift of the and set .˛I / -geodesic representative of ˛ to H is asymptotic to a pair of rays in LC , there are unique points u, v in these rays corresponding via affine duality in Lemma p 2.11 of z D hu; vi. Chapter 1 to the decoration, and the lambda lengths are given by .˛I / Theorem 2.5. For any ideal triangulation  of Fgs with s  1, the natural mapping ƒ W Tz .Fgs / ! R >0 ;

z 7! .˛ 7! .˛I //; z 

is a real-analytic homeomorphism. Proof. We must produce the inverse to ƒ and so suppose there is a positive real number assigned to each arc in . Let Fz ! F denote the topological universal cover z of arcs decomposing Fz of F D Fgs . The ideal triangulation  lifts to a collection  into triangular regions, and to each such arc is associated the real number assigned to its projection. We shall construct a corresponding collection of decorated geodesics in D, or equivalently, a collection of pairs of points in LC . Choose one of these topological triangles in Fz , call it t0 , and choose an ideal N v, N w, N where u, v, w is the standard triangle in D, say the triangle T0 spanned by u, light-cone basis for Minkowski space. The orientation of Fz induced from that on F gives a cyclic ordering to the vertices of t0 which corresponds to a cyclic ordering of u, v, w as a positively oriented basis for Minkowski space. According to Corollary 4.8 of Chapter 1, there are unique points in u, v, w realizing the numbers assigned to the edges of t0 as corresponding lambda lengths. This describes a lift of t0  Fz to the triangle T0  D plus a lift of the ideal points of t0 to a triple of points in LC covering the vertices of T0 , and this completes the basis step of our inductive construction of the inverse to ƒ . For the inductive step, consider one of the three triangular regions t adjacent to t0 in Fz . The common edge t \ t0 has already been lifted to D and its vertices to LC in the basis step. By Lemma 4.14 of Chapter 1, there is a unique lift T of t to D and its vertices to LC that agrees with the lift of t0 so that the lift to D of the vertex of t disjoint from t0 is separated from T0 by the lift of t \ t0 and the specified numbers on the edges of t are realized as lambda lengths. We may likewise uniquely lift to D and LC the other two triangular regions of Fz adjacent to t0 . Continue recursively in this way to define a mapping  W Fz ! D together with lifts z to LC . This mapping  is continuous and injective by of the ideal points of  D ./ construction (since we always choose the point in Lemma 4.14 of Chapter 1 separated from what has previously been constructed), and we claim it is also surjective. To this end, suppose that z 2 D and consider the sequence of triangular regions t0 ; t1 ; : : : defined recursively as follows: if z … .ti /, then there is a unique adjacent triangular region tiC1 so that .tiC1 / and z lie in the same component of D.ti /. This sequence of triangles either terminates with z in the image of  as desired or else it continues indefinitely. Passing from triangle ti to triangle tiC1 there are two cases depending upon whether we turn left or right, and there are thus two basic cases for a possible semi-infinite

2 Decorated Teichmüller spaces

59

sequence: either there are infinitely many left and infinitely many right turns in the sequence, or else the sequence ends with an infinite sequence of consecutive left turns or ends with an infinite sequence of consecutive right turns. In the latter case for instance if the sequence ends with an infinite sequence of common turns tk ; tkC1 ; : : : , for some k  0, then all of the triangles .tk /; .tkC1 /; : : : share a common ideal point. Now, there are only finitely many arcs in  and therefore only finitely many possible lambda lengths, hence there are only finitely many possible h-lengths of sectors of triangles complementary to  by Lemma 4.9 of Chapter 1. In particular, these h-lengths are bounded below. Conjugating by the Cayley transform and an appropriate Möbius transformation to guarantee that the common ideal point of the triangles .tk /; .tkC1 /; : : : is the point at infinity in U, it follows by hypothesis that the sequence of common turns must terminate, which is a contradiction. In the former case, there are infinitely many pairs of consecutive triangles tk ; tkC1 so that the type, right or left, of the turn tk1 ; tk is different from the type of tk ; tkC1 . The pair of triangles .tk /; .tkC1 / determines an ideal quadrilateral, and again since there are only finitely many possible lambda lengths, there are bounds above and below on the cross ratio of this ideal quadrilateral by Corollary 4.16b of Chapter 1. Since the distance between opposite sides of an ideal quadrilateral of bounded cross ratio is itself bounded below by Corollary 4.16d of Chapter 1, it follows that there can be only finitely many such pairs tk ; tkC1 , which is again a contradiction. The mapping  W Fz ! D is therefore a continuous bijection and indeed a homeomorphism. Let us now define a homomorphism  W 1 .F / ! PSL2 .R/ which leaves invariant the collection  of geodesics. Choose any fundamental domain D for the action of z The action of 1 .F / on Fz which is a union of triangular regions complementary to . 1 .F / identifies various pairs of frontier edges of D, and we may consider such a pair of edges e0 and e1 in the frontier of D, say with .e1 / D e0 for  2 1 .F /. There is z with ej in its frontier, for j D 1; 2, so that a unique triangle tj complementary to  t0  D and t1 6 D. There is then a unique Möbius transformation . / mapping .t1 / to .t0 / and mapping .e1 / to .e0 / by Corollary 4.2 of Chapter 1. We may define ./ in this manner for each such pairing  2 1 .F / of edges of D to define a homomorphism  W 1 .F / ! PSL2 .R/, where there are no relations to check since 1 .F / is a free group for a punctured surface F . It follows by induction that the representing group  D .1 .F // < PSL2 .R/ leaves invariant the collection  of geodesics by the uniqueness statement in Lemma 4.14 of Chapter 1. Thus, the conjugacy class of the representation  is independent of the choice of fundamental domain D, and  W Fz ! D is equivariant for the actions of 1 .F / on Fz and  on D. Furthermore, the representing group  is discrete since a sequence of Möbius transformations accumulating at the identity could not leave  invariant, the representation  is faithful since  is injective, and  maps peripherals to parabolics by construction. Thus,  indeed determines an element of T .F /, and it remains only to observe that the construction furthermore determines a -invariant collection of points in LC

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2 Lambda lengths in finite dimensions

1 lying over the parabolic fixed points of  in S1 , which descends to a decoration on the hyperbolic surface. This construction provides a two-sided inverse to the function ƒ . Just to be clear, the real-analytic structure of Teichmüller space was defined in the first section in terms of entries of the representing matrices for a set of generators, and the decorated bundle over Teichmüller space is given the induced real-analytic structure. That ƒ and its inverse are real-analytic follows from the fact that matrices in PSL2 .R/ representing covering transformations can be explicitly computed real analytically in terms of lambda lengths from the Basic Calculation 4.13 of Chapter 1 as was described above in the construction of  from a choice of fundamental domain.

Example 2.6. An ideal triangulation of the once-punctured torus F D F11 is illustrated in Figure 1.4 of Chapter 1. Indeed, it is not hard to confirm that any ideal triangulation is related to this fixed one (or indeed, therefore, to any other) by a homeomorphism of F , which is a special property of the once-punctured torus. In any case, the Basic Calculation 4.13 of Chapter 1 gives an explicit fundamental domain for the action of the underlying Fuchsian group, which can also be simply calculated from the lambda lengths in this case. Several remarks are in order. One might think of the specification of ideal triangulation  of F as a kind of choice of “basis” for these lambda length coordinates. It is also worth saying explicitly that the basis step of this inductive proof, i.e., the choice of triangular region t0 in the universal cover and the choice of ideal triangle spanned by u, N v, N w, N corresponds to normalizing to “kill” the quotient by conjugacy in the definition of Teichmüller space. We hope that the reader, much as the author, comes away from this result with a firm understanding of what is Teichmüller space: fixing a pattern of gluing triangles to get the specified topological surface F , the Teichmüller space of F corresponds to all possible consistent ways of gluing ideal triangles in the specified pattern. The consistency conditions on the gluings arise from the requirement that the resulting metric be complete and can be understood by considering the local picture of the glued triangles near a puncture. Namely, choose a point on an oriented ideal arc incident on a puncture and traverse the horocycle centered at the puncture passing through this chosen point in a given triangle to determine a specified point on the next consecutive arc incident on this puncture; this point, in turn, determines another horocycle in the next triangle and hence a specified point on the next consecutive arc, and so on. After a finite number of such steps, we return to the initial oriented ideal arc, and the point so determined may or may not agree with the initial choice of point on this arc depending on the nature of the gluings. One can see without difficulty that these points agree if and only if the resulting metric is complete near the puncture. Thus, the triangles cannot be glued together willy-nilly: there is one consistency condition for each puncture imposed by completeness of the resulting metric. It is worth emphasizing that a real convenience of decorated Teichmüller theory is that

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61

there are no such consistency conditions on lambda lengths as we have just seen. We shall further study these conditions on (undecorated) Teichmüller space in Theorem 4.2. Lemma 2.7. Lambda lengths are natural for the action of the mapping class group, z denotes the push-forward of metric and decoration, i.e., if  2 MC.Fgs /, and  ./ then z z D ..e/I  .// for any ideal arc e. .eI / Proof. This follows directly from the definitions of lambda lengths and push-forward.

Corollary 2.8. Suppose that  is an ideal triangulation of Fgs and ƒ is an assignment of lambda lengths to the ideal arcs in . If a mapping class on Fgs leaves  invariant and preserves ƒ, then this mapping class is an isometry of the corresponding hyperbolic surface. Proof. Manipulating the formula in the previous lemma for some mapping class , z D . 1 .e/I /. z Thus, if  preserves  and respects lambda we have .eI  .// z z i.e.,  acts by (decorated) lengths, then the coordinates of  ./ agree with those of , hyperbolic isometry. As was mentioned already, the proof of Theorem 2.5 in particular gives an effective algorithm for calculating holonomies, i.e., matrices representing . / for  2 1 .F /, in terms of lambda lengths (see the proof of Theorem 4.4 for another more elegant algorithm to this end), and there is the following special case of particular interest. According to Lemma 3.4 of Chapter 1, the modular group PSL2 .Z/ leaves invariant the Farey tesselation, so any subgroup must also leave it invariant. In particular, if  < PSL2 .Z/ is a finite-index subgroup without elliptic elements, then the Farey tesselation descends to an ideal triangulation on the surface D= , and from the very definition of the Farey tesselation using horocycles, this surface admits a decoration with all lambda lengths equal to one. Since a punctured arithmetic surface corresponds precisely to a finite-index subgroup of the modular group without elliptics, we have: Corollary 2.9. The collection of punctured arithmetic surfaces corresponds to the set of all ideal triangulations with lambda lengths identically equal to one. Furthermore, the topological symmetry group of the ideal triangulation is the hyperbolic isometry group of the corresponding surface. Theorem 2.10. For any surface F D Fgs with s  1, the action of MC.F / on lambda lengths with respect to a fixed ideal triangulation is described by permutation followed by finite compositions of Ptolemy transformations. In particular, only the operations of addition, multiplication and division are required, i.e., there are no minus signs 1 . 1

Thus, these formulas can be “tropicalized”, cf. Section 5.4 of Chapter 5.

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2 Lambda lengths in finite dimensions

z D . 1 .e/I / z for Proof. As before, naturality of lambda lengths gives .eI  .// z 2 Tz .F /. Given an ideal triangulation, we may any  2 MC.F /, ideal arc e, and  thus consider the ideal triangulation  1 ./ and simply pull-back the lambda length of e 2  to  1 .e/ to assign lambda lengths on the ideal arcs in  1 ./ which describe z By the Whitehead’s Classical Fact 1.24 of Chapter 1, there is a finite sequence  ./. of flips beginning with  1 ./ and ending with . In general, there is a one-to-one correspondence between the ideal arcs of the ideal triangulations before and after a flip, where one identifies the arcs not involved in the flip, and thus in particular a one-to-one correspondence between ideal arcs in  1 ./ and . Together with the identification of ideal arcs in  and  1 ./ induced by , there is thus an overall permutation of the arcs in  itself. Since the effect of a flip on lambda lengths is given by a Ptolemy transformation by Corollary 4.16a of Chapter 1, the result follows. There is a two form simply expressed in the coordinates of Theorem 2.5 which is moreover independent of the chosen ideal triangulation as follows. Fix some ideal triangulation  of F D Fgs and define a two form in coordinates with respect to  by X ! D 2 d log a ^ d log b C d log b ^ d log c C d log c ^ d log a; where the sum is over all triangles complementary to  in F whose edges have lambda lengths a, b, c in this clockwise cyclic ordering as determined by the orientation of F . Proposition 2.11. This two form is well defined independent of ideal triangulation, i.e., if ; 0 are two ideal triangulations of F D Fgs , then ! D !0 as two forms on Tz .F /. Proof. The computations follow those of Proposition 1.5 using Whitehead’s Classical Fact 1.24 of Chapter 1. In fact, this two form is the pull-back to Tz .F / of the Weil–Petersson Kähler two form on T .F / as we shall prove in the next section; it follows from general facts that it is therefore invariant under MC.F /, but as we have just seen, it enjoys the more general invariance under flips. Since the Weil–Petersson two form is non-degenerate on Teichmüller or moduli space, it furthermore follows that the tangent vectors to the fibers of the forgetful map Tz .F / ! T .F / span the vector subspace of the tangent space to Tz .F / whose contractions with our pull-back two form vanish; in other words, scaling all of the lambda lengths meeting a given puncture (where if an arc has both its endpoints at the puncture, then we scale by the square) leaves invariant this two form, and the corresponding s  1 vector fields span the degeneracies of the two form. A direct proof of this fact is described in Remark 4.14. 2.2 Partially decorated surfaces. (Extrapolated from the addendum to [128].) Consider a surface F D Fgs with s  1 as before, and choose among the punctures of F a distinguished non-empty set P . Obviously, there is only one such choice when s D 1.

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2 Decorated Teichmüller spaces

Definition 2.12. A partial decoration of F on P is the specification of one horocycle centered at each puncture in P . Define the P -decorated Teichmüller space TzP .F / to be the trivial bundle over T .F / where the fiber over a point is the space of all tuples of horocycles, one horocycle centered at each puncture in P . Definition 2.13. Define a quasi triangulation based at P to be the isotopy class of a collection of disjointly embedded ideal arcs so that each complementary region is either a triangle or a once-punctured monogon with its vertices in P . There are 6g  6 C 2s C #P ideal arcs in a quasi triangulation based at P , s  #P complementary once-punctured monogons, and 4g  4 C s C #P complementary triangles. It is not true that flips act transitively on quasi triangulations of a fixed surface, however, flips together with an additional combinatorial move do act transitively. The additional move is defined as follows: Definition 2.14. If an ideal arc a in a quasi triangulation  based at P decomposes a 1 into a triangle and a once-punctured monogon, bordered once-punctured bigon F0;.2/ then the quasi flip along a removes and replaces a with the unique ideal arc b ¤ a in 1 1 F0;.2/ so that b also decomposes F0;.2/ into a triangle and a once-punctured monogon as illustrated in Figure 2.1; again, we denote the resulting quasi triangulation a D  [ fbg  fag. c

c bD

a

.cCd /2 a

quasi flip

d

d

Figure 2.1. Quasi flip on a quasi triangulation.

z 2 TzP .F / and any ideal arc e connecting points of P , there is again Given  z defined as before. It is a small exercise using Ptolemy’s a lambda length .eI / 1 equation to verify that if F0;.2/ has boundary arcs c, d , then b D .c C d /2 =a, and we shall refer to this as a quasi Ptolemy transformation. We have the following result summarizing the material of this section for partially decorated surfaces: Theorem 2.15. For any quasi triangulation  of Fgs based at P , the natural mapping ƒ W TzP .Fgs / ! R >0 ;

z 7! .e 7! .eI //; z 

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2 Lambda lengths in finite dimensions

is a real-analytic homeomorphism. Furthermore, the action of PMC.F / on lambda lengths with respect to a fixed quasi triangulation is described by permutation followed by finite compositions of Ptolemy transformations and quasi Ptolemy transformations. Finally, the two form ! defined exactly as before, summing only over the triangles complementary to  in F and ignoring the once-punctured monogons, is well defined independent of the quasi triangulation . Proof. We may extend the quasi triangulation to an ideal triangulation by adding the unique ideal arc from undecorated puncture to decorated puncture in each complementary once-punctured monogon, and we may assign any lambda length to each of these added ideal arcs. Follow through the proof of Theorem 2.5 verbatim and notice that by the last sentence in Lemma 4.14 of Chapter 1, the resulting representation of the fundamental group in the Möbius group is independent of the choices of lambda lengths on the added ideal arcs. The proof that the action of PMC.F / is as stated follows exactly as before. Note that instead of the pure mapping class group, we could actually have taken the subgroup of MC.F / that fixes the distinguished punctures P setwise. To see that the two form is invariant, adopt the notation in the definition of quasi Ptolemy transformations, where ab D .c C d /2 , and compute as in the proof of Proposition 2.11: dQ cQ C .cQ  dQ /



.cCd /2 a



Q

Q d D dQ cQ  cQ aQ  aQ dQ C 2.cQ  dQ / c cCd cCd

D dQ cQ C aQ cQ C dQ aQ C 2cQ dQ D cQ dQ C aQ cQ C dQ a; Q as desired. The proof that finite compositions of flips and quasi flips together act transitively on all quasi triangulations of Fgs will be given in Section 5.1 of Chapter 4. A case of particular interest, which we shall further discuss later (cf. Theorem 4.2), is when P is a singleton, i.e., among the punctures, we have chosen a distinguished one. In this case, let us note that changing the partial decoration simply scales each lambda length by a common amount; indeed, moving the horocycle a hyperbolic distance d changes the hyperbolic length along geodesics by the amount 2d , and hence scales the lambda lengths by an amount ed . This gives: Corollary 2.16. For any quasi triangulation  of Fgs based at a single puncture, projective classes of lambda lengths of ideal arcs in  give a real-analytic parametrization of Teichmüller space T .Fgs / Tzfpg .F /=R>0 ; the action of the mapping class group is given by permutation together with Ptolemy and quasi Ptolemy transformations, and the two form described before is well defined independent of quasi triangulation.

2 Decorated Teichmüller spaces

65

More generally, it is often useful to consider the projectivization TzP .F /=R>0 , where R>0 acts on lambda lengths by homothety, or in other words, by translating each horocycle in the decoration of F by some fixed hyperbolic length. Example 2.17. For a multiply punctured sphere F D F0s , with s  4 and a distinguished puncture p, we compute the action of the subgroup of MC.F / fixing p on the Teichmüller space T .F / D Tzfpg .F /=R>0 , i.e., we compute the representation of the braid group of the sphere in projectivized lambda length coordinates. Let fci gs1 denote a collection of ideal arcs based at p disjointly embedded in F 1 each bounding a once-punctured monogon. The complement of these regions in F is an ideal .s  1/-gon R, which we suppose has consecutive edges c1 ; : : : ; cs1 in this order. Let fdi gs1 denote the diagonals of R where di separates ci ; ciC1 from the other 1 z 2 Tzfpg .F / edges and regard the indices as cyclic so that c1 D cs and so on. Fix some  and let z i D .ci I /; z

i D .di I /; for i D 1; : : : ; s  1. z The proof We claim that this collection of lambda lengths uniquely determines . is by induction, and the claim is trivial for s D 4; 5; 6 since in these cases the collection fci ; di gs1 contains an ideal triangulation of F . For the inductive step, let ei be the 1 diagonal of R separating ci1 , ci , ciC1 from the rest of the boundary of R. By the Ptolemy equation, we have z D 1 i D .ei I / i . i1 i  i1 iC1 /; for i D 1; : : : ; s  1. Cutting R along d1 produces a triangle and an .s  2/-gon whose edge lambda lengths are given and whose diagonal lambda lengths are either given or among the fi gs1 completing the proof of the claim. i Our technique is to keep track of the over-determined set i ; i of lambda lengths, for i D 1; : : : ; s  1. Let j denote the half right Dehn twist along the non-trivial curve in F that is homotopic to dj [ fpg, for j D 1; : : : ; s. These half Dehn twists generate the subgroup of mapping classes fixing p. is easily computed as a quasi Ptolemy transformation. The action of j˙1 on fi gs1 1 Only j ; j C1 are affected, and indeed, we have 2 jC1 W .j ; j C1 / 7! .j1 C1 .j C j / ; j /;

jC1 W .j ; j C1 / 7! .j C1 ; j1 .j C1 C j /2 /: . Likewise, the action of j˙1 affects only the parameters j 1 and j C1 among f i gs1 i ˙1 Neither ej nor ej C1 are affected by j , so the effect on j ˙1 can again be determined

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using Ptolemy transformations, namely, jC1 W . j 1 ; j C1 /   2 1 2 7! j1 j j1 C1 .j C j / C j 1 j ; j j C j C1 j C2 .j C j / ; j1 W . j 1 ; j C1 /   7! j1 j j C1 C j 1 j1 .j C1 C j /2 ; j C1 j1 .j C1 C j /2 C j C1 j C2 : This completes our discussion of the braid group of the sphere. Definition 2.18. For each non-empty subset P  f1; : : : ; sg of decorated punctures, define the perimeter functions `i W TzP .F / ! R0 for each i D 1; : : : ; s, ´ zP / D the hyperbolic length of horocycle at i 2 P , `i . 0 if i … P , and let  D

Ps

iD1 `i :

Undecorated punctures are thus regarded as having vanishing perimeter. Definition 2.19. We shall let †t denote the open t -dimensional simplex with closure x t for t  0. † Corollary 2.20. The Cartesian product of the forgetful map and the map that assigns the projectivized vector of perimeters `i =, for i 2 P , induces a natural homeomorphism TzP .F /=R>0 T .F /  †#P 1 . Proof. The very definition of partial decoration gives an identification TzP .F / ! T .F /  R#P >0 , where the projection onto the first factor is the forgetful map, and the projection onto the second factor is given by the tuple of perimeter functions. This descends to projective classes since homothety of lambda lengths preserves fibers of the decorated bundle, corresponding to homothety in Minkowski space, and h-lengths are homogeneous functions of lambda lengths of degree minus one (with an explicit formula given by Lemma 4.9 of Chapter 1) because a perimeter function is simply a sum of h-lengths by definition. Thus, the perimeter functions are themselves homogeneous of degree minus one in the lambda lengths. Finally, the projectivization of #P 1 R#P with explicit barycentric coordinates given >0 is given by the open simplex † by `1 =; : : : ; `p =, where  D `1 C    C `p . Definition 2.21. Define the partially decorated Teichmüller space Tz .F / of all partially or fully decorated surfaces to be the Cartesian product of T .F / with all possible specifications of horocycles about the decorated punctures in a set P  f1; : : : ; sg, for all non-empty subsets P . This space inherits a topology from T .F /  Rs0 , where a puncture is undecorated if the length of its horocycle is zero.

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67

Each TzP .F / and Tz .F / inherits a subspace topology from Tz .F / which agrees with its usual topology. Again, R>0 acts on partial decorations by homothety with quotient Tz .F /=R>0 , and we have the immediate: Corollary 2.22. For any surface F D Fgs , the space Tz .F /=R>0 has the natural structure x s1 Tz .F /=R>0 D T .F /  † of the product of T .F / with a closed simplex. The analogous result in the conformal case was first proved in [102]. 2.3 Bordered surfaces. We next treat surfaces with boundary excerpted from [139], where each boundary component is required to contain at least one distinguished point or cusp. In contrast, we shall discuss surfaces with smooth cusp-less boundary in Section 4. s Consider a surface F D Fg;r with boundary components @1 ; : : : ; @r , where r  1, and choose on each @j a non-empty set Dj  @j of distinguished points. By way of notation, if Dj consists of ıj  1 points, for j D 1 : : : ; r, then we define the vector s ıE D .ı1 ; : : : ; ır / and let Fg; ıE denote a fixed smooth surface F with this extra data. Let us remove from F the distinguished points D D D1 [  [Dr on the boundary and double the resulting surface along its boundary arcs so that each point of D gives rise to a puncture of the doubled surface F 0 . This double supports an involution W F 0 ! F 0 defined in the natural way. The arcs in the boundary of F connecting consecutive points of D give rise to a family of ideal arcs in F 0 which we denote by B, where #B D #D D ı1 C    C ır , and the distinguished points in the boundary of F give rise to a family D of punctures of F 0 . s 0 Definition 2.23. A decoration on Fg; ıE is a decoration on the punctures of F arising from the distinguished points on the boundary of F , which we identify with D itself. There is the natural involution W F 0 ! F 0 interchanging the two copies of F , and we s s 0 define the Teichmüller space T .Fg; ıE / of Fg;ıE to be the -invariant subspace of T .F /. s s The decorated Teichmüller space of the bordered surface Tz .Fg; ıE / of Fg;ıE is defined to be the -invariant subspace of TzD .F 0 / in the notation of the previous section, i.e., s the trivial bundle over T .Fg; ıE / where the fiber over a point is the space of all tuples of horocycles, one horocycle about each puncture in D.

Definition 2.24. A quasi triangulation  of the bordered surface F is the restriction to F of any -invariant quasi triangulation of F 0 based at D, so a quasi triangulation of F automatically contains B. There are thus 6g  6 C 3r C 2s C 2.ı1 C    C ır / ideal arcs in a quasi triangulation of F . We may perform flips or quasi flips on suitable ideal arcs in   B, and we may imagine performing the corresponding -equivariant flips or quasi flips in F 0 . Again,

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2 Lambda lengths in finite dimensions

we shall prove (cf. Section 5 of Chapter 4) that flips and quasi flips on ideal arcs in   B act transitively on quasi triangulations of F and have: s Theorem 2.25. For any quasi triangulation  of F D Fg; ıE , the natural mapping

ƒ W Tz .F / ! R >0 ;

z 7! .e 7! .eI //; z 

is a real-analytic homeomorphism. Furthermore, the action of MC.F / on lambda lengths with respect to a fixed quasi triangulation is described by permutation followed by finite compositions of Ptolemy transformations and quasi Ptolemy transformations. Finally, the two form ! defined exactly as before, summing only over triangles complementary to  in F , is well defined independent of the quasi triangulation . Proof. In light of our definitions, this follows directly from Theorem 2.15 and transitivity of flips and quasi flips. We shall give a more intrinsic description of the geometry of bordered surfaces in Section 5 of Chapter 4. Each of these variants, partially decorated surfaces and bordered surfaces, has been of utility; see [92], [116], [102] for examples of the former and [82], [139], [141], [52], [84] for examples of the latter. There is the common generalization of these two cases, which has also been useful, cf. [35], and is treated in the next section. s 2.4 General case. Consider a surface F D Fg; ıE and choose some subset P of the punctures. Let us also choose some subset Q of the distinguished points on the boundary of F , where we demand that each boundary component of F contain at least one point of Q. Equivalently, it is natural to regard a boundary component disjoint from Q as an undecorated puncture in the subsequent discussion. s z Definition 2.26. Fix some surface F D Fg; ıE and suppose P [ Q ¤ ;. Let TP;Q .F / #P C#Q denote the total space of the R>0 -bundle over T .F /, where the fiber over a point is the collection of all tuples of horocycles, one about each point of P and one about each point of Q. Ideal and quasi triangulations of F as well as flips and quasi flips on them are defined as before. s Theorem 2.27. For any quasi triangulation  of F D Fg; ıE based at P [ Q, the natural mapping

ƒ W TzP;Q .F / ! R >0 ;

z 7! .e 7! .eI //; z 

is a real-analytic homeomorphism. Furthermore, the action of PMC.F / on lambda lengths with respect to a fixed quasi triangulation is described by permutation followed by finite compositions of Ptolemy transformations and quasi Ptolemy transformations. Finally, the two form ! defined exactly as before, summing only over triangles complementary to  in F , is well defined independent of the quasi triangulation .

3 Weil–Petersson Kähler two form

69

3 Weil–Petersson Kähler two form This section is dedicated to the derivation [131] of the Weil–Petersson Kähler two form in lambda lengths for punctured surfaces starting from the elegant formulas [169] discovered by Scott Wolpert for this two form in terms of the geometry of the underlying surface. This calculation depends not only on Wolpert’s work but also on Fenchel–Nielsen coordinates, cf. the discussion in Section 1.2 of Chapter 1 or [171] for instance, and on a convexity property of hyperbolic length functions [89] proved by Steve Kerckhoff. Theorem 3.1. Suppose that  is an ideal triangulation of F D Fgs and consider lambda length coordinates on Tz .F / with respect to . The Weil–Petersson Kähler two form pulls back to X 2 d log a ^ d log b C d log b ^ d log c C d log c ^ d log a; where the sum is over all triangles in F  whose edges have lambda lengths a, b, c in an order compatible with that determined by the orientation induced by the clockwise traversal of the boundary of the triangle. Proof. The strategy is as follows. Remove from F small horoball neighborhoods of the punctures to produce the surface H with horocyclic boundary, and double the result along its boundary to produce a closed topological surface S with Teichmüller space T .S/. Let C denote the curves in S arising from the horocycles, one for each puncture. We shall consider limits in T .S/ as the geodesic representative  of C is pinched to a point. According to [170], the WP Kähler two form on T .S / extends in the limit to the desired WP Kähler two form on T .F /. In order to relate lambda lengths in F to hyperbolic lengths of closed geodesics in S, observe that in pinching , nearby hypercycles of  limit to horocycles in the induced structure on F . Indeed, this follows from the evident fact that geodesics in S which are orthogonal to  must limit to geodesics asymptotic to the punctures of F . Now, suppose that  is sufficiently small and consider the annular neighborhood A of  in F bounded by the hypercycles hL , hR so that each of the two annular components of S  .hL [  [ hR / has hyperbolic area unity. If c is a closed geodesic in S with hyperbolic length ` meeting  in two points, then c \ .S  A/ consists of two arcs with endpoints in hL [ hR , and we let `L and `R denote the respective hyperbolic lengths of these arcs in S. Define “generalized lambda lengths” p X D `X for X 2 fL; Rg; so that d ` D d `L C d `R D 2.d log L C d log R /: In light of earlier remarks, d log X limits to the exterior derivative of the logarithm of the lambda length, with respect to the horocycles about the punctures corresponding

70

2 Lambda lengths in finite dimensions

to the limiting hypercycles, of the ideal arc in the pinched surface F corresponding to cX , for X 2 fL; Rg. We recognize the Weil–Petersson Kähler two form by relying on one of Wolpert’s descriptions: Theorem 3.2 (Theorem 3.3 and Lemma 4.5 of [169]). If the geodesic length functions f`i gn1 of geodesic curves fci gn1 give local coordinates on T .S / for the closed surface S , then X !ij D !.d `i ; d `j / D cos ˛p ; p2ci \cj

where ˛p is the angle between ci and cj at the point p of intersection. Furthermore, if .! ij / D .!ij /1 , then we have X ! ij d `i ^ d `j : !D 1ij n

In order to apply this, suppose that  is an ideal triangulation of F . In fact, F inherits a pants decomposition fci gN 1 by doubling the arcs  \ H in the natural way, where N D 6g  6 C 3s. Furthermore, if e 2  and e  is the ideal arc in F that arises from a flip along e, then H \ e  similarly doubles to a closed curve in S . If ci  S arises from e 2   F , then we let ci  S denote the curve so derived from e  and set Q D fci ; ci gN 1 . Using Fenchel–Nielsen coordinates [171] on T .S / with respect to the pants decomposition fci gN 1 and convexity of hyperbolic length functions under Fenchel–Nielsen deformations [89] (together with the fact that ci \ cj D ; unless i D j ), we conclude that the hyperbolic lengths of curves in Q give local coordinates almost everywhere on T .S/, so Wolpert’s Theorem 3.2 is applicable. If x and y are oriented geodesics in S or oriented ideal arcs in F , then we let X cos ˛p : .x; y/ D p2x\y

By the first part of Wolpert’s Theorem, the matrix .!ij /2N i;j D1 of Kähler pairings has the  0 D N -by-N block form D , where 0 denotes the N  N matrix of zeroes, D is the A diagonal matrix with entries Di i D .ci ; ci /, and A D .Aij / with Aij D .ci ; cj /,  1 1 D for i; j D 1; : : : ; N . The inverse matrix is given by D DAD , so by the second 1 0 part of Wolpert’s Theorem, we conclude that the Kähler form ! on T .S / is given by ! D

X

.ci ; cj /

1ij N

.ci ; ci / .cj ; cj /

dci ^ dcj 

N X iD1

1 dci ^ dci ; (2.1) .ci ; ci /

where we identify a closed geodesic curve in S with its hyperbolic length for convenience.

71

3 Weil–Petersson Kähler two form

As before, the geodesic  separates each c 2 Q into a left half cL and right half cR , and we now pinch  retaining the notation cL and cR for the corresponding ideal arcs in F . Of course, if c; d 2 Q, then .c; d / limits to .cL ; dL / C .cR ; dR /.

bX

fX

fX

T aX eX gX

eX

cX g X

aX0 eX

bX

fX cX g X

aX0 eX0

dX

dX

dX

(a)

(b)

(c)

bX

cX

Figure 3.1. Notation near eX .

Now, suppose that eX 2 fcL ; cR W c 2 Qg, for X 2 fL; Rg, adopt the notation of Figure 3.1 for the ideal arcs near eX , and identify an ideal arc with its lambda length for convenience. By the Ptolemy equation in Corollary 4.16a of Chapter 1, we have

 ac.d log a C d log c/ C bd.d log b C d log d / (2.2) d log eX C d log eX D ac C bd X in the obvious notation. Of course, since ! limits to the WP Kähler two form ! 0 on F , the coefficient of deX ^ dfY in ! 0 must vanish whenever e; f 2 Q and fX; Y g D fL; Rg; see the remark following this proof. Furthermore, ci \ cj D ; unless i D j , and ci \ cj D ; unless ci and cj lie in the frontier of a common component of S  [fci gN 1 . Inspection of eqs. (2.1) and (2.2) shows that the coefficient of deX ^ dfX in ! 0 must vanish unless eX ; fX lie in the frontier of a common triangle complementary to  in F , and indeed, there is a contribution to ! 0 from each such triangle. Continuing with the notation of Figure 3.1 with X D R for example, we find from eq.(2.1) that the contribution to 14 ! 0 from the triangle T has the following projection into the subspace spanned by the two form d log aR ^ d log eR :



 1 1 ef ac .a ; e  / C : (2.3)  D .a; a /.e; e  / .a; a / ef C bg R .e; e  / ac C bd R It follows from the expressions for cosines in Corollary 4.16b of Chapter 1 that



  bg  ef ac  bd   /D and .eR ; eR /D ; .aR ; aR bf C ef R ac C bd R

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2 Lambda lengths in finite dimensions

and we claim that   ; eR /D .aR



acef  abcg  bdbg  bdef .ac C bd /.bg C ef /

 : R

To see this, consider the flips indicated in Figure 3.1 and adopt the notation indicated there. By the Ptolemy equation, we have

 bg C ef 0 aR D ; a

0 R  d a d C cg 1 0 cg C .bg C ef / : D Œ4pt eR D e eR a R R Again using the equation for the cosine in Corollary 4.16b of Chapter 1, we find

 cf  be 0   0 0  .aR ; eR / D .aR ; aR /D cf C be R 

cf  be fcg C da .bg C ef /g ; D cf C be fcg C da .bg C ef /g R and the claim follows upon clearing denominators. Define

X D .ac C bd /X and X D .bg C ef /X

for X 2 fL; Rg;

and identify F with the limit of the right side of S (making in this way a global sign convention). Thus, for instance, 1 1 D   .e; e / .eR ; eR / C .eL ; eL / D  bgef  bgCef R

D

1 

 bgef  bgCef L

1 L R : 2 eL fL bR gR  bL gL eR fR

Similarly, we find 1 1

L R ; D .a; a / 2 bL dL aR cR  aL cL bR dR



  1 acef acef   : .a ; e/ D 2

 R

 L Finally plugging these expressions into eq. (2.3), we may compute that 2 D ı , where ı D .bL dL aR cR  aL cL bR dR /.eL fL bR gR  bL gL eR fR /

4 Shear coordinates on Teichmüller spaces

73

and  D L L Œacef R  R R Œacef L  Œef R L .bL dL aR cR  aL cL bR dR / C Œac R L .eL fL bR gR  bL gL eR fR /

 D Œacef R   bd.bg C ef /  bg.ac C bd / L

 Œacef L Œ   bdef  abcg R C Œabcg L Œbdef R C Œbdef L Œabcg R D ı; so that in fact  D 12 . Since the pair a, e was arbitrary (subject, however, to this ordering induced from the orientation on the triangle) and since  was defined to be the coefficient in 14 ! of d log aR ^ d log eR , the proof is complete. Remark 3.3. One can indeed similarly compute directly that the coefficient of deX ^ dfY in ! 0 vanishes for fX; Y g D fL; Rg as was asserted before.

4 Shear coordinates on Teichmüller spaces As an alternative to the parametrization of T .Fgs / given by projectivized lambda lengths on a quasi triangulation based at a single puncture given in Corollary 2.16, we shall in this section discuss using instead cross ratios of adjacent pairs of triangles com1 plementary to an ideal triangulation. To this end, suppose that points j 2 S1 , for j D 1; 2; 3; 4, are the vertices at infinity in this counter-clockwise order of an ideal quadrilateral in D, and triangulate this quadrilateral by the diagonal connecting 1 to 3 . We may conjugate by a conformal map in the two distinct ways: 1) sending 1 7! 0,

2 7! 1, 3 7! 1, which maps 4 to some negative real number 1 ; and 2) sending

3 7! 0, 4 7! 1, 1 7! 1, which maps 2 to some negative real number 2 . It follows from Corollary 4.16c of Chapter 1 that Z D log 1 D log 2 is the signed hyperbolic distance between the orthogonal projections of 2 and 4 to the specified diagonal of the quadrilateral. Furthermore, choosing any decoration on the quadrilateral, say with points uj 2 LC covering j , for j D 1; 2; 3; 4, Corollary 4.16b of Chapter 1 gives ZD

1 hu2 ; u3 i hu1 ; u4 i log 2 hu1 ; u2 i hu3 ; u4 i

D log

.h.u2 /; h.u3 // .h.u1 /; h.u4 // : .h.u1 /; h.u2 // .h.u3 /; h.u4 //

Definition 4.1. We shall call Z the (Thurston) shear coordinate associated to the triangulated quadrilateral.

74

2 Lambda lengths in finite dimensions

The shear coordinate evidently depends only on the cyclic ordering on the vertices of the quadrilateral and the choice of diagonal triangulating the quadrilateral but is independent of decoration as one sees directly from the formula. Likewise given a once-punctured monogon triangulated by an edge e connecting the puncture to the distinguished point on the boundary, consider the lifts of this region to the universal cover of the surface; the edge e lifts to the diagonal of a quadrilateral, and the corresponding shear coordinate is found to vanish. Thus in any case, to an arc e in an ideal triangulation  of Fgs with specified hyperbolic structure  2 T .Fgs /, there is an associated shear coordinate Z .eI /, which happens to vanish if e does not separate distinct triangles complementary to . Theorem 4.2. For any ideal triangulation  of Fgs with s  1, the natural mapping T .Fgs / ! R ;

 7! .e 7! Z .eI //;

is a real-analytic homeomorphism onto the linear subspace determined by the following constraints: for each puncture p of Fgs , we have X

Z .eI / D 0;

where the sum is over all arcs e of  which are asymptotic to p counted with multiplicity, i.e., if e has both endpoints at p, then Z .eI / occurs twice in the sum. Proof. The necessity of the asserted constraints on shear coordinates can be checked directly using the formula Corollary 4.16b of Chapter 1 relative to any decoration on F . Alternatively, shear coordinates are interpreted in Corollary 4.16c of Chapter 1 as distances along geodesics between orthogonal projections, and there is the natural construction of gluing ideal triangles using these shears to produce a hyperbolic structure which is complete if and only if the constraints hold. The remaining argument is entirely analogous to the proof of Theorem 2.5, where we lift  to an ideal triangulation of the topological universal cover Fz of F D Fgs , z in Fz and an ideal triangle in D to begin choose a triangular region complementary to  the recursive construction of the mapping  W Fz ! D determined by the putative shear coordinates, which are again invariant under Möbius transformations. The inductive step of the recursion depends upon the fact that the cross ratio is a 1 complete invariant of ordered four-tuples of points in S1 . This mapping  is again a continuous injection by construction. In the earlier proof that  was surjective, there were two cases for a semi-infinite sequence of turns: either there were infinitely many left-followed-by-right (and hence right-followed-by-left) turns, or else the sequence ended with a consecutive semi-infinite sequence of left or right turns. The former case is again impossible since there are only finitely many distinct shear coordinates, so there is a bounded geometry by Corollary 4.16d of Chapter 1. To rule out the latter case, consider uniformizing in the upper half-plane U so that the consecutive triangles all share the point 1 at infinity as a common vertex. Letting

4 Shear coordinates on Teichmüller spaces

75

xj 2 R, for j  0, denote the consecutive further vertices of these triangles, we find x xj 1 that the sequential shear coordinates are given by ˙ log xjj C1 x using Corollary 4.6 of j Chapter 1. However, since there are only finitely many arcs in  incident on any given puncture, there is some n  1 so that for each k  1, we have xk nC1  xk n D x1  x0 . This is absurd, and so again the mapping  W Fz ! D is a surjective homeomorphism. The induced tesselation of D is invariant by a subgroup  < PSL2 .R/ defined in analogy to Theorem 2.5 again using that the cross ratio is a complete invariant of 1 ordered four-tuples of points in S1 . Returning to the discussion following Theorem 2.5, we wish to emphasize that a gluing of ideal triangles produces a complete hyperbolic structure on the punctured surface precisely when the constraints of Theorem 4.2 hold. Again, it is worth emphasizing that passing from Teichmüller space to decorated Teichmüller space, these kinds of constraints disappear, and lambda lengths give global unconstrained coordinates. Since the linear constraints on shear coordinates were used only to guarantee that the constructed mapping  W Fz ! D was surjective and that peripheral elements were represented by parabolics, this suggests that removing these constraints may correspond to dropping this restriction on peripheral elements. Indeed, this is the case as we next discuss. Fix some surface F D Fgs with s  1 and consider the space Homdfh D Homdfh .1 .Fgs /; PSL2 .R//=PSL2 .R/ of conjugacy classes of discrete and faithful representations  of 1 .F / in PSL2 .R/ so that if  2 1 .F / is a peripheral element, then the absolute value of the trace of . / is at least two. A peripheral element  2 1 .F / is therefore represented by either a parabolic or a hyperbolic element. In this context, we shall refer to the punctures of Fgs as holes, where a hole is a “-puncture” if the trace has absolute value two and is a “-boundary” if the trace has absolute value greater than two for  2 Homdfh , where these attributes are actually associated with holes rather than just peripheral elements by invariance of trace under conjugacy. Definition 4.3. Define the 2s -fold branched cover … W Ty .F / ! Homdfh ; called the holed Teichmüller space, where the fiber over a point of Homdfh is the collection of all tuples of orientations on the components of the -boundary. In particular, T .Fgs / is identified with a subspace of Ty .F /, and if r1 Cs1 D s where r1 is the number of boundary components, then there are 2r1 canonical embeddings of s1 / into Ty .F / corresponding to the possible orientations. T .Fg;r 1 y 2 Ty .F / and an arc e in In order to associate a shear coordinate to a point  F connecting holes, we may remove a small neighborhood of the holes of F that are

76

2 Lambda lengths in finite dimensions

y …./-boundary components in order to consider e as an arc in a surface with punctures y there is furthermore an orientation on each and boundary. Associated with the point , y …./-boundary component, and we may “spin” e around each boundary component in the sense determined by its orientation. y lifts More precisely, each geodesic boundary component of the surface D=…./ z to a geodesic in D that lies in the frontier of the universal cover F  D, and we may define a lift of e to Fz by sliding its endpoint along this geodesic to infinity in the sense determined by the specified orientation, finally straightening to a geodesic for y Thus, the specification of  y 2 Ty .F / and an ideal triangulation  of Fgs gives a …./. well-defined collection of arcs indexed by  decomposing F into ideal triangles, and y defined as before. each such arc e 2  has a well-defined shear coordinate Z .eI / Theorem 4.4 (Thurston–Fock). For any ideal triangulation  of the surface F D Fgs with s  1, the natural mapping Ty .F / ! R ;

y 7! .e 7! Z .eI // y 

is a real-analytic homeomorphism onto. Furthermore, the hyperbolic length of a boundary component is given by the absolute value of the sum of shear coordinates of the edges it traverses counted with multiplicity. Proof of Theorem 4.4. We must again provide an inverse to the mapping defined by taking shear coordinates and shall accomplish this by directly constructing a corresponding class of representations in Homdfh from the putative coordinates. As in the earlier discussion, suppose that Z is the shear coordinate of the triangulated quadrilateral in D with ideal vertices corresponding to 0, 1, 1 and eZ triangulated by the geodesic connecting 0 to 1. Consider the Möbius transformation ! Z 0 e 2 XZ D  Z : 0 e 2 One sees directly that XZ interchanges 0 with 1 and 1 with eZ , and indeed XZ2 D 1 2 PSL2 .R/; thus, XZ describes the unique Möbius transformation fixing the geodesic asymptotic to 0; 1 and mapping 1 7! eZ , which also happens to map eZ 7! 1 and hence is an involution. There are two further elliptic Möbius transformations of immediate interest, namely,     1 1 0 1 RD and L D ; 1 0 1 1 and again one sees directly that R maps 0 7! 1 7! 1 7! 0, L maps 0 7! 1 7! 1 7! 0, and so each of R; L are order 3 elements of PSL2 .Z/ with L D R1 . Definition 4.5. We modify the trivalent fatgraph G D G./ as in [126] (which is recalled in some detail in Section 5.4 of Chapter 5) to produce the corresponding

77

4 Shear coordinates on Teichmüller spaces

freeway G 0 by replacing each vertex of G by a corresponding triangular region in G 0 as illustrated in Figure 4.1b where now G 0 is regarded as a branched one-manifold, i.e., there is a well-defined one-sided tangent line at each vertex of G 0 . A freeway is thus a hybrid of a fatgraph and a train track [144]. We shall refer to a frontier edge of the added triangular regions as short edges of G 0 and to the other edges, which are in one-to-one correspondence with the edges of G itself, as the long edges of G 0 . An orientation of a short edge either “turns right”, i.e., agrees with the counter-clockwise orientation on the boundary of a small triangular region, or it “turns left”, i.e., disagrees with this orientation.

or

or d

d

 e

"

e

cDd

ı

cDd

c

c

" 

ı e

e a) Fatgraph from triangulation

b) Freeway from fatgraph

Figure 4.1. Fatgraphs and freeways.

We next directly define a representation  W 1 .F / ! PSL2 .R/ given putative shear coordinates on the ideal triangulation . We may regard the shear coordinates as associated to the long edges of the freeway G 0 derived from the trivalent fatgraph G D G./ dual to , and we choose as basepoint for the fundamental group of F a vertex of G 0 . Since G is a spine of F , any closed curve  in F based at is homotopic to a closed edge-path in G 0 starting at . Such an edge-path is uniquely described by a corresponding sequence e1 ; : : : ; enC1 of consecutive oriented edges of G 0 , and we define ./ D M.enC1 / : : : M.e1 / 2 PSL2 .R/; where

8 ˆ <XZ M.e/ D R ˆ : L

if e is long and has shear coordinate ZI if e is short and turns rightI if e is short and turns left:

Insofar as R3 D L3 D RL1 D XZ2 D 1 2 PSL2 .R/, it follows that  is well defined and indeed a representation of 1 .F /. We must prove that  is discrete, faithful,

78

2 Lambda lengths in finite dimensions

determines orientations on the -boundary components, and realizes the putative shear coordinates. To these ends, consider a hole of F so that the sum of coordinates on incident edges is non-zero, where the coordinates are counted with multiplicity as before. Remove a topological open disk neighborhood of each puncture to produce a surface F 0 with boundary. Restrict the ideal triangulation  of F to a collection of arcs in F 0 , and spin these arcs around each such boundary component, spinning to the right if the sum of coordinates is negative and to the left if the sum of coordinates is positive. This produces a finite family 0 of arcs decomposing F 0 into triangular regions and determines an orientation on each boundary component of F 0 , where the orientation agrees with the direction of this spinning. As in the proof of Theorem 2.5, consider the topological universal cover Fz 0 of F 0 and z 0 of 0 to Fz 0 . We may choose a triangular region complementary to  z 0 in Fz 0 the lift  and an ideal triangle in D to begin the recursive construction of the mapping  W Fz 0 ! D determined by the putative shear coordinates as before. Again by construction,  is a continuous injection which conjugates the action of  2 1 .F / on Fz to the action of ./ on .Fz 0 /, and furthermore, .1 .F // < PSL2 .R/ leaves invariant the collection z 0 / of geodesics in D. . We claim that each boundary component of F 0 is a -boundary. To see this for a fixed such boundary component, we may change base point and conjugate by a Möbius transformation so that the first edge of the corresponding edge-path is a small edge of the freeway lying in the triangle chosen in Fz 0 to begin the recursive construction of . As in the proof of Theorem 4.2, we may uniformize in the upper half-plane U so that the geodesic connecting x0 D 0 to 1 lies in the frontier of the ideal triangle chosen in D to begin the recursive construction of . There are two cases depending upon the orientation on the fixed boundary component. Suppose first that the arcs in 0 twist to the left along this boundary component of F 0 , so the consecutive lifts to U of these arcs have ideal points 1 and x0 < x1 < x2 <    . Let Zj denote the shear coordinate of x xj 1 the arc connecting 1 to xj , so we have as before that Zj D log xjj C1 x for j  1. Let j n denote the number of arcs spinning around this hole again counted with multiplicity, so that for all k  0, we have Zk nCj D Zj . It follows from a geometric P sequence that xj converges to x1 D xn .1  ez /1 as j tends to infinity, where z D jnD1 Zj and z > 0 by our construction. Furthermore on the based curve going once around this hole in its specified orientation,  takes the value ! ! Z1 Z1 Zn Zn e 2 e 2 e 2 e 2 .XZn L/ : : : .XZ1 L/ D ::: Zn Z1 0 e 2 0 e 2   z e 2 2x1 sinh z2 ; D z 0 e 2 where the final off-diagonal entry follows from the fact that x1 is invariant under this Möbius transformation as the limit of the invariant sequence x1 ; x2 ; : : : .

4 Shear coordinates on Teichmüller spaces

79

The P argument for the case that the arcs twist to the right is entirely analogous except that jnD1 Zj < 0 and we interchange the matrices L and R in the earlier calculation. This completes the proof that the metric completion of the constructed surface has each boundary component a -boundary of the asserted length since the absolute value of the trace of a Möbius transformation is twice the hyperbolic cosine of half its translation length. It remains only to prove that the group  D .1 .F // is discrete. To see this, we may choose a fundamental domain for  consisting of a finite collection of ideal triangles since .0 / is invariant under  by construction. It follows that each element of  must map each such ideal triangle to a disjoint ideal triangle, and discreteness follows. Definition 4.6. The previous proof gives an elegant description of the holonomy . / for each  2 1 .F / in terms of the representing edge-path on any corresponding freeway, which we shall refer to as the path-ordered product expansion of the holonomy. In fact, though Theorem 4.4 was well known among the Thurston school in the 1980s, cf. [159], this path-ordered product aspect of the proof we have presented was Fock’s more recent innovation [43], and it is the basis for the quantization [32], [78] of Ty .F /, where the shear coordinates are replaced by appropriate operators on a Hilbert space. The action of flips on shear coordinates is calculated using lambda lengths in Theorem 4.7. The other key ingredient for quantization beyond the combinatorial description of the mapping class group given in Section 4 of Chapter 4 is the Poisson structure on Ty .F / (i.e., a skew-symmetric pairing on smooth functions on Ty .F / that satisfies the Jacobi and Leibnitz identities) corresponding to the Weil–Petersson Kähler two form described in Proposition 2.11, and this Poisson structure is described in Theorem 4.11 and its center in Theorem 4.13. The analogue of Ptolemy transformations for shear coordinates is given by: y 2 Ty .F /. Suppose Theorem 4.7. Suppose that  is an ideal triangulation of F and  that e 2  separates two triangles complementary to , where these two triangles have frontier arcs a; b; e 2  and c; d; e 2  in these correct clockwise orders. Perform a flip on e to produce the ideal triangulation 0 D e , let e 0 denote the y and X 0 D Z0 .xI /, y for x D a; : : : ; d , set unique arc in 0  , set X D Z .xI / 0 0 y E D Z0 .e I /, y and define ˆ.X / D log.1 C eX /. Provided a, b, c, E D Z .eI /, d are all pairwise distinct, then E 0 D E and A0 D A C ˆ.E/; B 0 D B  ˆ.E/; C 0 D C C ˆ.E/; D 0 D D  ˆ.E/: In the following special cases, these formulas are to be modified as follows: a D c H) A0 D A C 2ˆ.E/I b D d H) B 0 D B  2ˆ.E/I c 2 fb; d g H) C 0 D C C E: a 2 fb; d g H) A0 D A C EI

80

2 Lambda lengths in finite dimensions

Proof. Begin with the case that a, b, c, d are all pairwise distinct and adopt the notation that arcs bounding a triangle together with x other than a, b, e and c, d , e are x1 , x2 , x in this clockwise cyclic order, for x D a; b; c; d . It is easiest to simply calculate in lambda lengths for some fixed decoration, where we identify an arc with its lambda ac length for simplicity. Thus, E D log bd , a1 b ; a2 e c1 d C D log ; c2 e A D log

b1 e ; b2 a d1 e ; D D log d2 c B D log

D E, and so E 0 D log bd ac a1 e 0 a2 d b1 c B 0 D log b2 e 0 c1 e 0 C 0 D log c2 b d 1a D 0 D log d2 e 0 A0 D log

a1 b a2 e b1 e D log b2 a c1 d D log c2 e d1 e D log d2 c D log

ac C bd bd ac ac C bd ac C bd bd ac ac C bd





ac ; bd   bd ; D B  log 1 C ac   ac ; D C C log 1 C bd   bd ; D D  log 1 C ac D A C log 1 C

using Ptolemy’s equation ee 0 D ac C bd , as was claimed. In the special cases, we likewise compute: 



e02 bd ac C bd 2 a D c H) A D log D A C 2ˆ.E/I D log 2 bd e bd   2 ac ac e2 b D d H) B 0 D log 0 2 D log D B  2ˆ.E/I ac ac C bd e c b ac a D b H) A0 D log D log D A C EI d a bd b ac a a D d H) A0 D log D log D A C EI d c bd c d ac c D b H) C 0 D log D log D C C EI b a bd d ac a c D d H) C 0 D log D log D C C E:  b c bd Corollary 4.8. In the notation of Theorem 4.7, consider the sub-surface Fx of F defined as the closure in F of the union of the two triangles with frontier edges a, b, e and c, d , e. Let ˛ be the boundary of a regular neighborhood in Fx of one of the vertices of these triangles. Then the sum of shear coordinates of arcs in  meeting ˛ agrees with the sum of the shear coordinates of arcs in 0 meeting ˛ counted with multiplicity as before. 0

4 Shear coordinates on Teichmüller spaces

81

Proof. In particular, if a, b, c, d are all pairwise distinct, then Fx is a quadrilateral embedded in F , the assertion is that A0 C D 0 D A C D C E; B 0 C C 0 D B C C C E;

A0 C B 0 C E 0 D A C B; C 0 C D 0 C E 0 D C C D;

and these relations follow from the formulas in Theorem 4.7 using the identity ˆ.E/  ˆ.E/ D log

1 C eE D log eE D E: 1 C eE

In the special cases, we likewise compute from Theorem 4.7: if a D c (and similarly if b D d ), then C 0 C 2D 0 C E 0 D C C 2D C E;

A0 C 2B 0 C E 0 D A C 2B C EI

if a D b (and similarly if c D d ), then A0 D A C E;

A0 C C 0 C D 0 D A C C C D C 2E;

C 0 C D 0 C E 0 D C C DI

if a D d (and similarly if b D c), then A0 D A C E;

A0 C B 0 C C 0 C 2E 0 D A C B C C;

B 0 C C 0 D B C C C EI

if a D c and b D d , then A0 C B 0 C E 0 D A C B C EI if a D d and b D c (and similarly if a D b and c D d ), then A0 D A C E;

B 0 D B C E;

A0 C B 0 C 2E 0 D A C B:

Suppose that  is an ideal triangulation of F D Fgs , for s  1, with dual trivalent fatgraph spine G. If a; b 2  are distinct, then let "ab be the number of components of F  .[ [ G/ whose frontier contains points of a and points of b counted with a sign that is positive if a and b are consecutive in the clockwise order (arising from the orientation of F ) in the corresponding region and with a negative sign if a and b are consecutive in the counter-clockwise order. Setting "aa D 0 for each a 2 , "ab takes the possible values 0, ˙1, ˙2 and comprise the entries of a skew-symmetric matrix indexed by . Definition 4.9. For any a 2 , regard the corresponding shear coordinate Za D y as a real-valued coordinate function defined on Ty .F /, and define a Poisson Z .aI / structure by setting fZa ; Zb g D "ab , the constant function on Ty .F / with value "ab , where we extend linearly using the Leibnitz rule to an appropriate class of functions in the shear coordinates on Ty .F /.

82

2 Lambda lengths in finite dimensions

Remark 4.10. An important property for the canonical quantization is that these structure constants "ab are in fact constant on Ty .F /. The Poisson bracket f; g of two functions depends only upon their differentials, so there is a corresponding skew-symmetric two-tensor  called the “Poisson bivector” so that for any two functions f; g, we have hdf ˝ dg; i D ff; gg, where h ; i is induced by the pairing between cotangent and tangent vectors. From the definition of our Poisson structure on Ty .F /, the corresponding Poisson bivector is given by  D

X @ @ @ @ @ @ ^ C ^ C ^ ; @Za @Zb @Zb @Zc @Zc @Za

where the sum is over all triangles complementary to  in F whose edges in this clockwise cyclic ordering as determined by the orientation of F have shear coordinates Za , Zb , Zc , and this evidently restricts to the Poisson structure induced on T .Fgs /  Ty .F / by the symplectic two form ! described in Proposition 2.11 (using also the remarks about degeneracies of this two form following the same result). Theorem 4.11. The Poisson structure f; g on Ty .F / is well defined independent of the choice of ideal triangulation  of F D Fgs . Proof. By Whitehead’s Classical Fact 1.24 of Chapter 1, flips act transitively on ideal triangulations. It therefore suffices to consider the case that two ideal triangulations differ by a single flip, then the Poisson structures for these two ideal triangulations coincide. In the notation of Theorem 4.7, suppose first that the arcs a, b, c, d are @ all pairwise distinct. We may compute with the bivector and set @X D @X for X D @ 0 0 0 0 0 0 0 A; B; C; D; E and @X D @X 0 , for X D A ; B ; C ; D ; E , so @X D @X for X D A; B; C; D, and @E D

  eE  0 eE  0 0 0 C @ @ C @B C @D  @0E A C E E 1Ce 1Ce

by Theorem 4.7. The relevant contribution to  is thus given by @A ^ @B C @B ^ @E C @E ^ @A C @C ^ @D C @D ^ @E C @E ^ @C 0 0 0 0 D @A0 ^ @B C @0C ^ @D C .@B C @D  @A0  @0C /



eE eE 0 0 0 ^ .@ C @ / C .@0 C @D /  @0E C 1 C eE A 1 C eE B 0 0 D @A0 ^ @B C @0C ^ @D 0 0 0 0 C @0E ^ .@B C @D  @A0  @0C / C .@B C @D / ^ .@A0 C @0C / 0 0 0 0 C @D ^ @0E C @0E ^ @A0 C @0C ^ @B C @B ^ @0E C @0E ^ @0C ; D @A0 ^ @D giving the relevant contribution to the bivector 0 , as required, where we have used eE eE the identity 1Ce E C 1CeE D 1 in the second equality.

4 Shear coordinates on Teichmüller spaces

83

0 In the special cases, we always have @X D @X for X D A; B; C; D, and in the various cases, the calculation is performed analogously with: if a D c (and similarly if b D d ), then

@0E C @E D

2eE eE @A C .@B C @D /I E 1Ce 1 C eE

if a D b (and similarly if c D d ), then @0E C @E D @A C

eE eE @ C @D I C 1 C eE 1 C eE

if a D d (and similarly if b D c), then @0E C @E D @A C

eE eE @ C @C I B 1 C eE 1 C eE

if a D c and b D d , then @0E C @E D

2eE 2eE @ C @B I A 1 C eE 1 C eE

if a D b and c D d (and similarly if a D d and b D c), then @0E C @E D @A C @C : Example 4.12. For the surface F11 , there is only one combinatorial type of ideal triangulation, as illustrated on the left in Figure 1.4 of Chapter 1. Letting A, B, C denote the three shear coordinates, we have fA; Bg D fB; C g D fC; Ag D 2, and the matrix of Poisson brackets is given by 0 1 0 2 2 @2 0 2 A: 2 2 0 The center of the Poisson algebra is generated by the kernel of this matrix, which is spanned by 2.A C B C C / corresponding to the unique hole. Theorem 4.13. Fix an ideal triangulation  of F D Fgs and index the holes of F by p D 1; : : : ; s. Consider the sum Cp of the shear coordinates of the arcs in  incident on the hole p counted with multiplicity as before. Then the center of the Poisson algebra is freely generated by fCp W p D 1; : : : ; sg. Proof. We begin by proving that each Cp is indeed central and suppose that e is an y There are two cases depending upon whether e arbitrary arc in  with E D Z .eI /. triangulates a once-punctured monogon or a quadrilateral complementary to   feg, and in the former case, let f 2  denote the frontier of this once-punctured monogon

84

2 Lambda lengths in finite dimensions

y It may be that the hole p corresponds to the puncture in this and set F D Z .f I /. monogon, in which case Cp D E, so of course fE; Cp g D 0 by skew-symmetry. If p corresponds to another hole of F , then we may write Cp D C C ıF , where ı D 0; 1 and fE; C g D 0. Since fE; F g D 0 by definition of the bracket, we conclude that fE; Cp g D 0 as required. In the latter case, let us adopt the notation of Theorem 4.7 for the nearby edges y and suppose first that a, b, c, d x D a; b; c; d and shear coordinates X D Z .xI /, P are all pairwise distinct. Any element Cp is of the form Cp D C C j4D1 ıj Dj , where D1 D A C B, D2 D C C D, D3 D A C D C E, D4 D B C C C E, fE; C g D 0, and each ıj D 0; 1. One checks using the definition of the bracket that fE; Dj g D 0, for j D 1; 2; 3; 4, so indeed fE; Cp g D 0 in this case. There are again special cases if a, b, c, d are not distinct, which we may summarize as follows: if a D c (and similarly if b D d ), then Cp D C C ı1 .A C 2B C E/ C ı2 .A C 2D C E/I if a D b (and similarly if c D d ), then Cp D C C ı1 A C ı2 .C C D/ C ı3 .A C C C D C 2E/I if a D d (and similarly if b D c), then Cp D C C ı1 .A C E/ C ı2 .A C B C C / C ı3 .B C C C E/I if a D c and b D d , then Cp D 2.A C B C E/I if a D b and c D d , then Cp D A; C; or A C C C 2EI if a D d and b D c, then Cp D A C B; A C E; or B C E; where fE; Cg D 0, the ı’s are equal to zero or one, and in each case the bracket of E with each possible term again vanishes by definition as one verifies with several small calculations. Thus, the asserted elements are indeed central. According to Corollary 4.8, these elements are furthermore invariant under flips, and by Theorem 4.11, the center of the Poisson algebra is likewise invariant under flips. Using the Classical Fact that flips act transitively on ideal triangulations (cf. Corollary 1.2 of Chapter 5), we may thus prove the result for any particularly convenient ideal triangulation, and it then follows for any ideal triangulation. We shall assume that 2g C 2s > 5 since the unique surface F11

85

4 Shear coordinates on Teichmüller spaces

of negative Euler characteristic ruled out by this inequality has already been handled separately in Example 4.12. We may equivalently describe the dual fatgraph to this convenient ideal triangulation, and it is comprised of various “building blocks” of the two types depicted in Figure 4.2 joined together in a line (where the cyclic ordering in the fatgraph structure is inherited from the plane of projection of our figure); for the surface Fgs , there are g copies of the building block in Figure 4.2a and s  1 copies of the building block in Figure 4.2b. It is clear that the s elements Cp for this ideal triangulation are linearly independent, and we have already shown that they are indeed central. We complete the proof by calculating that the center of the Poisson algebra for the ideal triangulation dual to this trivalent fatgraph has rank s.

Dj

Nj

Ej

Cj

Bj

Aj

Aj Xj

Xj C1

Xj

Xj C1 a) Increase genus

b) Add a punture

Figure 4.2. Fatgraph building blocks.

Adopting notation for the shear coordinates A, B, C , D, E in Figure 4.2a, we have the corresponding matrix of Poisson brackets between these variables given by Aj

Bj

Cj

Dj

Ej

Aj

0

1

1

0

0

Bj

1

0

1

1

1

Cj

1

1

0

1

1

Dj

0

1

1

0

2

Ej

0

1

1

2

0

:

Add the last row to the next-to-the-last row, add the last column to the next-to-the-last column, add the third row to the second row, and finally add the third column to the

86

2 Lambda lengths in finite dimensions

second column to obtain

0

0

0

B B0 0 B B B1 1 B B0 0 @ 0 2

1 1 0 0 1

0

0

1

C 2C C C 0 1C C 0 2C A 2 0 0

which evidently has rank 4 and can be further reduced (without adding the first column or row to any other) to yield 0 1 0 0 0 0 0 B C B0 0 C1 0 C 0 B C B C B0 1 0 0 0 C: B C B0 0 0 0 C2C @ A 0 0 0 2 0 We may thus erase from the matrix of Poisson brackets of all edges those columns and rows corresponding to the variables Bj , Cj , Dj , and Ej without changing the rank of this matrix. Likewise, adding a building block in Figure 4.2b creates exactly one degeneracy since the variable Nj Poisson commutes with all other variables as we have already shown. It remains only to calculate the rank of the matrix corresponding to a graph with edges Xj and Aj remaining after erasing all B-, C -, D-, E- and N -variable rows and columns. The corresponding Poisson bracket matrix has odd dimension 2gC2s5 > 0 and the block-diagonal form 0 1 1

1 0 1 0 0

1 1 0 1 1

0 0 1 0 1 0

0 0 1 1 0 1





0 1 0

0 1 1

0

1

1

0 :: :



: ::

:

::

:

Add each even-index row to its predecessor and add each even-index column to its predecessor to produce a matrix whose only non-zero elements are C1 on the main super-diagonal and 1 on the main sub-diagonal. This matrix has rank 2g C 2s  6, which completes the proof.

4 Shear coordinates on Teichmüller spaces

87

(The last part of this proof comes from joint work [33] with Leonya Chekhov with a small correction.) Remark 4.14. The exactly parallel discussion and identical calculations show that tangent vectors to Teichmüller space which contract to zero with the invariant two form discussed in the previous section are tangent to the fibers of the projection to Teichmüller space as was discussed before, and in particular, these vector fields are linearly independent. The same remark applies in the setting of partially decorated s or bordered surfaces. For a bordered surface F D Fg;.ı , however, there is a 1 ;:::;ır / further variant which is sometimes studied as follows. An ideal triangulation  of F in particular contains the arcs B lying in the boundary of F , and we may consider the two form defined in analogy to the previous section, summing over triangles complementary to  in F as before, but including only summands d log x ^ d log y for pairs of arcs lying in   B, where we have as usual identified an arc with its lambda length for convenience; put another way, we may consider the lambda lengths on the arcs in B as fixed a priori and pull-back the two form considered before. In this case, the degeneracies of this modified two form are spanned by the following vectors. For each j D 1; : : : ; r with ıj even, enumerate in their correct cyclic order induced from the orientation on the boundary the distinguished points p1 ; : : : ; pıj occurring on the j th boundary component. Enumerate the arcs of   B incident on PLk @ pk by a`k , for k D 1; : : : ; ıj and ` D 1; : : : ; Lk , and form the sums k D `D1 k, @ log a`

where again this sum is taken with multiplicity (so if a`k is asymptotic to pk in both directions, then there are two contributions to the sum). Finally forming the alternating sum ƒj D 1  2 C     ıj for ıj even, the degeneracies of the modified two form are freely spanned by fƒj W ıj is eveng. The proof is again exactly parallel to the proof of Theorem 4.13, now using also nonsingularity of an even-dimensional square matrix whose non-zero entries are C1 on the main super-diagonal and 1 on the main sub-diagonal.

3 Lambda lengths in infinite dimensions

In this chapter, we identify the space of all “tesselations” of the Poincaré disk with the collection of all orientation-preserving homeomorphisms of the circle and extend lambda length coordinates to a suitable decoration of this space. We study a “pinched” condition on these coordinates which guarantees that the tesselation corresponds to a quasi-symmetric homeomorphism of the circle. A universal object, the “punctured solenoid”, is likewise parametrized by profinite suitably quasi-periodic assignments of lambda lengths. Finally, lambda length deformations are studied as piecewise Möbius maps of the circle, and the closure of their natural Lie algebra is calculated.

1 Tesselations As we shall see in the next several sections, the previous considerations have applications to circle homeomorphisms and harmonic analysis. In effect, a decorated surface is tantamount to a “tesselation” of the universal cover which is invariant under a Fuchsian group with parabolic fixed points given by the ideal points of the tesselation, and we shall now ignore the group and concentrate instead on the decorated tesselation itself. This section is principally based on [132]. Definition 1.1. A tesselation  of D is a countable collection of geodesics decomposing D into ideal triangles, where  is required also to be locally finite, i.e., any point of D admits a neighborhood meeting only finitely many geodesics in  . We shall let 1  0  S1 denote the collection of ideal points of all the geodesics in  and  2 denote the set of ideal triangles complementary to . 1 . Note that  is not locally finite in the closed disk D [ S1 A principal example is the Farey tesselation  which has already been discussed at length. As we have seen, 0 is the image of the rational numbers plus infinity under the Cayley transform, and we shall simply identify 0 with this set denoted by 1 1 x D Q[f1g  S1 x is identified with the point piq 2 S1 0 D Q . Thus, pq 2 Q C pCiq 0 thereby establishing a bijection between  and the set of all points in the unit circle whose Cartesian coordinates in the plane are rational. Definition 1.2. For another example of a tesselation, we might begin with the triangle spanned by 01 , 10 , 11 and recursively define the ideal triangle adjacent to an edge with

1 Tesselations

89

1 endpoints ;  2 S1 already constructed to have its other vertex at the Euclidean 1 midpoint of the appropriate circular segment in S1 with endpoints ; . We shall call 0 this the dyadic tesselation d since its set d of ideal vertices is the collection of all points of the form e2 i , where  D 2pn is a dyadic rational, for some n 2 Z and some odd p 2 Z.

Definition 1.3. A distinguished oriented edge or simply doe on a tesselation  is the specification of an orientation on some particular geodesic in  . We shall take the edge connecting 01 to 10 as the standard doe on  and d letting 0 and d0 , respectively, denote the Farey and dyadic tesselations with this choice of doe. Let T ess0 denote the collection of all tesselations with doe of D. A key point is that tesselations with doe are “combinatorially rigid” in the following sense. Choosing 0 as a kind of basepoint for T ess0 , given any other  0 2 T ess0 , there is a canonically defined mapping f D f00 W 0 !  0 given recursively as follows. Start by defining f . 01 / and f . 10 /, respectively, to be the initial and terminal points of the doe of  0 . There is a unique ideal triangle complementary to  lying to the right of the doe, and f . 11 / is defined to be the ideal vertex of this triangle distinct from f . 01 / and f . 10 /. This defines f on the vertices of an ideal polygon P with frontier in  . 1 If e is a geodesic in D and f is a homeomorphism of S1 , then we let f .e/ denote the geodesic spanned by its endpoint images and similarly let f .P / denote the ideal polygon spanned by the ideal vertex images of an ideal polygon P  D. We recursively extend f in this same manner as before: a geodesic e in the frontier of P also lies in the frontier of a unique ideal triangle complementary to  whose interior is disjoint from P , there is likewise a unique ideal triangle complementary to  containing f .e/ in its frontier whose interior is disjoint from f .P /, and we extend f by mapping the vertex of the former triangle distinct from the endpoints of e to the vertex of the latter triangle distinct from the endpoints of f .e/. Definition 1.4. Given  0 2 T ess0 , we have thus defined the mapping f00 W 0 ! . 0 /0 , and by construction, this mapping is order-preserving. An elementary argument shows that an order-preserving mapping defined on a dense subset of the circle interpolates a unique orientation-preserving homeomorphism of the circle. The homeomorphism of the circle induced in this way by f00 is denoted by 1 1 ! S1 f 0 W S1

and is called the characteristic mapping of  0 2 T ess0 . Example 1.5. The characteristic mapping of the dyadic tesselation d0 with doe is especially interesting and is called the Minkowski question mark function; see [18] and the extensive bibliographies online. It is an exercise using Corollary 4.6 of Chapter 1 to compute the lambda lengths for d0 with the underlying decoration of the Farey tesselation. See also Remark 1.3 of Chapter 6 for the connection with the Thompson group T .

90

3 Lambda lengths in infinite dimensions

See Remark 1.5 of Chapter 6 for further explicit examples. Lemma 1.6. Suppose that  is a collection of geodesics in D that is in bijection with the Farey tesselation  by an order-preserving map 0 !  0 . The collection  is 1 locally finite in D and hence a tesselation if and only if  0 is dense in S1 . Proof. Suppose that i 2  accumulates in D. Taking a subsequence, there is a consecutive chain of triangles in 2 whose frontiers accumulate necessarily to a geodesic 1 , and the endpoints of  determine an open set in U  S1 disjoint from the doe. 1 0 Density in S1 implies that there is some y 2  which maps to U , but any possible 1 location of y 2 S1 violates that 0 !  0 is order-preserving. 1 open, construct a chain of triangles in the natural way, Conversely, given U  S1 where each added edge separates U from the previous chain starting from the doe. By compactness, there are subsequences of endpoints converging to distinct points, and  0 fails to be locally finite at any point lying on the geodesic they span. 1 Thus, given any countable dense subset S  S1 plus a bijection ˇ W N ! S , there is a corresponding tesselation defined as follows. Begin with the ideal triangle spanned by ˇ.1/, ˇ.2/, ˇ.3/ and recursively define the ideal triangle adjacent to an edge with 1 endpoints ;  2 S1 already constructed to have its other vertex at ˇ.i /, where i 2 N is the least index with ˇ.i / in the appropriate circular segment with endpoints ; . In fact, any tesselation of D clearly arises in this way as well, thus giving a natural identification of enumerated countable dense subsets of the circle with tesselations of D.

Definition 1.7. Let HomeoC .S 1 / denote the topological group of all orientation1 preserving homeomorphisms of S1 with the compact-open topology. The assignment 0 0 0  7! f gives a mapping of T ess into HomeoC .S 1 /. Lemma 1.8. The mapping T ess0 ! HomeoC .S 1 /;

 0 7! f 0 ;

is a bijection. Proof. Suppose that f 2 HomeoC .S 1 /, and define a collection  D ff .e/ W e 2  g, where as before, if e 2  has ideal points ; , then f .e/ is the geodesic with ideal points f . /; f ./.  is evidently a countable family of geodesics. Density of 0 and continuity of f imply density of f .0 /, from which it follows that  is a tesselation by Lemma 1.6. We take as doe the oriented geodesic connecting f . 01 / to f . 10 / to produce f0 2 T ess0 . The assignment f ! f0 is a two-sided inverse to  0 7! f 0 by construction. We induce a topology on T ess0 using the previous lemma. The Möbius group PSL2 .R/ acts continuously on the left on HomeoC .S 1 / by composition, and there is an induced topology on the quotient. Furthermore, f 2 HomeoC .S 1 / also acts on

1 Tesselations

91

 0 2 T ess0 in the natural way, where f ./ D ff .e/ W e 2  g in the previous notation, with the doe on f ./ given by the image under f of the doe for  . The assignment of characteristic mapping is evidently equivariant for these actions. Definition 1.9. Define the quotient space T ess D T ess0 =PSL2 .R/; which is thus homeomorphic to HomeoC .S 1 /=PSL2 .R/, the object of our ongoing principal interest. We say that  0 2 T ess0 is normalized if the doe connects the point 01 to the point 10 and if the other vertex of the triangle to the right of the doe is the point 11 . Likewise, we say that f 2 HomeoC .S 1 / is normalized if f fixes each of the points 01 , 1 1 , . 0 1 Since a Möbius transformation is determined by its values at three points, we conclude that T ess is homeomorphic to the collection of all normalized tesselations with doe and that HomeoC .S 1 /=PSL2 .R/ is homeomorphic to the collection of all normalized orientation-preserving homeomorphisms of the circle. In particular, T ess as well as HomeoC .S 1 /=PSL2 .R/ are infinite-dimensional Hausdorff spaces. In effect, we introduced distinguished oriented edges only to kill them by taking the quotient by the Möbius group, and this little manipulation was performed in order to produce a space of orbits which is Hausdorff. Given a tesselation  0 with doe representing a point of T ess, there is a well-defined shear coordinate assigned to each geodesic in  as given in the previous section. We may use the characteristic mapping f 0 to index these shear coordinates by edges of  in the natural way, i.e., given e 2  , the geodesic f .e/ 2  triangulates an ideal quadrilateral complementary to   ff .e/g, and the negative of the logarithm of the cross ratio of this quadrilateral computed as before is called the shear coordinate of  0 on e. Proposition 1.10. The assignment of shear coordinates gives a continuous injection T ess HomeoC .S 1 /=PSL2 .R/ ! R for the weak topology on the target. (Correcting [132], the image is not open.) Proof. Since the cross ratio is a complete invariant of a four-tuple of points under the action of the Möbius group, the assignment of shear coordinates is indeed an injection as before, and continuity follows easily. Myriad interesting questions and problems arise: Choose your favorite class of homeomorphisms of the circle (smooth, Hölder of some exponent, really any class), and ask for a topological characterization of the corresponding tesselations or for the characterization in terms of shear coordinates. In particular, characterize the image of

92

3 Lambda lengths in infinite dimensions

this embedding. Describe the inverse of a circle homeomorphism in terms of tesselations or in coordinates. Describe composition of circle homeomorphisms in terms of tesselations or coordinates. Among the first class of questions, we presently know just one partial answer1 , which we shall present in Theorem 2.8. Definition 1.11. A decoration on a tesselation  is the assignment of one horocycle centered at each point of  0 . The space of decorated tesselations with doe admits a 0 natural topology as a R! >0 -bundle over T ess , where ! is the first infinite ordinal and 0 the fiber is given the weak topology. T ess again admits a natural continuous left action of the Möbius group acting not only on tesselation with doe as before but also on horocycles, and we define the quotient

e

e e By definition, the map Te ess ! T ess which forgets decoration is a continuous T ess D T ess0 =PSL2 .R/:

surjection. Definition 1.12. The collection Mod of all countable dense subsets of the circle with a distinguished triple modulo the action of PSL2 .R/ is the universal moduli space, and the natural mapping T ess ! Mod is called the very forgetful mapping.

e e denote the collection of all W Qx ! L so that N W Qx ! S Remark 1.13. Let Mod is order-preserving, where N W L ! S is the natural projection, modulo the action e which in effect forgets the tesselation, of PSL .R/. There is a mapping Te ess ! Mod x so Mod e is a kind of universal decorated which we regard as a “marking” on the set .Q/, C

C

1 1

1 1

2

moduli space. The space Mod arises as the pushout T ess

e

/ T ess

 Mod

 / Mod

e

which explains by analogy with the classical case why we think of Mod as a universal moduli space. Let us also mention that in [132], we imposed the further constraint on T ess and Mod that the corresponding subsets of LC were discrete in LC [ f0g and radially dense. This is a natural restriction because points of T ess arising from finite-type surfaces or from the punctured solenoid enjoy these properties as we shall see (in Corollary 1.2 of Chapter 4 for surfaces and in Theorems 2.5 and 3.10 for the solenoid). In any case, this version of universal moduli space or perhaps its profinite analogue as in Section 3 may be related to Yuri Manin’s program [108] on quantum tori and real multiplication.

e

e

e

1 Dragomir Šari´c [151] has recently given complete solutions for the classes of general orientationpreserving, symmetric and quasi-symmetric homeomorphisms.

93

1 Tesselations

e

Given a decorated tesselation Q 0 with doe representing a point of T ess, there is a well-defined lambda length assigned to each geodesic in  defined just as in the classical case: the decoration on the triangle to the right of the doe determines lambda lengths on the frontier edges of this triangle by Lemma 4.8 of Chapter 1, and lambda lengths on consecutive edges are then uniquely determined according to Lemma 4.14 of Chapter 1. We may again use the characteristic mapping f 0 to index these lambda lengths by edges of  in the natural way. Proposition 1.14. The assignment of lambda lengths gives a continuous injection

e

 T ess ! R>0 :

(Again correcting [132], the image is not open.) Proof. The proof closely parallels the classical case, where the lambda lengths are used to uniquely determine a mapping 0 ! LC which is again evidently continuous. The corresponding myriad questions and problems as in the undecorated case are likewise relevant in the decorated case. Remark 1.15. The natural inclusion of diffeomorphisms induces Diff C .S 1 /=PSL2 .R/  HomeoC .S 1 /=PSL2 .R/; and one can show [132] that the familiar expression for the Weil–Petersson Kähler two form (namely, twice the sum over all triangles complementary to the tesselation of the wedge-products of consecutive pairs around the frontier of dlog lambda lengths as before) formally pulls back to the Kirillov–Kostant form [90], [95] given by .Lm ; Ln / D 2 i.m3  m/ım;n , where ı is Kronecker’s delta function and @ Lm D eim @ are the standard vector fields on the circle, for m 2 Z  f0; ˙1g. In fact, these forms converge for instance on the subspace of homeomorphisms of the circle which are 32 C " differentiable, cf. [132]. There is an associated basic deformation of lambda lengths described as follows. 1 1 The four points 01 , 10 , ˙ 11 2 S1 decompose S1 into four component arcs, one in each 2 of the quadrants I, II, III, IV in R (enumerated in counter-clockwise order starting from quadrant I where both coordinates are positive). For s 2 R>0 , define a piecewise 1 Möbius mapping on S1 by  8 1 ˆ s s  s ˆ

for in quadrant II ˆ ˆ ˆ s 1  ˆ 0 ˆ ˆ ˆ s 1 0 for in quadrant II; ˆ < 1 s s  s ƒ.s/. / D  1 ˆ s 0 ˆ

for in quadrant III; ˆ ˆ ˆ ˆ s 1  s s   ˆ ˆ 1 ˆ ˆ : s s  s for in quadrant IV. 0 s 1

94

3 Lambda lengths in infinite dimensions

See Figure 1.1. The four points 01 , 10 , ˙ 11 span an ideal quadrilateral with one frontier geodesic in each quadrant, and one can check from the formula that the oneparameter family in each quadrant is a family of hyperbolic transformations with invariant geodesic in D given by the corresponding frontier geodesic of this quadrilateral as illustrated in Figure 1.1. In particular, ƒ.s/ fixes each point 01 , 10 , ˙ 11 and hence is a homeomorphism of the circle for each s. 

 0 s

1

s

1

s

s

1 1



s 0

s

0 1

 s

s

1

1

s

s s

1

s

1

1



1 0

 0 s



s 0

s

1

s



1 1

Figure 1.1. Multiplicative group ƒ.s/.

These four one-parameter families of isometries are “tuned” to guarantee that each 1 . Put another way, on ƒ.s/ is furthermore once-continuously differentiable on S1 0 1 C the rays in L which project to 1 ; 0 , the function ƒ.s/ acts by multiplication by s, and on the rays in LC which project to ˙ 11 , the function ƒ.s/ acts by multiplication by s 1 . We may thus think of ƒ.s/ as acting on LC itself in the natural way by a piecewise SOC .2; 1/ homeomorphism giving a continuous one-parameter family of homeomorphisms of LC . As such, each ƒ.s/ acts on the lambda length of pair h.u/; h.v/ of horocycles by .h.u/; h.v// 7! .h.ƒ.s/u/; h.ƒ.s/v//, for u; v 2 LC , and hence ƒ.s/ acts on T ess0 . Summarizing, we have

e

Lemma 1.16. ƒ.s/ is a multiplicative subgroup of HomeoC .S 1 /, where each ƒ.s/ is once-continuously differentiable on S 1 with the four fixed points 01 , 10 , ˙ 11 . If  0 is any tesselation with doe connecting 01 to 10 with complementary triangles spanned by 01 , 10 , 1 and 01 , 10 ,  11 , then ƒ.s/ leaves invariant every lambda length coordinate on every 1 edge of  0 except that it scales the lambda length of the doe by the factor s. Proof. In light of the remarks above, only the last sentence requires comment. Each edge e 2  0 other than the doe has both its endpoints in some common quadrant in R2 , and therefore Möbius invariance of lambda lengths shows that its lambda length

2 Pinched lambda lengths

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is invariant under ƒ.s/ for each s. On the other hand, the doe has endpoints 01 and 10 , the points u1 ; u2 2 LC lying over these points map by ƒ.s/.uj / D suj , for j D 1; 2, and hence .h.u1 /; h.u2 // 7!.h.su1 /; h.su2 // p D hsu1 ; su2 i p Ds hu1 ; u2 i Ds.h.u1 /; h.u2 //:



2 Pinched lambda lengths This section is dedicated to investigating the following condition on lambda lengths. Definition 2.1. We say that a function  W  ! R>0 is pinched if there is some K > 1 so that for each e 2  , we have K 1 < .e/ < K. Applying the inductive construction of Theorem 1.14, we in any case uniquely lift  W  ! R>0 to a function denoted by D  W 0 ! LC and start by analyzing the topology of the image .0 /  LC using the triangles .t /, for t 2 2 . To this end, we begin with certain generalities which amount to a combinatorial treatment of continued fractions for a general tesselation. Definition 2.2. Fix a tesselation  of D. Define a left fan  of  to be an ordered collection fTi gn1 of ideal triangles in D including their frontiers together with a distinguished (unoriented) frontier edge e of T1 , so that: i) Ti is the closure in D of a triangle .t / for some t 2 2 and each i D 1; : : : ; n; ii) Ti \ Tj D ; unless ji  j j D 1, and in this case, Ti \ Tj is a common frontier edge of Ti and Tj for distinct i; j D 1; : : : ; n; iii) suppose that N > 1 and set T0 D e. All the triangles Ti share a common vertex 1 , and TiC1 \ Ti lies counter-clockwise from Ti \ Ti1 for i D 1; : : : ; N . y 2 S1 An example of a left fan is given in Figure 2.1, and one similarly defines a right fan 1 using the clockwise orientation in iii). N is the length and y 2 S1 is the pin of the fan . A frontier geodesic of some triangle which is asymptotic to v is called a spine of , and any other frontier geodesic is called a slat. The distinguished edge of T1 is the initial spine of , and if  has finite length, then the spine in the frontier of TN which is distinct from TN 1 \ TN (and distinct from the distinguished edge e in case N D 1/ is called the final spine of . The fan 2 is contiguous with the fan 1 provided the final spine of 1 coincides with the initial spine of 2 and the two fans lie on opposite sides of this common spine.

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3 Lambda lengths in infinite dimensions

final spine

T3

contiguous right fan

T2

slats

T1

initial spine

spines on a left fan

pin on a left fan

Figure 2.1. Terminology and notation for a left fan of length three.

A collection of triangles satisfying only conditions i) and ii) is called a chain of triangles with the length defined as for fans. It is obvious that any chain of triangles of finite length greater than one admits a unique decomposition into a sequence 1 ; : : : ; M of fans, where iC1 is contiguous to i and the types (left or right) of i ; iC1 are different, for i D 1; : : : ; M  1; a chain of length one admits a unique such decomposition, but the type of 1 D M is undetermined. Lemma 2.3. If W 0 ! LC is the normalized representative of some pinched  W  ! RC , then the image .0 / is discrete in LC . Proof. Suppose the statement is not true. There is then some sequence ui 2 .0 /, for E Adopting cylindrical coordinates i  1, which accumulates at a point u1 2 LC [ f0g. on Minkowski space, we may write ui D ri .cos i ; sin i ; 1/, for 1  i < 1, and u1 D r1 .cos 1 ; sin 1 ; 1/. Either r1 D 0 or ri converges to r1 ¤ 0 and i converges to 1 . Identify 0 with the Farey numbers, so that each ui is the image under of some rational number. Taking a subsequence, we may assume that the generation of the Farey number corresponding to uiC1 exceeds the generation of the Farey number 1 corresponding to ui and that (the principal branch of) the argument of N i 2 S1 is monotone. It is elementary to produce a chain of triangles for so that each uN i occurs as an ideal vertex of some triangle in the chain. This chain is unique and decomposes into a collection of contiguous fans with two possibilities: either the chain is a semi-infinite sequence 1 ; 2 ; : : : of contiguous fans of finite length, or the chain is a finite sequence of contiguous fans 1 ; 2 ; : : : ; M , where M is an infinite fan and j is a finite fan for 1  j < M . In either case, let us enumerate the ideal points of the slats of 1 (in the natural linear ordering: counterclockwise for left and clockwise for right fans), then the vertices of the slats of 2 ,

2 Pinched lambda lengths

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then 3 , and so on. Denote the ideal points of the slats in this linear order by v1 ; v2 ; : : : . 1 Now, consider a slat in some fan of this chain with ideal points uN i ; vNj.i/ 2 S1 and let yNi denote the pin of the corresponding fan. Since  is uniformly bounded below, so too are the Minkowski inner products hui vj.i/ i, and it follows that in any case, the height of vj.i/ must diverge with i . Since  is uniformly bounded above, the height of yi therefore tends to zero with i . Finally, since ui converges to u1 , we must conclude that hui ; yi i tends to zero with i, and this contradicts that  is uniformly bounded below. Lemma 2.4. Suppose that W 0 ! LC is the normalized representative of some  W  ! R>0 . If .0 / is discrete in LC and  is uniformly bounded above, then

.0 / is radially dense in LC . 1 Proof. Suppose not, so that U  S1 is an open interval disjoint from the projection 0 1

. N  /  S1 . Choose some geodesic .e/ N for e 2   , set c 0 D e and define a sequence ui in the image of N as follows. Set j D 0 and let e j 2  satisfy .e N j / D c j . There 2 is a unique triangle t 2  with ej in its frontier so that .t N / lies in the component of D  c j which contains U in its ideal boundary. To complete the recursive definition, let xj C1 2 0 be the ideal vertex of .t N / to which c j is not asymptotic and let c j C1 1 . be the frontier edge of .t N / separating .t N / from a neighborhood of U in D [ S1 This construction determines a chain of triangles in the natural way, and we decompose this chain into contiguous fans. By definition, U separates the projection to 1 S1 of the pin of each such fan from the ideal points of its slats. Consider uj D .xj / 2 LC , for j  1, and write uj D rj .cos j ; sin j / in cylindrical coordinates. The sequence j contains a convergent subsequence by construction, and rj diverges by the assumed discreteness. To finish the argument, we shall show that  is not uniformly bounded above. To this end, let vj denote the lift to LC of the pin of a fan containing uNj as an ideal point of one of its slats. Since U separates uNj from vNj , the difference of their arguments is uniformly bounded below. By discreteness, neither uj nor vj can accumulate at the origin, and since rj diverges, we conclude that huj ; vj i diverges as well, which contradicts that  is bounded above.

As an immediate consequence of the previous two lemmas, we have:  Theorem 2.5 ([132]). If  2 R>0 is pinched, then then there is a decorated tesselation whose lambda lengths are given by , and the corresponding subset .0 /  LC is discrete and radially dense.

Definition 2.6. An orientation-preserving mapping f W D ! D is said to be quasiconformal if jfz j C jfzN j sup z2D jfz j  jfzN j is finite.

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3 Lambda lengths in infinite dimensions

In fact, any smooth f maps infinitesimal circles to infinitesimal ellipses, and the eccentricity of such an ellipse is the expression whose supremum we take here; in particular, f is conformal if and only if it maps infinitesimal circles to circles, and the expression is identically equal to one. Definition 2.7. Every quasi-conformal mapping of D extends to a homeomorphism of 1 S1 which is orientation-preserving, and such boundary values of a quasi-conformal mapping are called quasi-symmetric homeomorphisms of the circle. Conjugating by the Cayley transform, a more intrinsic characterization of a quasisymmetric mapping f W R ! R is that there exists an M such that M 1 < lim

f .x C t /  f .x/ <M f .x/  f .x  t /

for all x, t .

We refer the reader to [1], [2], [65] for a detailed discussion of quasi-conformal and quasi-symmetric mappings including the facts just mentioned and to [16] for the quasisymmetric version of a universal Teichmüller space called Bers’ universal Teichmüller 1 space. In practice here, we shall recognize quasi-symmetric mappings of S1 as boundary values of quasi-conformal mappings of D. Our next result, the goal of this section, gives a partial answer to one of the myriad questions. As noted before, Dragomir Šari´c has recently made substantial progress on such questions in [151]. Theorem 2.8 (joint with Dennis Sullivan in [132]). Suppose that the function  W  ! R>0 is pinched. Then there is a decorated tesselation realizing  as its lambda length coordinates, and the corresponding homeomorphism of the circle is quasi-symmetric. Proof. For the first part, since the lambda lengths are pinched, so too are the corresponding h-lengths (defined in analogy to Section 4.2 of Chapter 1 as the lengths along horocycles between geodesics) uniformly bounded above and below. The mapping from lambda lengths to cross ratios is again described by Corollary 4.16b of Chapter 1, so the corresponding cross ratios are likewise uniformly bounded above and below. The proof of the first part is verbatim the same as the proof in Theorem 2.5 that the mapping from the universal cover of the surface to D is a homeomorphism. Indeed, one uses the lambda lengths to recursively define a mapping D ! D pointwise fixing the triangle spanned by 01 , 10 , 11 which is a continuous injection by construction, and the facts just mentioned are used in the same manner as before to prove that this mapping is in fact a homeomorphism  W D ! D. This homeomorphism thus extends to an order1 preserving mapping from 0 to another countable dense subset of S1 and therefore 1 1 interpolates a normalized homeomorphism f W S1 ! S1 of the circle corresponding to a normalized tesselation  0 of D. Furthermore,  0 comes equipped with a decoration Q 0 determined by the lambda lengths as before. To see that f is quasi-symmetric, we shall show that f is given by the boundary values of a quasi-conformal homeomorphism ˆ W D ! D, and we must construct ˆ differently from  to see that it is quasi-conformal.

2 Pinched lambda lengths

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To this end, since the lambda lengths are pinched, we may scale them all by some overall factor to guarantee that they are all greater than two. It suffices to prove the result for the scaled lambda lengths, for then the original lambda lengths describe different decorations on the same underlying tesselation, so the two corresponding homeomorphisms of the circle coincide. Consider the Farey tesselation  decorated so that each lambda length is equal to 2. The complementary regions to the union of all the horocycles in this decoration with [ are of one of two types: either “hexagons” whose alternating sides are geodesic segments of length 2 log 2 (by Lemma 4.5 of Chapter 1) and horocyclic segments of length 1=2 (by Lemma 4.9 of Chapter 1), or “strips” bounded by a pair of asymptotic geodesic rays and a horocyclic segment of length 1=2. Let H denote the union of all the hexagons and  denote the union of all the strips. Likewise, the horocycles in the decoration of Q 0 are disjoint because of our scaling by Lemma 2.3. The complementary regions to the union of these horocycles with [ are again of two types: either hexagons whose alternating sides are geodesic segments of length between 2 log 2 and 2 log 2 C 4 log K and horocyclic segments of length between .2K 4 /1 and 2K 4 , or strips bounded by a pair of asymptotic geodesic rays and a horocyclic segment of length between these same latter bounds. Let H denote the union of all these hexagons for  and  denote the union of all these strips. There is a natural one-to-one correspondence between the hexagons in H and the hexagons in H , and because of the bounds on lengths of geodesic and horocyclic sides, there are quasi-conformal homeomorphisms uniformly near the identity on the boundaries between corresponding hexagons which combine to give a quasi-conformal mapping ˆ W H ! H . In order to extend to the strips, consider a strip whose horocylic segment has length h. Such a strip is conformal to the region in the upper half-plane U described by fz D x C iy W 0  x  1 and h1  yg. There is thus a quasi-conformal homeomorphism from the collection of strips in  lying inside a common horoball, i.e., the region containing the center bounded by the horocycle for Q 0 to a consecutive collection of regions fz D x C iy W n  x  n C 1 and y D h1 n g in U, where n 2 Z and the hn are uniformly bounded above and below. The corresponding region in  is conformal to fz D x C iy W y D 2g, and it is easy to extend ˆ across this region preserving quasi-conformality. Perform this extension for each horoball to finally construct the desired quasi-conformal mapping ˆ W D ! D with boundary values given 1 1 by f W S1 ! S1 . Remark 2.9. This corrects the original statement in [132] which posited also dif1 ferentiability at the ideal points in analogy to the extension to S1 of a lift to the universal cover D of a homeomorphism of the underlying surface and in analogy to the Minkowski question mark function. To see that smoothness does not hold, use the formula in Corollary 4.6 of Chapter 1 for lambda lengths in the upper halfspace model to produce pinched lambda lengths so that the two one-sided derivatives at infinity disagree. The error was pointed out by Dragomir Šari´c. See also Remark 3.14.

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3 Lambda lengths in infinite dimensions

3 Punctured solenoid This entire section is excerpted from joint work [147] with Dragomir Šari´c. Let G < PSL2 .Z/ be any torsion-free finite-index subgroup and choose a basepoint in the quotient surface M D D=G. Consider the category CM of all pointed finitesheeted unbranched covers  W F ! M , where F is a punctured Riemann surface with base-point. CM is a directed set, where 1  2 if there is a finite-sheeted unbranched cover 2;1 W F2 ! F1 of pointed Riemann surfaces so that the following diagram commutes: 2;1 / F1 F2 B BB | | BB || B || 1 2 BB | ~| M. In other words by covering space theory, if i < G < PSL2 .Z/ uniformizes Fi for i D 1; 2, then 1  2 if and only if 1 is a finite-index subgroup of 2 . Definition 3.1. A topological space, the punctured solenoid, is defined (in analogy to the closed case [156]) to be the inverse limit HM D lim CM I a point of HM is thus determined by choices of points yi 2 Fi for each cover i W Fi ! M , where the choices are “compatible” in the sense that if 1  2 , then we have in the notation above 2;1 .y2 / D y1 . Since punctured surface groups are cofinal in the set of punctured orbifold groups, we could have equivalently considered the category of orbifold covers of M in the definition of HM . Furthermore, if  < G is of finite-index, then HD= is naturally homeomorphic to HD=G , and we may thus think of the punctured solenoid H D HD=PSL2 .Z/ based on the modular curve. One can from first principles develop the Teichmüller theory of H along classical lines [2] as has been done [156], [123], [124] for the solenoid of closed surfaces. Instead, we next introduce an explicit space homeomorphic to H following [123], [124], and we shall then define the Teichmüller space representation theoretically in analogy to punctured surfaces. G has characteristic subgroups GN D \f < G W Œ W G  N g; for each N  1, and these are nested GN C1 < GN . Define a metric G  G ! R by   ı 7! minfN 1 W  ı 1 2 GN g; y of G as a space to be the metric completion of and define the profinite completion G G, i.e., suitable equivalence classes of Cauchy sequences in G. Termwise multiplicay the structure of a topological group, and termwise tion of Cauchy sequences gives G y multiplication by G gives a continuous action of G on G.

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3 Punctured solenoid

Definition 3.2. For any sub-group G < PSL2 .Z/ of finite-index, we may define the quotient y D .D  G/=G; y HG D D G G where  2 G acts by y ! D  G; y W DG

.z; t / 7! .z; t  1 /:

Lemma 3.3 ([123], [124]). H is homeomorphic to HG for any subgroup G < PSL2 .Z/ of finite-index. The topological picture of H is easy to understand in the model HG as we next explain. Example 3.4. Take G to be a Fuchsian group uniformizing a once-punctured torus for definiteness, say with fundamental domain the ideal quadrilateral Q. Let the frontier edges of Q in the counter-clockwise order be e, f , g, h and let ; ı 2 G be the edge y is a fundamental domain for the identifications .e/ D g, ı.f / D h. The set Q  G y action of G on D  G. Furthermore, the sides of these polygons are identified in the pattern: y e  t g  t  1 and f  t h  t ı 1 for t 2 G; y Thus, the punctured solenoid is and these are the only identifications on Q  G. obtained by “sewing” the Cantor set of polygons along their boundary sides according y to the action of G on D and on G. This immediately shows that path components of H , also called leaves of H are unit disks (by residual finiteness of G) and that each leaf is dense in H (by the density y of G in G). Definition 3.5. Let us now for definiteness simply fix G D PSL2 .Z/ and consider the y PSL2 .R// of all functions  W G  G y ! PSL2 .R/ satisfying collection Hom0 .G  G; the following three properties: Property 1:  is continuous; y we have Property 2: for each 1 ; 2 2 G and t 2 G, .1 ı 2 ; t / D .1 ; t21 / ı .2 ; t /I y there is a quasi-conformal mapping  t W D ! D so Property 3: for every t 2 G, that for every  2 G, the following diagram commutes: y DG

.z;t/7!. z;t 1 /

 t  1 id

 t id

 y DG

/DG y

. t .z/;t/7!. t 1 ı .z/;t 1 /

 / D  G. y

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3 Lambda lengths in infinite dimensions

y for the common refinement of the C 1 Furthermore,  t varies continuously in t 2 G topology of uniform convergence on compacta in D and the usual topology of Bers’ universal Teichmüller space on the extension of  t to the circle at infinity. As to Property 1, notice that since G is discrete,  is continuous if and only if it is so in its second variable only. Property 2 is a kind of homomorphism property of  mixing the leaves; notice in particular that taking 2 D I shows that .I; t / D I for y Property 3 mandates that for each t 2 G, y  t conjugates the standard action all t 2 G. y of G on D  G at the top of the diagram to the action  W .z; t / 7! ..; t /z; t  1 / at the bottom, and we let G D f W  2 Gg G. Notice that the action of G on 1 y y extends continuously to an action on S1  G. DG Definition 3.6. The solenoid with marked hyperbolic structure  is y D .D  G/=G y H D .D  G/

: y thus induces a homeomorphism H ! H . The collection  t , for t 2 G, y PSL2 .R// to be the collection of all continuous maps Define the group Cont.G; y PSL2 .R// is taken y ! PSL2 .R/, where the product of two ˛; ˇ 2 Cont.G; ˛W G y PSL2 .R// acts on pointwise .˛ˇ/.t / D ˛.t / ı ˇ.t / in PSL2 .R/. In fact, ˛ 2 Cont.G; y PSL2 .R// according to  2 Hom0 .G  G; .˛/.; t / D ˛ 1 .t  1 / ı .; t / ı ˛.t /; and this action is continuous. Theorem 3.7. There is a natural homeomorphism of the Teichmüller space of the solenoid H with y PSL2 .R//=Cont.G; y PSL2 .R//: T .H / D Hom0 .G  G; Rather than define the Teichmüller space of the punctured solenoid abstractly and describe the proof here, we shall for simplicity simply take this as the definition of the Teichmüller space T .H / by analogy with punctured surfaces. Again with an eye towards simplicity here, rather than defining punctures of solenoids intrinsically (as suitable equivalence classes of ends of escaping rays), we can more simply proceed as follows. Each  t W D ! D extends continuously to a quasiy is a -puncture symmetric mapping  t W S 1 ! S 1 . We say that a point .p; t / 2 S 1  G 1 if  t .p/ 2 Q, and a puncture of H itself is a G -orbit of -punctures. A -horocycle at a -puncture .p; t / is the image under  t of a horocycle in D centered at  t1 .p/. Definition 3.8. A decoration on H , or a decorated hyperbolic structure on H , is a y  Q ! PSL2 .R/  LC , where function Q W G  G .; Q t; q/ D .; t /  h.t; q/

3 Punctured solenoid

103

y PSL2 .R//, which satisfies the following further conwith .; t/ 2 Hom0 .G  G; ditions: y the image h.t; Q/  LC is discrete and radially dense; Property 4: for each t 2 G, y ! LC is continuous; Property 5: for each q 2 Q, the restriction h.; q/ W G Property 6: Q is G-invariant in the sense that if ı 2 G, then ı ı .; Q t; q/ D .ı; Q t ı 1 ; ıq/; where ı acts diagonally ı W .; q/ 7! .ı; ıq/ on PSL2 .R/  LC with ıq the natural action of G D PSL2 on LC . y  Q; PSL2 .R/  LC / denote the space of all decDefinition 3.9. Let Hom0 .G  G orated hyperbolic structures satisfying the properties above, and define the decorated Teichmüller space of the solenoid as the quotient y  Q; PSL2 .R/  LC /=Cont.G; y PSL2 .R//; Tz .H / D Hom0 .G  G y  Q; PSL2 .R/  LC / by y ! PSL2 .R/ acts on Q 2 Hom0 .G  G where ˛ W G     .˛ /.; Q t; q/ D ˛ 1 .t  1 / ı .; t / ı ˛.t /  h.t; ˛.t /q/ : y R / to be those f 2 Cont.G; y R / so that f .t  1 ; .e// D Finally, define ContG .G; >0 >0 y f .t; e/, for each  2 G, e 2  and t 2 G. y  G There is a natural lambda length mapping  W Tz .H / ! .R>0 / which assigns to a y  Q ! PSL2 .R/  LC the lambda length for the G metric of the function Q W G  G -horocycles determined by h at the endpoints of the geodesic in H labeled by  .

Theorem 3.10. The assignment of lambda lengths determines a real-analytic homeomorphism y R /; Tz .H / ! ContG .G; >0  y R /. where we take the strong topology on R>0 and on Cont G .G; >0

Proof. To prove the mapping is injective, we must again define the construction of y  G decorated hyperbolic structure from a continuous  t 2 .R>0 / . To this end, begin the definition of Q D   h on the triangle to the right of the doe in  with lambda lengths given by  t . As usual according to Lemma 4.1 of Chapter 1, we can uniquely lift to a triple of points in LC lying over ˙1; i . In effect, we shall guarantee that Property 2 holds by construction using a variant of the path-ordered product of holonomy with a requisite re-scaling. It is easily seen from the identification of G D PSL2 .Z/ with the oriented edges of  illustrated in Figure 3.3 in Chapter 1 that any  2 PSL2 .Z/ can be written uniquely in the one of the following forms:

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3 Lambda lengths in infinite dimensions

i)  D I ; ii)  lies in the free semi-group generated by either U , T or U 1 , T 1 ; iii)  arises from either i) or ii) by addition of the prefix S. Let us adopt the notation of Figure 3.3 in Chapter 1, where the label A lies next to the oriented edge eA D eI A in the figure. To define .; t / 2 PSL2 .R/, we shall specify an ideal triangle-doe pair .0 ; e 0 / in D, where e 0 is oriented with 0 to its right. There is then a unique  2 PSL2 .R/ mapping to the vertices of 0 the vertices ˙1; i of the triangle  to the right of the doe eI in  and mapping eI to e 0 . Let us write  2 G in one of the forms i-iii) above. Of course, if  D I , then .; t/ D I as follows from the functional equation in Property 2, and we take .0 ; e 0 / D .; eI /.  on the edges of  If  D S , then let us begin with the lambda lengths  t 2 R>0 and employ Lemma 4.1 of Chapter 1 to uniquely realize a lift to LC of the vertices of this decoration on the triangle . On the triangle to the left of the doe, consider the lambda lengths  tS 1 D  tS . It need not be that  t .eI / D  tS .eI /, and we re-scale, taking lambda lengths  t .eI /  tS ./  tS .eI / on the edges eU , eT . This defines lambda lengths on the edges of the quadrilateral in  triangulated by the doe. Again using Lemma 4.1 of Chapter 1, there is a unique lift of i to LC realizing the lambda lengths, and the projection of this point 1 is one of the vertices of 0 . The other two vertices of 0 are ˙1, and the to S1 doe is e 0 D eS , completing the definition in this case. Notice that this element of PSL2 .R/ that maps .; eI / ! .0 ; eS / is necessarily involutive, cf. the beginning of Section 4. The case of any word in one of the semi-groups in ii) is handled by induction on the length, where for instance for any such  that has a prefix U , one begins from the lambda lengths  t on  and relies on analogously re-scaled lambda lengths  tU 1 on the edges eU ; eT ; the edge eU plays the role of the doe in the first case. The remaining case iii) of a word from one of the semi-groups with prefix S is y ! handled in exactly the same manner completing the construction of  W G  G PSL2 .R/. The functional equation in Property 2 on  follows by construction. Furthermore,  since  t 2 R>0 depends continuously on t (because of the definition of the topology), .; t/ is also continuous in t as required in Property 1; indeed, the entries of .; t / 2 PSL2 .R/ are algebraic functions of finitely many lambda lengths. As for Property 3 in the definition of T .H /, we have y R /, then  t 2 R is pinched, for each t 2 G. y Claim 3.11. If  2 ContG .G; >0 >0 y ! R  To see this, note that the very definition of continuity of the function  W G >0

3 Punctured solenoid

105

means that for every K there exists an N with 1 C K 1 

 t .e/ 1CK  t .e/

for all e 2  and all  2 GN .

Take say K D 12 and its corresponding N . A fundamental domain for GN has only a finite collection of values of lambda lengths, and any other lambda length is at most a factor of 3/2 times a lambda length in this finite set, and at least a factor of 1/2 times a lambda length in this finite set.  t is therefore pinched, proving the claim. y  t W  ! R>0 is necessarily pinched. By TheoIt follows that for each t 2 G, rem 2.8, there is a corresponding quasi-conformal homeomorphism  t W D ! D. Commutativity of the diagram and continuity in Property 3 follow by construction, and this y PSL2 .R// completes the proof that the function  constructed above lies in Hom0 .GG; and hence determines a point of T .H /. To define a decoration on the -punctures, each  t determines a decoration on y as required. Property 4 is guaranteed by the claim and Theorem 2.5.   ftg  D  G, Property 5 holds as before in the strong sense that the Euclidean coordinates of each h.t; q/ are algebraic functions of finitely many lambda lengths, and Property 6 holds by invariance of lambda lengths under PSL2 .R/. Remark 3.12. The prescience for the solenoid of the earlier study of pinched lambda lengths was a pleasantly surprising development in its day. y R /. Then for each t 2 G, y  t W  ! Proposition 3.13. Suppose that  2 Cont.G; >0 R>0 corresponds to a quasi-conformal homeomorphism  t W D ! D whose quasi1 1 ! S1 is differentiable at each point of Q with derivative symmetric extension  t W S1 uniformly near unity. Proof. As above, continuity of  t in t implies that each  t is pinched, which gives y by Theorem 2.8. Using the upper halfspace a quasi-symmetric map  t , for t 2 G, model, normalize  t such that it fixes 0 and 1, whence the geodesics of  that limit to 1 are mapped by  t onto geodesics which likewise limit to 1. Again by continuity, we conclude that for each " > 0, e 2  and  2 PSL2 .Z/ fixing 1, we have j t .e/   t . n e/j < " for n sufficiently large. The differences an D  t .n/   t .n  1/ are then "1 close using continuity of the assignment of decorated ideal triangles given lambda lengths. We show that limn!1 n1  t .n/ exists and is bounded, which proves the proposition. To this end since jai  ankCi j < "1 for all i , k, we find ˇ ˇ1 ˇ .a1 C    C an /  1 .a1 C    C ank /ˇ n nk  1=n

n X ˇ ˇai 

1 .ai nk

ˇ C aiCn C    C aiCn.k1/ /ˇ  "1 ;

iD1

and it follows that

1 n

 t .n/ is a Cauchy sequence, as desired.

106

3 Lambda lengths in infinite dimensions

Remark 3.14. Differentiability at the rational points, which holds in the case of punctured surfaces thus also holds for the solenoid according to Proposition 3.13 but does not hold for general decorated tesselations with pinched lambda lengths however, cf. Remark 2.9. According to Theorem 3.10 and in contrast to surfaces or homeomorphisms of the circle, now we have lambda length functions y ! R>0 e W G

for each e 2 

instead of numbers on the edges of an ideal triangulation or tesselation, one such continuous function for each (unoriented) edge e of the Farey tesselation, giving coordinates on Tz .H /. This has the important consequence that the decorated Teichmüller space Tz .H / of the solenoid is identified with the function space y R / Tz .H / D Cont G .G; >0 just as for surfaces

Tz .F / D R>0 ;

for an ideal triangulation  of F D Fgs , and in contrast to circle homeomorphisms, where we only have an embedding

e

 T ess  R>0

e

into the function space. Notice that this passage from T ess to Tz .H / amounts to requiring a kind of quasi invariance under PSL2 .Z/ on the circle homeomorphisms. Furthermore, rather than the awkward weak topology on the function space for T ess, we have the strong topology for the solenoid. Because of these and other parallels with punctured surfaces, we believe that the solenoid provides a most natural universal Teichmüller theory. In further parallel to the case of punctured surfaces, there is a mapping class group MC.H / of the solenoid H given by all isotopy classes of quasi-conformal homeomorphisms modulo isotopies that are bounded in the hyperbolic metric on the leaves. This mysterious uncountable group contains the countable subgroup M CBLP .H / of socalled baseleaf-preserving homeomorphisms, which setwise fix the leaf of H arising from the identity id 2 G, i.e., the quotient of D  id in HG . Just as flips generate MC.F /, so too an equivariant version of flips turn out to generate M CBLP .H / as follows.

e

Definition 3.15. Let K be a finite-index subgroup of G, let  be a tesselation of D x and suppose that e is a fixed which is invariant under K with ideal points  1 D Q, unoriented edge of  with distinct endpoints. Define another tesselation  0 as follows: for each f 2   Ke, there is an identical edge f 2  0 ; for each f 2 Ke, consider the quadrilateral P with diagonal f comprised of the triangles on either side of f

3 Punctured solenoid

107

complementary to  , and let f 0 denote the other diagonal of P ; for each edge f 2 Ke, there is a corresponding dual edge f 0 2  0 . The resulting tesselation  0 is clearly also invariant under K. We say that  0 arises from  by the (K-equivariant) flip or Whitehead move along e 2 . Furthermore, if  is a tesselation with doe d 2  , then we may induce a doe d 0 on 0  as follows: if d … Ke, then d 0 D d as oriented edges, while if d 2 Ke, then there is the unique orientation on d 0 so that correctly oriented tangent vectors to d; d 0 give a positive basis for the tangent space at d \ d 0 . Thus, for any tesselation  invariant by a group K, there is a corresponding equivariant flip for each edge e of , and there is exactly one distinct equivariant move for each K-orbit of edges of . An equivariant flip acts not only on invariant tesselations but also on invariant tesselations with doe. Lemma 3.16. Suppose that K is a finite-index subgroup of G and  is a K-invariant tesselation with doe d . Perform an equivariant flip along the edge e to produce the invariant tesselation  0 with doe d 0 . Let h D h.; d /; h0 D h. 0 ; d 0 / denote the characteristic maps and define k D h0 ı h1 . Then k is a quasi-symmetric map, and k. / D  0 with k.d / D d 0 . Proof. The characteristic maps h, h0 are quasi-symmetric by Theorem 2.8 since lambda lengths for a tesselation are pinched by Lemma 6.1, hence the composition k is quasisymmetric as well. That k maps  to  0 and d to d 0 follow from the definition of characteristic maps, completing the proof. Remark 3.17. In fact, M CBLP .HG / is also identified with the group of all so-called virtual automorphisms of G, namely, the group of all isomorphisms between finite-index subgroups of G which may have different indexes; see [123], [124], [122]. Furthermore, the homeomorphism k in the previous lemma is in fact a virtual automorphism of PSL2 .Z/, cf. [147]. Definition 3.18. We call k D k.; K; e/ 2 ModBLP .HG / the flip homeomorphism or Whitehead homeomorphism associated with the K-equivariant flip along e 2  for any K-invariant tesselation  with doe. Notice that by definition if f 2 Ke, then k.; K; e/ D k.; K; f /, so we regard flip homeomorphisms as indexed by K-orbits of edges Ke rather than by edges e. In contrast to the case of punctured surfaces where only certain sequences of flips give rise to mapping classes (namely, the sequence must begin and end with combinatorially identical ideal triangulations), for the punctured solenoid, each equivariant flip gives rise to a mapping class. In fact, these equivariant flips generate M CBLP .H / as shown in [147], and a complete set of relations for them is given in [21]. We shall likewise give relations on flips and hence presentations for mapping class groups of surfaces at the end of the next chapter.

108

3 Lambda lengths in infinite dimensions

In still further analogy to the case of punctured surfaces, there is an invariant two form whose avatar is the pull-back of the Weil–Petersson Kähler two form given in Theorem 3.1 as follows. A fundamental polygon P for the once-punctured torus group G < PSL2 .Z/ on D consists of the ideal triangle T1 with vertices ˙1; i and T2 with vertices ˙1; i. We shall define tangent vectors on Tz .HG / in terms of the lambda length parametrization. y R / is a subset of vector space Cont G .G; y R / of all continuous Note that Cont G .G; >0  y onto R  that are invariant under the action of G. The norm on functions from G y R / is given by ContG .G; kvk D

sup

jv.t; e/j

y e2 t2G;

y R /: for v 2 Cont G .G;

y R / is an open subset of the vector space With respect to this norm, ContG .G; >0 G y  y R / induced by the norm on Cont .G; R /, and the subspace topology on ContG .G; >0 y R / coincides with the compact-open topology. It follows that the tangent ContG .G; y R /. space at any point of Tz .HG / is identified with ContG .G; To define the Weil–Petersson two form on the tangent space at an arbitrary point y R /, let u; v 2 ContG .G; y R / be two arbitrary vectors in the tangent  2 ContG .G; >0 i space at . Furthermore, let ;u;v .t / be the evaluation of the two form 2.d log a ^ d log b C d log b ^ d log c C d log c ^ d log a/ y and the usual on the triangle Ti  ft g and vectors u.t; / and v.t; /, for i D 1; 2,t 2 G, notation for a, b, c. We define the Weil–Petersson two form by Z ! .u; v/ D Œ1;u;v .t / C 2;u;v .t / d m.t /; y G

y The actual formulas are where m is the Haar measure on G. 1;u;v .t/ D 2. 2;u;v .t/ D 2.

u.t; e0 /v.t; e1=2 / u.t; e1=2 /v.t; e2=1 / u.t; e2=1 /v.t; e0 / C C /; .t; e0 /.t; e1=2 / .t; e1=2 /.t; e2=1 / .t; e2=1 /.t; e0 /

u.t; e0 /v.t; e2=1 / u.t; e2=1 /v.t; e1=2 / u.t; e1=2 /v.t; e0 / C C /: .t; e0 /.t; e2=1 / .t; e2=1 /.t; e1=2 / .t; e1=2 /.t; e0 /

In the case when  is a lift of a decoration on the punctured torus D=G and the tangent vectors represent deformations in Tz .D=G/, the two form ! .u; v/ is equal to the classical Weil–Petersson two form by Theorem 3.1. If  comes from a lift on some higher genus surface and the tangent vectors represent deformations for that surface, then the two form is a positive multiple of the classical Weil–Petersson form. y it is Since .t; e/, for fixed e 2  , is a continuous positive function in t 2 G, y bounded below away from zero. Moreover, u.t; e/ and v.t; e/ are continuous in t 2 G, i for fixed e 2  . Thus, ;u;v .t /, for i D 1; 2, are bounded and continuous real

109

3 Punctured solenoid

functions in t , and the Weil–Petersson two form ! .u; v/ is well defined, namely, the y is compact. integral converges since G Restricting attention to the base leaf, let ;u;v jT be the evaluation of the two form 2.d log a^d log bCd log b^d log cCd log c^d log a/ on the triangle T D .a; b; c/ in D , where d log a D da=a, and so on, are given their corresponding values in terms of .id; a/ and u.id; a/ on the baseleaf D, over which we shall average . Consider any sequence of characteristic subgroups Gn of PSL2 .Z/ such that \n Gn D fid g. Let Pn be a sequence of fundamental polygons for Gn such that the frontier edges of Pn belong to the Farey tesselation  , and the Pn are nested. y R / and let u; v 2 ContG .G; y R / be two Proposition 3.19. Let  2 ContG .G; >0 tangent vectors based at . Then 2 X ;u;v jT ; n!1 k.n/

! .u; v/ D lim

T 2Pn

where the sum is over all triangles T complementary to  in the fundamental polygon Pn , and k.n/ is the number of triangles in Pn . 2 P Proof. We first show that limn!1 k.n/ T 2Pn ;u;v jT does not depend on the choice of the fundamental polygons Pn . Indeed, for n large enough the quantities , u and v are almost invariant under the group Gn . Thus, the quantity ;u;v jT is also almost invariant under Gn . Namely, given " > 0 there exists n such that j;u;v jT  ;u;v j T j < " for all triangles T complementary to  whenever  2 Gn . Hence, if the limit exists it does not depend on the choice of Pn . For a fixed sequence Pn of fundamental polygons, the sum over triangles 2 P T 2Pn ;u;v jT is a Cauchy sequence. This immediately follows from the above k.n/ observation of almost invariance of ;u;v jT . It remains to show the equality between ! .u; v/ and the limit of the sum. In the case when , u and v arise from a decorated hyperbolic structure on some surface Fgs , the equality follows from the definition of the Haar measure and the observation that 2 our normalization factor k.n/ is chosen correctly. The general case follows from the i uniform continuity of ;u;v .t / and the fact that ;u;v j T D i;u;v .t / in the case when y is an element of G < G y and T is the corresponding triangle Ti  f g. t D 2G The solenoid thus provides a useful universal object in Teichmüller theory not only for the reasons mentioned before but also because of the nice expression for the invariant two form on Tz .H /, for which the regularization difficulties on T ess disappear, cf. Remark 1.15.

e

Remark 3.20. In fact, this two form on decorated Teichmüller space Tz .H / is invariant under M CBLP .H /, and it descends to a non-degenerate form on Teichmüller space T .H / as shown in [147].

110

3 Lambda lengths in infinite dimensions

4 Lie algebra of tesselations This section is based on [107], which is joint work with Fedya Malikov, and on [138]. As spaces, we may identify T ess with the collection Homeon .S 1 / of all normalized orientation-preserving homeomorphisms of the circle, namely, orientation-preserving homeomorphisms of the circle fixing the three points 01 , 10 , 11 . Since an element of PSL2 .R/ is determined by its values at these (or any distinct) three points, we might think that R  sl2 is a sensible kind of tangent space to HomeoC .S 1 / itself, where sl2 is the Lie algebra of the group SL2 .R/, the double cover of PSL2 .R/, and R is given its natural Fréchet structure in shear coordinates. This is of course not strictly speaking correct since the image of the embedding in Proposition 1.14 is not open. In fact to employ lambda lengths, we shall go a bit further and define the group Homeon .S 1 / of decorated and normalized homeomorphisms of the circle to be the set of all pairs .fQ; f / with f 2 Homeon .S 1 / and fQ W T ess ! T ess a homeomorphism covering f with the obvious group structure. In particular, there is a surjective topological group homomorphism Homeon .S 1 / ! Homeon .S 1 / gotten by projecting onto the second factor. There is furthermore an isomorphism Homeon .S 1 / T ess of Homeon .S 1 /-spaces gotten by assigning to .id; id/ 2 Homeon .S 1 / the Farey tesselation  equipped with its canonical decoration, where all lambda lengths are unity. As in the previous discussion for Homeon .S 1 /, we seek a natural Lie algebra structure on the product RC sl 2 , where RC is given its natural Fréchet structure in lambda lengths, which is again not quite right since the image of the embedding in Proposition 1.14 is not open. Recall from Lemma 3.4 of Chapter 1 that PSL2 .Z/ acts simply transitively on the oriented edges of the Farey tesselation  , and let eA denote the oriented edge of  which is the image eI A under the right action of A on the standard doe eI connecting 0 1 to 10 . Likewise, let pq A denote the image of the point pq 2 S1 under A. 1 We have already in Lemma 1.16 described the basic deformation ƒ.s/ of the single lambda length on the doe, and we may extend this to other edges of the Farey tesselation as follows. For each A 2 PSL2 .Z/, define the corresponding

B

B

B

e

B

e

B

e

ƒA .s/ D A1 ƒ.s/A; which we again may regard as a family of piecewise SOC .2; 1/ homeomorphisms of LC . As before, each ƒA .s/ acts on the lambda length of pair of horocycles, and hence ƒA .s/ acts on T ess0 .

e

Lemma 4.1. For each A 2 PSL2 .Z/, ƒA .s/ is a one-parameter multiplicative subgroup of HomeoC .S 1 /, where each ƒA .s/ is once-continuously differentiable on S 1 with exactly four fixed points, which are given by 01 A, 10 A, ˙ 11 A. Furthermore, for any decoration on the Farey tesselation  , ƒA .s/ leaves invariant the lambda length on every edge of  except that it scales the lambda length of the unoriented edge underlying eA by the factor s.

111

4 Lie algebra of tesselations

Proof. All the assertions for ƒA .s/ follow from the corresponding assertions for ƒ.s/ itself in Lemma 1.16.

B

Thus, these one-parameter families ƒA .s/ are regarded as the coordinate deformations at the identity of Homeon .S 1 /, and we are led to consider the corresponding vector fields on S 1 .   Lemma 4.2. The vector field on S 1 corresponding to ac db 2 sl2 is given by f.b C c/ cos  C .a  d / sin  C .c  b/g

@ ; @

where  is the usual angular coordinate on the circle.  ˇ.t/ Proof. For the proof, we shall let ˛.t/ denote the corresponding one-parameter .t/ ı.t/ 0 subgroup of SL2 .R/, so a D ˛ .0/; : : : ; d D ı 0 .0/, where ˛.0/ D 1 D ı.0/, ˇ.0/ D 0 D .0/, ˛ 0 .0/ ı 0 .0/  ıCˇ  D 0, and the prime denotes the derivative with respect to t . The action of  ˛ corresponds to the right action of the Möbius group, and we conjugate its action on the upper half-plane by the Cayley transform to compute that ei 2 S 1 maps to Œ.ı  ˛/i  . C ˇ/ C Œ.ı C ˛/i  .  ˇ/ ei : Œ.ı C ˛/i C .  ˇ/ C Œ.ı  ˛/i C . C ˇ/ ei Take

d j dt tD0

of i times the logarithm of this expression to derive the asserted formula.

We refer to a vector field as in Lemma 4.2 as a global sl2 vector field and shall take the standard basis for sl2 given by       0 1 0 0 1 0 eD ; f D ; hD ; 0 0 1 0 0 1 with the usual Lie bracket Œh; e D 2e, Œh; f D 2f , Œe; f D h. In this notation, the derivative of ƒ.s/ is directly calculated to be 8 ! ˆ 1 2 ˆ ˆ D h C 2e on quadrant I; ˆ ˆ ˆ 0 1 ˆ ˆ ! ˆ ˆ ˆ 1 0 ˆ ˆ ˆ < 2 1 D h C 2f on quadrant II; ! #D ˆ 1 0 ˆ ˆ D h  2f on quadrant III; ˆ ˆ ˆ 2 1 ˆ ˆ ! ˆ ˆ ˆ 1 2 ˆ ˆ on quadrant IV. ˆ : 0 1 D h  2e

112

3 Lambda lengths in infinite dimensions

@ Writing # D #./ @ , we graph this function #. / in Figure 4.1 and remark that though it “looks like” the usual sine function, it is once- but not twice-continuously differentiable.

0    2

Figure 4.1. Elementary vector field #I .

Definition 4.3. The vector field # lives naturally in the space psl 2 defined to consist 1 of all piecewise sl2 vector fields on the circle S1 with finitely many pieces and with x  S 1. breakpoints in the piecewise structure among the rational points Q Though # is itself actually defined and continuous at its breakpoints, we do not require this of vector fields in psl2 , which are regarded as undefined at their breakpoints. The reason for allowing these more general vector fields in psl 2 is that brackets of conjugates of # will fail to be defined at their breakpoints as we shall see. Given two elements of psl2 , there is a natural bracket defined by taking the crudest common refinement of their pieces and taking the usual bracket from sl 2 on each such piece. Thus, psl2 is naturally a Lie algebra containing sl 2 as the sub-algebra of global sl2 vector fields. Remark 4.4. It is an interesting open problem to find an honest Fréchet Lie group whose honest Lie algebra is psl 2 . Perhaps this is related to the Thompson group, cf. Section 1 of Chapter 6. Having thus defined the very special element # 2 psl2 , we proceed to define #A D A1 #A

for A 2 PSL2 .Z/;

113

4 Lie algebra of tesselations

using the adjoint action on each piece. A short calculation shows that #S D #, where we here and below adopt the standard notation of Lemma 3.4 of Chapter 1 for elements of the modular group PSL2 .Z/. Thus, if e 2  is an unoriented edge, then we may associate the well-defined element #e D #A D #SA 2 psl2 , where A 2 PSL2 .Z/ maps the unoriented edge underlying the doe of  to the edge e 2  . Definition 4.5. The vector field #e is called the elementary vector field associated with e 2  .   Figure 4.2 illustrates the elementary vector field #A for A D ac db , with the ideal points indicated in their Farey enumeration, where the matrix near an interval indicates the corresponding element of sl 2 (and where we have tacitly assumed in drawing the figure that the entries of ˙A are non-negative with jcj  jaj).



ad bc 2ab 2b.bCd / 2a.aCc/ ad CbcC2ab b a



bCd aCc

eA



ad CbcC2cd 2c.aCc/



d c

2d.bCd / ad bc 2cd



ad Cbc 2cd 2d.b d / 2cd ad bc 2c.c a/



d b a c



2ab ad bc 2b.b d / 2a.c a/ ad Cbc 2ab



Figure 4.2. Elementary vector field #A . 1 An elementary vector field #A is defined everywhere on S1 , even at its breakpoints since # itself has this property, so it makes sense to define the normalization #NA D #A  x 2 psl2 of #A , where x 2 sl2 is chosen so that #A agrees with x at the points 01 , 1 1 , . For some examples of normalization, #N D # since it already vanishes at these 0 1 three points (and at  11 as well), and a short calculation furthermore shows that

#N U n D #U n C h  2.n  1/f;

for n > 1.

114

3 Lambda lengths in infinite dimensions

We shall preview the next definitions with their motivation. We wish to calculate the Lie algebra closure in psl 2 of sl 2 and the normalized elementary vector fields. Exploratory calculations of brackets of these vector fields lead to the vector fields on the circle we next introduce, which have remarkable algebraic properties. After studying these properties, we shall then prove that the Lie algebra closure we seek is actually all of psl 2 and derive a surprisingly simple additive basis. Definition 4.6. For each oriented edge eA D eI A of  with A 2 PSL2 .Z/, define the corresponding fan vector field A D

X

#N U n A

n0

and hyperfan vector field A

D

X

n#N U n A :

n0

The initial point of the oriented edge eA is called the pin of the corresponding fan or hyperfan. Because of the normalization, these infinite sums converge pointwise to vector fields on the circle except perhaps at the pin, and the convergence is uniform on each compactum not containing the pin. (It is for this reason, to get such convergent sums, that we normalized the elementary vector fields.) There is, however, no reason for these vector fields to live in psl2 , i.e., have only finitely many pieces, and yet: Lemma 4.7. We have the equalities 8 ˆ on quadrant II <2e U D 2h  2f on quadrant III ˆ : 0 on quadrants III and IV; ´ 2e on quadrants I and III I D 0 on quadrants III and IV:

Proof. Letting AdX .x/ D X 1 xX for x 2 sl 2 and X 2 PSL2 .Z/, we first calculate that AdU .e/ D h C e  f; AdU .f / D f; AdU .h/ D h  2f: The equality for the fan U is proved separately in each quadrant, where one confirms that AdU .h  2e/ C AdU 2 .h  2f / C 2h  2f D 2e for quadrant I, and quadrants III and IV are immediate. Using the expression given before for the normalizaton #N U n ,

115

4 Lie algebra of tesselations

the identity in quadrant II amounts to 2h  2f D

n1 hX

i AdU j .h C 2f / C AdU n .h C 2e/

j D1

C AdU nC1 .h  2e/ C

h nC1 X

i h  2.j  1/f ;

j D1

which is proved by induction on n  2. Likewise for the hyperfan I , the equality in quadrants III and IV is immediate, and the equality in quadrants I and II is tantamount to the identity f ehD

n1 hX

i AdU j .h  f /  AdU n .e/;

j D1

which again follows by induction on n  2. Corollary 4.8. For any A in the modular group, the fan A and hyperfan psl2 .

A

lie in

Proof. Any two fans or hyperfans are conjugate modulo an element of sl 2 , i.e., AB  B 1 A B;

AB

 B 1

AB

2 sl2

for A; B 2 PSL2 .Z/,

and the reason these differences do not vanish identically is that fans or hyperfans are linear combinations of normalized vector fields. Nevertheless, it follows from this observation and Lemma 4.7 that fans and hyperfans indeed have only finitely many pieces in their piecewise Möbius structure. Whereas the elementary vector fields are once continuously differentiable on the circle, fans are continuous but not differentiable, and hyperfans are not even continuous, here using that any fan is conjugate to I and any hyperfan conjugate to I modulo sl 2 . Furthermore, I integrates to the square of the primitive parabolic transformation in the modular group fixing the endpoint 10 of the doe eI and rotating counter-clockwise about this point in positive time. In general, consider the hyperfan A and let VA be the vector field which likewise integrates to the primitive parabolic modular transformation fixing the endpoint of eA . If the triangle complementary to  on the right of the doe lies to the right of the oriented edge eA , then A vanishes on the right of eA and agrees with VA on the left; if the triangle complementary to  to the right of the doe lies to the left of the oriented edge eA , then A vanishes on the left of eA and agrees with VA on the right. Thus, hyperfans are “piecewise parabolic” maps on the circle (in contrast to the “earthquakes” introduced by Thurston [160], which are “piecewise hyperbolic”). Theorem 4.9. The Lie algebra closure of sl2 together with the collection of all elementary vector fields f#N e W e 2  g is the entire Lie algebra psl2 . Furthermore, psl 2 is generated as a vector space by sl2 and the collection of all hyperfans.

116

3 Lambda lengths in infinite dimensions

Proof. The ray from the origin in R2 through the point  12 decomposes the second quadrant II into two regions denoted by II1 , which contains  11 in its closure, and II2 , which contains 01 in its closure. Applying the formulas in Figure 4.2 to the case A D U , we find that 8 2e on quadrant I; ˆ ˆ ˆ <4h  4f C 2e on region II ; 1 #N U D ˆ 4f on region II2 ; ˆ ˆ : 0 on quadrants III and IV. Calculating brackets, we have 8 ˆ on quadrants I, III and IV. <0 Œ#N U ; e D 8e C 4h on region II1 ; ˆ : 4h on region II2 , ´ 16f on region II2 , ŒŒ#N U ; e ; h D 0 else, 1 so ŒŒ#N U ; e ; h is supported on a single circular segment C  S1 . Since sl 2 is simple, we may likewise realize any element of sl2 on C , and in particular, both T U and ST U , which are supported on C , lie in the Lie algebra generated by the elementary vector fields and sl 2 . Since this Lie algebra is by definition closed under the adjoint action of SL2 .R/, it follows that every hyperfan lies in this algebra. To complete the proof, we must show that sl2 and hyperfans span psl2 as a vector 1 space. To this end, consider the circular segment  ainb S 1 whose endpoints are given by the ideal points of eA , where the entries of A D c d 2 SL2 .R/ are non-negative. Let C denote the circular sub-segment bounded by the ideal points of eUA . By the previous characterization of hyperfans, A and TA take common values on the intersection of their supports, UA , SUA , . A  TA / all have support C , and furthermore, these vector fields restricted to their common support C are given by UA SUA

. The matrix

A



TA /

0

D 2.b C d /2 e C 2.a C c/2 f  2.a C c/.b C d /h; D 2b 2 e  2a2 f C 2abh; D 2d 2 e C 2c 2 f  2cdh:

2.b C d /2 @ 2b 2 2d 2

2.a C c/2 2a2 2c 2

1 2.a C c/.b C d / A 2ab 2cd

is found to have determinant 8.ad  bc/3 D 8 ¤ 0, and hence any vector field supported on C lies in the span of these. A similar argument using the hyperfans SA , STA , S UA likewise handles the circular sub-segment bounded by the ideal points

117

4 Lie algebra of tesselations

of eTA , and indeed the analogous argument for a circular segment lying in the lower half-plane completes the proof. Remark 4.10. In fact, one can give a complete set of additive relations among hyperfans, namely, for any A 2 PSL2 .Z/, we have U 1 A

2

A

C

UA

D

U 1 SA

2

SA

C

USA :

Indeed from the definitions of fans and hyperfans, we find A  UA D #NA D SA  USA ; A  UA D A ; SA  USA D SA arising from the identity #A D #SA discussed before, which thus yields the asserted relations among hyperfans. Furthermore, these relations are linearly independent, and they span the space of linear dependencies among hyperfans, cf. Proposition 6 of [107]. One can use this result together with further calculations to give an additive basis for psl2 , namely, e, f , h and f

A

W A 2 fI g [ U P [ U 1 N [ S T P [ S T 1 N ;

where P is the monoid of products of T , U and N is the monoid of products of T 1 , U 1 , cf. Theorem 2.1 of [138]. We shall not prove these facts here since the arguments in print are somewhat involved and yet are the simplest we know. In contrast, we have included the proof of Theorem 4.9 here since it is much simpler than the previously published proof. This completes our algebraic discussion of the Lie algebra psl 2 that arises by considering lambda lengths as coordinate deformations of the space Homeon of decorated normalized homeomorphisms of the circle, and we finally turn our attention to the harmonic analysis of these deformations. Let Diff C .S 1 /  HomeoC .S 1 / denote the subgroup of infinitely differentiable orientation-preserving homeomorphisms of the unit circle, whose Lie algebra diff C @ consists of all infinitely differentiable real vector fields f . / @ of the circle, which we may Fourier expand X an cos n C bn sin n f ./

B

n0

D

X

cn ei n ;

with cn D cNn .

n2Z

Just as for homeomorphisms, let Diff n .S 1 / D Diff C .S 1 / \ Homeon .S 1 /

118

3 Lambda lengths in infinite dimensions

denote the subgroup of normalized diffeomorphisms fixing 01 ; 10 ; 11 with its Lie algebra diff n of all infinitely differentiable vector fields which vanish at these three points. @ defined on S 1 , we define its normalization Given any vector field V D f ./ @ @ @ D ff ./ C ˛ cos  C ˇ sin  C g ; Vx D fN./ @ @ N where ˛; ˇ;  2 R are chosen so that f ./ D 0 for  D 0; ; 3=2. For example with f . / D ei n , a calculation gives the following normalized trigonometric fields for n  2: ¯ d 1® n @ D i.z C z n / C ˛i.z C z 1 / C ˇ.z  z 1 / C 2 i z ; cos n @ 2 dz with 2˛ D .1/n  1; 2 D 1  .1/n ; 2ˇ D .1  .1/n /  .1/nC1 .i n C i n /; and sin n

¯ d @ 1® n D z  z n C ˛i.z C z 1 / C ˇ.z  z 1 / C 2 i z ; @ 2 dz

with ˛ D 0 D ;

2ˇ D i.1/nC1 .i n  i n /;

which constitute a natural basis for diff n that we shall require later. Put another way, the exponential functions ei n admit the normalization n i xe i n D ei n  Œb0n C b1n ei C b1 e ;

where

8 C1; n  0.4/; ˆ ˆ ˆ < 0; n  1.4/; b0n D ˆ C1; n  2.4/; ˆ ˆ : 0; n  3.4/I

8 8 0; n  0.4/; 0; n  0.4/; ˆ ˆ ˆ ˆ ˆ ˆ 1, the Fourier expansion P @ #NA D n2Z cn ei n @ is given by

 .b  d /  i.a  c/ n  i.n3  n/ cn D Œ.c  a/2 C .b  d /2 .b  d / C i.a  c/



n  d  ic b  ia n C 2.c 2 C d 2 / C 2.a2 C b 2 / d C ic b C ia

n 2 .b C d /  i.a C c/ 2  Œ.c C a/ C .b C d / ; .b C d / C i.a C c/ and the Fourier modes c0 ; c˙1 are chosen to guarantee that the expansion is normalized.

119

4 Lie algebra of tesselations

Proof. Since the normalized elementary vector field #NA is once continuously differentiable on the circle, we may twice integrate by parts the usual expression for Fourier coefficients to conclude that cn D

X 1 1 zj ei n j@Ij ; 2 i n3  n

for n2 > 1;

j

where Ij , for j D 1; 2; 3; 4, are the intervals in the psl2 structure of #NA and #NA D ˚ @ xj cos  C yj sin  C zj @ on Ij . We may calculate these zero-modes zj explicitly from Figure 4.2 for A ¤ I; S , where there are actually four separate cases depending upon in which quadrant lies eA . In each case, the explicit not-entirely-painless calculation yields the asserted exprespiq 1 sion using that the Farey point pq 2 S1 corresponds to the complex number pCiq . Notice that for n D 0; ˙1 both sides of the asserted expression vanish identically. In particularP for the elementary vector field #I , the analogous direct calculation gives 1 i n @ #I 8i , which likewise agrees with the asserted formula. n2.4/ n3 n e @ Remark 4.12. Notice that since #N satisfies a Lipschitz condition of exponent 2, we have jcn j D O.jnj2 / by a standard result in analysis. This satisfactorily describes the harmonic analysis of psl 2 , and we next wish to understand the reverse inclusion. In order to employ lambda lengths to this end, define Diff n .S 1 / to be the collection of all pairs .fQ; f / , where f 2 Diff n .S 1 / and fQ is a homeomorphism of T ess covering f ; there is natural group structure on Diff n .S 1 / as before and a natural surjective group homomorphism Diff n .S 1 / ! Diff n .S 1 /. We define a section  W Diff n .S 1 / ! Diff n .S 1 /

e

e

e

e

e

as follows. If f 2 Diff n .S 1 /, we may conjugate by the Cayley transform C W U ! D to produce a diffeomorphism f C D C 1 ı f ı C W R ! R, where f C fixes 0; 1 2 R. A horocycle in U tangent to x 2 R is determined by its Euclidean diameter d , and we c define the Euclidean diameter of the horocycle for  .f / at f C .x/ to be j df .x/jd , dx where this definition is motivated by Corollary 4.6 of Chapter 1. Likewise, define the evolution under f C of a horocycle in U centered at infinity so that its Euclidean height scales by the reciprocal of the derivative of f C at infinity. This defines a natural action of Diff n .S 1 / on decorations and hence defines the desired section  . The chain rule then shows that  is furthermore a group homomorphism. This section  allows us to regard

e

B

Diff n .S 1 /  .Diff n .S 1 //  Diff n .S 1 /  Homeon .S 1 / so as to calculate with lambda lengths, that is, we shall finally calculate diff n using normalized trigonometric vector fields as follows.

120

3 Lambda lengths in infinite dimensions

Let w.z; t/ denote the one-parameter family of diffeomorphisms of S 1 that arise @ @ by integrating the normalized trigonometric fields cos n @ and sin n @ discussed @w before. In particular, we have @z j tD0 D 1, an identity we shall serially apply in the sequel. To calculate  , define si for s 2 R; sCi 1Cz 1 s D C 1 .z/ D i ; for z 2 S1 1z where C is the Cayley transform, so that z D C.s/ D

W .s; t/ D C 1 ı w.C.s/; t / Di

si 1 C w. sCi ; t/ si 1  w. sCi ; t/

WR!R

ˇ ˇ D 1. and in particular @W @s tD0 By Corollary 4.6 of Chapter 1, the lambda lengths are jy  xj .x; y/ D p dx dy if the horocycles at x; y 2 R have respective diameters dx , dy . By definition of the section , the lambda length .x; y/ evolves under W .; t / to ˇ ˇ ˇ W .y; t /  W .x; t / ˇ W .x; y; t / D qˇ ˇ  ˇ ˇ ; ˇ @W ˇ dx ˇ @W ˇ dy @s x @s y and so we seek ˇ ¯ @ ˇˇ ® log W .x; y; t / D ˇ @t tD0 D

ˇ

@ˇ @t tD0

˚

W .x; y; t /

W .x; y; 0/ ˇ ˚ @ˇ W .x; y; t / @t tD0

.x; y/ ˇ³ ˇ ² ˇˇ W .y; t /  W .x; t /ˇ 1 @ ˇˇ ˇ Dˇ : qˇ ˇ qˇ ˇ ˇy  x ˇ @t ˇ tD0 ˇ @W ˇ ˇ @W ˇ @s x

@s y

ˇ ˇ D 1 gives Direct calculation using the identity @W @s tD0 ˇ ¯ @ ˇˇ ® log W .x; y; t / ˇ @t tD0 ˇ ˇ

³ ² ˇ @W ˇ 1 @W ˇˇ ˇˇ @2 W ˇˇ 1 @2 W ˇˇ ˇ ˇ ˇ D ˇ  C :  ˇy  x ˇ ˇ @t y @t x ˇ 2 @s@t x @s@t y tD0

4 Lie algebra of tesselations

121

We may substitute into the previous expression the known derivatives @w s  i @W 2i .s; t / D . ; t /; si @t Œ1  w. sCi ; t / 2 @t s C i so at t D 0, we find ˇ ˇ ˇ i dz ˇˇ 2i @W 2 dz ˇ D  .s C i / ; .s; 0/ D ˇ si 2 si 2 dt ˇzD si @t Œ1  sCi dt zD sCi sCi d where dz is the coefficient of dz in the normalized trigonometric fields described dt before. In case one of the points, say the point y, lies at infinity, a parallel calculation to that above again using Corollary 4.6 of Chapter 1 leads to ˇ ³ˇ ² ¯ @ ˇˇ ® 1 @2 W ˇˇ @2 W ˇˇ ˇˇ log W .x; 1; t / D  C ; @t ˇ tD0 2 @s@t x @s@t 1 ˇ tD0

which is the limit of the earlier formula for fixed x 2 R as y ! 1. Theorem 4.13. For each n 2 Z, we have the expansion ei n

¯ ® @ n n i @ ei C b1 e D b0n C bC1 @ @ ± 1 X° n C n C n. C n / C .  n / #N e . /; 4 e2  

where e 2  has ideal points ;  2 S 1 and the constants bjn are given before Theorem 4.11 Proof. A series of calculations in the various cases of the residue of n modulo four lead from the previous expression to the formulas o @ C n 1 X . n n C 1/ n n n n n. C  / C   / #N e ;

. @ 8 e2

n n   o @ C n i X . n n  1/ n n n n n. cos n C  / C   / #N e ;

. @ 8 e2

n n  sin n



which turn out to be independent of the case and combine to give the asserted formulas for exponentials. @ with the corresponding function Let us identify the vector field #N e D #N e . / @ N #e ./ of the same name. Motivated by harmonic analysis, it is natural to try to expand a general function on the circle using this topological basis. Fundamental questions

122

3 Lambda lengths in infinite dimensions

include the convergence and uniqueness of such expansions. Because of the supports N we see that any linear combination P N of the functions #, c e2 e #e . / necessarily takes 1 x well-defined values at the points Q  S , so any expansion converges at least in this weak sense. On the other hand, for any such expansion and P any R-valued function F defined N x we likewise see that the alternate expansion on Q, e2 .ce C F . / C F .//#e . / 1 x takes the same values on Q, where ;  2 S again denote the ideal points of e 2  . The expansion given in Theorem 4.13 is thus not the simplest possible; for instance, we x However, could simply omit the term n. n C n / without affecting the values on Q. the expression in Theorem 4.13 is geometrically natural and provides for good analytic control as we next describe. The basic estimates   from [138], expressed in terms of the invariant A D ac C bd of a matrix A D ac db , are that there are constants const independent of A and n so that: if gne is the coefficient of #N e in the expansion in Theorem 4.13 of ei n , then we have jgne j < const n2 jA j1 and jgneI j < jnj=2 (thus explaining our choice of solution in Theorem 4.13); P jA j2 < 1, where the sum is over all A 2 PSL2 .R/  fI; S g; the L2 -norm of #NA is less than const jA j 2 ; for every , we have j#NA . /j < const jA j1 . 5

Various results about the function theory of expansions in f#N e g are derived from these facts in [138]. In particular, the basic quest undertaken there was to express the “Hilbert transform” J in terms of lambda lengths. Definition 4.14. The Hilbert transform is the operator which takes a square integrable real-valued function on the circle and returns the boundary values of the harmonic conjugate of its extension to the disk, namely, J is linearly induced by J.ei n / D i sgn.n/ei n . In the language of harmonic Beltrami differentials, it is almost a tautology that the suitably normalized Hilbert transform gives the complex structure on Teichmüller space, a fact first articulated by Steve Kerckhoff. Of course, coupled with our expression for the WP Kähler two form in Theorem 3.1, this gives in principle a complete expression for the Weil–Petersson hermitian metric itself. However, we have found no useful consequences of this in practice since the Hilbert transform is reasonably complicated as follows:   Theorem 4.15. If f D eA 2  , where A D ac db 2 PSL2 .Z/, then 1 X de N Jx.#Nf / D #e ; 4 i e2   

4 Lie algebra of tesselations

123

where ,  denote the endpoints of e 2  , de D

4 X

hj j .Nj  /.Nj  / log

j D1

N N  .j  /.  / ; .Nj  /.  / N

and .b  d /  i.a  c/ ; .b  d / C i.a  c/ d  ic h2 D 2.c 2 C d 2 /; 2 D ; d C ic b  ia h3 D 2.a2 C b 2 /; 3 D ; b C ia .b C d /  i.a C c/ h4 D Œ.c C a/2 C .b C d /2 ; 4 D : .b C d / C i.a C c/ h1 D Œ.c  a/2 C .b  d /2 ;

1 D

A fundamental point about function theory using these wavelets is that Pbecause of the supports of the functions #N e , the coefficients ce in an expansion f . / e2 ce #N e . / x in can be recursively calculated from the values of the function f . / at the points Q their Farey enumeration. Furthermore, because the harmonic expansion of the functions #N e ./ is known from Theorem 4.11, this provides a viable alternative method of harmonic analysis based on this method of Farey sampling. We have actually patented these methods of signal processing [142] whose practical application depends upon data sampled at the Farey points.

4 Decomposition of the decorated spaces

We return from the more analytic considerations of the previous chapter to finally present the mapping class group-invariant ideal cell decomposition of the decorated Teichmüller space Tz .F / in detail first for a fully decorated punctured surface F D Fgs ; we again separately present the decomposition for partially decorated and/or bordered surfaces so as not to lose sight of the forest for the trees. The argument depends upon a “convex hull construction” in Minkowski space from [41] for complete but noncompact finite-volume hyperbolic manifolds of any dimension that gives a canonical decomposition of the manifold into polyhedra, which is sometimes called the canonical or Epstein–Penner decomposition. z to a There is a canonical assignment of an ideal cell decomposition (i.c.d.) ./ 1 z z decorated structure  2 T .F / as we shall explain. In dimension two it is natural to “solve the equation” for any i.c.d. of F and consider z 2 Tz .F / W ./ z  g: C ./ D f In fact, it turns out to follow by construction that fC ./ W  is an i:c:d: of F g is an MC.F /-invariant decomposition of Tz .F /. The only truly difficult point amounts to solving an extremal problem to prove that each piece of the decomposition is actually a cell. (In fact, transitivity of flips and the presentation of the mapping class groups in Corollary 1.2 of Chapter 5 logically depend only upon the existence of the decomposition and not on this more difficult result that the putative cells are cells.)

1 Convex hull construction We present the basic convex hull construction, which is the special case for punctured surfaces of a general construction [41]. We furthermore formulate the existence of the cell decomposition of decorated Teichmüller space in Theorem 1.8, which we spend the next several sections reformulating and finally proving in Section 4. 1 In higher dimensions, the convex hull construction still applies, but owing to Mostow Rigidity [161], it cannot be promoted to a decomposition of a Teichmüller-like space of complete finite-volume hyperbolic metrics. Perhaps one could relax completeness in the convex hull construction and pursue a corresponding theory in higher dimensions.

1 Convex hull construction

125

The key purpose of this section is to describe the canonical assignment via a convex z of F to a point  z 2 Tz .F /, and we begin with a hull construction of an i.c.d. ./ series of lemmas upon which the construction depends. Lemma 1.1. Suppose that  < SOC .2; 1/ corresponds to a point of Tz .Fgs / and u 2 LC . Then the orbit  u is discrete in LC [ f0g if and only if u is a fixed point of a parabolic element of . In particular, 0 is an accumulation point of u if and only if u is not a parabolic fixed point. Proof. Suppose that u 2 LC is a fixed point of some parabolic in . Discreteness of  u is equivalent to discreteness of .t u/ for any t 2 R, so by Lemma 2.3 of Chapter 3, we may assume that u corresponds to an embedded horocycle in Fgs . The Euclidean radius of a lift of this horocycle to D is therefore bounded away from one, so by the last part of Lemma 2.11 of Chapter 1, the height of any point in u is bounded away from zero, i.e., 0 is not an accumulation point of u. Conversely, suppose that 0 is not an accumulation point of u for some u 2 LC . Let K be the complement in Fgs of a family of embedded horoballs, one horoball about each puncture of Fgs , whose existence is again guaranteed by Lemma 2.3 of Chapter 2. 1 Choose a lift Kz of K to D, so Kz is disjoint from some neighborhood of S1 in D. Again using the last part of Lemma 2.11 of Chapter 1, choose ˇ > 0 so that if the height of a point in LC exceeds ˇ, then the corresponding horocycle in D lies in this neighborhood and hence is disjoint from Kz with projection to Fgs therefore disjoint from K. Since  u does not accumulate at 0, there is some t 2 R so that the height of each point of .tu/ exceeds ˇ, and so the horocycle on Fgs corresponding to t u lies inside Fgs  K. It follows that t u and hence u itself corresponds to parabolic fixed points. To complete the proof, we must show that if u accumulates at any point u1 of 1 LC [ f0g, then it accumulates at 0. The collection of parabolic fixed points of  in S1 1 is dense in S1 by the proof of Theorem 2.5 of Chapter 2, and it follows from this that 1 1 the -orbit of any point in S1 is likewise dense in S1 . In particular, the collection 1 . We may of attracting fixed points of hyperbolic transformations is also dense in S1 C therefore choose contracting eigenvectors in L of hyperbolic transformations j 2  which converge to the ray in LC containing u1 . Furthermore, since there is a shortest closed -geodesic in Fgs , there is a bound below to the absolute value of the eigenvalue of any hyperbolic transformation in . If uj 2 u accumulates to u1 , the sequence jj uj 2  u therefore accumulates to 0 as required. (In fact, the “length spectrum is discrete” in the sense that there are only finitely many isotopy classes of closed curves in F whose hyperbolic lengths are less than a fixed bound, so taking the j th powers jj is not really necessary.) z 2 Tz .Fgs / by definition determines one horocycle in Fgs about each puncA point  ture, and we may choose a point ui 2 LC corresponding via affine duality Lemma 2.11 of Chapter 1 to each such horocycle, for i D 1; : : : ; s. Let B D  u1 [    [ us

126

4 Decomposition of the decorated spaces

denote the collection of -orbits of these points, which is evidently independent of our choice of lifts ui . The crucial facts about B in this section are that it is discrete (as we have just shown) and that it is radially dense in the sense that for any open set of rays in LC , there is a point of B whose corresponding ray lies in this open set. Corollary 1.2. B is discrete in LC [ f0g and radially dense. Proof. Discreteness of B follows from the previous result, and radial density follows 1 . again from the fact that the parabolic fixed points of  are dense in S1 Remark 1.3. This is the avatar of Theorem 2.5 of Chapter 2 for tesselations and Property 4 in the definition of the solenoid. Let C denote the closed convex hull of B in Minkowski three-space, i.e., the smallest convex set in the underlying vector space structure on R3 containing B. Lemma 1.4. Each ray from the origin inside LC meets the frontier of C exactly once. Furthermore, for each u 2 B, the ray ft u W t  1g lies in C . Proof. Suppose that r is a ray from the origin lying strictly inside LC . Since B  LC is discrete and radially dense by the previous result, r meets the convex hull of some set fu1 ; u2 ; u3 g  B and hence meets the frontier of C at least once. Again by discrete radial density, there are respective points v1 ; v2 ; v3 2 B of arbitrarily large height whose corresponding rays are arbitrarily near those of u1 , u2 , u3 . It follows that every intersection point of r beyond the first does indeed lie in C . Likewise for the second part, if u 2 B, then there are points of arbitrarily large height whose rays are arbitrarily near the ray of u, so the asserted ray does indeed lie in C . Lemma 1.5. The frontier of C strictly inside LC consists of a countable collection of codimension-one faces, each of which is the convex hull of a finite collection of points in B. Each face lies in an elliptic plane, and the set of faces is locally finite inside LC . Proof. A “support plane” for a convex body C is a plane containing a point of C so that C is disjoint from one of the corresponding open half-spaces. Suppose that z0 lies in the frontier of C strictly inside LC and let S be a support plane for C through z0 . S cannot be a hyperbolic plane since such a plane separates LC and B is radially dense. If S D fx 2 R3 W hx; si D 1g were parabolic, then s 2 LC [ LC , and indeed s … LC since hz0 ; si D 1 and z0 lies inside LC . Thus, s 2 LC cannot be a multiple of any point of B since hs; si D 0 and C lies above S . By the last part of Lemma 1.1, there are j 2  so that j s ! 0, but for z 2 B, we have by SOC .2; 1/-invariance of h  ;  i that 1  hj1 z; si D hz; j si ! 0; which is absurd. We have thus shown that support planes to C are indeed elliptic. It follows from discreteness of B that there are only finitely many points of B below any fixed support plane. After at most two rotations of any support plane through a

1 Convex hull construction

127

given point in the frontier of C , we can therefore construct a support plane containing three affinely independent points in B. Finally for local finiteness, suppose that a compactum K meets infinitely many faces ˆ1 ; ˆ2 ; : : : . Choose points xj 2 ˆj \ K converging to a point x1 2 K. The plane of ˆj converges to a limiting plane containing x1 , and this limiting plane is also a support plane of C as the limit of such. Discreteness of B shows that not all the faces ˆj could be distinct, as required. An edge of the frontier of C connects two points of B by construction, which determines a corresponding pair of horocycles in H, and there is a unique geodesic in H connecting their centers. Project to Fgs each such geodesic arising from an edge in z of arcs in Fgs connecting punctures. the frontier of C to get a collection ./ z 2 Tz .Fgs /. z is an i.c.d. of Fgs for any  Theorem 1.6. ./ z is embedded, and we proceed by conProof. Let us first show that each arc in ./ z tradiction supposing that e 2 ./ is not simple. There are then lifts eQ1 , eQ2 of e to 1 D so that the ideal points of eQ1 separate the ideal points of eQ2 in S1 , and hence there are edges e1 , e2 in the frontier of C whose endpoints respectively project to the ideal points of eQ1 , eQ2 . By discrete radial density of B, there is a fifth point of B at arbitrarily large height whose ray is arbitrarily near that of an endpoint of e1 . The closed convex hull of these five points of B cannot contain e1 and e2 in its frontier, so e1 , e2 cannot z are simple, and occur in the frontier of C either, which is absurd. Thus, arcs in ./ the analogous argument shows that they are disjointly embedded as well and hence finite in number. If an element  2  < SOC .2; 1/ leaves invariant a face of the frontier of C , then it must preserve the support plane of that face, which is elliptic by Lemma 1.5, since  acts linearly. According to Lemma 2.8 of Chapter 1,  must then itself be elliptic, and that contradicts that the group  is a surface group. By Lemma 1.4, it follows z are indeed polygons, so ./ z is an i.c.d. as was that complementary regions to ./ claimed. z 7! ./ z canonically assigns an i.c.d. to a Thus, the convex hull construction  z on a punctured surface F . It is natural to define the decorated hyperbolic structure  following subspaces of decorated Teichmüller space associated to an arbitrary i.c.d.  of F : ı

z 2 Tz .F / W ./ z D g  C ./ D f z 2 Tz .F / W ./ z  g; C ./ D f where equality and inclusion of i.c.d.’s are understood only up to proper isotopy fixing the punctures. Since the convex hull construction is natural in the sense that z D ..// z for any  2 MC.F / by definition, it follows that . .// fC ./ W  is an i:c:d: of F g

128

4 Decomposition of the decorated spaces

is an MC.F /-invariant decomposition of Tz .F /. With rather hard work in the next several sections, we shall prove that each C ./ is a cell and in fact has the natural structure of the product of R>0 with an open simplex together with certain of its faces coming from simplicial coordinates, and many basic corollaries follow from this. The first step is to give an algebraic formulation in terms ı

of lambda lengths for membership in C ./ or C ./, and then prove cellularity as a purely algebraic fact by solving a variational problem. z 2 Tz .Fgs / with corresponding Lemma 1.7. Let  be an ideal triangulation of Fgs and  ı  z 2 C ./ is lambda lengths ƒ 2 R>0 . Then a necessary and sufficient condition for  that ƒ have positive simplicial coordinate on each edge of . Furthermore if 0   is an i.c.d., then a necessary and sufficient condition for ı z 2 C .0 /  C ./ is that ƒ have positive simplicial coordinates on 0 and vanishing  simplicial coordinates on   0 . Finally, if 0  i , for i D 1; 2, are ideal triangulations containing 0 , then the simplicial coordinate of an edge in 0 is well defined independent of the choice of 1 , 2 . Proof. Since the simplicial coordinate is proportional to the signed Euclidean volume of a tetrahedron with its vertices in LC by Lemma 4.17 of Chapter 1, necessity of the conditions is clear, i.e., global convexity implies local convexity. For the converse, we claim that also local convexity implies global convexity, i.e., given a collection of (possibly degenerate) tetrahedra with vertices in LC meeting pairwise along common triangular faces, each tetrahedron with non-negative signed volume, the union of these tetrahedra comprise a convex body. To this end, we proceed by induction on the number of tetrahedra where the basis step is trivial. For the induction, consider two adjacent such tetrahedra corresponding to a pentagon with sides a, b, c, f , g in this clockwise cycle order and orientation which is triangulated by an arc d joining the initial point of a to the initial point of f and an arc e joining the initial point of a to the initial point of c. A flip on the edge d produces an arc with lambda length .cg Cef /=d by Ptolemy’s relation Corollary 4.16a of Chapter 1, where we identify an arc with its lambda length as usual. Non-negativity of the simplicial coordinates for the edges e and d before the flip give the a priori inequalities a2 C b 2  e 2 c 2 C d 2  e2 C ; abe cde c 2 C e2  d 2 f 2 C g2  d 2 0  C ; cde dfg 0 

which we assume in order to prove 0 

/2  e 2 g 2 C . cgCef a2 C b 2  e 2 d : C eg.cgCef / abe d

1 Convex hull construction

129

Clearing denominators in the latter and factoring, we must show 0  g 2 .ac C bd /.ad C bc/ C 2abcefg  cde 2 g 2 C e 2 ab.f 2  d 2 / C defg.a2 C b 2  e 2 /: Using the first a priori inequality in the form .ac C bd /.bc C ad / > e 2 .ab C cd /; the right-hand side of the putative inequality is thus bounded below by g 2 e 2 .ab C cd / C 2abcefg  cde 2 g 2 C e 2 ab.f 2  d 2 / C defg.a2 C b 2  e 2 /; and canceling like terms and dividing by e, it evolves that we must prove 0  eab.f 2 C g 2  d 2 / C 2abcfg C dfg.a2 C b 2  e 2 /: Now using the first a priori inequality in the form a2 C b 2  e 2  and multiplying through by

c , ab

ab 2 .e  c 2  d 2 / cd

we must show

0  ec.f 2 C g 2  d 2 / C 2c 2 fg C fg.e 2  c 2  d 2 / D ec.f 2 C g 2  d 2 / C c 2 fg C fg.e 2  d 2 / D ec.f 2 C g 2  d 2 / C fg.c 2 C e 2  d 2 /; which follows from the second a priori inequality. For the final assertion in the previous notation and by definition of simplicial coordinates, we must check that c 2 C e2  d 2 a 2 C h2  d 2 D ; cde adh i.e., we must show that ah.c 2 C e 2  d 2 / D ce.a2 C h2  d 2 / if the simplicial coordinate of e (or equivalently h) vanishes, where h arises from the flip on e. To this end, recall from the discussion of quadrilaterals following Corollary 1.10 of Chapter 2 that we have s .ac C bd /.ad C bc/ eD ; .ab C cd / s .ac C bd /.ab C cd / hD : .ad C bc/

130

4 Decomposition of the decorated spaces

The desired identity follows upon substituting these values using the fact that .ab C cd /e D .ad C bc/h. This proves the final assertion if the i.c.d. 0 happens to have exactly one complementary quadrilateral with the rest of the complementary regions triangles. The general result then follows from this case together with the transitivity result Lemma 1.4 of Chapter 2 for flips on polygons. (The first part corrects the proof of Proposition 2.7 of [128], where there were algebra mistakes and an unnecessary assumption.) Here is the statement of the main decomposition theorem, whose proof will occupy most of this chapter: Theorem 1.8. For any surface F D Fgs of negative Euler characteristic with s  1 punctures, ˚ C ./ W  is an i.c.d. of F is an MC.F /-invariant cell decomposition of Tz .F /, which descends to an ideal simplicial decomposition of Tz .F /=R>0 . In fact, projectivized simplicial coordinates on arcs ı

as barycentric coordinates give C ./=R>0 the natural structure of an open simplex, and this simplicial structure extends to C ./=R>0 by adding certain faces, namely, those faces corresponding to sub-arc families of  with simplicial coordinates whose support is an i.c.d. of F . The set of all such C ./ is indeed some sort of MC.F /-invariant decomposition of Tz .F / by naturality of the convex hull construction (since the convex hull construction is performed without reference to any base surface) as noted before. Since the convex hull construction is evidently equivariant for homothety in the vector space R3 underlying Minkowski space as well, this decomposition descends to the projective level of Tz .F /=R>0 . Thus, the only word in the first sentence of Theorem 1.8 that requires further proof is the word “cell”. The second part of the theorem explains how we shall prove this cellularity, and it is this that subsumes the next several sections. We shall reformulate the second sentence of Theorem 1.8 in Section 2 as the “arithmetic problem” of decorated Teichmüller theory and discuss it from several points of view. We shall then prove unique solvability of this arithmetic problem to verify the second sentence.

2 Arithmetic problem The material in this section is based on [128], [134]. Using the fatgraph formalism, we next translate the second sentence of Theorem 1.8 into an explicit system of algebraic equations called the arithmetic problems, one such system of equations for each trivalent fatgraph; indeed, let us emphasize that the arithmetic problem depends only upon the graph underlying the fatgraph. As was mentioned in the Introduction, the arithmetic problem is in a sense the decorated Teichmüller version of the Beltrami equation in the conformal theory.

131

2 Arithmetic problem

Because of the basic isomorphism between fatgraph spines and ideal cell decompositions in Theorem 1.25 of Chapter 1, we shall feel free to pass back and forth between the i.c.d. and fatgraph formalisms to whichever is most convenient. We may thus assign the lambda length or simplicial coordinate to an ideal arc or to its corresponding edge in the dual fatgraph. Furthermore for a trivalent fatgraph G, the sectors (as defined in Section 4 of Chapter 1) of the dual ideal triangulation are naturally identified with pairs of half-edges of G consecutive for the fattening, which will also be called sectors. Finally, the cell C ./ corresponding to an i.c.d.  may sometimes be written C.G/, ı

where G is the fatgraph dual to , and likewise for C .G/.

ı ı 2 ı1 d

2 

c 1

a  ˇ ı ˛



˛1 ˛ ˛2

e

flip

f

" ˇ1 b ˇ 2 ˇ Figure 2.1. Standard notation for flips.

Our standard notation for the edges and sectors near an edge e is indicated in Figure 2.1. We may imagine the figure in the universal cover, so the edges a, b, c, d need not be distinct, and likewise for the sectors of G. In case e 2 E is a loop, i.e., the endpoints of e are not distinct, then our standard notation is illustrated in Figure 2.2 (no flip on e is possible but see [46] for a nice elaboration that includes a notion of flips on loops). The simplicial coordinate of a loop is defined by interpreting the formula in terms of nearby lambda or h-lengths in the universal cover; namely, if e is a loop and a is the other edge of a trivalent fatgraph G incident on the common endpoints of e, then the simplicial coordinate for G of the edge e is 2a=e 2 .

a

" "

˛

e

Figure 2.2. Standard notation near a loop.

132

4 Decomposition of the decorated spaces

We continue with two basic lemmas, the first of which gives a fundamental telescoping property of simplicial coordinates, and the second of which uncovers further aspects of the relationship between lambda lengths and simplicial coordinates. Lemma 2.1. Let G be a trivalent fatgraph, and suppose that  is an efficient closed edge-path in G serially traversing edges ei alternating with sectors ti for i D 1; : : : ; n. Let Ei denote the P Pnsimplicial coordinate of ei and ˛i the h-length of the sector ti . Then n iD1 Ei D 2 iD1 ˛i . Proof. The proof follows from the definition of simplicial coordinates in terms of hlengths, namely, the simplicial coordinate of edge e is given by ˛ C ˇ  " C  C ı   in the notation of Figure 2.1. Definition 2.2. Consider a trivalent fatgraph G with set E of edges with F .G/ Fgs together with an assignment of lambda lengths  W E ! R>0 . We say that  satisfies the no vanishing cycle condition provided that all the corresponding simplicial coordinates are non-negative and there is no essential cycle in G all of whose simplicial coordinates vanish. We may likewise say that an abstract assignment of (putative) simplicial coordinates satisfies this condition. Lemma 2.3. The no vanishing cycle condition implies that the lambda lengths at any vertex of a trivalent fatgraph G satisfy the three strict triangle inequalities. Proof. Adopt the notation of Figure 2.1 for the half-edges near an edge e (again in the universal cover if the edges a, b, c, d are not distinct or even if e does not have distinct endpoints). If c C d  e, then c 2 C d 2  e 2  2cd , so the assumed non-negativity of the simplicial coordinate E of e gives 0  cd Œ.a  b/2  e 2 , and we find a second vertex of G so that the triangle inequality fails. This is a basic fact about simplicial coordinates. It follows that if there is any such vertex so that the triangle inequalities fail for the lambda lengths of incident half-edges, then there must be a non-trivial efficient closed edge-path  passing through such triangles. Letting ei denote the consecutive edges of G serially traversed by  and bi denote the half-edge of G incident on the common endpoint of ei and eiC1 , we find ejP C1  bj C ej for j D 1; : : : n. Upon summing and canceling like terms, we find 0  jnD1 bj , which is absurd since lambda lengths are positive. Notice that Theorem 1.8 actually implies Lemma 2.3 since support planes of the convex hull for a decorated hyperbolic structure are elliptic by Lemma 1.5, which implies the triangle inequalities on lambda lengths by Lemma 4.11 of Chapter 1. It is not that we shall need Lemma 2.3 in the proof of Theorem 1.8, rather, we have presented the direct algebraic proof here for the insight it provides on the algebraic relationship between lambda lengths and simplicial coordinates. Problem 2.4 (Arithmetic Problem). Fix a trivalent fatgraph G with set E of unoriented edges and assign capital variables Xe , one for each e 2 E. If the Xe satisfy the

133

2 Arithmetic problem

no vanishing cycle condition, then find corresponding lower-case variables xe > 0 interpreted as lambda lengths, so that the Xe are derived from the xe as their simplicial coordinates. In fact, the formula in Definition 4.18 of Chapter 1 for simplicial coordinates in terms of lambda lengths shows that the arithmetic problem is well-posed on the underlying trivalent graph independent of fattening. Definition 2.5. Furthermore for any trivalent fatgraph G, the formula Lemma 4.9 of Chapter 1 for h-lengths in terms of lambda lengths (opposite over product of adjacent) shows that in the notation of Figure 2.1, we have ˛ˇ D e 2 D  ı if e is embedded, and " D " in the notation of Figure 2.2 if e is a loop, a condition on h-lengths that we shall call the coupling equation in either case. Notice that we may z conveniently identify the set E.G/ of oriented edges of G with its set of sectors and thereby consider the abstract real-algebraic quadric variety z

V .G/  RE .G/ determined by the coupling equations on the trivalent fatgraph G. The next result follows immediately from the definitions and from Theorem 2.5 of Chapter 2: Theorem 2.6. The assignment of h-lengths to sectors gives an embedding Tz .F / ! z RE .G/ , which induces a homeomorphism z .G/ z  RE .G/ : Tz .F / ! V .G/ \ RE >0

We shall prove unique existence of solution to the arithmetic problem and prove z Theorem 1.8 by studying in the next section a dissipative flow on RE .G/ which limits z

.G/ to V .G/ \ RE >0 . Let us immediately give several examples of solutions to the arithmetic problem:

Example 2.7. Consider the labeled fatgraph Ga in Figure 2.3a with the indicated notation for h-lengths and lambda lengths, and let capital roman letters denote corresponding simplicial coordinates. According to our conventions, the simplicial coordinates are AD

2.b 2 C e 2  a2 / ; abe

BD

2.a2 C e 2  b 2 / ; abe

ED

2.a2 C b 2  e 2 / ; abe

and the h-lengths are equal in pairs and given by ˛D

a D ı; be

ˇD

b D ; ae

"D

e D : ab

134

4 Decomposition of the decorated spaces

a ˛

a ˛ ˛C ˛  d ı ıC  f ı e ˇC ˇ ˇ  

" ˇ ˛ e

a

b

 ı 

ı

b

ıC ı  f  C  b

c

a)

e

˛C ˛  d ˇ ˇC ˇ

c

b)

c)

Figure 2.3. Examples of arithmetic problem.

There are closed edge-paths on Ga corresponding to the boundary components with respective total lengths (i.e., sums of simplicial coordinates counted with multiplicity) given by 1 1 .B C E/; Qb D .A C E/; 2 2 and the h-lengths are evidently given by Qa D

Qe D

1 .A C B/; 2

1 1 1 Qa D ı; ˇ D Qb D ; " D Qe D : 2 2 2 Using the coupling equations, one finally calculates the lambda lengths: ˛D

a2 D

4 16 D ; Qb Qe .A C B/.A C E/

b2 D

4 16 D ; Qa Qe .A C B/.B C E/

e2 D

4 16 D : Qa Qb .A C E/.B C E/

Example 2.8. There are closed edge-paths on the fatgraph Gb in Figure 2.3b whose total lengths are given by 1 P D E C F C .B C C /; 2 1 Q D D C F C .A C C /; 2 1 R D D C E C .A C B/; 2

1 A; 2 1 T D B; 2 1 U D C; 2 SD

135

2 Arithmetic problem

and the solution to the arithmetic problem for Gb is given by ˛ D S;

ˇ D T;

 D U;

ıD

˛C D ˛ D

1 PQR ; 2P P C Q C R

ˇC D ˇ D

C D  D

1 PQR ; 2R P C Q C R

ıC D

ı D

PQR P ; 2QR P C Q C R 1 PQR ; 2Q P C Q C R R PQR ; 2PQ P C Q C R

Q PQR : 2PR P C Q C R

Example 2.9. Performing a flip on the edge of Gb labeled e, we obtain the fatgraph Gc in Figure 2.3c, and the total lengths of the corresponding paths on Gc are given by 1 P D F C .B C C C E/; 2 1 Q D D C E C F C .A C C /; 2 1 R D D C .A C B C E/; 2

1 A; 2 1 T D .B C E/; 2 1 U D C: 2 SD

The solution to the arithmetic problem for Gc is given by ˛ D S; ˇD

PT ; P CR

 D U; ıD ˇ D

˛C D ˛ D ˇC D

QR.2P C 2R  2T  Q/ ; 4.P C R/.P C R  T /

R.2P C 2R  2T  Q/2 ; 4.P C R/.P C R  T /

C D  D

PQ.2P C 2R  2T  Q/ ; 4.P C R/.P C R  T /

RT ; P CR

ıC D

PQ2 ; 4.P C R/.P C R  T /

RQ2 ; 4.P C R/.P C R  T /

ı D

P .2P C 2R  2T  Q/2 : 4.P C R/.P C R  T /

In fact, all examples of trivalent graphs with at most four vertices have the same character: the h-lengths are given by rational functions of the simplicial coordinates whose linear factors in numerator and denominator correspond to closed edge-paths on the graph. In effect, the telescoping property Lemma 2.1 gives linear constraints on sums of h-lengths along closed paths in terms of simplicial coordinates, and for

136

4 Decomposition of the decorated spaces

these graphs, these “linear samplings” of h-lengths give an invertible system, as one discovers with computer arithmetic to derive the solutions above. On the other hand, for the simplest trivalent graph with six vertices, the solution to the arithmetic problem is not rational, but it is Galois, cf. [134] for these explicit computations. Here is another point in decorated Teichmüller theory beyond the closely related discussion at the end of Section 1 of Chapter 2 where we have long hoped for solid connections with Galois theory. There is the following purely geometric reformulation of the arithmetic problem in terms of finite configurations of points in horocycles as follows. Suppose that z 2 Tz .F / and  is a fixed ideal triangulation of F D Fgs straightened to geodesics  for  2 T .F /, and let hi , for i D 1; : : : ; s, denote the specified horocycles, which we assume are disjointly embedded for convenience. Identify each hi with an abstract circle (without basepoint), and consider the configuration  \ hi  hi S 1 , for i D 1; : : : ; s, i.e., we have an s-tuple of finite configurations of points in an abstract circle, called the “geodesic configuration”. z is the canonical i.c.d. arising from the convex hull Now, suppose that  D ./ construction, so by Lemmas 4.11 and 1.5 of Chapter 1 or Corollary 1.10 of Chapter 2, z on  satisfy the strict triangle inequalities on each triangle t the lambda lengths for  complementary to . Thus, the hypotheses of Lemma 4.12 of Chapter 1 hold, and so for each t , there is a unique point equidistant to the decoration on the vertices of t . Furthermore, projection to horocycles in the sense of Lemma 4.12 of Chapter 1 of these equidistant points determines a second configuration of points in each hi S 1 , and this s-tuple of finite configurations in an abstract circle is called the “equidistant configuration”. Since the simplicial coordinates determine the equidistant configuration according to Lemma 4.12 of Chapter 1, and the h-lengths determine the geodesic configuration, we have proved: Theorem 2.10. The arithmetic problem amounts to solving for the geodesic configuration from the equidistant configuration. It follows that the decomposition in [22] agrees pointwise with the one given here, which predates it. There is another interpretation of the arithmetic problem from [134] in terms of a “fermionic integral” which shows that log lambda lengths and simplicial coordinates are Legendre dual for the associated partition function, cf. Theorem 6.2. This fermionic interpretation of the arithmetic problem depends in essence only on the following simple but remarkable little lemma. Lemma 2.11. Suppose that G is a trivalent fatgraph with set E of edges and with corresponding lambda lengths xe and simplicial coordinates Xe , for e 2 E. If  denotes the sum of the hyperbolic lengths of all the horocycles in a decoration, i.e.,  is the sum of all the perimeter functions, then @=@ log xe D Xe . Proof. By definition,  is given by the sum of the h-lengths of all sectors of G. Among all the h-lengths of sectors of G, only ˛; ˇ; : : : ;  have any dependence on xe in the

137

2 Arithmetic problem

notation of Figure 2.1 by Lemma 4.9 of Chapter 1. We compute directly that @ @  D ˛ C ˇ C  C ı C " C / @ log e @ log e   @ b c d e e a D C C C   @ log e be ae de ce ab cd D ˛  ˇ    ı C " C  D Xe ; as was claimed. Lemma 2.12. In the notation of Lemma 2.11, the Hessian H of  as a function of flog xe ge2E has absolute determinant j det H j  .3=N /N . Proof. Define a matrix AN 2N whose rows are indexed by the edges and columns by the sectors of G as follows. If the `th sector of G lies between the half edges eOj and eOk and is opposite half-edge ei , where eOi , eOj , eOk are contained in the respective distinct edges ei , ej , ek of G, then define 1 D Ai` D Aj ` D Ak` and Am` D 0

if m 62 fi; j; kg:

In the remaining case of a loop with j D k, then Ai` D 1 and Aj ` D 2, while if i D j , for example, then Ai` D 0 and Ak` D 1. N N N Adopting the notation LOG.xi /N iD1 D .log xi /iD1 and EXP.xi /iD1 D .exp xi /iD1 , we have by definition t N .Xi /N iD1 D A EXP A LOG.i /iD1 ;

where the superscript t denotes the transpose, so by the Chain Rule H D ADAt ; with D the diagonal whose entries are given by the h-lengths. In particular, if Xi is given by j k i ` m i C  C C  ; Xi D i k i j j k i m i ` ` k then one finds that Hi i D

j k i ` m i C C C C C : i k i j j k i m i ` ` k

Since the determinant is the product and the trace the sum of the eigenvalues, by the dominance of geometric over arithmetic mean, we conclude 

j det H j  as required.

tr H N

N



D

3 N

N

138

4 Decomposition of the decorated spaces

Suppose that we perform a flip on the edge e of the trivalent fatgraph G to produce the trivalent fatgraph G 0 . Of course, the lambda lengths of edges of G 0 are determined by those on the edges of G, where only the lambda length of e changes as governed by Ptolemy’s relation. This calculation is “local” in the sense that lambda lengths of edges of G 0 depend only upon lambda lengths of nearby edges in G. In contrast for simplicial coordinates, we have: Lemma 2.13. Suppose that we perform a flip on the edge e of a trivalent fatgraph G with set E of edges to produce the trivalent fatgraph G 0 , and adopt the notation of Figure 2.1. Let Xi denote the simplicial coordinate with respect to G of i 2 E, and let Xi0 denote the simplicial coordinate of i 2 E  feg with respect to G 0 further letting Xe0 denote the simplicial coordinate of f with respect to G 0 . Then we have: X t0 Xe0 0 Xb0 C Xc0 Xa0 C Xb0 C Xe0 0 Xa0 C Xd0 Xc0 C Xd0 C Xe0 0

D X t ; if t   \ 0 and t ¤ a; b; c; d , D Xe ; D Xb C Xc C Xe ; D Xa C Xb ; D Xa C Xd C Xe ; D Xc C Xd :

Proof. The first two formulas follow from the definitions, and the latter four are consequences of the telescoping property Lemma 2.1. One checks directly that the linear equations in Lemma 2.13 do not have full rank, rather, they are “co-rank 1”, and non-locality of the flip transformation law for simplicial coordinates follows (as will be explicitly demonstrated in the next example). Lemma 2.14. In the notation of Lemma 2.13, we in fact have Xe C Xe0 D 0; ac Xe D Xc  Xc 0 ; ac C bd bd Xb  Xb 0 D  Xe D Xd  Xd 0 : ac C bd

Xa  Xa0 D 

Proof. This follows directly from the definitions and Ptolemy’s relation. Remark 2.15. Among a small group, this is called the “Shinkansen Lemma” since it was proved on the fast train from Tokyo to Kyoto, and it implies the previous lemma. It is actually kind of amazing that Xa  Xa0 D Xc  Xc 0 and Xb  Xb 0 D Xd  Xd 0 even away from the common face Xe D 0, and there are no corresponding formulas in the conformal case. Expressions such as these may be useful for example in recognizing invariant differential forms.

139

2 Arithmetic problem

Problem 2.16 (Related Arithmetic Problem). If G and G 0 are trivalent fatgraphs that differ by an elementary move, then calculate the simplicial coordinates on G 0 from those on G. It is not even a priori clear that this problem is well-posed, and we interpret it as follows. Given the simplicial coordinates on G, we might solve the arithmetic problem on G to calculate the corresponding lambda lengths on G. We may then calculate the lambda lengths on G 0 using Ptolemy’s relation, and finally use the definitions to express the simplicial coordinates on G 0 in terms of those on G. We are effectively “arithmetically continuing” the simplicial coordinates using Ptolemy transformations and the alleged solution to the arithmetic problem. Example 2.17. Consider the graphs Gb ; Gc illustrated in Figure 2.3, adopt the notation of Figure 2.3b for the edges of Gb , and further adopt the notation given in Example 2.8 for the total lengths of paths on Gb ; we also adopt the notation of Figure 2.3c for the edges of Gc , except that the label i on an edge in Figure 2.3c will be denoted by i 0 here. According to Example 2.8, the arithmetic problem on Gb is solved by a2 D

1 2.P C Q C R/ D ; ˛˛C QRS

c2 D

1 2.P C Q C R/ D ; C PQU

eD

2.P C Q C R/ 1 D ; ˇC PR

b2 D

1 2.P C Q C R/ D ; ˇˇC PRT

dD

2.P C Q C R/ 1 D ; ˛C QR

f D

2.P C Q C R/ 1 D : C PQ

We may compute the corresponding lambda lengths on Gc using Ptolemy’s equation, namely identifying an arc with its lambda length as usual, we have i 0 D i if i ¤ e, and r b.d C f / P CR 2.P C Q C R/ 0 D : e D e Q PRT Finally, using the definition of the simplicial coordinates in terms of lambda lengths on Gc , we calculate P QCR ; 2 R.P C Q C R C 2T / D 0 D S  T C ; 2.P C R/

A0 D 2S;

B0 D T C

C 0 D 2U; E0 D T C

QP R ; 2

F 0 D T  U C

P .P C Q C R C 2T / 2.P C R/

completing our solution to Problem 2.16 for Gb and Gc .

140

4 Decomposition of the decorated spaces

Using Whitehead’s result Fact 1.24 of Chapter 1 that flips act transitively on ideal triangulations of a fixed surface, it follows that a solution to Problem 2.16 for all pairs of fatgraphs on the surface which are related by a flip together with the solution to Problem 2.4 for any single fatgraph on the surface provides the solution to Problem 2.4 for any fatgraph G.

3 Putative cells are cells This section is based on [128] and corrects several typos (for instance, in the Jacobians in Claim 3.6). Suppose that G is a trivalent fatgraph with set E D E.G/ of edges and set Ez D z E.G/ of oriented edge. Identify Ez with the collection of all sectors  D .G/ of G. Thus, z .G/ E.G/; in the natural way, where an oriented edge points towards a vertex, and we take the sector of this vertex “opposite” the oriented edge; for instance, the edges a, b, c, d of G oriented pointing towards e are respectively identified with sectors ˛, ˇ, , ı in Figure 2.1. We shall variously index by  or Ez as convenient. If e 2 E has distinct endpoints, again in the notation of Figure 2.1 for the nearby edges of G, where a, b, c, d need not be distinct, define vectors BÅe ; CÅe 2 R , where BÅe , CÅe have their only non-zero entries in the .˛; ˇ; ; ı/ subspace with the respective entries of BÅe given by .1; 1; 1; 1/ and those of CÅe given by .1; 1; 1; 1/. These vectors are illustrated on the left in Figure 3.1. Notice that this determination of CÅe depends up to sign on a choice of orientation on e 2 E. 1 1 1 0 1

BÅe

e

1

1 1

0

0

1

Åe C

1

e

1

0

0

e

e BÅe

1

1

2

Åe C

Figure 3.1. The vectors BÅe and CÅe .

Likewise if the endpoints of e are not distinct, then there are corresponding vectors z Å Be ; CÅe 2 RE defined by interpreting Figure 2.1 in the universal cover of G and adding entries corresponding to coinciding sectors; these vectors are illustrated on the right in Figure 3.1. We shall often handle “non-embedded cases”, i.e., either e is a loop

141

3 Putative cells are cells

or the edges a, b, c, d are not distinct, in this same manner of “superposition”, i.e., interpreting in the universal cover and summing. Given any zE 2 R , we may define the formal h-length of a sector ˛ of G as the ˛th coefficient of zE and the formal simplicial coordinate of e 2 E as the linear combination of formal h-lengths ˛ C ˇ  " C  C ı   perhaps interpreted in the sense of superposition. If the formal h-lengths of zE are all positive, i.e., if zE 2 R>0 , then we shall write simply zE > 0. It follows from Theorem 2.6 that zE > 0 satisfies the coupling equations if and only if zE 2 V.G/ \ R>0 Tz .F .G//. The formal h-lengths of any such zE thus determine lambda lengths on the edges of G via the coupling equation, and hence expressions for formal simplicial coordinates, which may however be negative, e.g., if zE … C .G/. z

Lemma 3.1. fBÅe ; CÅe W e 2 E.G/g is a basis of RE .G/ for any trivalent fatgraph G. Furthermore, if X xe BÅe C ye CÅe > 0; zE D e2E.G/

then the formal simplicial coordinate of e is 4xe . z Proof. Let E D E.G/ and Ez D E.G/. It is clear that the span of the vectors z E Å Å fBe ; Ce W e 2 Eg in R agrees with the span of the vectors f.BÅe ˙ CÅe /=2 W e 2 Eg, and among these, there are exactly three which have non-zero projection into the subspace corresponding to the three sectors about a common vertex of G. These three projections .1; 1; 0/, .1; 0; 1/, .0; 1; 1/ are illustrated in Figure 3.2 and are linearly independent proving the first part.

0

1 1

1

0

1

1

1 0

Figure 3.2. Vectors 12 .BÅe ˙ CÅe /.

For the second part again considering supports of the vectors BÅe , CÅe , for e 2 E, and in the notation of Figure 2.1 (perhaps by superposition in the non-embedded case), we find that the formal h-lengths are given by ˛ D xe C ye C xb C yb ;  D xe  ye C xd C yd ; " D xa C ya C xb C yb ; so indeed ˛ C ˇ C  C ı  "   D 4xe .

ˇ D xa C y a C x e C y e ; ı D xe  ye C xc C yc ;  D xc C y c C x d C y b ;

142

4 Decomposition of the decorated spaces

Fix a trivalent fatgraph G with sets E; Ez of unoriented and oriented edges, respecz tively, and define the following linear subspaces of RE : XD

nX e2E

Y D

nX

o xe BÅe W xe 2 R ; o ye CÅe W ye 2 R ;

e2E

and the following subspaces of X : Xx D

nX

o

xe BÅe W xe  0 and some xe > 0

e2E

[ ı

XD

nX

o xe BÅe W xe > 0 for all e :

e2E

The space Xx has the natural structure of a deleted cone on a simplex, where open faces of the simplex correspond to subsets A  E on which xe > 0 for e 2 A. A face corresponding to A  E is said to be finite if the smallest subgraph of G containing G  A is a forest (i.e., each component is a tree), or equivalently, A is dual to an i.c.d. of the punctured surface F .G/ corresponding to G. Define the subspace ı

X C D X C .G/ DX .G/ [ ffinite faces of Xx .G/g  Xx .G/: z

Let … W RE .G/ X  Y ! X denote projection onto the first factor, and regard z the cell C.G/ in Tz .F .G// as a subset C .G/  V .G/  RE .G/ of the domain of …. Notice that by the second part of Lemma 3.1, the formal simplicial coordinates are constant on fibers of …. Theorem 3.2. For each trivalent fatgraph G, the projection … restricts to a homeomorphism … W C .G/ ! X C ı

ı

mapping C .G/ onto X . Furthermore, if G 0 arises from G by collapsing a forest in G, ı

then … maps C .G 0 / onto the corresponding open finite face of X C . It is in this form that we shall prove the second sentence of Theorem 1.8 based on a series of four claims. The argument subsumes the rest of this section and depends upon an “energy function” KW

z .G/ RE >0

! R>0 ;

zE 7!

X  e2E.G/

˛ˇ log ı

2

;

143

3 Putative cells are cells

in the notation of Figure 2.1, or more conveniently on the left in Figure 3.3 near the edge e 2 E with distinct endpoints; if e 2 E is a loop, then the corresponding term in K.Ez / (calculated by superposition) is given by .log " ="/2 in the notation of Figure 2.2, which is more conveniently depicted on the right in Figure 3.3. ı ı1

d 

2

˛1 ˛ ˛2 a

ı2 

eˇ

ı

˛

"

c

 b ˇ2

 1

ˇ1

a

ı

" ˛ "

e

ˇ

Embedded case

Loop case

Figure 3.3. Standard notation near an edge e of G.

Making a choice of orientation of e 2 E determines CÅe (by having positive projections to the sectors adjacent to the terminal point), which changes sign under the change of orientation. Further define ´ ˛ˇ if e is embedded, ı e D " if e is a loop, " in the notation of Figures 3.3, so log e also changes sign under the change of orientation on e, and .log e /2 is the contribution to the energy from e 2 E. Furthermore by definition, K.Ez / D 0 if and only if zE satisfies the coupling equations and hence describes a point of V .G/; thus, provided zE > 0 and ….EzP / 2 X C , zE describes a point of C .G/  Tz .F .G//. On the other hand since K D e2E .log e /2 , we compute directly that for any coordinate , X log @K 2 D @ e e2E

e

@ e : @

It thus follows from the definition of the energy function that if K.Ez / D 0 for zE > 0, then the gradient vector field rK.Ez / D 0 (calculated in the natural vector space structure z of RE ) also vanishes at zE, i.e., zE is a critical point of K. We shall prove an appropriate converse to this in Claim 3.5. z Suppose that zE 2 RE and consider the affine subspace ˚ z YzE D yE C zE 2 RE W yE 2 Y and yE C zE > 0 :

144

4 Decomposition of the decorated spaces

The gradient rK restricts to a vector field rKjYzE on YzE for each zE, and we take the negative-time flow of this restricted gradient   zEt D rKjYzE t .Ez /; so energy is decreasing along trajectories (the system is “dissipative”). z

Claim 3.3. Fix zE 2 RE . If ….Ez / 2 X C and zE1 is an accumulation point of zEt , then zE1 > 0. Proof. K is homogeneous of degree zero and … is homogeneous of degree one in the formal h-lengths of zE. Thus some projective limit ŒEz1 of zEt must always exist for z t ! 1. It remains to show ŒEz1 2 RE >0 =R>0 , i.e., no coordinate of a representative zE1 vanishes. To this end, we may normalize so that supe2E fEz t .e/g D 1 for each t . Suppose first that e has distinct endpoints and ˛ t ! 0 in the notation on the left of Figure 3.3, where the h-lengths are now formal. Since the energy decreases along trajectories and h-lengths are bounded above, we must have  t ! 0 or ı t ! 0. It follows that there is an efficient closed edge-path in G alternately traversing edges and sectors of G with the consecutive respective formal simplicial coordinates Si and h-lengths ˛i ! 0, for i D 1; : : : ; n.PBy Lemma 2.1, P the sum of formal simplicial coordinates telescopes, and we have niD1 Si D 2 niD1 ˛i ! 0, which contradicts that ….Ez t / 2 X C . Likewise, Se D 2˛ in the notation on the right of Figure 3.3 if e is a loop. Claim 3.4. If ….Ez / 2 X C , then YzE ¤ ;. Proof. Let xE D ….Ez / 2 X C and zE D xE C yE for yE 2 Y . We may induct on the number of vanishing formal h-lengths for zE, and the basis step where there are no vanishing h-lengths is trivial. For the induction, suppose that e has distinct endpoints and ˛ D ˛.Ez / D 0 in the notation of Figure 3.3. If  ı D .Ez /ı.Ez / > 0, then a small multiple of CÅe added to zE produces a vector with all coordinates ˛ , ˇ,  , ı positive. It follows that if ˛ D 0, then also either  D 0 or ı D 0. As before, there is a cycle in G with vanishing formal hlengths, hence vanishing formal simplicial coordinates contradicting that ….x/ E 2 X C. Likewise if e is a loop in the notation of Figure 3.3, then 0 D 2˛ D Se contradicts that ….x/ E 2 X C . Furthermore, a small multiple of CÅe added to zE decreases j"  " j, so the induction proceeds unless " D " D 0 or "; " > 0. In the former case, again ı D 0 leads to a cycle in G with vanishing formal simplicial coordinates contradicting ….x/ E 2 X C , and in the latter case as before, a small multiple of CÅa renders ı positive. Sadly, not all of our calculations can be done by superposition, so we must finally distinguish between the various cases of topology near an edge in a fatgraph. In fact, the arithmetic problem and energy function both descend to the level of the underlying unfattened trivalent graph.

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3 Putative cells are cells

We must therefore delineate between the cases of an edge in a trivalent graph, and the basic dichotomy is whether an edge e is a loop or has distinct endpoints. In the latter case, there are four possibilities depending upon identities among the edges a, b, c, d in the notation on the left of Figure 3.3. These four non-loop non-embedded cases for an edge e are illustrated in Figure 3.4, where notation for nearby edges and h-lengths as well as orientations are specified for each relevant edge in the figure. Recall that the orientation on any edge f uniquely determines both CÅf and a square root log f of the contribution to the energy from f 2 E.  1 2 c ı   e

ı ı1 ı 2 d 

 e ˇ " ı ˛

aHb

c

aHd

ˇ

˛ " b ˇ2 ˇ 1

2 1 

ˇ

First non-embedded case

Second non-embedded case

aHd



 e ˇ " ı ˛

 cHd

bHc Third non-embedded case

 e ˇ " ˛ ı

aHb

Fourth non-embedded case

Figure 3.4. Notation for non-embedded cases.

In the remaining discussion, we must often separately treat six cases: the neighborhood of e on the left of Figure 3.3 is embedded in the fatgraph; e is a loop as on the right in Figure 3.3; and the four cases of Figure 3.4. Claim 3.5. zE > 0 is a zero of K if and only if it is a critical point of KjYzE . Proof. As in the earlier discussion, we shall let Figure 3.3, and have already remarked that X log @K D 2 @yf e e2E

e

@ e @yf

e

D

z/ e .E

D

for f 2 E;

˛ˇ ı

in the notation of

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4 Decomposition of the decorated spaces

so a zero zE of K is therefore indeed a critical point of KjYzE . Conversely, if K.Ez / ¤ 0, then e .Ez / ¤ 1 for some e 2 E, and we may take such an edge with .log e /2 greatest. In the embedded case in the notation of Figure 3.3 and from the definitions, we compute that

    .˛ C t /.ˇ C t / 2 ˇ1 ˇ2 2 @ 1 @K log C log D .  t /.ı  t / .˛ C t /" 2 @ye @t  2  2  2  ˛1 ˛2 ı1 ı2 1 2 C log C log C log .ˇ C t /" .  t / .ı  t / tD0 ˛ˇ D .˛ 1 C ˇ 1 C  1 C ı 1 / log ı ˇ1 ˇ2 ˛ 1 ˛2 ı1 ı2  1 2 C ˛ 1 log  ˇ 1 log C  1 log C ı 1 log : ˛" ˇ"  ı Since .log e /2 is greatest by assumption, the previous expression is non-zero unless .log a /2 D .log b /2 D .log c /2 D .log d /2 D .log e /2 are all greatest. We may assume without loss that ˛ˇ >  ı, which furthermore implies ˛1 ˛2 > ˇ"; ˇ1 ˇ2 > ˛";

ı > 1 2 ;  > ı1 ı2 :

These inequalities for e 2 E are incompatible with the corresponding inequalities @K @K for a; b; c; d 2 E, and hence @y .Ez / ¤ 0, .Ez / D 0 and K.Ez / ¤ 0 implies that @y e f for f D a; b; c; d , provided at least one of a, b, c, d likewise has an embedded neighborhood as on the left of Figure 3.3. In the contrary case, each of a, b, c, d must be as in the first non-embedded case of Figure 3.4, so the argument devolves to that case which is treated later. Next consider the case that e is a loop in the notation of Figure 3.3, and compute

     1 @K @ " C t 2 ." C t /."  t / 2 D log C log 2 @ye @t "t ı tD0 " " "  1 1 1 1 D ." C " / log C ."  " / log : " ı 1 and .log e /2  .log Insofar as log." ="/ has the opposite sign from "1  " by assumption, this expression can vanish only if " D " as required. We may calculate the expression

@ e 1 @K ˛ˇ 1 @K  D  .˛ 1 C ˇ 1 C  1 C ı 1 / log 2 @ye @ye 2 @ye ı

a/

2

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3 Putative cells are cells

in the four respective cases and notation of Figure 3.4 to be ı 1 ı2 1 2 ˛ C .˛ 1  ˇ 1 / log ;  ı 1 log ı ˇ  ˇ1 ˇ2 1 2 ˇ" ˛ 1 log C ı 1 log C .ˇ 1 C  1 / log ; ˛" ı  ˇ" .ˇ 1 C  1 / log ;   1 log

and .ˇ 1  ˛ 1 / log

ˇ ı C . 1  ı 1 / log : ˛ 

Only the second case of Figure 3.4 is non-trivial and requires comment, and for this, we argue in analogy to an earlier proof as follows. Again, the prohibited vanishing implies .log f /2 D .log e /2 , for f D a; b; c. Without loss, assume ˛ˇ >  ı, which implies ˇ1 ˇ2 > ˛", ı > 1 2 , and  > ˇ". The corresponding inequalities for the @K @K edge a are incompatible with these, so @y .Ez / D 0 and K.Ez / ¤ 0 implies @y .Ez / ¤ 0. e a Claim 3.6. Each critical point zE > 0 of rKjYzE with ….Ez / 2 X C is non-degenerate with Euler–Poincaré index one, i.e., the Hessian of KjYzE is positive definite and the index of the gradient vector field rKjYzE at a zero of this vector field is one. Proof. If K.Ez / D 0, then i D 1; : : : ; n, so

i

D 1 for each i 2 E now enumerating edges of G by

n 1 @2 K @ X log .Ez / D 2 @yk @yj @yk i iD1

X @ i D @yj n

i

iD1





@ i .Ez / @yj



@ i .Ez / : @yk

Thus, the Hessian of K at zE is 2At A, where At denotes the transpose of AD



@ i @yj



for i; j D 1; : : : ; n:

Both parts of the claim follow provided A is non-singular. Namely, the inner product z of .At A/E v with vE agrees with the norm squared of AE v , for vE 2 RE .G/, by definition of the transpose, which is non-vanishing for vE ¤ 0 provided A is non-singular. Likewise as an isomorphism, A induces a map of degree one. To get rid of annoying factors in our proof that A is invertible, define for i D 1; : : : n in the notation of Figure 3.3 ´ ˛ˇ D  ı; if e has distinct endpointsI i i D the lambda length a; if e is a loop with incident edge a;

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4 Decomposition of the decorated spaces

and set

A0 D .i i / A;

so the desired invertibility of A follows from that of A0 . To see that A0 is non-singular, suppose that n   X @ i n 0E D

i i i @yj j D1 iD1

is a relation among the rows for some i 2 R. We show that each coefficient i must vanish, and there are cases depending on the topology of the graph underlying G near e 2 E. Begin with the embedded case of Figure 3.3 where all of the illustrated edges are distinct. Just once to be explicit, our orientation conventions give: ˛1 ˛2 ; ˇ" ˇ1 ˇ2 D .1; 1; 1; 1/ in .ˇ1 ; ˇ2 ; ˛; "/ space with b D log ; ˛" 1 2 D .1; 1; 1; 1/ in .1 ; 2 ; ı; / space with c D log ; ı ı1 ı2 D .1; 1; 1; 1/ in .ı1 ; ı2 ; ; / space with d D log ;  ˛ˇ D .1; 1; 1; 1/ in .˛; ˇ; ; ı/ space with e D log : ı

CÅa D .1; 1; 1; 1/ in .˛1 ; ˛2 ; ˇ; "/ space with CÅb CÅc CÅd CÅe

a

D log

Notice that both e and CÅe depend upon a choice of orientation on e, and hence the Jacobian .@ i =@yj / is independent of this choice. This Jacobian is given by 1 0 ˇ 0 0 " A C ˛ C ˛ C B ˛ B C ˇ C ˇ 0 0 " C B C; B ı  0 0 C C  C  C B A @ 0 0  D C ı C ı  ˛ ˇ  ı E C"C where the capital roman letters denote the simplicial coordinates of the same lower-case roman lettered edges. Subtract the second row from the first and the fourth from the third, and then add the second and subtract the fourth from the fifth to get: 1 0 0 0 0 A C ˛ .B C ˇ / B ˛ B C ˇ C ˇ 0 0 "C C B B 0 0 C C  .D C ı / 0 C C: B @ 0 0  D C ı C ı  A 0 .D C ı / E 0 B C ˇ

3 Putative cells are cells

149

Directly expanding the determinant along the first column, one finds a sum of nonnegative terms at least one of which is positive since ˛; : : : ; "; ˛ ; : : : ; " > 0 and A; : : : ; E  0. We leave as easier exercises the analogous confirmations of non-singularity of Jacobian in the other cases, but nevertheless here give these Jacobians for the four respective non-embedded cases in Figure 3.4 in the analogous notation: 0 1 ˛Cˇ 0 0 ˛ˇ B 0 C C C  C  ı  B C; @ 0 A   D C ı C ı ˛Cˇ  ı E C"C 0 1 AC˛Cı ˇ  "C B C ˛ B C ˇ C ˇ 0 " B C; @ A ı 0 C C  C   ˛Cı ˇ  E C"C 0 1 AC˛Cı ˇC "C @ ˛Cı B CˇC " C  A; ˛Cı ˇC E C"C 0 1 ˛Cˇ 0 ˇ˛ @ 0 B C Cı   ı A: ˇ˛  ı E C"C In particular for a loop e, we find @ e =@ye D " C " > 0 in the notation of Figure 3.3 using the top-left entry of the Jacobian in the first case of Figure 3.4. Proof of Theorem 3.2. Fix some xE 2 X C and consider the negative-time restricted gradient flow .rKjYxE /jt on YxE , which is non-empty by Claim 3.4 and clearly a z

convex open set in RE >0 . Define z VxxE D fxE C yE 2 RE E 2 Y and xE C yE  0g: >0 W y

We wish to apply the Poincaré–Hopf index theorem to conclude that .rKjYxE /jt has a unique attracting fixed point in VxxE and in fact in YxE by Claim 3.3. To this end, we must consider the behavior of rKjYxE near the boundary VxxE  YxE . As before, we may assume that the formal h-lengths of xE are bounded above. Consider a point zE 2 YxE lying on a level set K D const, for some large constant. Since h-lengths are bounded above (by our choice of deprojectivization), there must be some formal h-length of zE which is small. First suppose that e 2 E has distinct endpoints in the notation of Figure 3.3 and it is ˛ which is small. We can add to zE a multiple of CÅe to increase ˛ unless one of ; ı is also small using the definition of CÅe . Likewise if e is a loop in the notation of Figure 3.3, then 2˛ D Se cannot be small, and if " or " is small, then  or ı is small.

150

4 Decomposition of the decorated spaces

The usual telescoping properties of h-lengths and simplicial coordinates show that there is a thus a cycle in G the sum of whose simplicial coordinates is small, but this contradicts that such sums are a priori bounded below since the formal simplicial coordinates are constant on YxE by the second part of Lemma 3.1. It follows that rKjYxE .Ez / is transverse to the level set. We may therefore define a map  W X C ! C .G/;

xE 7! lim t!1 .rKjYxE /jt .xE 0 /;

where xE  xE 0 2 Y and xE 0 > 0. By Claims 3.3-3.6 and the previous paragraph, this section  is well defined, and it is moreover clearly a continuous right inverse to …. We conclude by proving that  is surjective and suppose that zE > 0 with K.Ez / D 0 E so K.Ez 0 / D 0 by Claim 3.5 and zE 0 > 0 by and xE D ….Ez / 2 X C . Let zE 0 D  .x/, Claim 3.3. Claim 3.6 finally gives zE D zE 0 as required.

4 First consequences In the last sections, we have essentially proved the following basic result: Theorem 4.1. For any F D Fgs with s  1, ˚

z 2 Tz .F / W ./ z D g W  is an i.c.d. of F f



is an MC.F /-invariant cell decomposition of Tz .F / that descends to an ideal simplicial decomposition of Tz .F /=R>0 . Dually, cells in this decomposition are in one-to-one correspondence with isotopy classes of fatgraph spines of F . Furthermore, projectivized simplicial coordinates give natural barycentric coordinates on projectivized cells. Proof. It remains only to prove that each i.c.d. of F actually occurs for some point of Tz .F /, and for this, it is enough to prove that each ideal triangulation occurs. To this end, simply take the lambda length on each edge of an ideal triangulation to be one and observe that the formula in Definition 4.18 of Chapter 1 guarantees positivity of each associated simplicial coordinate. Definition 4.2. For any ideal triangulation  of any surface F D Fgs with s  1, we define the center of the cell C ./ to be the point of C ./  Tz .F / which has all its lambda lengths equal to one on . By Lemma 2.9 of Chapter 2, each such center of a cell is an arithmetic punctured surface, and by Lemma 2.8 of Chapter 2, the finite group of symmetries of  is realized as the group of isometries of the corresponding decorated hyperbolic metric.

4 First consequences

151

Equivalently, the center of the cell is determined by specifying that all the simplicial coordinates are constant and equal to two. Likewise for a more general i.c.d. of F , we may define the center of the cell in z T .F / to correspond to unit lambda lengths using Corollary 1.10 of Chapter 1 and ask for the number field associated to its corresponding Fuchsian group. Alternatively and giving a different point in Tz .F /, we might take equal simplicial coordinates on the edges using the solution to the arithmetic problem and ask the same question. The remainder of this section, which is based on [128], [130], [131], derives the more or less direct consequences of Theorem 4.1. Just as for the action of a Fuchsian group on D since MC.F / acts with finite isotropy and leaves invariant an ideal simplicial decomposition which is clearly locally finite, we immediately conclude: Corollary 4.3. The quotient Tz .F /=MC.F / is an orbifold for any surface F D Fgs with s  1. It is often the case in examples that it is easier to study the quotient Tz .F /=R>0 of Tz .F /, for F D Fgs , by the diagonal multiplicative action on lambda lengths corresponding to homothety in Minkowski space. Furthermore as we have observed before, the cell decomposition descends to an ideal simplicial decomposition of this projectivization of Tz .F /. In particular for s D 1, the projectivization is naturally identified with T .F / itself, and the next result again follows immediately: Corollary 4.4. If F D Fg1 , then the cell decomposition in Theorem 4.1 descends to an ideal simplicial decomposition of T .F / itself. Example 4.5. For F D F11 , recall that T .F / and MC.F / denote the quotients by the hyperelliptic involution. This quotient T .F / D Tz .F /=R>0 is naturally identified with the Poincaré disk D, MC.F / is the modular group PSL2 .Z/, and the invariant ideal simplicial decomposition of T .F /=R>0 is exactly the Farey tesselation discussed at the beginning of Section 3 of Chapter 1 as we shall now see. There is only one combinatorial type of trivalent fatgraph corresponding to F , namely, the fatgraph G with edges a, b, c as indicated in Figure 4.1, and there is likewise only one combinatorial type with one four-valent vertex, which arises by collapsing any one of the edges of G, as is also illustrated in the figure. On the other hand, there are different isotopy classes of embeddings of G as spine in F , and a flip corresponds to collapsing an edge passing to a codimension-one face of the cell (four points in the light-cone become coplanar in the convex hull construction) and expanding it differently passing into an adjacent codimension-zero cell with its different isotopy class of embedding of G. Collapsing the edge labeled c moves up in Figure 4.1, collapsing a moves to the lower right, and collapsing b moves to the lower left. Each flip begins and ends with two different such isotopy classes whose corresponding cells agree along a common codimension-one face, and one imagines combinatorially “taking a walk in Teichmüller space” in this manner.

152

4 Decomposition of the decorated spaces

c

b

a G

Figure 4.1. Ideal simplicial decomposition of T .F11 /.

To describe a natural identification of T .F / with D explicitly, let us begin with the cell C.G/, which we shall explain how to identify with the ideal triangle t whose 1 vertices are given by the projections u; N v; N w x 2 S1 of the standard light-cone basis C vectors u; v; w 2 L described in Section 4 of Chapter 1. The convex hull construction applied to a point in the interior of C.G/ determines two identical decorated ideal triangles in Minkowski space, and we identify the underlying ideal triangle with t using Corollary 4.2 of Chapter 1. The decorated triangle has its lambda lengths a, b, c, say which respectively correspond to the edges of t connecting uN with w, x vN with w, x and uN with v, N satisfying the three strict triangle inequalities by Lemmas 1.5 and 4.11 of Chapter 1, and so by Lemma 4.12 of Chapter 1 there is a unique equidistant point to the horocycles. Furthermore, this equidistant point is given by  1  2 D p .a C c 2  b 2 /u C .b 2 C c 2  a2 /v C .a2 C b 2  c 2 /w 2 H 2 K

4 First consequences

153

as computed in the proof of Lemma 4.12 of Chapter 1, where K D .a C b C c/.a C b  c/.a C c  b/.b C c  a/: Furthermore, the respective simplicial coordinates of a, b, c are given by 2.b 2 C c 2  a2 / 2.a2 C c 2  b 2 / 2.a2 C b 2  c 2 / ; ; abc abc abc by definition, and since each is non-negative on C.G/, we conclude that the projection p  2  K b  a2 c 2  a2  b 2 N D p 2D ; p p K C a2 C b 2 K K actually lies in t . This assignment of equidistant point N plus the identification with t of the elliptic face in the convex hull construction gives a mapping from the interior of C .G/ to the interior of t , and this mapping extends to a homeomorphism from C .G/ to t , i.e., the lambda lengths are smoothly calculable from N 2 t as well. ba In particular when c D 0, the formula gives N D . bCa ; 0/ 2 t , so performing the 0 similar construction for the fatgraph G arising from the flip along c and identifying the elliptic face in the convex hull construction for points of C.G 0 / with the ideal triangle t 0 whose vertices are u; N v; N  w in the natural way, these maps combine to give a homeomorphism from C .G/ [ C .G 0 / to t [ t 0 . Continuing in this manner gives a geometrically natural homeomorphism T .F / ! D as desired. The hyperelliptic involution W F ! F is manifest as the fatgraph automorphism that reverses each oriented edge of each fatgraph. One furthermore recognizes the standard generator T of the modular group PSL2 .Z/ of order 3 as Aut.G/= , where Aut.G/ is of order 6. Likewise, the standard involution S 2 PSL2 .Z/ corresponds to the automorphism group of the fatgraph arising from the collapse of c, which has automorphism group of order 4, modulo the hyperelliptic involution. Notice that for more complicated surfaces, the equidistant points may lie outside the corresponding triangles, and it is not clear what might be the analogue of this construction. We had hoped at one point to induce a complex structure on Teichmüller space in this manner using the complex structure on the surface but never with a satisfactory result. The vertices of the Farey decomposition lie in the circle at infinity, i.e., this is only an ideal triangulation of D. This is because any two edges of G span a cycle, so the common vanishing of their simplicial coordinates would violate the no vanishing cycle condition. It follows that there cannot be five coplanar points in the light-cone z z 2 Tz .F 1 /. in Minkowski space lying in a common -orbit for  1 The representation of the mapping class group as rational maps on lambda lengths in Theorem 2.10 of Chapter 2 is especially simple for the surface F , where the flip along c in Figure 4.1 produces an edge with lambda length d where cd D a2 C b 2 by Ptolemy’s relation. This is related to the classical Markoff equation x 2 C a2 C b 2 D 3abx

154

4 Decomposition of the decorated spaces

of number theory [28] as follows. The sum of the two roots of this quadratic equation in x is given by 3ab according to the quadratic formula, and yet immediately 3ab D x C .a2 C b 2 /=x for a solution to this equation. It follows that if .a; b; x/ is a Markoff triple, i.e., a solution to the Markoff equation, then so too is .a; b; .a2 Cb 2 /=x/. Insofar as .1; 1; 1/ is a Markoff triple and is the punctured arithmetic surface corresponding to the center of the cell, centers of adjacent cells are given by other Markoff triples and so on. In fact by well-known results about Markoff triples, the centers of cells for F11 are exactly the diophantine solutions to Markoff’s equation. This completes our detailed discussion for the once-punctured torus. The upshot of Theorem 4.1 is that the classical relationship between the Farey tesselation and the Teichmüller space of the once-punctured torus exhibited in Example 4.5 extends to any once-punctured surface, and indeed to projective classes of decorated multiply-punctured surfaces (and even partially decorated bordered surfaces as discussed in the next section). Thus, top-dimensional cells in the decomposition of Tz .F /, which has real dimension 6g  6 C 3s for F D Fgs , correspond to trivalent fatgraphs. More generally, the dimension of C .G/ is given by the number #E.G/ of edges of G. Dually on the level of ideal cell decompositions, codimension-zero cells correspond to ideal triangulations of F , codimension-one to ideal cell decompositions of F with one complementary quadrilateral and all the rest triangles, and codimension-two to ideal cell decompositions of F with either one complementary pentagon or two complementary quadrilaterals and all the rest triangles, and so on. More generally letting vk D P vk .G/ denote the number of k-valent vertices of the P fatgraph G, we have 2#E.G/ D k3 kvk and 2g C s  2 D 12 k3 .k  2/vk . The codimension of C .G/ in Tz .F / is thus given by X 1X 6g  6 C 3s  kvk D .k  3/vk : 2 k3

k3

Definition 4.6. As a point of notation, we shall let G D G .F / denote the ideal simplicial decomposition of Tz .F /=R>0 . Just as we consider the duality between ideal cell decompositions and fatgraph spines in a surface, we take the dual fatgraph complex Gy D Gy.F / to G , i.e., Gy has a vertex for each isotopy class of trivalent fatgraph spine, an edge connecting vertices if the corresponding spines differ by a flip, and in general, a cell of dimension d in Gy for each cell of codimension d in G . In the special case of the once-punctured torus, define G .F11 / and Gy.F11 / to be the quotients of these complexes by the hyperelliptic involution. Notice that unlike the ideal cell decomposition G .F /, the dual Gy.F / is an honest cell complex. For example for F D F11 , the dual Gy.F / of the Farey decomposition G .F / is the infinite tree with all vertices trivalent called the Farey tree. Corollary 4.7. For any surface F D Fgs with s  1, the mapping class group MC.F / acts with finite isotropy on the contractible complex Gy.F / of dimension 4g C 2s  5.

5 Partially decorated bordered surfaces

155

Proof. The natural action of MC.F / on G .F / Tz .F / induces an MC.F /-action on the dual Gy.F /. This action has finite isotropy since the isotropy subgroup of a cell C .G/ in Tz .F /=R>0 is the fatgraph automorphism group Aut.G/ of G (modulo the hyperelliptic involution for F D F11 ) and hence finite. As the dual of a contractible space G .F / Tz .F /=R>0 by Theorem 2.5 of Chapter 2, the fatgraph complex Gy.F / is itself contractible. The maximum codimension in G .F / occurs for an i.c.d. with one complementary .4g C 2s  2/-gon, which is triangulated by 4g C 2s  2  3 arcs. In fact, [57] gives an MC.F /-equivariant deformation retraction of G .F / onto Gy.F / using coordinates from [127] thus giving another more difficult proof of contractibility of Gy.F /

5 Partially decorated bordered surfaces In this section, we extend the cell decomposition of the decorated Teichmüller space of a multiply-punctured surface to the setting of partially decorated, bordered, and more general hybrid surfaces. In essence, the surface comes equipped with some collection of horocycles, and the convex hull of the corresponding orbits in the light-cone again gives rise to an appropriate decomposition of the surface. The topological type of this decomposition of the surface again determines an ideal simplex in its decorated Teichmüller space as we shall see. There are, however, particulars in the partially decorated and bordered cases both in execution and result. 5.1 Partially decorated surfaces. This part of the section is an extrapolation of the addendum to [128] armed with the extra formalism of “punctured fatgraphs”, which we next develop. To begin, suppose that F D Fgs with s  1, that P is a non-empty subset of f1; : : : ; sg, and that TzP .F / is its P -decorated Teichmüller space. Recall that a quasi triangulation of F based at P is (the isotopy class of) a collection of arcs in F so that each complementary region is either a triangle or a once-punctured monogon with 1 vertices at points of P , and TzP .F / R >0 for any quasi triangulation 1 of F based at P by Theorem 2.15 of Chapter 2. Definition 5.1. A subset  of a quasi triangulation 1 is a quasi cell decomposition or q.c.d. of F if each complementary region is either a polygon or an exactly oncepunctured polygon. z 2 TzP .F /, there is again the Fuchsian group  of the underlying hyperbolic Given  structure acting on Minkowski space, and there is again a subset BP of the open positive light-cone corresponding to the decorated punctures. BP is again discrete and radially dense, and we may as before take the closed convex hull of BP to determine a invariant convex body C in Minkowski space. The proof of Lemma 1.4 is unchanged,

156

4 Decomposition of the decorated spaces

but Lemma 1.5 no longer holds: faces of C cannot be hyperbolic just as before, but now they can be parabolic as well as elliptic; in the former case, they project to exactly once-punctured complementary regions by Lemma 2.8 of Chapter 1. Following the proof of Theorem 1.6, discrete radial density of BP implies that the z of F , which is based at P extreme edges of its convex hull C project to a q.c.d. ./ by construction, and we have: z 2 TzP .F /, the convex Theorem 5.2. For any partial decoration P of F D Fgs and  z of F based at P . hull construction produces a q.c.d. ./ Definition 5.3. A punctured fatgraph is a fatgraph in the usual sense where there are two colors of vertices, called punctured and unpunctured, but now we allow also uniand bi-valent vertices as long as they are punctured. A punctured fatgraph with only punctured univalent and unpunctured trivalent vertices will be called a uni-trivalent fatgraph. We shall distinguish fatgraphs without punctured vertices from punctured fatgraphs by referring to the former as ordinary fatgraphs. We conveniently depict punctured fatgraphs as before by drawing planar projections and using the usual icon for an unpunctured vertex and the icon for a punctured vertex. For example, the various combinatorial types of punctured fatgraphs for the pair of pants F03 are illustrated in Figure 5.1.  e



e a b

e

e

b



a

a b

b b

a

b e

b

b a e

 a

a

a e

a e

e e

b a

e a b

a

e b

b  

Figure 5.1. Punctured fatgraphs for the pair of pants.

Definition 5.4. Given any q.c.d.  of F D Fgs , we may build its dual punctured fatgraph G embedded in F as follows. There is one vertex for each component of F [, and the vertex is colored punctured if and only if the corresponding component

5 Partially decorated bordered surfaces

157

contains a puncture of F . There is furthermore one edge of G for each arc in , where an edge of G joins vertices if the corresponding arc lies in the common frontier of the corresponding complementary regions. The cyclic ordering on the half-edges is that induced by the orientation of F as before. Dual to the construction of G from , one can likewise construct a dual q.c.d. from such an isotopy class of punctured fatgraph spines of F . Replacing each punctured vertex by a small cycle with incident edges in the natural way, the punctured fatgraph G dual to a q.c.d. produces a spine in F itself. The reader might imagine the q.c.d. associated with each punctured fatgraph in Figure 5.1. In particular, a punctured fatgraph is dual to a quasi triangulation if and only if it is uni-trivalent. Notice that the Euler characteristic of a punctured fatgraph G in F D F .G/, or equivalently of F itself, is simply given by the number of unpunctured vertices minus the number of edges. Definition 5.5. Suppose that e is an edge with distinct endpoints in a punctured fatgraph G, where at least one endpoint of e is unpunctured. A (Whitehead) collapse of G along e collapses e to a vertex which is fattened as before, where the resulting vertex is punctured if and only if one of the endpoints of e is punctured. Let G=e denote the resulting punctured fatgraph. For example if one endpoint of e is univalent and therefore punctured, and the other endpoint is k-valent and necessarily not punctured, then the resulting .k  1/-valent vertex of G=e is punctured, for k  3. This gives a partial ordering on punctured fatgraphs of fixed topological type F as before, the set of q.c.d.’s of F is partially ordered by inclusion, and we have: Theorem 5.6. Fix a surface F D Fgs with s  1 and a non-empty set P  f1; s; : : : ; sg. The partially ordered set of all q.c.d.’s of F based at P is isomorphic to the partially ordered set of all punctured fatgraph spines of F with punctured vertices corresponding to f1; : : : ; sg  P . Definition 5.7. Suppose that G is a uni-trivalent punctured fatgraph with corresponding quasi triangulation  of F D F .G/ and e is an edge of G with distinct endpoints. If neither endpoint of e is punctured, then we may perform a flip as usual along e in  or G. If one endpoint of e is punctured, then we may perform a quasi flip, which was discussed in Section 2 of Chapter 2 and is illustrated again for convenience on the bottom of Figure 5.2. Notice that a quasi flip along e in a punctured fatgraph simply moves the edge e from one side of the nearby edges to the other as illustrated on the bottom of Figure 5.2. The top of the figure further depicts the quasi flip in the universal cover in the obvious notation under the action of the parabolic fixing the undecorated puncture. Recall again from Section 2 of Chapter 2 that the partially decorated Teichmüller space Tz .F / is the union of TzP .F / over all non-empty subsets P  f1; : : : ; sg, with

158

4 Decomposition of the decorated spaces

a d

1

c1 e1 d1

a0

1

e c

a1

1

e0

1

c0

d0

c

e " " a



 ı

quasi

˛



flip

d

Figure 5.2. Notation for quasi flips.

the subspace topology on each TzP .F /, where a puncture becomes undecorated as the length of its horocycle tends to zero, and that Tz .F /=R>0 has the natural structure of x s1 . T .F / times a closed simplex † z 2 Tz .F / thus determines a q.c.d. ./ z of F by taking the convex hull A point  only of the decorated punctures, and we may define an MC.F /-invariant decomposition of Tz .F / by again taking cells z  g z 2 Tz .F / W ./ C./ D f for each q.c.d.  of F , and again the difficult point is to prove that putative cells are cells. To formulate the arithmetic problem for a punctured fatgraph, we must first define what is the simplicial coordinate of an edge with univalent endpoint. In the notation on the bottom-left of Figure 5.2, the simplicial coordinate of edge e is given by " C " C  C ı   D

2 c 2 C d 2  e2 C ; e cde

i.e., the h-length ˛ is taken to zero in the usual interpretation of simplicial coordinate, or in other words the lambda length a is taken as infinity. Definition 5.8. A closed edge-path in any punctured fatgraph is quasi-efficient if whenever it twice consecutively traverses some edge, then the intervening vertex is

159

5 Partially decorated bordered surfaces

punctured. A formal assignment of putative simplicial coordinates to the edges of a punctured fatgraph satisfies the no quasi-vanishing cycle condition if there is no nontrivial quasi-efficient closed edge-path each of whose simplicial coordinates vanish. We must prove unique existence of the solution to the arithmetic problem for partially decorated surfaces: uniquely calculate lambda lengths realizing given non-negative putative simplicial coordinates satisfying the no quasi-vanishing cycle condition. Suppose that  is the quasi triangulation dual to a uni-trivalent fatgraph spine G of F , and consider a once-punctured monogon complementary to . This monogon contains a unique isotopy class of arc connecting the puncture to the cusp, and we add this arc to  as in the bottom-left of Figure 5.2 for each complementary monogon to produce an ideal triangulation C of F from . Let G C and G denote the respective dual ordinary trivalent fatgraph and punctured uni-trivalent fatgraph. Turning to consideration of the cell decomposition of partially decorated Teichmüller space, let  denote the sectors of G C that are not adjacent to undecorated punctures, and take the basis BÅe , CÅe as before for R , for e 2 . We furthermore omit from the energy function K of G C the terms corresponding to edges of C  , and this amounts to demanding a priori that the formal h-lengths ", " on the bottom-left of Figure 5.2 are equal. Since this condition is invariant under the dissipative flow in Section 3 by construction, we have: Theorem 5.9. For any F D Fgs with s  1, ˚

z 2 Tz .F / W ./ z D g W  is a q.c.d. of F f



is an MC.F /-invariant cell decomposition of Tz .F /, which descends to an ideal simx s1 . Dually, cells in this decomplicial decomposition of Tz .F / =R>0 T .F /  † position are in one-to-one correspondence with isotopy classes of possibly punctured fatgraph spines of F . Furthermore, projectivized simplicial coordinates give natural barycentric coordinates on projectivized cells. Corollary 5.10. For any surface F D Fgs with s  1, finite compositions of flips and quasi flips act transitively on the set of all quasi triangulations of F . Proof. This again follows immediately from the existence of the triangulation and connectivity of Tz .F / since codimension-one cells arise from either one complementary quadrilateral or one complementary once-punctured bigon. Alternatively, a unitrivalent fatgraph is simply an underlying ordinary fatgraph with added edges each of which has a univalent vertex. It is not hard to see that flips and quasi flips allow one to move these added edges around the ordinary fatgraph at will, and together with the earlier transitivity of flips on ordinary fatgraphs, the result follows. At the expense of choosing one among the punctures of a surface Fgs , the constructions and projectivized coordinates descend to the undecorated Teichmüller space, so

160

4 Decomposition of the decorated spaces

this case of a partial decoration with a single decorated puncture is of special significance. Transformations on lambda lengths for different choices of punctures are of course determined by Ptolemy’s equation, and the associated transformations on simplicial coordinates gives an interesting special case of the related arithmetic problems. 5.2 Bordered surfaces. This section comes from [139], where we study a bordered r s surface F D Fg; ıE with distinguished points D in the boundary @F D [f@i giD1 of r F , where ıi 2 Z>0 denotes #.D \ @i /  1, for i D 1; : : : ; r  1. Let B denote the collection of component arcs of @F  D, and let F  D F  D; so @F  D [B. Recall from Section 2 of Chapter 2 that F 0 is the double of F along the arcs in B, so each point of D gives rise to a puncture of F 0 . Indeed, the punctures of F 0 arise either two-to-one from the punctures of F or one-to-one from the points of D, which we simply identify with these particular punctures of F 0 . Furthermore, F 0 supports the involution W F 0 ! F 0 interchanging the two copies of F  F 0 and pointwise fixing each arc in B. The decorated Teichmüller space Tz .F / of the bordered surface is defined to be the -invariant subspace of the partially decorated Teichmüller space TzD .F 0 /. z 2 TzD .F 0 /, there is again an underlying Fuchsian group  0 acting on the Given  Poincaré disk D with quotient F 0 D D= 0 , and  0 is a Fuchsian group of the first kind in the sense that  0 -orbits of points in D accumulate everywhere in the circle at infinity. A geodesic in F 0 connecting punctures in D corresponding to an arc in B lifts to a geodesic in D. The union of all such lifts of all such elements of B decomposes D into components, and each such component gives the developing image of a new kind of hyperbolic structure on F itself: namely, a hyperbolic metric of finite area on F  , where F  does not have geodesic boundary, but rather has totally geodesic boundary @F  D [B. Thus, each point of D gives rise to a cusp on @F  as illustrated in Figure 5.3, which also depicts the unique geodesic @i in F in the homotopy class of the corresponding boundary component. Notice that F   F 0 by construction as a fundamental domain for the action of .

F

F ³i

³i Figure 5.3. F  with totally geodesic boundary.

5 Partially decorated bordered surfaces

161

Choose a complementary region R in D to the union of all lifts of elements of B, and let  be the subgroup of  0 which fixes R setwise. R is convex hence simply connected, and is thus isometric to the metric universal cover of F with the new kind of hyperbolic structure with covering group ;  is called a Fuchsian group of the second kind since -orbits fail to accumulate on a collection of open intervals in the circle at infinity. Notice that the involution acts by reflection in the frontier arcs of R. This gives the more intrinsic description promised in Section 2 of Chapter 2 of a -invariant element of TzD .F 0 / as a Fuchsian group  of the second kind uniformizing a hyperbolic metric of finite area on F  with totally geodesic boundary together with a specification of horocyclic arcs, one arc about each point in D. Definition 5.11. An ideal arc (in the bordered surface) F is (the isotopy class of) an arc connecting points of D which is essential in the sense that it is neither null homotopic nor isotopic into @F  by a proper isotopy. The union of B with a disjointly embedded collection of ideal arcs (or its proper isotopy class) is a quasi cell decomposition or q.c.d. of F provided that each complementary region is either a polygon or a oncepunctured polygon. A quasi triangulation is a q.c.d. each of whose complementary regions is either an ideal triangle or a once-punctured monogon. Thus, ideal arcs in a bordered surface are not allowed to have endpoints at punctures. We may alternatively view an ideal arc as a geodesic in F  which is asymptotic to the cusps on @F  . Definition 5.12. A bordered fatgraph is an ordinary or punctured fatgraph together with a new kind of edge called a tail edge which has one or two univalent vertices which are unpunctured. We may perform Whitehead collapses, Whitehead moves or flips, and quasi flips exactly as for punctured fatgraphs but only along non-tail edges. The tail edges are simply there as markers for where are the components of B, which occur in cycles along the boundary components of F , and the punctured vertices, if any, are treated as before. Definition 5.13. Suppose that  is a q.c.d. of the bordered surface F , and let 0 be the -invariant q.c.d. of F 0 which restricts to  on F  F 0 . Take the dual (possibly punctured) fatgraph G 0 to 0 in F 0 and restrict it to G  F , where the edges of G 0 which meet [B are taken as the tail edges of G. This bordered fatgraph G in F is the dual bordered fatgraph to . Conversely, given a bordered fatgraph spine G embedded in F 0 with its univalent vertices on tail edges lying in [B, we may double G in F in the natural way erasing the bivalent vertices resulting from the tail edges to get a (possibly punctured) fatgraph G 0 in F 0 . The dual q.c.d. 0 to G 0 in F 0 restricts to F to give the q.c.d. of F dual to G. As before, Whitehead collapse along non-tail edges gives a partial ordering on bordered fatgraph spines of F , inclusion gives a partial ordering on q.c.d.’s of F , and we have:

162

4 Decomposition of the decorated spaces

s Theorem 5.14. Fix a bordered surface F D Fg;ı with r  1 boundary components, s  0 punctures and set D ¤ ; of boundary distinguished points. The partially ordered set of all q.c.d’s of the bordered surface F is isomorphic to the partially ordered set of all bordered fatgraph spines of F with #D tails and s punctured vertices.

Now turning to the convex hull construction, suppose that  is the Fuchsian group z 2 Tz .F / for some of the second kind discussed above corresponding to a point  bordered surface F . We may regard  as acting on Minkowski space and regard the decoration at the points of D as providing #D  1 many -orbits of points in the light-cone whose union we denote B, which is no longer radially dense but is still discrete. Taking the convex hull of B, the only possible hyperbolic support planes z of ideal must contain a lift of some element of B. We again produce a family ./ arcs in the bordered surface F with endpoints at D. The proof of Theorem 1.6 again analogously gives: s z 2 Tz .F /, the convex hull and  Theorem 5.15. For any bordered surface F D Fg;ı z of F . construction produces a q.c.d. ./

In order to formulate the arithmetic problem and prove that z 2 Tz .F / W ./  g C ./ D f s is a cell for each q.c.d.  of F D Fg; ıE , first define the simplicial coordinate for e 2 B 2

2

2

e to be 2 a Cb if the triangle in F complementary to  and containing e has edges abe a, b, e, so we are simply calculating simplicial coordinates as usual in the double F 0 . Suppose that  is a quasi triangulation of F and let G denote the dual bordered fatgraph with tail edges in one-to-one correspondence with B, with non-tail edges in one-to-one correspondence with , and possibly punctured vertices as well.

Definition 5.16. An edge-path in a bordered fatgraph G is quasi-efficient if it never twice consecutively traverses the same edge unless the intervening vertex is punctured, so in particular, no quasi-efficient edge-path can twice traverse a tail. We say that there are no quasi-vanishing cycles provided there is no quasi-efficient cycle on G with vanishing sum of simplicial coordinates, and now we also say there are no quasivanishing arcs if there is no quasi-efficient arc with terminal and final edges given by tails again with vanishing sum of simplicial coordinates. s To explicate this definition, fix a quasi triangulation  of F D Fg; ıE with boundary distinguished points D  @F , so there are #D D ı1 C    C ır edges comprising the frontier @ of F  D with component arcs B. Define the subspace

Cz ./ D f.y; E x/ E 2 RB  RB W there are no vanishing cycles or arcsg; 0 where the coordinate functions are taken to be the simplicial coordinates. The no vanishing cycle condition means that there is no essential simple closed curve C  F

5 Partially decorated bordered surfaces

163

meeting a representative of  transversely and efficiently so that X xp ; 0D p2C \[

where p 2 C \ a, for some arc a 2 , contributes to this sum the coordinate xp of a. The no vanishing arc condition means that there is no essential simple arc A  F meeting  transversely and efficiently and properly embedded in F with its endpoints disjoint from D so that X X yp C xp ; 0D p2A\@

p2A\[

where xp ; yp again denote the simplicial coordinate at an intersection point. There are always exactly two terms in the former sum. We may think of this as a convex condition on yE given x. E Furthermore, if the strict triangle inequalities on lambda lengths hold on each triangle complementary to  which contains and edge in the frontier @ , then non-negativity of the interior simplicial coordinates implies strict triangle inequalities on all the triangles complementary to  as in Lemma 2.3. Definition 5.17. The arithmetic problem for bordered surfaces is then as follows: uniquely calculate lambda lengths for any putative assignment of simplicial coordinates to edges of the bordered fatgraph G with no quasi-vanishing cycles and no quasivanishing arcs, where we allow possibly negative simplicial coordinates only on the tail edges of G. Remark 5.18. We mention parenthetically, that there is a related arithmetic problem for bordered surfaces, where one fixes the positive lambda lengths of the arcs in B and the non-negative simplicial coordinates on edges of  and asks for lambda lengths on edges of . This is a generalization of the Euclidean version discussed in Section 1 of Chapter 2. We do not know if this related problem admits (unique) solution. This is in a sense analogous to an issue for cluster algebras “with parameters”, cf. [47], [46], [31]. We claim that the dissipative flow in Section 3 for the double F 0 restricts to the desired flow for F . To see this, note that the coupling equation automatically holds on any arc in B for any -invariant assignment of formal h-lengths to sectors of F 0 , so this term simply drops out from the energy function for F 0 , and the dissipative flow is -equivariant and leaves invariant this subspace. The proofs of Claims 3.3 and 3.4 apply because of the no quasi-vanishing cycle or arc condition. The analysis of Claims 3.5 and 3.6 is unchanged, and we have: s Theorem 5.19. For any bordered F D Fg; ıE with r  1 boundary components, ˚ z 2 Tz .F / W ./ z D g W  is a q.c.d. of F f

164

4 Decomposition of the decorated spaces

is an MC.F /-invariant cell decomposition of Tz .F /, which descends to an ideal simplicial decomposition of Tz .F /=R>0 . Dually, cells in this decomposition are in one-to-one correspondence with isotopy classes of possibly punctured bordered fatgraph spines of F . Furthermore, projectivized simplicial coordinates give natural coordinates on projectivized cells, and finite compositions of flips and quasi flips acts transitively on the set of all quasi triangulations of F . See [91], [163] for other treatments of the ideal simplicial decomposition in the bordered case. s Corollary 5.20. For any bordered surface F D Fg; ıE with r  1, the group MC.F / is z torsion free, and T .F /=MC.F / is a manifold Eilenberg–MacLane space K.MC.F /; 1/.

Proof. MC.F / acts cellularly on the decomposition of the space Tz .F /. Suppose that some cell C ./, for  a q.c.d. of F , is left setwise invariant by an element  2 MC.F /. The arcs in  come in a canonical linear ordering since they occur in order in the counter-clockwise orientation of F about each point of D starting from the tangent to the curve @F with orientation induced from that of F . It follows that the stabilizer of any cell is trivial from which the remaining assertions follow. This linear ordering on the ideal arcs in a q.c.d. of a bordered surface is a basic distinction from punctured surfaces, where the ideal arcs come only in a canonical cyclic ordering about each puncture. 5.3 General case. We close this section by simply stating an omnibus theorem in the general case that subsumes and extends several of the previous results in this section: s Theorem 5.21. Suppose that F D Fg; ıE is a possibly bordered surface, P is a possibly empty subset of the punctures and Q a possibly empty subset of the collection of distinguished points on the boundary, where we demand that P [ Q ¤ ; and Q intersects each boundary component of F . Given  2 T .F / and a decoration on the points in Q, if any, as well as a decoration on the punctures in P , if any, the convex hull construction applied to these -orbits produces a q.c.d. of F based at P [ Q. Furthermore, ˚ z 2 Tz .F / W ./ z D g W  is a q.c.d. of F f

is an MC.F /-invariant cell decomposition of Tz .F /, which descends to and ideal simplicial decomposition of Tz .F /=R>0 . Dually, cells in this decomposition are in one-to-one correspondence with isotopy classes of possibly punctured fatgraph spines of F possibly with tails. Furthermore, projectivized simplicial coordinates give natural barycentric coordinates on projectivized cells, and finite compositions of flips and quasi flips act transitively on the set of all quasi triangulations of F based at P [ Q.

165

6 Fermionic formulation of the arithmetic problem

6 Fermionic formulation of the arithmetic problem In this completely independent and optional section taken from [134], we formulate the arithmetic problem as a fermionic functional integral and compute various related super Lie algebras. A nice reference on the background statistical physics is [72]. 6.1 Arithmetic problem as fermionic integral. Suppose that  is an ideal triangulation of F D Fgs , let G denote the dual trivalent fatgraph with corresponding freeway  as in Definition 4.5 of Chapter 2; recall that there are short edges of  (contained in a triangle complementary to ) and long edges (each meeting a unique arc in ). Define a (fermion) site of  to be a half-edge of  which is contained in a small edge, so there are two sites corresponding to each vertex of  . If u is a vertex of  , where  is given by a planar projection, let U be a smooth neighborhood of u in the plane of projection. We shall identify the components of U   with the half-edges of  in such a way that the two components of U   with smooth frontier are identified with the fermion sites of  incident on u. Enumerate the sites of  by i D 1; 2; : : : ; S. Consider the S S matrices E G (and U G , Uz G , respectively), where we set the entry G Eij (and UijG , UzijG ) to unity if sites i and j are configured as indicated in Figure 6.1a (and 6.1b, 6.1c) and to zero otherwise, for i; j D 1; 2; : : : ; S.

j

j

i

(a)

i

(b)

j

i

(c)

Figure 6.1. Fermion sites on a freeway.

z 2 Tz .F / of decorated Teichmüller space, we assign corresponding Given a point  h-lengths to the sectors and lambda lengths to the edges of G and then induce h-lengths on the fermion sites and lambda lengths on the long edges of  . Suppose that site i is incident on vertex v of  , that the two small edges of  incident on v have respective h-lengths ˛; ˇ, and that the other small edge of  incident on these two edges has h-length ". Introduce a formal (thermodynamic) parameter and assign the quantity z D e Œ 2 .˛Cˇ "/ ; i . I / 

to the site i D 1; : : : ; S (which the reader will recognize as a simple transform of the hyperbolic invariant in Lemma 4.12 of Chapter 1).

166

4 Decomposition of the decorated spaces

If yi are real variables indexed by the sites i D 1; 2; : : : ; S, then we let D.yi / denote the diagonal matrix with entries y1 ; : : : yS , and define the S  S matrices z D D.i .I // z E G D.i .I // z M G . ; I /   G z U C Uz G D.i . I // z C D.i . I // and z K G D M G .0; 0I /: Thus, K G D E G C U G C Uz G depends only on the fatgraph G and indeed only on the z underlying graph because of the symmetry in the definition of M G . ; I /. "

2 1

5 ˛

ˇ

6

b

10 9 e

11 12

a



4 3

ı

7 8



Figure 6.2. Notation for fermion sites and h-lengths on the freeway  .

Example 6.1. Adopt the notation of Figure 6.2 for the freeway  (which arises from the fatgraph G depicted in Figure 2.3a), where the lower-case Roman letters a, b, e denote lambda lengths, the upper-case Roman letters A, B, E denote the corresponding simplicial coordinates, the Greek letters denote h-lengths, the index i D 1; 2; : : : ; 12 enumerates the sites of . According to the definitions, we have 1 0 0

EG

B0 B0 B B0 B B0 B B0 D B0 B B0 B B0 B B0 @0 0

0 0 1 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0C C 0C 0C C 0C C 0C C; 0C 0C C 1C C 0C 0A 0

167

6 Fermionic formulation of the arithmetic problem

0

UG D

0 B0 B0 B B0 B B0 B B0 B0 B B0 B B0 B B1 @0 0

0

Uz G D

and so

0 B0 B0 B B0 B B0 B B1 B0 B B0 B B0 B B0 @0 0

0

KG D

0 B0 B0 B B0 B B0 B B1 B0 B B0 B B0 B B1 @0 0

1

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0 0

0 0C C 0C 0C C 0C C 0C C; 0C 0C C 0C C 0C 0A 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1 0 0 0 0

0 0C C 0C 0C C 0C C 0C C; 0C 0C C 0C C 0C 0A 0

0 0 1 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1 0 0 0 1

1 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0 1

0 0 0 0 1 0 0 0 0 0 0 0

0 1 0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 1 0

0 0 0 1 0 0 0 1 0 0 0 0

0 0C C 0C 0C C 0C C 0C C: 0C 0C C 1C C 0C 0A 0

1

1

z is the diagonal matrix with entries Moreover, D.i . I // 











.e 2 .ˇ C"˛/ ; e 2 .ˇ C"˛/ ; e 2 . Cı/ ; e 2 . Cı/ ; e 2 .˛C"ˇ / ; e 2 .˛C"ˇ / ; 











e 2 .ıC / ; e 2 .ıC / ; e 2 .˛Cˇ "/ ; e 2 .˛Cˇ "/ ; e 2 . Cı/ ; e 2 . Cı/ /

168

4 Decomposition of the decorated spaces

z and we find that M G . ; I / z is given by with a similar description of D.i .I //, 1 0 A 0

B 0 B B 0 B 0 B B 0 B B e" B B 0 B B 0 B B 0 Beˇ B @ 0 0

0 0

e

A 2

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

e 0 0 0 e

e

2

0 0 0 0 0 0 0 0 0 0 0

0 e" 0 0 0 0 0 0 0 e˛ 0 0

0 0 0 0 0 0

e

B 2

0 0 0 0 0

0 0 0

e 0 0 0 0 0 0 0 eı

0 0 0 0

e

B 2

0 0 0 0 0 0 0

0 eˇ 0 0 0 e˛ 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

e

E 2

0

0 0 0

e 0 0 0 eı 0 0 0 0

0 0 0 0 0 0 0 0

C C C C C C C C C C C C E C 2 e C C 0 C A 0 0

completing our example. Let 0 denote the S  S zero matrix, t denote the transpose, and define the 2S  2S block matrix   z 0 M G . ; I / z D ; AG . ; I / z t ŒM G . ; I / 0 which we regard as a skew quadratic form defined on the 2S-dimensional space spanned by fermionic variables 1 ; : : : ; S ; N1 ; : : : ; NS . Taking this quadratic form as an action, we find the partition function Z 1 G t 1 z G S z 4 d d N e 2  ŒA .; I /  ; Z . ; I / D .1/ where

D . 1 ; : : : ; S ; N1 ; : : : ; NS /; d d N D d 1 : : : d S d N1 : : : d NS ; and the integral is in the sense of Berezin. Noting that S=4 is the number of edges of G, a standard calculation (cf. 2.1 of [72]) gives that z D .1/ 14 S det M G . ; I /; z ZG . ; I / where det denotes the determinant. Moreover, the phase space is given by f˙1gV , where V denotes the set of vertices of G, which we may interpret geometrically as follows. There are two cyclic orders on a set with three elements, and if v is a vertex of G with  .v/ D 1, where  2 f˙1gV , then we change the cyclic order at v to differ from that of G while if  .v/ D C1, then we retain the cyclic order of G at v. Modify each vertex of G in this way to produce

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6 Fermionic formulation of the arithmetic problem

another fatgraph G , say with s boundary components. The boundary components of G give rise to corresponding closed edge-paths on G itself which we denote @j , for j D 1; : : : ; s . z (using Furry’s Theorem), we find Expanding the exponential in ZG . ; I / z D ZG . ; I /

X

s Y





.1/`.@j / e.C / .@j / ;

2f˙1gV j D1

where `.@/ denotes the combinatorial length (i.e., the number of edges comprising @ with multiplicity) and .@/ denotes the geometric length (i.e., half the sum of the corresponding simplicial coordinates again with multiplicity, cf. Lemma 2.1) of the closed edge-path @ on G. We find that z D 2#V e.C / ; ZG . ; I /

P where  D jsD1 .@j / is independent of  2 f˙1gV . The two thermodynamic parameters and  were introduced in order to treat the small and large edges of  on an equal footing, however, now we specialize and define z D ZG .1; 0I / z D 2jV j e ; ZG ./ z D AG .1; 0I /: z AG ./ The main result of this section follows directly from the calculation above and Lemma 2.11: Theorem 6.2. Let xj (and Xj respectively) denote the lambda length (and simplicial coordinate) of edge i D 1; : : : ; N of the trivalent fatgraph G. Then for each i D 1; : : : ; N , we have z @ log ZG ./ D Xi ; @ log xi z that is, log xi and Xi are Legendre duals for the action AG ./. 6.2 Related super Lie algebras. Fix a trivalent fatgraph G with corresponding freeway  once and for all. As is customary given a partition function, we are interested z In fact, we cannot explicitly calculate this in the spectrum of the matrix AG . ; I /. spectrum, but as a first step, we shall describe the spectrum of   0 K z D ; A0 D AG .0; 0; I / K t 0 where K D K G ; one might subsequently try to perturbatively analyze the spectrum of z itself. AG . ; I / The complex vector space W spanned by the sites of  admits a natural Z=2-grading as follows. Insofar as  is itself a fatgraph, its various boundary components induce

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4 Decomposition of the decorated spaces

a canonical orientation on each small edge of  . A given site corresponds to either an initial or terminal point of its corresponding small edge as in Figure 6.3, and this dichotomy gives rise to the grading on W . Thus, W is canonically a super vector space of type .S=2; S=2/, where S is the number of sites of  . Let Œ;  denote the super commutator of operators on W .





initial

terminal

Figure 6.3. Z=2 grading on sites of a freeway.

K is clearly an odd operator on W , and we next introduce further odd operators F , R, T , each of which is symmetric and idempotent. To this end, suppose that e is a large edge of , and enumerate the sites of  adjacent to e by i D 1; 2; 3; 4 as illustrated in Figure 6.4a, and let wi 2 W denote the corresponding basis vector. Each of F , R, T preserves the subspace of W spanned by w1 , w2 , w3 , w4 for each large edge e, and in this basis, the restriction of the operators are given by 0 1 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1 B1 0 0 0C B C B C C ; R D B0 0 0 1C ; T D B0 0 1 0C F DB @0 0 0 1A @1 0 0 0A @0 1 0 0A 0 0 1 0 0 1 0 0 1 0 0 0 completing the definition of F , R, T . One calculates directly that ŒR; T D 2F;

ŒF; T D 2R;

ŒF; R D 2T;

so F , R, T span twice the tautological representation of s`2 . We furthermore compute directly that FKF D K t ; and so



0 A0 D K t

K 0





0 D FKF

which is one ingredient for analyzing the spectrum of A0 .

K 0



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6 Fermionic formulation of the arithmetic problem

6 5 3

2

4

1

1

4 2

3

(a)

(b)

Figure 6.4. Numeration of sites of a freeway.

If e is a large edge of , then we continue with the notation of Figure 6.4a for the nearby sites and define the two associated vectors

e D w1 C w3 ;

e D w1  w3 ;

where wi 2 W is the basis vector corresponding to the site i . Moreover, if v is a vertex of G, then there is a corresponding triangle complementary to  , and we adopt the notation of Figure 6.4b for the nearby sites and define the three associated vectors

v D w1 C w2 C w3 ;

v D w1 C !w3 C ! 2 w5 ;

 v D w1 C ! 2 w3 C !w5 ;

where wi 2 W is again the basis vector corresponding to the site i and ! is a primitive cube root of unity. Lemma 6.3. The spectrum of KF is described as follows. For each edge e of G, e (and e respectively) is an eigenvector of KF of eigenvalue C1 (and 1). For each vertex v of G, v (and v ,  v respectively) is an eigenvalue of KF of eigenvalue C2 (and 1, 1). The proof is a routine verification. Theorem 6.4. If is an eigenvector of KF of eigenvalue  as described in Lemma 6.3, then    

and i F i F are eigenvalues of A0 of respective eigenvalues i  and i . Proof. Calculation gives   

0 A0 D i F FKF  D

i  F 

K 0





i F

 Di





i KF D FKF



; i F



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4 Decomposition of the decorated spaces

where the first equality follows from the previous expression for A0 D with an analogous computation for the other eigenvector.



0 K FKF 0



Though it is not usual to consider the spectrum of an odd operator, we have also investigated the spectrum of K itself as we finally discuss. Provided G is connected, one can easily show that K is (super) primitive irreducible, so that the spectrum is governed by the Perron–Frobenius Theorem. In fact, there is some integer k > 1 so that for any i; j D 1; 2; : : : ; S, either Kijk > 0 or KijkC1 > 0 (where Kijk denotes the ij th entry of the k th iterate of K). Our approach is standard for understanding the spectrum of K: construct a tractable super Lie algebra containing K. In so doing, we shall find an interesting class of representations of super Lie algebras, one for each trivalent graph. To define the new operators on W , recall the phase space f˙1gV considered before. We define a finite-order unitary operator Z on W for each  2 f˙1gV as follows. We have defined a fatgraph G corresponding to  in the previous section and may regard the boundary components of G as a family of oriented closed edge-paths @1 ; : : : ; @s on  itself. Such a closed edge-path @ on  consecutively visits various sites i0 ; i1 ; : : : ; iP of S, and we define a corresponding operator Z.@/ mapping wij to wij C1 , for j D 0; : : : ; P , where wi 2 W is as usual the basis vector corresponding to site i, the subscripts are taken modulo P C 1, and the support of Z.@/ is the span of wi0 ; : : : ; wiP . Define the desired operator Z D

s Y

Z.@j /;

j D1

where the product denotes the composition of these commuting operators. Example 6.5. Adopt the notation for the sites of the freeway in Figure 6.2 and take  2 f˙1gV to have constant value C1. We find 0 1 0 0 0 1 0 0 0 0 0 0 0 0 B0 0 0 0 1 0 0 0 0 0 0 0 C B C B0 1 0 0 0 0 0 0 0 0 0 0 C B C B C B0 0 0 0 0 0 0 0 0 0 1 0 C B C B0 0 0 0 0 0 0 1 0 0 0 0 C B C B0 0 0 0 0 0 0 0 1 0 0 0 C B C: Z D B C B0 0 0 0 0 1 0 0 0 0 0 0 C B0 0 1 0 0 0 0 0 0 0 0 0 C B C B C 0 0 0 0 0 0 0 0 0 0 0 1 B C B C B1 0 0 0 0 0 0 0 0 0 0 0 C B C @0 0 0 0 0 0 0 0 0 1 0 0 A 0 0 0 0 0 0 1 0 0 0 0 0

6 Fermionic formulation of the arithmetic problem

173

Lemma 6.6. In the notation of Lemma 6.3, the spectrum of Z F , for  2 f˙1gV , is described as follows. For each edge e of G, e (and e ) is an eigenvector of KF of eigenvalue C1 (and 1 respectively). Given a vertex v of G, suppose that the associated triangle complementary to the universal freeway corresponding to G is as illustrated in Figure 6.4b. Then v (and v ,  v ) is an eigenvalue of KF of eigenvalue C1 (and ! 2 , ! respectively). Again the proof is a routine verification. If  2 f˙1gV , we shall write  for the element of f˙1gV so that . /.v/ D .v/ for each v 2 V . In analogy to FKF D K t , we may calculate directly that t F Z F D Z ;

and together with Lemmas 6.3 and 6.6, we conclude Corollary 6.7. fKF g [ fZ F W  2 f˙1gV g is an abelian self-adjoint family of operators acting on W . Now, the super Lie algebra generated by fKg [ fZ W  2 f˙1gV g satisfies interesting “universal” relations (i.e., relations independent of the particular fatgraph G), as follows, where I denotes the identity operator on W : Proposition 6.8. For any  2 f˙1gV , we have: • .F Z /6 D .Z F /6 D I ; • ŒK; Z D K 2 C Z2 ; • ŒZ ; Z D K 2 ; • ŒK; R D I C K 2 and ŒZ ; R D I C Z2 . Again the proofs are routine calculations. One presumably wants to “close” the universal algebra. The eigenvalues of K which are fourth roots of unity are highly degenerate: Proposition 6.9. Suppose that Z D  for 4 D 1 and some  2 f˙1gV . Then .I  2 T / is an eigenvector of K itself with the same eigenvalue . Again, the proof is routine and omitted. It is tempting to conjecture that these eigenvectors actually span the eigenspaces of K corresponding to fourth roots of unity.

5 Mapping class groupoids and moduli spaces

We first give presentations of mapping class groups and groupoids for both punctured and bordered surfaces. A general discussion of methods of integration over Riemann’s moduli space is followed by several sample calculations of Weil–Petersson volumes. We then give a comprehensive treatment of fatgraphs in the general case of possibly non-orientable surfaces. Fatgraph tables are presented for orientable once-punctured surfaces of genera two and three. Finally, aspects of compactifying Riemann’s moduli spaces are discussed.

1 Ptolemy and mapping class groupoids Definition 1.1. Define the Ptolemy groupoid Pt.F / of F D Fgs , for s  1, to be the fundamental path groupoid of the fatgraph cell complex Gy.F / in Definition 4.6 of Chapter 4, i.e., the objects are the vertices of Gy.F / and the morphisms are homotopy classes of paths in Gy.F / connecting vertices. In effect, Pt.F / is the fundamental path groupoid of the space Tz .F /=R>0 , which is discretized by passing to edge-paths in the complex Gy.F /. Since Gy.F / is connected and simply connected, there is a unique morphism in Pt.F / between any two objects. In fact, Pt.F / of course only depends on the two-skeleton of Gy.F /, i.e., on the codimensiontwo skeleton of G .F /, whose cells will be explained presently. As a point of notation, if G is a trivalent fatgraph and e 2 E.G/ is an edge of G with distinct endpoints, then we shall write the flip along e as We W G ! G 0 , where G 0 is the result of flipping G along its edge e, and we can extend the definition to any edge of G by conventionally taking the flip along a loop to be the identity. Let f 2 E.G 0 / arise from e 2 E.G/ by the flip We . Since the edges of G and G 0 other than e and f are naturally identified and e can be associated with f , given ordered edges e1 ; : : : ; en of G, we may write Wen ı    ı We1 for the composition of flips read from right to left, i.e., first perform We1 and last perform Wen , without explicit reference to the domain and range of each flip. On the other hand, it is more subtle than meets the eye since a sequence of flips may change the character (loop or distinct endpoints) of an edge. Corollary 1.2. The Ptolemy groupoid Pt.F / of any surface F D Fgs with s  1 is generated by flips, i.e., finite sequences of flips act transitively on the set of all (isotopy classes of ) trivalent fatgraph spines, or equivalently on the set of all (isotopy classes of )

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175

ideal triangulations, of F . Furthermore, a complete presentation of Pt.F / is provided by our convention that We D 1 if e is a loop plus the following relations: Involutivity: if e 2 E.G/, then We ı We D 1; Commutativity: if e1 ; e2 2 E.G/ do not share a common endpoint, then We1 ıWe2 D We2 ı We1 ; Pentagon: if e1 ; e2 2 E.G/ are distinct and do share exactly one common endpoint, then We1 ı We2 ı We1 ı We2 ı We1 D 1. Proof. Since Tz .F / is path connected, there is a path beginning and ending at any two arbitrary centers of top-dimensional cells, and we may put this path into general position with respect to the codimension-one faces of the cell decomposition. Since codimension-one faces correspond precisely to flips (i.e., to coplanarity of four points in the light cone), the transitivity result follows. Turning to the relations, consider a homotopy of two paths in Tz .F / sharing common endpoints. Since Tz .F / is homeomorphic to an open ball, any two such paths are homotopic. This homotopy can be put into general position with respect to the codimension-two skeleton of the cell decomposition. The fatgraph corresponding to a two-dimensional cell in Gy.F / has either one five-valent or two four-valent vertices with the rest trivalent (i.e., either five coplanar points in the light cone or two distinct four-tuples of coplanar points). The links in Gy.F / of these two types of cells are illustrated as i.c.d.’s in Figure 1.1 and respectively correspond to the commutativity and

Commutativity relation

Pentagon relation

Figure 1.1. Links of codimension-two cells.

pentagon relations. The involutivity relation furthermore corresponds to a degenerate two-cell of Gy.F /, i.e., crossing a codimension-one face and then turning around and crossing it again.

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This is our promised alternative proof of Whitehead’s classical transitivity result 1.24 of Chapter 1 which also provides all the relations of the Ptolemy groupoid. Some pretend this result is obvious. It is tantamount to the existence of the ideal cell decomposition of decorated Teichmüller space in codimension less than two using simple connectivity of this space and general position. Not only that, but codimension-three cells, i.e., three-dimensional cells of Gy.F /, can likewise easily be enumerated and correspond to homotopies among homotopies, and so on. Note that the proof does not depend on the fact that putative cells are cells, just on the existence of the decomposition of Tz .F / indexed by fatgraph spines. For the combinatorial presentation of MC.Fgs /, let us consider a labeled fatgraph, by which we mean a trivalent fatgraph spine G of F D Fgs together with a bijection ` from the set of edges of G to the set f1; 2; : : : ; 6g  6 C 3sg called a labeling on G. In fact, MC.F / acts freely on the set of all labeled fatgraphs G in F since if a mapping class fixes an oriented edge of G, then it fixes also each of the edges to the left and right of its initial and terminal points (and the only orientation-preserving homeomorphism of a surface fixing each unoriented edge of trivalent fatgraph spine is the hyperelliptic involution). See [120], where an analogous required freeness is assured by specifying a single distinguished oriented edge of the fatgraph. Definition 1.3. Define the mapping class groupoid M .F / to be the category whose objects are MC.F /-orbits of labeled fatgraph spines in F (modulo the hyperelliptic involution for F D F11 ) and whose morphisms are pairs of labeled fatgraphs modulo the diagonal action of MC.F / together with the obvious composition. More explicitly, given two morphisms in M .F /, i.e., two MC.F /-orbits of pairs ..G1 ; `1 /; .G2 ; `2 // and ..G3 ; `3 /; .G4 ; `4 //, we may compose only if .G2 ; `2 / and .G3 ; `3 / are in the same MC.F /-orbit; in this case, we can arrange by applying mapping classes to the pairs that .G2 ; `2 / D .G3 ; `3 / and define the composition to be the MC.F /-orbit of the pair ..G1 ; `1 /; .G4 ; `4 //. Definition 1.4. Consider the flip on an edge e in some labeled fatgraph .G; `/. The edge e has its label `.e/, so we may alternatively index the flip on e by its label. Furthermore, we induce a labeling on the resulting fatgraph given by the identity on the labeling of each arc other than e and inducing the label of e on the edge arising from e resulting from the flip. This combinatorial move on labeled fatgraphs is called a labeled flip or labeled Whitehead move. The MC.F /-orbit of a labeled flip is called the class of the flip. Thus, labeled flips act on labeled ideal triangulations in the natural way. Theorem 1.5. The mapping class groupoid M .F / of F D Fgs , for s  1, admits the following presentation. Generators are given by MC.F /-orbits of permutations of labels and classes of labeled flips. Relations are given by our convention that the class of a flip on a loop is the identity plus those of the symmetric group together with the following:

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177

Involutivity: for any class of labeled flip Wi , we have Wi ı Wi D identity; Commutativity: for any class of labeled flips Wi ; Wj where the edges with labels i , j do not share a common endpoint, we have Wi ı Wj D Wj ı Wi ; Pentagon: for any class of labeled flips Wi , Wj where the edges with labels i , j share exactly one common endpoint, we have Wi ıWj ıWi ıWj ıWi D .i; j /, where .i; j / denotes the transposition of labels i and j ; Naturality: for any class of labeled flip Wi and any permutation  , we have  ı Wi D W .i/ ı  . Furthermore, for any surface F D Fgs other than F11 and F03 , classes of labeled flips alone generate M .F /. Proof. Labeled involutivity, commutativity, and naturality clearly hold for sequences of labeled flips, and tracing through the labeling of edges in Figure 1.2, we find that the labeled pentagon relation holds as well.

f e f

f

e

e

e f

e

f

Figure 1.2. Labeled pentagon relation.

Let U denote the two-skeleton of the fatgraph complex Gy.F /. In particular, U is simply connected. We shall build a connected and simply connected complex L together with a natural projection p W L ! U . In the upcoming construction of L for the surface F11 , let us tacitly take the quotient by the appropriate involution. There is by definition a vertex of L for each labeled fatgraph spine .G; `/ of F . We construct the 1-skeleton of L in two stages as follows. For each labeled flip from .G1 ; `1 / to .G2 ; `2 /, there is an edge in L connecting .G1 ; `1 / to .G2 ; `2 /, and we refer to these edges as “horizontal”. There is furthermore an edge in L connecting .G; `/ to .G;  ı `/ for each labeled fatgraph .G; `/ and each permutation  of labels, and we refer to these edges as “vertical”.

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5 Mapping class groupoids and moduli spaces

The 2-skeleton of L has four types of 2-cells as follows. For each labeled fatgraph .G; `/ supporting a labeled commutativity relation Wi ı Wj ı Wi ı Wj D 1, there is one 2-cell starting from .G; `/ with boundary edges Wi Wj Wi Wj ; for each labeled fatgraph supporting a labeled pentagon relation Wi ı Wj ı Wi ı Wj ı Wi ı .i; j / D 1, there is one 2-cell starting from .G; `/ with boundary edges Wi  Wj  Wi  Wj  Wi  .i; j /, and we refer to these two types of 2-cells as “horizontal”. For each relation 1 ı    ı n D 1 in the symmetric group and each labeled fatgraph .G; `/, there is furthermore a 2-cell with boundary edges n      1 starting from .G; `/, and we refer to these 2-cells as “vertical”. Finally, for each labeled fatgraph .G; `/ and each permutation  , there is a 2-cell starting from .G; `/ with boundary  Wi  1 W.i/ , and we refer to these 2-cells as “semi-vertical”. This completes the definition of the 2-complex L. The map p W L ! U projects out the vertical cells and forgets the labeling mapping semi-vertical 2-cells to edges in U . More explicitly, p simply forgets the labeling on vertices, maps horizontal edges of L to corresponding edges of U , and maps each vertical edge of L to the common unlabeled fatgraph given by its endpoints. Horizontal 2-cells of L map to corresponding 2-cells of U , each vertical 2-cell of L is collapsed to the corresponding vertex of U , and a semi-vertical 2-cell of L maps to the degenerate 2-cell of U corresponding to consecutively traversing the corresponding edge of U with opposite orientations. The projection p is evidently continuous. The complex L is obviously connected, and we claim that it is simply connected as well. To prove this, suppose that O is a closed curve in L, so p.O/ is a closed curve in U . If p.O/ is a vertex v of U , then O lies in the fiber p 1 .v/. This fiber is homeomorphic to the 2-skeleton of the simplex of dimension .6g  6 C 3s/Š  1 and is therefore simply connected, so O is null homotopic in L. If p.O/ is an edge e of U , then O lies in the pre-image p 1 .e/. This pre-image is the union of two open sets A D p 1 .e  u/ and B D p 1 .e  v/, where u; v are the endpoints of e in U . Since each of A and B is homotopy equivalent to the fiber over a vertex of U , each is simply connected by the previous case. Owing to the semi-vertical cells, the intersection of A and B is connected, so it follows from the Seifert–Van Kampen Theorem that p 1 .e/ is simply connected as well. Thus, O is null homotopic in L in this case as well. If p.O/ is a tree T in U , then we may iterate the argument in the previous case to conclude that p 1 .T / is simply connected, so again O is null homotopic in L. Suppose that p.O/ is the boundary of a 2-cell c in U corresponding to an unlabeled commutativity or pentagon relation. Let e be an edge of U in p.O/ and let T be the union of the other edges in p.O/. By construction of U , T is a tree, so A D p 1 .e/ and B D p 1 .T / are simply connected by the previous two cases. Furthermore, the intersection of A and B is connected owing to the horizontal 2-cells of L mapping to c (and actually any one such 2-cell would suffice). Again by the Seifert–Van Kampen Theorem, p 1 .c/ is simply connected, and O is null homotopic in L.

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179

Taken together with Theorem 1.5 and simple connectivity of U , these claims imply the asserted simple connectivity of L. Furthermore by construction, the corresponding labeled involutivity, commutativity, pentagon, and naturality relations on flips (rather than classes of flips) together with those of the symmetric group and our convention on loops give a complete set of relations for the fundamental path groupoid of the complex L. The mapping class group MC.F / acts cellularly on L in the natural way induced by the action on the underlying fatgraph spine and labeling, and we next show that this action is without isotropy. Since there is no automorphism of a trivalent fatgraph G fixing each edge of G for F ¤ F11 , vertices of L have no isotropy. A vertical 1-cell corresponding to two different labelings on G can have no isotropy since the required mapping class would at once have to fix each edge of G and yet permute them. Vertical 2-cells are therefore also without isotropy. A horizontal 1-cell corresponding to a flip along an edge e in a labeled fatgraph G has no isotropy since the corresponding mapping class would have to not only fix the edges near e but also preserve the fattening of G, and likewise for the horizontal 2-cells since a mapping class must preserve the vertical or horizontal character of 1-cells. Finally, consider a semi-vertical 2-cell corresponding to  ıW.m/ ı ıWm , where m is the label of an edge of the labeled fatgraph .G; `/ with distinct endpoints and .G 0 ; `0 / denotes the result of the flip Wm . A stabilizer  2 MC.F / of this 2-cell must permute its set of vertices, and in light of the previous remarks, the only tenable possibility is that  permutes .G; `/ with .G 0 ;  ı `0 / and permutes .G 0 ; `0 / with .G;  ı `/, i.e.,  interchanges diagonally opposite vertices of the 2-cell. In particular,  2 fixes each such vertex and is therefore the identity, so that  is an involution. The permutation  is thus also an involution since  induces the same permutation of labels as  . Let h, i , j , k denote the labels of (the not necessarily distinct) edges incident on the edge labeled m occurring in this cyclic order, where edges labeled h; i are incident on one endpoint of the edge labeled m and edges labeled j; k are incident on the other. If .m/ D m, then since  induces the same permutation of labels as  and  preserves incidence and ordering of edges, we must have  .i /;  .j /;  .k/;  .h/ respectively equal to either h, i , j , k or j , k, h, i. In either case since  is an involution, we conclude that h D j and i D k, i.e.,  is the hyperelliptic involution on F11 . If .m/ D n ¤ m, then  .n/ D m since  is an involution. Let w, x, y, z denote the labels of (the not necessarily distinct) edges incident on the edge labeled n again occurring in this cyclic order, where edges labeled w; x are incident on one endpoint of the edge labeled n and edges labeled y, z are incident on the other. Since  maps .G; `/ to .G 0 ;  ı `0 / and induces the same permutation of labels as  , we conclude that .k/; .h/;  .i /;  .j / is respectively equal to either w, x, y, z or y, z, w, x. Likewise since  maps .G;  ı `/ to .G 0 `0 /, we conclude that  .h/;  .i /;  .j /;  .k/ is respectively equal to either w, x, y, z or y, z, w, x. In any of the four cases, we conclude that all of h, i , j , k must coincide, which is absurd. The action is therefore indeed without isotropy, and the fundamental path groupoid M .F / of L=MC.F / therefore has the presentation given in Theorem 1.5.

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5 Mapping class groupoids and moduli spaces

For the final assertion in the theorem, notice that the labeled pentagon relation shows that the transposition of labels on adjacent edges can be accomplished by a sequence of five labeled flips. Other than the stipulated surfaces F11 and F03 , there is a trivalent fatgraph spine G of F with no loops so that any two edges share at most one common vertex; for example, for the surface F04 , the Mercedes sign is such a fatgraph. By connectivity of G, any transposition of labels on G can thus be accomplished by a sequence of labeled flips, and since transpositions generate the symmetric group, any permutation of labels on G can likewise be realized. Since compositions of flips act transitively on trivalent fatgraph spines and thus carry any trivalent fatgraph to such a G, the claim follows. In light of the last sentence of Theorem 1.5, this presentation of M .F / is not optimally efficient, and the generators for permutations are included just to conveniently describe the relations as above. Furthermore, it is easy to see that various relations imply others, and we hope that what this presentation lacks in efficiency is compensated by simplicity. This particular presentation of the mapping class groupoid more or less starts with our proof [128] as interpreted (extended and nearly corrected) by J. Teschner [157]. Remark 1.6. Unpublished work [55] of K. Igusa would apparently give an alternative proof of the Madsen–Weiss–Tillman Theorem from the conjectured stable acyclicity of the quotient by the mapping class group of the cubical complex with the same oneskeleton as the complex L in the proof of Theorem 1.5 and with higher dimensional cells added only for commutativity and naturality but not for the pentagon relation. Indeed, the homology of this modified complex allegedly gives precisely the unstable homology of open moduli space. Since MC.F / acts with trivial isotropy on the connected and simply connected complex L constructed in the proof of the previous theorem and M .F / is the fundamental path groupoid of L=MC.F /, we have: Corollary 1.7. For any surface F D Fgs , with s  1, MC.F / is the stabilizer of any object in M .F /. This gives the combinatorial presentation for mapping class groups of punctured surfaces which was promised in Section 1 of Chapter 1. We now turn our attention to the parallel discussion for bordered surfaces. Definition 1.8. The mapping class groupoid M .F / of a bordered surface F D s z Fg; ıE is the fundamental path groupoid of the quotient manifold T .F /=MC.F /, i.e., the fundamental path groupoid of the simplicial complex dual to the ideal simplicial decomposition of .Tz .F /=R>0 /=MC.F /. in Chapter 4. Again, MC.F / is isomorphic to the stabilizer of any object in M .F /, and we have the following result (with no need for the further labeling required in the punctured case):

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181

s Corollary 1.9. The mapping class groupoid M .F / of the bordered surface F D Fg; ıE is generated by flips and quasi flips, and a complete set of relations for M .F / is given by involutivity of flips and quasi flips, commutativity of flips and quasi flips along pairs of arcs that do not lie in the frontier of a common triangle, the pentagon relation for flips along pairs of arcs that do lie in the frontier of a common triangle, and the hexagon relation between flips and quasi flips illustrated in Figure 1.3.

Figure 1.3. The hexagon relation of flips and quasi flips.

The hexagon relation arises from the link of a codimension-two cell corresponding to a punctured fatgraph where all vertices are either uni- or trivalent except for one trivalent punctured vertex.

2 Integration over moduli spaces The next consequence of Theorem 4.1 of Chapter 4 solved what at the time was a outstanding problem: describe a fundamental domain for the action of the mapping class group on the Teichmüller space sufficiently explicitly so as to be able to effectively integrate over moduli space. The idea is simple. Cells corresponding to the finite collection of (unmarked and unlabeled) trivalent fatgraph isomorphism classes of a fixed topological type F D Fgs contain a fundamental domain for the action of MC.F /

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5 Mapping class groupoids and moduli spaces

on Tz .F /. Let us choose an appropriate section of the forgetful map  W Tz .F / ! M.F / to pull-back the given differential form on M.F / and integrate along the image of the chosen section in the cell corresponding to each such fatgraph G weighting the contribution to the integral by the reciprocal of the order of the corresponding fatgraph automorphism group Aut.G/. Here is the formal statement of the integration scheme: Corollary 2.1. Suppose that F D Fgs with s  1 and Fgs ¤ F11 . Let  W T .F / ! Tz .F / be any MC.F /-equivariant section which is transverse to each face in the ideal cell decomposition of Tz .F / so that for each trivalent fatgraph G, the projection C.G/\ Image ./ ! M.F / is # Aut.G/-to-one over its image except over a locus in M(F) of zero measure. Let  W Tz .F / ! M.F / be the natural forgetful mapping to moduli space. If ! is any top-dimensional differential form on M.F /, then Z Z X 1 !D   !; # Aut.G/ M.F / C.G/\ Image./ where the sum is over the finite set of all isomorphism classes of unmarked and unlabeled trivalent fatgraphs G with F .G/ D F . In the special case of F D F11 , there is an additional factor 2 on the right-hand side owing to the hyperelliptic involution. Proof. By the change of variables formula for integration (i.e., if f W X ! R Y is any diffeomorphism and

is a top-dimensional differential form on Y , then Y D R   f

), the integral of ! over M.F / agrees with the integral of  ! over a region X  in the image of  which forms a fundamental domain for the action of MC.F /. Since  meets transversely each open face in the cell decomposition that it intersects by hypothesis, the integral over  coincides with the integral over the intersection of  with the top-dimensional cells of the cell decomposition. By MC.F /-equivariance, the integral over  \ C.G/ agrees with Aut.G/ times the integral over C.G/, and the result follows. To actually implement this integration scheme, we thus require appropriate sections of Tz .F / ! T .F /. Definition 2.2. Consider a surface F D Fgs with s  1. Let `i W Tz .F / ! R>0 denote the perimeter functions, and choose a collection of positive constants ci > 0, for i D 1; : : : ; s. There is a corresponding section Tz .F / ! T .F / determined by the equalities `i D ci , for each i D 1; : : : ; s, called the perimeter section corresponding to the tuple c1 ; : : : ; cs . If c1 D    D cs , then the corresponding perimeter section is called a constant perimeter section

2 Integration over moduli spaces

183

Remark 2.3. The reader may wonder how to effectively handle the enormous sum over fatgraph isomorphism classes on the right-hand side of the equation in Corollary 2.1, and the remarkable fact described in Theorem 4.19 is that there are tools from high energy physics called “matrix models” which are well-suited to this enumeration. This was the approach to calculating the virtual Euler characteristic in [130]. This was also the approach used in Maxim Kontsevich’s famous solution [92] of Ed Witten’s Conjecture relating intersection theory on moduli space with the KdV hierarchy. It is worth noting, though, that the matrix model corresponds to summing over all isomorphism classes of fatgraphs for a fixed surface F , and yet a constant perimeter section does not typically meet each top-dimensional cell in Tz .F /; for example, if there is a loop in a fatgraph G, then C .G/ cannot meet any constant perimeter section. On the other hand for once-punctured surfaces, a constant perimeter section does meet all top-dimensional cells. Lemma 2.4. Any constant perimeter section is MC.F /-equivariant. Furthermore for F D Fgs , any perimeter section is transverse to each face in the ideal cell decomposition of Tz .F /. Finally, for each trivalent fatgraph G, the projection C .G/ \ Image . / ! M.F / is # Aut.G/-to-one over its image except over a locus in M.F / of zero measure. Proof. Equivariance of a constant perimeter section is clear. For transversality, since the coordinate transformations on lambda lengths associated with changes of underlying fatgraph spines are diffeomorphisms (indeed, they are rational functions with no minus signs, cf. Theorem 2.10 of Chapter 2), we may choose any convenient fatgraph to prove transversality. Let G be a trivalent fatgraph spine of F , where G has s  1 loops (i.e., cycles of length one), say f`i gs1 1 , where `i has endpoint vi for i D 1; : : : ; s  1. There is a distinct puncture associated with each `i , and we let i denote the hyperbolic length of the corresponding horocycle, for i D 1; : : : ; s  1, also letting s denote the length of the remaining horocycle in the decoration. Let tOi denote the unique half-edge incident on v which is not contained in `i , and let ti denote the edge of G containing tOi , for i D 1; : : : ; s. Let E denote the collection of edges of G, and suppose that #E D N . According to Theorem 2.6 of Chapter 4, Tz .F / is identified with the real quadric determined by the coupling equations, and we let f˛i g2N 1 denote the various h-lengths. We may re-order the subscripts to arrange that ˛i is the h-length of the sector at vi P that is opposite tOi , so that i D 2˛i , for i D 1; : : : ; s  1, and s D 2 2N s ˛i by Lemma 2.1 of Chapter 4. As already noted, the face condition associated to an edge e of G is determined by positivity of the simplicial coordinate Xe , which is linear in h-lengths, and furthermore, membership in a face of codimension p  0 in the celldecomposition of Tz .F / corresponds to the vanishing of p distinct functions among fXe W e 2 Eg.

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5 Mapping class groupoids and moduli spaces

We prove transversality by showing that the gradients fri W i D 1; : : : ; sg [ frXe W e 2 X ¨ Eg are linearly independent in R2N . In fact, the only functions among fXe W e 2 Eg with non-trivial dependence on ˛i are X`i and X ti , for i D 1; : : : ; s  1, so we may simply project out the subspace spanned by fri W i D 1; : : : ; s  1g. Now consider a vertex v 62 fvi gs1 O fO, g, O respectively 1 , say with incident half edges e, contained in the edges e; f; g 2 E. Only rXe , rXf , rXg among fre W e 2 Eg have a non-zero projection into the subspace † spanned by the h-lengths of the sectors incident on v and opposite e, O fO, g, O respectively, taken in this order as a basis for †. In fact, these projections are respectively given by .1; 1; 1/, .1; 1; 1/, and .1; 1; 1/, and 12 rs has projection .1; 1; 1/. Thus, if the projection into † of arXe C brXf C crXg C d rs vanishes, then we must have 0 D b C c C d  a D a C c C d  b D a C b C d  c; so a D b D c D d . However, a face in the decomposition of Tz .F / is determined by a proper collection of functions among fXe W e 2 Eg, indeed, the collection must satisfy the no vanishing cycle condition, and the transversality follows. z 2 C .G/ \ Image . /, say with corresponding For the final claim, suppose that  lambda lengths  W E ! R>0 . Each 2 Aut.G/ induces a permutation  on E, and the action of MC.F / on C .G/ \ Image . / corresponds to  ı  by naturality of lambda lengths. Thus, the projection is indeed # Aut.G/-to-one unless there is a nontrivial 2 Aut.G/ so that  ı  D . If there is an oriented edge e 2 E which is not fixed by  , then this imposes the restriction that .e/ D  ı  .e/, so  lies in a hyperplane, z lies in a locus of measure zero in M.F / as was claimed. and ./ If every oriented edge of G is fixed by  , where 2 Aut.G/ is non-trivial, then an elementary argument, which is left to the reader, shows that is the hyperelliptic involution on F D F11 . In this case, simply repeat the argument above on the quotient by the hyperelliptic involution of the decorated bundle.

3 Weil–Petersson volumes Using the expression in lambda lengths for the Weil–Petersson Kähler two form in Theorem 3.1, we shall first illustrate Corollary 2.1 by calculating the WP volume of M.F11 /. Compare also Wolpert’s calculation in [168]. Adopt the usual notation that a fundamental domain is a quadrilateral with opposite sides a and b and diagonal c, occurring in this clockwise order and identify an edge with its lambda length for convenience. The WP volume form, which agrees with the two form in this case, is

3 Weil–Petersson volumes

185

given by ! D 4.d log a ^ d log b C d log b ^ d log c C d log c ^ d log a/ D

2.C dA ^ dB C B dC ^ dA C A dB ^ dC / ; .B C C /.A C C /.A C B/

where we have used the solution to the arithmetic problem in Example 2.7 of Chapter 4 to express the volume form in terms of the simplicial coordinates, which are again given by capital Roman letters with corresponding lower-case lambda lengths. The horocyle of unit length has A C B C C D 1=2, so the volume form is expressed by !D

dA ^ dB : .A C B/. 12  A/. 12  B/

Finally, # Aut.G/ D 6 and the domain of integration in Corollary 2.1 is given by ˚ D D .A; B/ 2 R2 W A; B > 0 and A C B  12 : Thus, the Weil–Petersson volume of the moduli space of F11 is Z Z 1 dA ^ dB 1 2 !D 6 D 3 D .A C B/. 12  A/. 12  B/   Z 1=2A Z 1=2 @ ACB 1 dA dB log 1 D 3 0 @B B 0 2 Z 1 dx log x 2 4 D D 3 0 1  x2 6 completing our detailed discussion of the classical case F11 . Definition 3.1. For a general surface Fgs , whose Teichmüller space has complex dimension 3g C s  3, the Weil–Petersson volume form is sg D

! 3gCs3 ; .3g C s  3/Š

where ! is the WP two form on Fgs In [131], we have employed Corollary 2.1 together with an explicit solution of the arithmetic problem to compute with some difficulty that the WP volume, i.e., the integral of 12 over M.F12 / is  4 =8. Since in general one cannot solve the arithmetic problem in closed form, it seems hopeless to compute higher WP volumes this way, but who knows? In any case, there are now the innovations [114] of M. Mirzakhani for effective calculations in the bordered case. See also [83], [109]. We next calculate the WP volume form in higher genera by deriving it for a special class of fatgraphs and then proving invariance under Ptolemy transformations.

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5 Mapping class groupoids and moduli spaces

1 Definition 3.2. For any fixed n  1, define a collection GnC1 of trivalent fatgraphs by pairwise identifying univalent vertices of the uni-trivalent fatgraph illustrated in Figure 3.1 subject to the following restrictions. Connect enC1 to some ei01 among 0 gfei01 g. Confe10 ; e20 ; : : : ; en0 g, and then connect e1 to some ei02 among fe10 ; e20 ; : : : enC1 0 0 0 0 tinue in this way inductively connecting ej to some ej C1 among fe1 ; e2 ; : : : ; enC1 g 1 fei01 ; ei02 ; : : : ; ei0j g, for j D 1; 2; : : : ; n. Let GnC1 denote the family of fatgraphs so constructed for n  1. Furthermore, relaxing the restrictions entirely, let G .n/ denote the collection of all fatgraphs arising from any pairing at all of univalent vertices except 0 that we still require that enC1 is identified with anything other than enC1 . 1 It is not hard to see that each G 2 GnC1 has its corresponding surface F .G/ oncepunctured of genus n C 1. More generally, for every g; s with 2g  2 C s > 0 and s / with its corresponding surface F .G/ of type Fgs . odd, there is some G 2 G . 2gCs3 2 Furthermore, every G 2 G .n/, for any n, has a unique cycle of length two comprised of a0 ; b0 , and hence # Aut.G/  2.

Lemma 3.3. Suppose that G 2 G .n/, set N D 6n  3, and let x1 ; x2 ; : : : ; xn denote the respective lambda lengths of the edges a0 ; b0 ,a1 , b1 , c1 , d1 ,: : : , an ; bn ; cn ; dn , f1 ; : : : ; fn , e1 ; : : : ; enC1 in the notation of Figure 3.1. Identifying an edge with its lambda length and setting xQ i D d log xi , for i D 1; : : : ; N , the WP volume form  on T .F .G// pulls back to  D 24nC2

N X

.1/i xQ 1 : : : xyQi : : : xQ N ;

iD1

where the hat denotes omission.

e1 b0 f1

e10 a0

e2 c1

b1

a1

d1 e 20

f2

en c2

b2

cn

bn f fn 3

a2 d2 e30

an

en0 C1

Figure 3.1. The class of graphs G .n/.

Proof. By Theorem 3.1 of Chapter 2, the WP two form pulls back to n  X !i ; ! D 2 !0 C j D1

dn

enC1

187

3 Weil–Petersson volumes

where !0 D !00 D eQ10 aQ 0 C 2aQ 0 bQ0 C bQ0 eQ10 C fQ1 aQ 0 C bQ0 fQ1 D ŒeQ 0 C fQ1  .aQ 0 C bQ0 / .aQ 0  bQ0 / 1

and !j D bQj aQj C aQj fQj C fQj bQj C aQj eQj0 C1 C eQj0 C1 dQj C dQj aQj C cQj fQj C1 C fQj C1 dQj C dQj cQj C bQj eQj C eQj cQj C cQj bQj D j C j C j ; with 1 0 C fQj  eQj C fQj C1  2dQj ŒdQj  aQj C bQj  cQj ; ŒeQ 2 j C1 1 j D ŒeQj0 C1  fQj C eQj C fQj C1  2dQj ŒdQj  aQj  bQj C cQj ; 2 j D ŒcQj  aQj ŒfQj C1  fQj C bQj  dQj ;

j D

for j D 1; : : : ; n, where one directly checks these identities of two forms. Furthermore, one computes directly that !j0 D j j j D .eQj0  fQj0 /.fQj C1  fQj /.2aQj bQj cQj dQj C where j

j /;

D .eQj0 C1 C fQj C1 /.bQj  cQj /.dQj  aQj /.cQj  aQj /:

Our overall goal is to compute  D ! 3nC1 =.3n C 1/Š, and since ! D !0 C

n X

. j C j C j /

j D1

is a sum of .3n C 1/ two forms, we are led to consider a monomial !0"0 1"1 "11 1"3 : : : n"3n in the .6n C 3/ variables fa0 ; : : : ; fn g. Notice that none of j , j , j contain any terms of degree greater than one in aQj , bQj , cQj , dQj , while none of i , i , i , for i ¤ j , have any dependence whatsoever upon these variables (nor does !0 ). Thus, a non-vanishing monomial of degree .6n C 2/ in the .6n C 3/ variables must have "i D 1, for all i , and we conclude that D

Y Y ! 3nC1 . j j j / D 23nC1 !j0 ; D 23nC1 !00 .3n C 1/1 n

n

j D1

j D0

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5 Mapping class groupoids and moduli spaces

Q where jnD1 j denotes the exterior product 1 ^    ^ n of forms f gjnD1 in this order. Substituting the expressions already computed for !j0 , we find that D2

3nC1

n Y 



.eQj0  fQj0 /.fQj C1  fQj /

!00

j D1

h

 2n

n Y

n X

.aQj bQj cQj dQj / C 2n1

j D1

3

i Q : : : aQj bQj cQj dQj : : : aQ n bQn cQn dQn ;

Q

Q 1 b1 cQ1 d1 ja

j D1

where the hat again denotes omission. Define n Y

D

.eQj0  fQj /.fQj C1  fQj /;

j D1

V1 D 2!o0 

n Y

.aQj bQj cQj dQj /;

j D1

V2 D !00 

n X

3

 j

 aQ 1 bQ1 cQ1 cQ1 : : : aQj bQj cQj dQj : : : aQ n bQn cQn dQn ;

j D1

so

24n  D V1 C V2 :

We shall separately compute the two expressions V1 ; V2 . First, let us compute n.n1/=2

.1/

D

n Y

.fQj  fQj C1 /

j D1

D

n hX

D

.eQk  fQk /

kD1

n i Y .1/j fQ1 : : : fQnj : : : fQn .fQn  eQn / .eQk  fQk /

j D0 n1 X

b

n Y

kD1

b

.1/j fQ1 : : : fQnj : : : fQn

j D0

h

 .1/

n

n Y kD1

DF

n hX

.1/ eQ1 j

i eQk C .1/n1 eQ1 : : : eQnj 1 fQnj eQnj C1 : : : eQn .fQn  eQnC1 / : : : eQ2 : : : eQ nC1j

nC1

j D0

where F D

Qn

i



n hX j D0

Q and E D QnC1 eQj . j D1

j D1 fj

b

i fQ1 : : : fQnj : : : fQn E;

189

3 Weil–Petersson volumes

One now readily computes V1 D 2

n Y

aQj bQj cQj dQj Œ2aQ 0 bQ0 C .bQ0  aQ 0 /.fQ1 C eQ10 / 

j D1

D4

n Y

aQj bQj cQj dQj ŒaQ 0 bQ0  C .1/n.n1/=2 .bQ0  aQ 0 /FE :

j D1

In order to compute V2 , we must introduce the three forms j D .bQj  cQj /.dQj  aQj /.cQj  aQj /; for j D 1; : : : ; n, and define the function  W f2; : : : ; n C 1g ! f1 : : : ; ng, where .`/ D j if and only if ej0 C1 D e` . Thus, we find

2

n n ° hX i X 0 .1/j eQ1 : : : eQnC1j : : : eQnC1 eQkC1 k .1/n.n1/=2 V2 D 2aQ 0 bQ0 F j D0

CF

kD1

n hX

2

.1/j eQ1 : : : eQnC1j : : : eQnC1 eQnC1 n

j D0



n2 hX

i

b

n1 i X ± fQkC1 k .1/nCj fQ1 : : : fQnj : : : fQn E

j D0

kC1

h D 2aQ 0 bQ0 FE

n1 X

.nC1j / C FEn C FE

j D0

D 4aQ 0 bQ0 FE

n X

n2 X

i nj 1

j D0

j :

j D1

To complete the proof, simply substitute these expressions for V1 , V2 into the expression for  and expand in terms of lambda lengths. Theorem 3.4. For any trivalent fatgraph G with corresponding lambda length coordinates x1 ; : : : ; xN , the WP volume form pulls back to  D ˙24gC2s4

N X

2

.1/i d log x1 ^    ^ d log xi ^    ^ d log xN

iD1

on Tz .F .G// . /, Proof. First notice that for any surface Fgs with s odd, there is some G 2 G . 2gCs3 2 so by the previous lemma, the putative formula indeed holds for at least one fatgraph

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5 Mapping class groupoids and moduli spaces

of type Fgs . Furthermore, we shall presently compute that it is likewise invariant under flips, and hence is invariant. Indeed in our usual notation for the flip along an edge e producing a new edge f , where e locally separates a, b from c, d with a opposite c and b opposite d , the Ptolemy equation gives fQ D

1 Œac.aQ C c/ Q C bd.bQ C dQ /  e; Q ac C bd

and we compute bQ cQ dQ fQ  aQ cQ dQ fQ C aQ bQ dQ fQ  aQ bQ cQ fQ C aQ bQ cQ dQ     bd bQ ac aQ Q Q Q  eQ  aQ cQ d  eQ D b cQ d ac C bd ac C bd     ac cQ bd dQ Q Q Q C aQ b d  eQ  aQ b cQ  eQ C aQ bQ cQ dQ ac C bd ac C bd ac C bd C ac C bd Q Q D aQ b cQ d C aQ bQ cQ dQ  bQ cQ dQ eQ C aQ cQ dQ eQ  aQ bQ dQ eQ C aQ bQ cQ eQ ac C bd D .bQ cQ dQ eQ  aQ cQ dQ eQ C aQ bQ dQ eQ  aQ bQ cQ eQ C aQ bQ cQ dQ / as required. For surfaces with an even number of boundary components, one can repeat the / by analysis of Lemma 3.3 for a particular fatgraph arising from G 2 G . 2gC.s1/3 2 adding two edges, one of which is a loop and likewise derive the general case. In order to give further insights into the details of the integration scheme, we assign ourselves the task of estimating the WP volume of a top-dimensional cell in Tz .Fg1 / asymptotically in g. To this end, fix any trivalent fatgraph G of type Fg1 and let further  W Tz .Fg1 / ! M.Fg1 / be the forgetful map as in Corollary 2.1. Suppose N D 6g  3 and let xi and Xi , respectively, denote the lambda lengths and simplicial coordinates of the edges of G in some order, for i D 1; : : : ; N . If  is a section of , then define the domain D.G/ D C.G/ \ Image . / ˚ D Xi > 0 for each i D 1; : : : ; N; and  D .X1 ; X : : : ; XN / D 12 ; P where  D 12 N iD1 Xi is the hyperbolic length of the horocycle, i.e., the sum of h-lengths of all sectors using Lemma 2.1 of Chapter 4. Notice that by Theorem 3.1 of Chapter 2, the pull-back   ! of the WP Kähler two form ! to decorated Teichmüller space is invariant under homothety of lambda lengths, hence so too is    invariant. Since simplicial coordinates are homogeneous functions of lambda lengths, a standard application of Stokes’ Theorem shows that Z Z  D 2 d ^   ; D.G/

D.G/

191

3 Weil–Petersson volumes

where D.G/ is the region D.G/ D fXi > 0 for each i D 1; : : : ; N; and 0 <

X

Xi < 1g:

i

On the other hand, by Proposition 2.11 of Chapter 4 and Theorem 3.4, we have d ^    D ˙24g2

N X

N X Xi xQ i ^ .1/j xQ 1 ^    ^ xyQi ^    ^ xQ N

iD1

D ˙24g2 

N Y

iD1

xQ i :

iD1

It follows that

Z

   D 24g1

Z  D.G/

D.G/

N Y

xQ i :

iD1

By Lemma 2.12 of Chapter 4, the Jacobian determinant of the transformation from simplicial coordinates to logarithms of lambda lengths is bounded below by .N=3/N , and a change of variables gives the estimate Z D.G/

 >

24g1 N N 3N

Z 1N D.G/

N Y iD1

dXi D

24g2 N N ; 3N .N /

and we have therefore derived the estimate: Theorem 3.5. A top-dimensional cell in the ideal simplicial decomposition of a once4g2 N N punctured surface Fg1 has WP volume at least 23N .N , where N D 6g  3. / Remark 3.6. In fact, we can combine with this estimate a computation of the asymptotic growth of the number of top-dimensional cells in the decomposition of Tz .Fg1 /, cf. the next section, in order to derive a lower bound on the WP volume of moduli space M.Fg1 / as in [131]. This result apparently implied the unsuitability of the perturbative approach and helped guide the early development of string theory [152]. Our further final encounter with WP geometry brings us finally to the cohomology of moduli spaces, as follows: Given the explicit ideal simplicial decomposition in Theorem 4.1 of Chapter 4, it is natural to try to describe cycles or cocyles on (decorated) moduli space in these terms. One essential difficulty in manipulating this ideal simplicial decomposition is that there are so darn many cells, cf. the next section. The cells of a given codimension are stratified by the valence tuple of the underlying fatgraph, and there is an extreme stratum which in codimension k has a corresponding i.c.d. with one complementary (k C 3)-gon and all the other complementary regions triangles. During a visit in 1991, E. Witten privately proposed that these might be cycles representing the MMM classes, cf. [117], [121], [113].

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5 Mapping class groupoids and moduli spaces

We proved in [133] that these cells in even codimension do admit orientations rendering them cycles, and using the integration scheme together with Stokes’ theorem on the fatgraph complex (cf. below), we proved that the first such cycle, i.e., the union of all fatgraphs with one five-valent vertex and all the rest trivalent, is the Poincaré dual of the Weil–Petersson Kähler two form. Subsequently, Kontsevich [94] showed that any valence tuple with only vertices of odd valence gives a cycle, and much later Igusa [68], [69] and independently Mondello [116] proved that the span of these combinatorial cycles agrees with the span of the MMM classes. We include as Appendix B our calculation [133] of the Poincaré dual of the WP Kähler form for several reasons. Parts of the computation (e.g., Stokes’ theorem on the fatgraph complex) are basic to decorated Teichmüller theory and not special to this calculation, but also, the appearance of the functional equation of the dilogarithm as a matter of course in the calculation is something we find profound. A rough sketch of Appendix B is as follows. Starting from our formula for the WP two form, we may naturally choose corresponding primitives, i.e., one-forms whose exterior derivatives agree with our expression, on each top-dimensional cell of G .F /. There is a natural analogue of Corollary 2.1 for integrating lower-degree forms across higher codimension cells, and differences of these primitives across codimension-one cells likewise lead to further primitive functions on codimension-two cells, which turn out to precisely agree with the classical Rogers’ function, a version of the dilogarithm, evaluated at the cross ratio of the corresponding quadrilateral. The codimension-two cells corresponding to commutativity relations drop out of the calculation, and the codimension-two cells corresponding to the pentagon relations turn out to have constant coefficients coming from the Abel–Spence functional equation on Rogers’ function. (Let us correct another mistake: in [133], it is not true that the WP two form agrees with the first MMM class, and in the paper, we compute the Poincaré dual of the former, not the latter.) We have long hoped that higher-codimension analogues of this computation might provide functional equations on higher polylogarithms, but a basic obstruction has been the absence of a lambda length or other global differential geometric expression for the higher MMM classes. On the contrary and at the heart of his solution [92] to Witten’s Conjecture, Maxim Kontsevich gave an explicit piecewise-linear two form for the Chern classes 2 s k 2 H .Fg ; Z/ of the fiberwise cotangent bundle of forgetting the puncture k W M.FgsC1 / ! M.Fgs /, for k D 1; : : : ; s C 1. Lemma 3.7. Suppose that G is a trivalent fatgraph spine of the surface F D Fgs whose edges have respective simplicial coordinates Xi , where i D 1; : : : ; N D 6g  6 C 3s. If the edge-path of the kth puncture is comprised of sequential edges with simplicial P coordinates X1 ; : : : ; X` , for `  3, and  D 12 `iD1 Xi denotes the perimeter of the kth horocycle, then the pull-back to Tz .Fgs / of the first Chern class of the fiberwise

4 Fatgraphs revisited

193

cotangent bundle of forgetting the kth puncture is   X  Xi  Xj D d ^ d ; k   where the sum is over all 1  i  j  `  1. Proof. There is truly nothing further to prove beyond Kontsevich’s explanation that this is the pull-back of the corresponding formula on the classifying space of the circle. That is, the simplicial coordinates have exactly the same required formal properties as the Strebel coordinates used in [92]. Remark 3.8. In the same way, all the further computations in [92] may equivalently be interpreted by replacing Strebel R mC1coordinates with simplicial coordinates. For example, the MMM classes m D are thus expressed in terms of simplicial coordinates.

4 Fatgraphs revisited Let us here begin anew with fatgraphs in order to systematically present the theory following [131], [145]. We shall treat the most general case of possibly non-orientable surfaces with general fatgraphs, though it is only the special case of orientable surfaces with so-called “untwisted fatgraphs” that is employed at several previous and future junctures. It is only in the applications to computational biology in Section 4 of Chapter 6 that the full generality of twisted fatgraphs is required. A compact and connected surface F is uniquely determined up to homeomorphism by the specification of whether it is orientable plus its genus g D g.F / and number r D r.F / of boundary components. Definition 4.1. The orientation double cover of F is the oriented surface Fz together with the continuous map p W Fz ! F so that for every point x 2 F there is a disk neighborhood U of x in F , where p 1 .U / consists of two components on each of which p restricts to a homeomorphism and where the further restrictions of p to the boundary circles of these two components give the two possible orientations of the boundary of U . Such a covering p W Fz ! F always exists, and its properties uniquely determine Fz up to homeomorphism and p up to natural equivalence. In particular, if F is connected and orientable, then Fz has two components with opposite orientations, each of which is identified with F by p. Furthermore provided F is connected, F is non-orientable if and only if Fz is connected, and a closed curve in F lifts to a closed curve in Fz if and only if a neighborhood of it in F is homeomorphic to an annulus.

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5 Mapping class groupoids and moduli spaces

4.1 Skinny surfaces associated to fatgraphs. Consider a finite one-dimensional CW-complex G, or in other words, a graph in the usual sense of the term comprised of vertices V D V .G/ and edges E D E.G/, which do not contain their endpoints and where an edge is not necessarily uniquely determined by its endpoints. Our standard notation will be v D v.G/ D #V and e D e.G/ D #E. To avoid cumbersome cases in what follows, we shall assume that no component of G consists of a single vertex or a single edge with distinct endpoints. Removing a single point from each edge produces a subspace of G, each component of which is called a half-edge. A half-edge which contains u 2 V in its closure is said to be incident on u, and the number of distinct half-edges incident on u is the valence of u. Definition 4.2. A fattening on G is the specification of a cyclic ordering on the halfedges incident on u for each u 2 V , and an X -coloring on G is a function E ! X , for any set X. A fatgraph G is a graph endowed with a fattening together with a coloring by a set with two elements, where we shall refer to the two colors on edges as “twisted” and “untwisted”. Example 4.3. A ladder fatgraph G is depicted in Figure 4.1, where on the left and right of G, there are the two standard neighborhoods shown on the far left and right in the figure, and in between, we choose one or the other of the two “building blocks” indicated in the middle of the figure. It is not hard to check that F .G/ has a single boundary component and genus n C 2, where we make n choices of building blocks

or

Figure 4.1. Ladder fatgraphs.

in between. By construction, G has exactly two cycles of length two, and these must be invariant under any automorphism  2 Aut.G/. As before,  ¤ 1 cannot fix any oriented edge of G, and furthermore,  ¤ 1 can fix neither of the cycles of length two since it respects the fattening. If  interchanges these two cycles, then the choices for the first and last building blocks must be the same. Imposing the constraint that the first and last building blocks are different gives 2n1 fatgraphs of genus g D n C 2 with one boundary component, and each of these fatgraphs is rigid in the sense that it has no automorphisms. In the same way, one sees that each of these fatgraphs is combinatorially distinct, so the number of (rigid) untwisted trivalent fatgraphs G with F .G/ D Fg1 is at least 2g3 . Now finally putting twists on various edges in the building blocks likewise gives examples in the general case.

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4 Fatgraphs revisited

A fatgraph G uniquely determines a surface F .G/ with boundary as follows. Construction 4.4. For each vertex u 2 V .G/ of valence k  2, let Pu be an oriented surface diffeomorphic to a polygon of 2k sides containing in its interior a single vertex of valence k; the half-edges incident on this vertex are also incident on univalent vertices contained in alternating sides of Pu and identified with the half-edges of G incident on u so that the induced counter-clockwise cyclic ordering on the boundary of Pu agrees with the fattening of G about u. For a vertex u of valence k D 1, the corresponding surface Pu contains u in its boundary. See Figure 4.2. The surface F .G/ is the quotient of tu2V Pu where the frontier edges, which are oriented with the polygons on their left, are identified by a homeomorphism if the corresponding half-edges lie in a common edge of G, and this identification of oriented segments is orientation-preserving if and only if the edge is twisted. The graphs in the polygons Pu , for u 2 V , combine to give a fatgraph embedded in F .G/ with its univalent vertices in the boundary, which is identified with G in the natural way so that we regard G  F .G/.

u Pu

u

u

Pu Pu Figure 4.2. The polygon Pu associated with a vertex u.

It is often convenient to regard a fatgraph as a planar projection of a graph embedded in three-space, where the cyclic ordering is given near each vertex by the counterclockwise ordering in the plane of projection and edges can be drawn with arbitrary under/over crossings. Untwisted edges are depicted as ordinary edges and twisted edges with an icon , which we more generally take as defined modulo two, i.e., an even number of icons  represents an untwisted edge and an odd number represents a twisted edge. Examples of fatgraphs and their corresponding surfaces are depicted in Figure 4.3, where the bold lines indicate the planar projection of the fatgraph and the dotted lines indicate the gluing along edges of polygon. The graph G is a strong deformation retract of F .G/, so the Euler characteristic is .F .G// D .G/ D v.G/  e.G/, and the boundary components of F .G/ are composed of the frontier edges of tu2V Pu which do not correspond to half-edges of G. Let r.G/ denote the number of boundary components of F .G/. Proposition 4.5. Suppose that G is a fatgraph and X; Y  E.G/ are disjoint collections of edges. Change the color, twisted or untwisted, of the edges in X and delete from

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5 Mapping class groupoids and moduli spaces

5

9

9

5

5

7

4

4 6

9

7

7 8

4 8

6

3

3

3

2

2 1

1

G1

8

6

2 1

G2

G3

Figure 4.3. The surface associated to a fatgraph.

G the edges in Y to produce another fatgraph G 0 , whose cyclic orderings on half-edges are induced from those on G in the natural way. Then jr.G/  r.G 0 /j  #X C #Y . Proof. According to the triangle inequality, it suffices to consider the case that X [Y D ff g, and we set r D r.G/. If f 2 E.G/ is incident on a univalent vertex, then changing the color or deleting f alters r, so we may assume without loss that this is not the case. Consider a properly embedded arc a meeting f in a single transverse intersection but otherwise disjoint from G. Instead of changing the color on f to produce G 0 , let us cut F .G/ along a and re-glue along the resulting copies of a reversing orientation in order to produce a surface homeomorphic to F .G 0 /. If the endpoints of the arc a occur in the same boundary component of F .G/, then the change of color on f must either leave r invariant or increase it by exactly one, while if they occur in different boundary components, then changing the color on f decreases r by exactly one. In the remaining case, rather than removing the edge f , let us consider instead cutting F .G/ along a to produce a surface homeomorphic to F .G 0 /. If the endpoints of a occur in the same boundary component of F .G/, then the cut along a either leaves r invariant or increases it by exactly one, while if they occur in different boundary components, then cutting on a decreases r by exactly one. Definition 4.6. A fatgraph G is untwisted if all of its edges are untwisted, and this is evidently a sufficient but not a necessary condition for F .G/ to be orientable. Definition 4.7. Two fatgraphs G1 and G2 are strongly equivalent if there is an isomorphism of their respective underlying graphs G1 and G2 that respects the cyclic orderings and preserves the coloring. G1 and G2 are equivalent if there is a homeomorphism F .G1 / ! F .G2 / mapping G1  F .G1 / to G2  F .G2 /. Strong equivalence evidently implies equivalence, and equivalence implies that the associated surfaces are homeomorphic. However, neither converse holds in general.

4 Fatgraphs revisited

197

Definition 4.8. The vertex flip of G at a vertex u reverses the cyclic ordering on the half-edges incident on u and adds another icon  to each half-edge incident on u. In particular, a vertex flip on a univalent vertex simply adds an icon  to the edge incident upon it. Proposition 4.9. Two untwisted fatgraphs are strongly equivalent if and only if they are equivalent. Two trivalent fatgraphs G1 and G2 are equivalent if and only if there is another fatgraph G so that G and G2 are strongly equivalent, and G arises from G1 by a finite sequence of vertex flips. In particular, G and G1 are equivalent if G arises from G1 by a vertex flip. Proof. If G1 and G2 are both untwisted, then a homeomorphism from F .G1 / to F .G2 / mapping G1 to G2 restricts to a strong equivalence. The converse follows by construction in any case thus proving the first assertion. The third assertion therefore follows since the flip on a vertex u of G1 corresponds to reversing the orientation of the polygon Pu in the construction of F .G/, i.e., remove the neighborhood of u from the plane of projection in our graphical depiction, turn it upside down in three-space, and finally replace it in the plane of projection and twist one further time each incident half-edge of G; this clearly extends to a homeomorphism of F .G1 / to F .G/ mapping G1 to G that does not preserve coloring. Insofar as strong equivalence implies equivalence by construction and equivalence of fatgraphs is a transitive relation, if there is such a fatgraph G as in the statement of the result, then G1 and G2 are indeed equivalent. Conversely, we assume that G1 and G2 are connected. Let G be a trivalent fatgraph with v vertices and e edges, and choose a maximal tree T in it. There are 1  .G/ D 1  v C e edges in G  T since we may collapse T to a point without affecting v  e. This is therefore the Euler characteristic of the collapsed graph with a single vertex and one edge for each edge of G  T . We claim there is a finite composition of flips on G that results in a fatgraph with any specified twisting on the edges in T . Indeed, consider the collection of all functions from the set of edges of G to Z=2, a set with cardinality 2e . Vertex flips act naturally on this set of functions, and there are clearly 2v possible compositions of such vertex flips. The simultaneous flip of all vertices acts trivially on this set of functions corresponding to reversing the cyclic orderings at all vertices, so only 2v1 such compositions could act non-trivially. Since 2e =2v1 D 21vCe and there are 1  v C e edges of G  T by the previous paragraph, the claim follows. Suppose finally that G1 and G2 are equivalent and that  W F .G1 / ! F .G2 / is a homeomorphism of surfaces restricting to a homeomorphism of G1 to G2 . Perform a vertex flip on G1 and identify edges before and after in the natural way to produce a fatgraph which is again equivalent to G2 and in which T remains a maximal tree by a homeomorphism still denoted by , mapping T to .T /  G2 . According to the previous paragraph, there is a composition of vertex flips on G1 producing a fatgraph G where an edge of the maximal tree T  G is twisted if and only if so too is its image under .

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5 Mapping class groupoids and moduli spaces

Adding an edge of G  T to T results in a unique cycle of G, a neighborhood of which in F .G/ is either an annulus or a Möbius band and likewise for edges of G2  .T /. Since  restricts to a homeomorphism of the corresponding Möbius bands or annuli in F .G/ and F .G2 /, an edge of G  T is twisted if and only if its image under  is twisted. Thus, G and G2 are indeed strongly equivalent. Lemma 4.10. The fatgraphs depicted in Figures 4.4a and 4.4b are strongly equivalent, and the fatgraphs depicted in Figures 4.4c, 4.4d, and 4.4e are pairwise equivalent.

w u c) a)

d)

b)

e)

Figure 4.4. Elementary equivalences of fatgraphs.

Proof. The strong equivalence of 4.4a and 4.4b is seen directly. Performing vertex flips on the vertices labeled u; w in 4.4c and erasing pairs of icons  on common edges produces 4.4d, which is therefore strongly equivalent to 4.4e by the first assertion. 4.2 Fatgraphs and permutations. Adopt the standard notation for a permutation on a set S writing .s1 ; : : : ; sk / for the cyclic permutation s1 7! s2 7!    7! sk 7! s1 on distinct elements s1 ; : : : ; sk 2 S , which is called a transposition if k D 2. We compose permutations ;  on S from right to left, so  ı  .s/ D  . .s//. An involution is a permutation  where  ı D 1S , with 1S denoting the identity map on S . Two permutations are disjoint if their supports are, so disjoint permutations evidently commute. Definition 4.11. A stub of a fatgraph G is a half-edge which is not incident on a univalent vertex of G.

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4 Fatgraphs revisited

There are exactly two non-empty connected fatgraphs with no stubs, namely, the two we have proscribed consisting of a single vertex with no incident half-edges and a single edge with distinct endpoints. A fatgraph G determines a triple . .G/; u .G/;  t .G// of permutations on its set S D S.G/ of stubs as follows. Construction 4.12. Suppose u is vertex of G of valence k  2 with incident stubs s1 ; : : : ; sk.u/ in some linear order compatible with the cyclic order given by the fattening on G, and consider the cyclic permutation .s1 ; : : : ; sk.u/ /. The cyclic permutations corresponding to distinct vertices of G are disjoint by construction, so the composition  .G/ D

Y

.s1 ; : : : ; sk.u/ /

fvertices u2V W u has valence2g

is well defined and likewise for the compositions of transpositions u .G/ D

Y

.h; h0 /;

fpairs of distinct stubs h;h0 contained in some untwisted edge of Gg

 t .G/ D

Y

.h; h0 /:

fpairs of distinct stubs h;h0 contained in some twisted edge of Gg

In particular,  .G/ has no fixed points since we have taken the product over vertices of valence at least 2, and u .G/ and  t .G/ are disjoint involutions whose fixed points are the stubs associated to the univalent vertices of G. Enumerating the stubs of the fatgraphs G1 , G2 , G3 as illustrated in Figure 4.3, we have for example:  .G1 / D  .G2 / D  .G3 / D .1; 2; 3/.4; 5; 6/.7; 8; 9/; u .G1 / D .2; 8/.3; 6/.4; 7/.5; 9/;  t .G1 / D 1S ; u .G2 / D .2; 8/.3; 6/.4; 9/.5; 7/;  t .G2 / D 1S ; u .G3 / D .2; 8/.3; 6/.5; 9/;  t .G3 / D .4; 7/: Remark 4.13. In another treatment of fatgraphs as triples of permutations on the set of all half-edges instead of stubs, the univalent vertices correspond to fixed points of the analogue of  ; furthermore, there is a transposition in the analogue of u ı  t corresponding to each half-edge. The formulation we have given here treating univalent vertices as “endpoints of half-edges instead of endpoints of edges” does not require these additional transpositions. Definition 4.14. Define a labeling on a fatgraph G with N stubs to be a linear ordering on its stubs, i.e., a bijection from the set of stubs of G to the set f1; 2; : : : ; N g.

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5 Mapping class groupoids and moduli spaces

Proposition 4.15. Fix a natural number N  2. The map G 7! . .G/; u .G/;  t .G// in Construction 4.12 induces a bijection between the collection of strong equivalence classes of fatgraphs with N stubs and the collection of conjugacy classes of triples .; u ;  t / of permutations on N letters, with  fixed point free and u and  t disjoint involutions. Proof. The assignment G 7! . .G/; u .G/;  t .G// gives a natural mapping from the collection of labeled fatgraphs with N stubs to the collection of triples of permutations on f1; 2; : : : ; N g. This induced mapping has an obvious two-sided inverse, where the labeled fatgraph is constructed directly from the triple of permutations (here using our convention that no component of G is a single vertex or a single edge with distinct univalent endpoints). A strong equivalence of labeled fatgraphs induces a bijection of f1; 2; : : : ; N g conjugating corresponding triples of permutations to one another and conversely, so the result follows. Construction 4.16. Let G be a fatgraph with triple .; u ;  t / of permutations on its set S of stubs determined by Construction 4.12. Construct a new set Sx D fNs W s 2 S g and a new permutation N on Sx with one k-cycle .Nsk ; : : : ; sN1 / in N for each k-cycle .s1 ; : : : ; sk / in  . Construct a new permutation Nu on SN from u , where there is one transposition .Ns1 ; sN2 / in Nu for each transposition .s1 ; s2 / in u ; construct yet another new permutation Nt on S t Sx from  t , where there are two transpositions .Ns1 ; s2 / and .s1 ; sN2 / in Qt for each transposition .s1 ; s2 / in  t . Define permutations on S t Sx by  0 D  ı N ;  0 D u ı Nu ı Nt I the order of composition on the right-hand side is immaterial since the permutations have disjoint supports. Proposition 4.17. Let Construction 4.12 assign the triple .; u ;  t / of permutations to the fatgraph G with set S of stubs, let  0 ;  0 be determined from them according to Construction 4.16, and consider the untwisted fatgraph G 0 determined by Construction 4.12 from the triple . 0 ;  0 ; 1StSx /. Then the surface F .G 0 / is the orientation double cover of F .G/, and the covering transformation is given by s $ sN . In particular, F .G 0 / is connected if and only if F .G/ is non-orientable provided F .G/ is connected. Moreover, there is a bijection between the boundary components of F .G 0 / and the set of orientations on the boundary components of F .G/, i.e., F .G 0 / has twice as many boundary components as F .G/. Proof. By construction, F .G 0 / has the required properties of the orientation double cover, so the first two claims follow from general principles. The final assertion follows as well since every boundary component of F .G/ has an annular neighborhood in F .G/.

4 Fatgraphs revisited

201

Proposition 4.18. Adopt the hypotheses and notation of Proposition 4.17 and let 0 D  0 ı  0. i) The orientations on the boundary components of F .G/ are in bijection with the cycles of 0 . Indeed, suppose s11 s12 s21 s22 : : : sn1 sn2 is the ordered sequence of stubs traversed by an oriented edge-path in G representing a boundary component of F .G/ with some orientation, where sj1 ; sj2 are contained in a common edge of G with perhaps sj1 D sj2 if they are contained in an edge incident on a univalent vertex, for j D 1; 2; : : : ; n. Erasing bars on elements from the corresponding cycle of 0 produces the sequence .s12 ; s22 ; : : : ; sn2 / of stubs of G sequentially traversed by the corresponding oriented boundary component of F .G/, called a reduced cycle of 0 . ii) There is the following algorithm to determine if G is connected in terms of the associated triple .; u ;  t / of permutations. For any linear ordering on S , let X be the subset of S in the reduced cycle of 0 containing the least stub. . / If X D S, then G is connected terminating the algorithm. If X ¤ S , then consider the existence of a least stub s 2 X  S with .s/ 2 X . If there is no such stub s, then G is not connected terminating the algorithm. If there is such a stub s, then update X by adding the subset of S in the reduced cycle of 0 which contains s. Go to . /. Proof. We shall first treat the case that  t D 1StSx , i.e., G is untwisted, setting  D u . Consider a stub s of G and the effect of  ı  on it. By construction, the stub s is contained in an edge incident on a univalent vertex if and only if s is a fixed point of . Furthermore, .s/ D  ..s// is in this case the stub following s in the cyclic ordering at the non-univalent endpoint. If s is not a fixed point of , then the stubs s and  .s/ must be half-edges contained in a common edge of G, and s;  .s/; .s/ D  . .s// is similarly a consecutive triple of stubs in an edge-path of G corresponding to a boundary component of F .G/ oriented with F .G/ on its left. Thus, a cycle of  ı  is comprised exactly of every other stub traversed by an edge-path in G corresponding to a boundary component of F .G/ oriented, proving the first part. Furthermore, the collection of stubs in X must lie in a single component of G in light of the previous remarks, so if at some stage of the algorithm X D S, then G is connected. If at some stage of the algorithm there is no stub s with  .s/ 2 X , then X is comprised of all the stubs in some component of G, so X ¤ S in this case implies that G has at least two components. Turning finally to the general case, F .G 0 / is the orientation double cover of F .G/, and the projection map on stubs just erases the bars by Proposition 4.17. The proof in this case is thus entirely analogous. To illustrate these constructions and results for the fatgraphs depicted in Figure 4.3, we find  .G1 / ı u .G1 / D .5; 7/.3; 4; 8/.1; 2; 9; 6/;  .G2 / ı u .G2 / D .1; 2; 9; 5; 8; 3; 4; 7; 6/:

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5 Mapping class groupoids and moduli spaces

It follows that r.G1 / D 3 and r.G2 / D 1, and since .G1 / D .G2 / D 1, the genera are g.G1 / D 0 and g.G2 / D 1. For G3 , according to Construction 4.16 and Proposition 4.17, the permutations for the orientation double cover are N 2; N 1/. N 6; N 5; N 4/. N 9; N 8; N 7/; N  0 D .1; 2; 3/.4; 5; 6/.7; 8; 9/.3; N 8/. N 3; N 6/. N 5; N 9/.4; N N 4; N 7/:  0 D .2; 8/.3; 6/.5; 9/.2; 7/. The untwisted fatgraph G30 corresponding to . 0 ;  0 ; 1S.G3 /tS.G x 3 / / is depicted in Figure 4.5, and it is connected in keeping with the fact that F .G3 / is non-orientable. The cycles of 0 D  0 ı  0 are N 3; N 5; N 8/; N .1; 2; 9; 6/; .1;

N 7; N 5; 7; 6/; N .3; 4; 9; N 4; N 8/ and .2;

corresponding to the oriented boundary cycles of G30 . The reduced cycles of 0 are therefore .1; 2; 9; 6/; .1; 3; 5; 8/;

and

.2; 7; 5; 7; 6/; .3; 4; 9; 4; 8/;

each pair corresponding to the two orientations of a single boundary component of F .G3 /. Thus r.G3 / D 2 and thus g.G3 / D 1 since again .G3 / D 1. 6

6x 4x

5x

1x

3x

9x

8x

3

1

2x 7x

5

4

2

G30

8

7 9

Figure 4.5. Example of the orientation double cover.

4.3 Matrix models. This section discusses results from [131]. We first explain an absolutely fundamental and purely combinatorial connection between fatgraph combinatorics and Gaussian integrals over Lie groups, or more physically, a field theory where the field variables are matrix-valued. Theorem 4.19. Consider the set of all unmarked isomorphism classes ŒG of (possibly disconnected) fatgraphs G with vk ŒG vertices of valence k, for k  3, and sŒG boundary components. Then for each finite tuple vk  0, with k  3, we have the

203

4 Fatgraphs revisited

following equality of formal power series in the variable N : X ŒG with vk .G/Dvk

D

N

1 Q

N sŒG # AutŒG Z

k3

vk Š

Y tr.H k / vk

HN

where tr denotes the trace, N D 2N=2  N Hermitian matrices, and dM.H / D

N Y

k

k3

dHi i

2 =2

2 =2

dM.H /;

, H N denotes the group of all N -by-N

Y

iD1

e tr H

.d ReHij /.d ImHij /

i<j

is the natural unitary-invariant measure. Proof. Adopt the convention that all indices k run from 1 to a maximum valence K and all indices i , possibly with super- and subscripts, from 1 to N , and compute that the integral in Theorem 4.19 is given by vk Z Y 1 X 1 2 Q dM.H /e tr H =2 Hi1 j2 Hi2 j3 : : : Hik i1

N k vk Š H N k i1 ;:::;ik

k

1 Q D

n k .vk Šk vk / 

Z

dM.H /e tr H

2 =2

HN

Y

X

vk Y

k

f.i / W  D 1; : : : ; vk

j D1

Hi j i j Hi j i j : : : Hi j i j 1 2

2 3

k 1

and D 1; : : : ; kg

1 Q D

n k .vk Šk vk / 

Z

dM.H /e tr H

2 =2

HN

vk XY ./ j D1

Hi j

j 1k i2k

Hi j

j 2k i3k

: : : Hi j

j kk i1k

;

where . / denotes the set  . / D f D fi k g W k;  D 1; : : : ; K and D 1; : : : ; vk g

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5 Mapping class groupoids and moduli spaces

 with each i k D 1; : : : ; N . The first expression follows from the definition of trace, the second from taking vk th powers, and the third from pulling the sum outside the product. j j j A collection of indices of the form  D .i1k i2k ; : : : ; ikk / occurring in a term of . / is called a “cycle” of length k, so each entry of . / consists of vk cycles of length k. This cycle contributes a factor

Hi j

j 1k i2k

Hi j

j 2k i3k

Hi j

j kk i1k

to the integrand, and we may associate the set of pairs j j j j j j S. / D f.i1k ; i2k /; .i2k ; i3k /; : : : ; .ikk ; i1k /g

of consecutive indicesR in .  tr H 2 =2 denote the Gaussian moment of , so Let h i D 1 N H N dM.H /  e h1i D 1 explains the constant N in Theorem 4.19. The following fundamental result devolves to explicit computation (cf. [17]): Wick’s Lemma. The Gaussian moment hHi1 j1 Hi2 j2 : : : Hin jn i vanishes for n odd and equals Y X hHi˛ j˛ Hiˇ jˇ i partition elements partitions into n=2 f˛;ˇg doubletons of f1;2;:::;ng

for n even, and in fact hHab Hcd i D ıad ıbc , where ı denotes the Kronecker delta function. The integral in Theorem 4.19 is therefore given by Q

X 1 v k k .vk Šk /

2./

X

Y

partitions of[ 2 S./ into doubletons

partition elements f.a;b/;.c;d /g

ıad ıbc :

Thus, the term in the sum corresponding to  2 . / and the partition P of [2 S. / into doubletons contributes either 0 or 1 to the integral, and we shall let x .P / denote this contribution. We may associate a fatgraph plus extra structure to each term .; P / in . / as follows. Begin with vk copies of a standard neighborhood of a k-valent vertex in the plane, and label the vertices 1; : : : ; vk , for k D 1; : : : ; K. As in Figure 4.6a, the boundary of each component decomposes naturally into k “sectors” and k “stubs”, j j j which we linearly order by labeling 1; 2; : : : ; k. Given a cycle j D .i1k ; i2k ; : : : ; ikk / j j j in  of length k, assign the integers i1k ; i2k ; : : : ; ikk 2 f1; : : : ; N g to the sectors about the j th vertex in order. Finally, assign the labels of adjacent sectors to stubs and use the orientation on the plane to associate one “incoming” and one “outgoing” index for each stub as in Figure 4.6b. The cycle j thus induces a labeling on the stubs about the j th vertex with each stub labeled by an ordered pair of integers between 1 and N .

205

4 Fatgraphs revisited j

j

i33

i13

stub

stub

j

sector

sector

stub

sector

j

i13

i33

j

j

i13

i33

j i23

j i23

a) Stubs and sectors

j i23

b) Incoming/outgoing

Figure 4.6. Labeled sectors and stubs.

Each element of P consists of a pair of elements of [2 S. /, and we may attach a band between the corresponding stubs being careful to preserve orientation in the sense that outgoing points to incoming. Adding such a band for each pair of pairs in P produces a fatgraph we shall denote G.P /. An N -labeling on a fatgraph G is a linear ordering on the k-valent vertices for each k together with a linear ordering on the sectors about each vertex plus an assignment of integers from 1 to N to the sectors of G. Thus, G.P / is 1-labeled by construction and inherits an N -labeling from . The N -labeling on sectors in a common boundary component of G.P / must be constant in order that x .P / ¤ 0 since by Wick’s Lemma, hHab Hcd i D ıad ıbc . It follows that the integral in Theorem 4.19 is given by Q

X 1 v k k .vk Šk /

2./

X partitions P

1 vk k .vk Šk /

x .P / D Q

X

N sŒG.P / :

partitions P

Let us explicate the sum over partitions P . For fixed v3 ; : : : ; vK , arrange in the plane vk standard k-valent vertices, so that there is an induced 1-labeling on any fatgraph with this vertex set. An element of the index set is a partition P of the stubs into doubletons, which in turn determines a fatgraph G.P /. It follows that there are P .1 C k kvk /ŠŠ terms in the sum, and we illustrate the 15 1-labeled fatgraphs in the index set for exactly two 3-valent vertices in Figure 4.7. Let ŒG denote the number of appearances of the isomorphism class ŒG of unlabeled fatgraphs in this index set. For example, the three fatgraphs in the first column of Figure 4.7 represent the three isomorphism classes and have respective ŒG given by 9, 3, 3 and # AutŒG given by 2, 6, 6. Aut.G/ acts transitively without fixed points on the collection of 1-labelings on G,

206

5 Mapping class groupoids and moduli spaces

Figure 4.7. 1-labeled fatgraphs with v3 D 2.

but the action is with isotropy, and we evidently have Y .vk Šk vk / .G/.# Aut.G// D k

D #f1  labelings on G W vk .G/ D vk for k D 1; : : : ; Kg: Finally, the integral in Theorem 4.19 is given by Q

1 vk k .vk Šk /

X

ŒG N sŒG D

ŒG with vk .G/Dvk for kD1;:::;K

X ŒG with vk .G/Dvk for kD1;:::;K

N sŒG # AutŒG

as was claimed. It follows that the virtual Euler characteristics [23] of Riemann’s moduli spaces are given by P

X

ˆ.I; N / D

ŒG with2.G/DI

D

1

N

X P

.k2/vk DI

vk .G/

N s.G/ # Aut.G/ Z N Y tr H k vk 1  tr H 2 =2 Q  dM.H /e ; k k vk Š H

.1/

k

k

and we define the generating function Z.x; N / D

X I 0

p ˆ.I; N /.i x/I :

207

4 Fatgraphs revisited

Care is required to correctly manipulate this expression [131], [73] and compute that R log Z.x; N / D log

D

HN



1

tr

P

H k xk k

k2 dM.H /e x2 R 2 1 2 tr H H N dM.H /e 2

;

N NY 1 .1 C px 2 /N p ;

.ex 2 /x .x 2 / p 2 2x

pD1

where log denotes the natural logarithm and  denotes Euler’s Gamma function. This is often called the “Penner matrix model” in the physics literature. Stirling’s formula can be used to extract the required coefficients in terms of Bernoulli numbers Bk . For example, the coefficient of N in log Z.x; N / is the asymptotic series 1 X

X B2g x 2k .1/k x 4g2 C BkC1 ; 2g.g  1/ k gD1 giving the value .1  2g/ D B2g =2g for the virtual Euler characteristic of the mapping class group MC.Fg1 / of a once-punctured surface in agreement with [60], where  is Riemann’s zeta function. 4.4 Fatgraph tables for F21 and F31 . By way of example, we present the fatgraph complexes for the once-punctured surfaces of genera two and three. These results, which depend upon computer enumeration, were given in [112], but the genus two data given there was flawed due to an error in manipulating files. We first give an overview of the fatgraph complex for F21 by specifying the number of isomorphism classes corresponding to each possible tuple v3 ; v4 ; : : : ; vK of vertex valences, where vk denotes the number of vertices of valence k, for k D 3; : : : ; K. We shall denote such a valence tuple by 3v3 : : : k vk : : : K vK . In each case, we list the total number of corresponding fatgraph isomorphism classes as well as the non-trivial fatgraph automorphism groups that arise together with the number of occurrences of each group in square brackets. For example, the first entry in the following table corresponds to a single vertex of valence 8, which occurs in codimension 5; there are a total of four distinct such fatgraphs, one of which has an automorphism group of order 2, one of which has an automorphism group of order 8, and the remaining two fatgraphs admit no symmetries. The fatgraph complex for F21 is summarized in this notation in Table 4.1. In greater detail, the isomorphism classes of fatgraphs for F21 are presented using the usual notation of pairs of permutations ; , where  describes the vertices and  the edges of an untwisted fatgraph, as in Section 4. For each fixed  , we enumerate the various isomorphism classes of fatgraphs by specifying the corresponding involutions  . For each such pair, we also indicate in square brackets the order of the corresponding fatgraph automorphism group, which in this case of a once-punctured surface is

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5 Mapping class groupoids and moduli spaces

Table 4.1. Summary of the fatgraph complex for F21 .

Valence

Codimension

Total

1

5

4

Z=2Œ1 ; Z=8Œ1

2

5

4

7

Z=2Œ4 ; Z=5Œ1 ; Z=10Œ1

1 1

4

7

Z=2Œ1

1 1

3 7

4

7

3

4

3

6

Z=2Œ3 ; Z=4Œ1

2 1

3 6

3

15

Z=2Œ2 ; Z=3Œ1 ; Z=6Œ1

1 1 1

3 4 5

3

24

2 2

2

33

3 1

2

19

4 1

3 4

1

29

Z=2Œ4 ; Z=4Œ1

6

0

9

Z=2Œ5 ; Z=3Œ1

8

4 6

3 4 3 5 3

Groups

Z=2Œ11

necessarily a cyclic group. The order of this cyclic group is given in square brackets following each entry  together with a superscript ˙ whenever the group is non-trivial, where the plus sign indicates that the generator of the automorphism group preserves the orientation of the corresponding cell, and the minus sign indicates orientation reversal. In this notation, the fatgraph complex of F21 is described as follows:  D .1; 2; 3; 4; 5; 6; 7; 8/ W (1,3)(2,5)(4,7)(6,8) [1]

(1,3)(2,4)(5,7)(6,8) [2C ]

(1,3)(2,6)(4,7)(5,8) [1]

(1,5)(2,6)(3,7)(4,8) [8 ]

 D .1; 2; 3; 4; 5/.6; 7; 8; 9; 10/ W (1,6)(2,4)(3,5)(7,9)(8,10) [2C ]

(1,6)(2,5)(3,7)(4,9)(8,10) [1]

(1,6)(2,5)(3,10)(4,8)(7,9) [2C ]

(1,6)(2,7)(3,8)(4,9)(5,10) [10C ]

C

(1,6)(2,4)(3,10)(5,8)(7,9) [2 ]

(1,6)(2,9)(3,7)(4,10)(5,8) [5C ]

C

(1,6)(2,7)(3,10)(4,8)(5,9) [2 ]  D .1; 2; 3; 4; 5; 6/.7; 8; 9; 10/ W (1,7)(2,4)(3,5)(6,9)(8,10) [1]

(1,7)(2,6)(3,8)(4,9)(5,10) [1]

(1,7)(2,4)(3,6)(5,9)(8,10) [1]

(1,7)(2,4)(3,10)(5,8)(6,9) [1]

(1,7)(2,8)(3,5)(4,10)(6,9) [1]

(1,7)(2,5)(3,6)(4,9)(8,10) [2 ]

(1,7)(2,5)(3,8)(4,10)(6,9) [1]

4 Fatgraphs revisited  D .1; 2; 3; 4; 5; 6; 7/.8; 9; 10/ W (1,8)(2,9)(3,5)(4,6)(7,10) [1] (1,8)(2,6)(3,9)(4,10)(5,7) [1] (1,8)(2,5)(3,9)(4,6)(7,10) [1] (1,8)(2,4)(3,9)(5,7)(6,10) [1]

(1,8)(2,7)(3,5)(4,10)(6,9) [1] (1,8)(2,5)(3,6)(4,9)(7,10) [1] (1,8)(2,5)(3,7)(4,10)(6,9) [1]

 D .1; 2; 3; 4/.5; 6; 7; 8/.9; 10; 11; 12/ W (1,5)(2,4)(3,6)(7,11)(8,9)(10,12) [2 ] (1,5)(2,4)(3,12)(6,10)(7,11)(8,9) [1] (1,5)(2,6)(3,7)(4,11)(8,9)(10,12) [2 ] (1,5)(2,7)(3,10)(4,11)(6,12)(8,9) [1] (1,5)(2,10)(3,6)(4,12)(7,11)(8,9) [2 ] (1,5)(2,4)(3,7)(6,11)(8,9)(10,12) [4C ]  D .1; 2; 3; 4; 5; 6/.7; 8; 9/.10; 11; 12/ W (1,12)(2,4)(3,5)(6,9)(7,10)(8,11) [1] (1,12)(2,4)(3,6)(5,9)(7,10)(8,11) [1] (1,11)(2,12)(3,5)(4,9)(6,8)(7,10) [1] (1,11)(2,6)(3,12)(4,8)(5,9)(7,10) [1] (1,12)(2,6)(3,8)(4,9)(5,11)(7,10) [1] (1,12)(2,8)(3,5)(4,11)(6,9)(7,10) [1] (1,12)(2,4)(3,11)(5,8)(6,9)(7,10) [1] (1,11)(2,4)(3,9)(5,12)(6,8)(7,10) [1]  D .1; 2; 3; 4; 5/.6; 7; 8; 9/.10; 11; 12/ W (1,9)(2,4)(3,5)(6,10)(7,11)(8,12) [1] (1,12)(2,4)(3,5)(6,10)(7,9)(8,11) [1] (1,12)(2,4)(3,9)(5,8)(6,10)(7,11) [1] (1,9)(2,4)(3,7)(5,12)(6,10)(8,11) [1] (1,9)(2,12)(3,5)(4,7)(6,10)(8,11) [1] (1,9)(2,4)(3,8)(5,12)(6,10)(7,11) [1] (1,8)(2,5)(3,9)(4,12)(6,10)(7,11) [1] (1,12)(2,8)(3,5)(4,9)(7,10)(8,11) [1] (1,7)(2,4)(3,11)(5,9)(6,10)(8,12) [1] (1,8)(2,11)(3,5)(4,12)(6,10)(7,9) [1] (1,11)(2,5)(3,12)(4,8)(6,10)(7,9) [1] (1,7)(2,4)(3,12)(5,9)(6,10)(8,11) [1]

(1,12)(2,5)(3,6)(4,9)(7,10)(8,11) [2 ] (1,7)(2,8)(3,9)(4,12)(5,10)(6,11) [2 ] (1,7)(2,12)(3,8)(4,11)(5,9)(6,10) [3C ] (1,7)(2,12)(3,9)(4,11)(5,8)(6,10) [6 ] (1,7)(2,11)(3,8)(4,9)(5,12)(6,10) [1] (1,11)(2,5)(3,12)(4,9)(6,8)(7,10) [1] (1,12)(2,5)(3,8)(4,11)(6,9)(7,10) [1]

(1,8)(2,4)(3,12)(5,11)(6,10)(7,9) [1] (1,9)(2,11)(3,12)(4,7)(5,8)(6,10) [1] (1,11)(2,12)(3,8)(4,9)(5,7)(6,10) [1] (1,9)(2,11)(3,7)(4,8)(5,12)(6,10) [1] (1,8)(2,9)(3,12)(4,7)(5,11)(6,10) [1] (1,7)(2,8)(3,11)(4,12)(5,9)(6,10) [1] (1,10)(2,11)(3,12)(4,8)(5,6)(7,9) [1] (1,9)(2,12)(3,8)(4,11)(5,7)(6,10) [1] (1,9)(2,7)(3,11)(4,8)(5,12)(6,10) [1] (1,12)(2,7)(3,9)(4,11)(5,8)(6,10) [1] (1,10)(2,11)(3,8)(4,12)(5,6)(7,9) [1] (1,12)(2,4)(3,11)(5,8)(6,10)(7,9) [1]

 D .1; 2; 3; 4/.5; 6; 7; 8/.9; 10; 11/.12; 13; 14/ W (1,14)(2,11)(3,7)(4,5)(6,8)(9,12)(10,13) [1] (1,5)(2,4)(3,13)(6,10)(7,11)(8,14)(9,12) [1] (1,5)(2.4)(3,11)(6,14)(7,10)(8,13)(9,12) [1] (1,13)(2,4)(3,8)(5,11)(6,14)(7,10)(9,12) [1] (1,14)(2,4)(3,8)(5,13)(6,10)(7,11)(9,12) [1] (1,14)(2,10)(3,11)(4,5)(6,8)(7,13)(9,12) [1] (1,13)(2,14)(3,10)(4,5)(6,8)(7,11)(9,12) [1] (1,9)(2,13)(3,11)(4,14)(5,12)(6,8)(7,10) [1] (1,9)(2,10)(3,11)(4,14)(5,12)(6,8)(7,13) [1] (1,14)(2,6)(3,7)(4,5)(8,11)(9,12)(10,13) [1]

(1,14)(2,7)(3,11)(4,5)(6,8)(9,12)(10,13) [2C ] (1,14)(2,4)(3,11)(5,13)(6,8)(7,10)(9,12) [2C ] (1,14)(2,4)(3,10)(5,13)(6,8)(7,11)(9,12) [2 ] (1,13)(2,4)(3,14)(5,11)(6,8)(7,10)(9,12) [2 ] (1,5)(2,6)(3,7)(4,11)(8,14)(9,12)(10,13) [2 ] (1,14)(2,6)(3,10)(4,8)(5,13)(7,11)(9,12) [2 ] (1,13)(2,6)(3,14)(4,8)(5,11)(7,10)(9,12) [2 ] (1,9)(2,10)(3,11)(4,8)(5,12)(6,13)(7,14) [2 ] (1,9)(2,13)(3,7)(4,14)(5,12)(6,10)(8,11) [2 ] (1,9)(2,10)(3,14)(4,8)(5,12)(6,13)(7,11) [2 ]

209

210

5 Mapping class groupoids and moduli spaces

(1,13)(2,10)(3,6)(4,7)(5,11)(8,14)(9,12) [1] (1,14)(2,11)(3,6)(4,7)(5,13)(8,10)(9,12) [1] (1,14)(2,8)(3,10)(4,5)(6,11)(7,13)(9,12) [1] (1,14)(2,11)(3,7)(4,5)(6,10)(8,13)(9,12) [1] (1,13)(2,10)(3,7)(4,5)(6,14)(8,11)(9,12) [1] (1,5)(2,7)(3,10)(4,11)(6,13)(8,14)(9,12) [1] (1,13)(2,8)(3,14)(4,5)(6,10)(7,11)(9,12) [1]

(1,9)(2,10)(3,7)(4,14)(5,12)(6,13)(8,11) [2 ] (1,9)(2,10)(3,14)(4,8)(5,12)(6,11)(7,13) [1] (1,9)(2,6)(3,10)(4,13)(5,12)(7,14)(8,11) [1] (1,9)(2,14)(3,10)(4,7)(5,12)(6,13)(8,11) [1] (1,9)(2,10)(3,14)(4,7)(5,12)(6,13)(8,11) [1] (1,5)(2,7)(3,14)(4,10)(6,11)(8,13)(9,12) [1]

 D .1; 2; 3; 4; 5/.6; 7; 8/.9; 10; 11/.12; 13; 14/ W (1,10)(2,4)(3,5)(6,9)(7,13)(8,14)(11,12) [1] (1,12)(2,13)(3,8)(4,14)(5,11)(6,9)(7,10) [1] (1,8)(2,4)(3,14)(5,13)(6,9)(7,10)(11,12) [1] (1,10)(2,13)(3,14)(4,7)(5,8)(6,9)(11,12) [1] (1,13)(2,5)(3,14)(4,8)(6,9)(7,10)(11,12) [1] (1,7)(2,8)(3,13)(4,14)(5,10)(6,9)(11,12) [1] (1,14)(2,4)(3,10)(5,8)(6,9)(7,13)(11,12) [1] (1,13)(2,14)(3,8)(4,10)(5,7)(6,9)(11,12) [1] (1,14)(2,8)(3,5)(4,10)(6,9)(7,13)(11,12) [1] (1,10)(2,13)(3,7)(4,8)(5,14)(6,9)(11,12) [1] (1,10)(2,4)(3,7)(5,14)(6,9)(8,13)(11,12) [1] (1,14)(2,7)(3,10)(4,13)(5,8)(6,9)(11,12) [1] (1,8)(2,13)(3,5)(4,14)(6,9)(7,10)(11,12) [1] (1,8)(2,10)(3,14)(4,7)(5,13)(6,9)(11,12) [1] (1,10)(2,14)(3,5)(4,7)(6,9)(8,13)(11,12) [1] (1,10)(2,7)(3,13)(4,8)(5,14)(6,9)(11,12) [1] (1,14)(2,4)(3,13)(5,8)(6,9)(7,10)(11,12) [1] (1,10)(2,14)(3,8)(4,13)(5,7)(6,9)(11,12) [1] (1,12)(2,13)(3,14)(4,8)(5,11)(6,9)(7,10) [1]  D .1; 2; 3; 4/.5; 6; 7/.8; 9; 10/.11; 12; 13/.14; 15; 16/ W (1,13)(2,4)(3,12)(5,8)(6,9)(7,16)(10,15)(11,14) [1] (1,9)(2,4)(3,7)(5,8)(6,13)(10,16)(11,14)(12,15) [1] (1,9)(2,4)(3,15)(5,8)(6,12)(7,13)(10,16)(11,14) [2C ] (1,14)(2,4)(3,12)(5,8)(6,13)(7,16)(9,11)(10,15) [1] (1,9)(2,4)(3,13)(5,8)(6,16)(7,12)(10,15)(11,14) [1] (1,9)(2,4)(3,6)(5,8)(7,13)(10,16)(11,14)(12,15) [2C ] (1,14)(2,15)(3,16)(4,9)(5,8)(6,12)(7,13)(10,11) [1] (1,14)(2,15)(3,9)(4,13)(5,8)(6,12)(7,16)(10,11) [1] (1,14)(2,15)(3,9)(4,13)(5,8)(6,16)(7,12)(10,11) [1] (1,10)(2,6)(3,7)(4,16)(5,8)(9,13)(11,14)(12,15) [1] (1,15)(2,12)(3,13)(4,7)(5,8)(6,9)(10,16)(11,14) [1] (1,10)(2,12)(3,13)(4,16)(5,8)(6,9)(7,15)(11,14) [1] (1,9)(2,10)(3,6)(4,16)(5,8)(7,13)(11,14)(12,15) [1] (1,9)(2,12)(3,13)(4,6)(5,8)(7,16)(10,15)(11,14) [1] (1,14)(2,15)(3,7)(4,13)(5,8)(6,12)(9,11)(10,16) [1] (1,9)(2,15)(3,10)(4,16)(5,8)(6,12)(7,13)(11,14) [1] (1,10)(2,16)(3,12)(4,15)(5,8)(6,9)(7,13)(11,14) [1] (1,13)(2,16)(3,12)(4,7)(5,8)(6,9)(10,15)(11,14) [1] (1,15)(2,7)(3,9)(4,12)(5,8)(6,13)(10,16)(11,14) [1] (1,9)(2,12)(3,7)(4,15)(5,8)(6,13)(10,16)(11,14) [1] (1,13)(2,7)(3,9)(4,16)(5,8)(6,12)(10,15)(11,14) [1] (1,16)(2,13)(3,7)(4,10)(5,8)(6,9)(11,14)(12,15) [2C ] (1,16)(2,13)(3,7)(4,10)(5,8)(6,12)(9,15)(11,14) [2C ] (1,10)(2,13)(3,6)(4,15)(5,8)(7,16)(9,12)(11,14) [1]

211

4 Fatgraphs revisited (1,16)(2,7)(3,13)(4,10)(5,8)(6,9)(11,14)(12,15) [4 ] (1,10)(2,15)(3,6)(4,16)(5,8)(7,12)(9,13)(11,14) [1] (1,14)(2,13)(3,15)(4,9)(5,8)(6,12)(7,16)(10,11) [1] (1,9)(2,13)(3,10)(4,15)(5,8)(6,16)(7,12)(11,14) [1] (1,13)(2,6)(3,12)(4,9)(5,8)(7,16)(10,15)(11,14) [1]  D .1; 2; 3/.4; 5; 6/.7; 8; 9/.10; 11; 12/.13; 14; 15/.16; 17; 18/ W (1,4)(2,5)(3,14)(6,15)(7,10)(8,11)(9,18)(12,17)(13,16) [2C ] (1,4)(2,5)(3,9)(6,11)(7,10)(8,15)(12,18)(13,16)(14,17) [2C ] (1,4)(2,5)(3,17)(6,11)(7,10)(8,14)(9,15)(12,18)(13,16) [2C ] (1,4)(2,5)(3,14)(6,16)(7,10)(8,15)(9,18)(11,13)(12,17) [1] (1,4)(2,5)(3,15)(6,11)(7,10)(8,18)(9,14)(12,17)(13,16) [1] (1,4)(2,17)(3,11)(5,15)(6,16)(7,10)(8,18)(9,14)(12,13) [2C ] (1,4)(2,14)(3,9)(5,17)(6,11)(7,10)(8,15)(12,18)(13,16) [2C ] (1,4)(2,17)(3,11)(5,15)(6,16)(7,10)(8,14)(9,18)(12,13) [1] (1,4)(2,15)(3,8)(5,17)(6,12)(7,10)(9,18)(11,14)(13,16) [3C ]

The nine combinatorially distinct trivalent fatgraph spines in this notation for the surface F21 are illustrated in Figure 4.8.

(1,4)(2,5)(3,14)(6,15)(7,10)(8,11)(9,18)(12,17)(13,16)

(1,4)(2,5)(3,14)(6,16)(7,10)(8,15)(9,18)(11,13)(12,17)

(1,4)(2,14)(3,9)(5,17)(6,11)(7,10)(8,15)(12,18)(13,16)

(1,4)(2,5)(3,9)(6,11)(7,10)(8,15)(12,18)(13,16)(14,17) (1,4)(2,5)(3,17)(6,11)(7,10)(8,14)(9,15)(12,18)(13,16)

(1,4)(2,5)(3,15)(6,11)(7,10)(8,18)(9,14)(12,17)(13,16)

(1,4)(2,17)(3,11)(5,15)(6,16)(7,10)(8,14)(9,18)(12,13)

(1,4)(2,17)(3,11)(5,15)(6,16)(7,10)(8,18)(9,14)(12,13)

(1,4)(2,15)(3,8)(5,17)(6,12)(7,10)(9,18)(11,14)(13,16)

Figure 4.8. The trivalent fatgraph spines of F21 .

212

5 Mapping class groupoids and moduli spaces

We finally simply give an overview of the fatgraph complex for F31 using the same notation as before in Table 4.2. Table 4.2. Summary of the fatgraph complex for F31 . Valence 121 72 61 81 51 91 41 101 31 121 52 61 41 62 41 51 71 42 81 31 61 71 31 51 81 31 41 91 32 101 42 52 43 61 31 53 1 1 1 1 3 4 5 6 31 42 71 32 62 32 51 71 32 41 81 33 91 45 1 3 1 3 4 5 32 41 52 32 42 61 33 51 61 33 41 71 34 81 32 44 33 42 51 34 52 34 41 61 35 71 34 43 35 41 51 36 61 36 42 37 51 38 41 310

Codimension 9 8 8 8 8 8 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 6 5 5 5 5 5 5 5 4 4 4 4 4 3 3 3 2 2 1 0

Total 131 183 357 381 425 495 781 802 1602 884 1735 1848 2030 1201 2186 1494 1539 9408 4980 2590 5267 5759 2225 510 10080 16170 16655 11669 12455 3606 10100 42385 11282 23228 5230 31152 39347 7238 31760 9548 12594 1726

Groups Z=2Œ10; Z=3Œ2; Z=12Œ1 Z=2Œ20; Z=7Œ2; Z=14Œ1 Z=2Œ9 Z=2Œ13 Z=2Œ29 Z=2Œ45; Z=4Œ1 Z=2Œ29; Z=4Œ2; Z=8Œ2

Z=2Œ29 Z=2Œ88 Z=2Œ41; Z=3Œ4; Z=6Œ1 Z=3Œ9

Z=2Œ89; Z=3Œ2; Z=6Œ1 Z=2Œ38 Z=3Œ4; Z=9Œ2 Z=2Œ72; Z=4Œ2 Z=2Œ147 Z=2Œ130

Z=2Œ53; Z=4Œ1 Z=2Œ250 Z=2Œ164 Z=2Œ81 Z=2Œ350; Z=3Œ19; Z=4Œ4; Z=12Œ1 Z=2Œ89; Z=3Œ9; Z=6Œ1 Z=2Œ380 Z=2Œ39; Z=4Œ2 Z=2Œ99; Z=3Œ11; Z=6Œ1

5 Compactifications

213

5 Compactifications There is little doubt that the further effective application of the cell decomposition of Tz .F / to computing topological or other invariants of moduli spaces M.F / or mapping class groups MC.F / depends upon understanding some compactification of M.F /. There are three natural approaches to compactifying the combinatorial description of moduli space M.F / provided by the cell decomposition of Tz .F /: • The combinatorial completion [112], [137], [102] of Tz .F /=R>0 descends to a z .F / D .Tz .F /=R>0 /=MC.F /, which is sometimes called compactification of M the “Penner compactification”. This relies on the natural simplicial completion of the ideal simplices to formal honest simplices and is most easily discussed using arc families. • The classical Thurston boundary [158], [144] of Teichmüller space T .F / is the space PL0 .F / of projective measured laminations of compact support on the surface F , a piecewise linear sphere that compactifies the open ball T .F / to a closed ball upon which MC.F / acts continuously. • The classical Deligne–Mumford compactification [37], [15] usually denoted by x .F /. A point of M x .F /  M.F / is described by a “stable curve”, which arises M as the limit of pinching to zero the hyperbolic lengths of a collection of disjointly embedded essential simple closed “pinch curves”, no two components of which are isotopic. In the limit, the curves are collapsed to hyperbolic punctures; this discussion in fact takes place in M.F /, i.e., on the level of MC.F /-orbits, and a copy of the moduli space of each such class of punctured surface is included in x .F /. M x .F / has the structure of a complex projective algebraic The DM compactification M variety [37] and arises differential geometrically as the mapping class group quotient of the augmented Teichmüller space [173], a (non locally compact) space stratified by loci with short families of disjointly embedded essential curves, no two of which are parallel. (It is illuminating to imagine this space in terms of the various Fenchel–Nielsen coordinates on each stratum.) We shall serially discuss all three approaches after first elaborating parenthetically on a condition already discussed in Section 2 of Chapter 4. 5.1 No vanishing cycles. In this section, we collect several estimates which are used to correlate divergences of simplicial coordinates and lambda lengths and shall often identify an ideal arc with its lambda length for convenience as usual. Given a fatgraph spine G of F whose edges have lambda lengths ei and simplicial coordinates Ei , for i D 1; : : : ; N , we say as before that there are no vanishing cycles (of simplicial coordinates) provided that each Ei  0 and any essential cycle in G traverses at least one edge with non-vanishing simplicial coordinate. Recall Lemma 2.3 in Chapter 4, which we repeat here for convenience:

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5 Mapping class groupoids and moduli spaces

Lemma 5.1. The no vanishing cycle condition implies that the lambda lengths at any vertex of a trivalent fatgraph G satisfy the three strict triangle inequalities. We proceed to uncover various consequences of this. Lemma 5.2. Provided there are no vanishing cycles, the product of the lambda length and the simplicial coordinate of an edge is bounded above by four. Proof. In the notation of Figure 5.1, we have eE D

c 2 C d 2  e2 a2 C b 2  e 2 C ; ab cd

so each summand is twice a cosine by Lemma 5.1 and the Euclidean law of cosines.

d  c

a 

ˇ

ı

˛ e

" b

Figure 5.1. Standard notation near edge e.

Lemma 5.3. Suppose there are no vanishing cycles and the lambda lengths are bounded below by L > 1. Then the h-lengths are bounded above by 2L1 , and the simplicial coordinates are bounded above by 8L1 . Proof. Adopt the notation of Figure 5.1. According to Lemma 5.1, the no vanishing cycle condition implies that a < b C e, and hence ˛D

a 1 1 1 2  .b C e/ D C  be be e b L

proving the bound on h-lengths. The bound on simplicial coordinates then follows from their expression as linear combinations of h-lengths. Lemma 5.4. Suppose that there are no vanishing cycles and the lambda lengths are bounded below by L > 1. In the notation of Figure 5.1, if ˛ is bounded above by " < 1, 1 then b and e are each bounded below by 12 " 2 .

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5 Compactifications

Proof. ˛ D

a be

 " and a  L give that " 

a be



L , be

and so

1 1 "  : b e L Furthermore, the no vanishing cycle condition gives jb  ej < a by Lemma 5.1, and dividing by be, we find a 1 1  ": j  j< e b be q q It follows that either b1  L" or 1e  L" . In the former case, we have 1  b

r

p p "  " < 2 "; L

and finally by j 1e  b1 j  ", we conclude that also r p p 1 1 "  C" C "  " C "  2 "; e b L as was claimed. Definition 5.5. We say that a collection X of edges of a fatgraph G is recurrent if for every edge e 2 X , there is an efficient closed edge-path e in G so that e traverses e and traverses only edges in X. In other words, X has no univalent vertices. Any subset X has a (possibly empty, e.g., in the case of a planar tree) maximal recurrent subset R.X /, namely, the set of edges of X traversed by some efficient closed edge-path in G. Consider a continuous one-parameter family xi .t / of lambda lengths on the edges of a trivalent fatgraph G, where t  0. We assume that xi .t / satisfies the no vanishing cycle condition for each t > 0 and shall next derive constraints on the limiting values xi .0/. Let us scale the lambda lengths so that xi .t /  1 for all t  0, i.e., all of the horocycles in the decoration are disjointly embedded. Let Xi .t /, for t > 0, denote the simplicial coordinate of edge i with limiting value Xi .0/, for i D 1; : : : ; N , so Xi .t / is bounded above for each t  0 by Lemma 5.3. Lemma 5.6 (IJ Lemma). In the notation above assuming no vanishing cycles for t > 0, define I D fi W xi .t / ! 1g and J D fi W Xj .t / ! 0g. Then I  J and R.J / D I , where R.X / denotes the maximal recurrent subset of X . Proof. If xi ! 1, then Xi ! 0 by Lemma 5.2, and so indeed I  J . To see that R.J /  I , suppose that  is a cycle in G˛ so that Xj ! 0 for each edge i traversed by  . By Lemma 2.1 of Chapter 4, the sum of the included h-lengths for  must tend to zero, and hence each included h-length must also tend to zero. Lemma 5.4 a shows that if the included h-length bc ! 0, then b ! 1 and c ! 1, so each edge of  lies in I as required.

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5 Mapping class groupoids and moduli spaces

For the reverse inclusion I  R.J /, suppose that xi ! 1, so Xi ! 0 by Lemma 5.2. We must show that there is cycle in J passing through edge i. To this end, if the endpoints of edge i coincide, then the existence of a cycle is established. If the endpoints are distinct, then another edge incident on an endpoint must also lie in I by Lemma 5.1. Continuing in this manner using Lemma 5.1, we find a cycle in I passing through edge i . Again by Lemma 5.2, this cycle must lie in J , as required. In order to elucidate the following lemma, we first discuss the geometry underlying a certain “hypercyclic renormalization procedure” (which was already encountered in the proof of Theorem 3.1 of Chapter 2). Recall from Definition 3.10 of Chapter 1 that an "-hypercycle is a curve at constant distance "  0 to a geodesic  in F , and the pair of "-hypercycles on either side of  for " > 0 have common constant geodesic curvature between the limiting values zero (geodesic) and unity (horocycle). Any short geodesic curve has a large hypercyclic neighborhood in F , cf. [24], and among those neighborhoods with fixed area, there is a distinguished one N" where the boundary hypercycles have common hyperbolic length. Now imagine pinching  to a node of a stable curve, and suppose that a is an arc connecting decorated punctures in F which meets  once. The lambda length p ı  D e of a thus diverges, and a gives rise to a pair of arcs a0 ; a1 in the pinched surface connecting a puncture to the new node; the boundary of N" gives rise to a pair of horocycles, one on either side of the node. On the other hand, the arc a restricts to 0 an arc in N" which is bisected by , so the common p p limiting lambda length  of each a0 ; a1 is given by 0 D e.ı=2/ . This gives 0 D , which is the basic renormalizing transformation. Lemma 5.7 (Renormalization Lemma). Fix a trivalent fatgraph G and index the edges of G by i D 1; : : : ; N . Suppose that xi is an assignment of lambda lengths to the edges p of G so that there are no vanishing cycles of simplicial coordinates where xi  2, for i D 1; : : : ; N . Suppose further that there are two sizes of lambda lengths: i) Small: 1  xi  L << K; ii) Large: xi  K. Define new lambda lengths xi0 on G by: i) Small: xi0 D xi , the lambda length is unchanged; p ii) Large: xi0 D xi . Then for K sufficiently large relative to L, the lambda lengths xi0 on G also have no vanishing cycles of simplicial coordinates; in fact, the modified simplicial coordinate of each large edge is strictly positive. Proof. Since the lambda lengths xi have no vanishing cycles, they must satisfy the triangle inequalities at each vertex by Lemma 5.1. It follows that for 2K > L, there cannot be a vertex with only one incident edge of large lambda length.

5 Compactifications

217

Consider an edge e of G with distinct endpoints, say with incident edges a, b at one endpoint and c, d at the other endpoint. The simplicial coordinate of the edge e 2 2 e 2 which we associated with the former endpoint and a is the sum of a term a Cb abe like term which we associate with the latter. To begin, we analyze the contribution to the simplicial coordinate separately from each such term in various cases. As a point of notation, we shall let lower case letters denote small and upper case letters denote large lambda lengths. Suppose that all three lambda lengths A, B, E are large. The contribution of the modified lambda lengths to the simplicial coordinate of edge E is thus A0 2 C B 0 2  E 0 2 ACB E D p : 0 0 0 AB E ABE This quantity is positive since A, B, E satisfy the triangle inequalities again by Lemma 5.1. Suppose that e is small and A, B are large. The contribution of the modified lambda lengths to the simplicial coordinate of edge B is thus e 2 C A0 2  B 0 2 e2 C A  B D p A0 B 0 e e 2AB eCAB  p e AB .e C A  B/ D ; p e AB where the inequality follows from the assumption that e  1. This quantity is again positive since A, B, e satisfy the triangle inequalities by Lemma 5.1. Furthermore, the 0 2 CB 0 2 e 2 , which is positive for contribution to the simplicial coordinate of edge e is A A 0B0e K > L. It follows from the two previous paragraphs that the modified simplicial coordinate of any large edge is indeed positive, and it remains only to analyze the case of small edges. There are various possibilities described in the notation of upper and lower case letters adopted above: i) a, b, c, d , e; ii) A, B, C , D, e; iii) A, B, c, d , e; and iv) a, b, C , D, e. In case i), the simplicial coordinate of e is unchanged and hence non-negative. In case ii), the modified simplicial coordinate is again positive for K > L as in the last sentence of the previous paragraph. The two cases iii) and iv) are identical, and we concentrate on case iii). The three small lambda lengths c, d , e uniquely determine a PSL2 .R/-orbit of a triple u, v, w of points in the open positive light-cone in Minkowski three-space realizing the given lambda lengths as the square roots of negative inner products. Since c, d , e satisfy the triangle inequalities by Lemma 5.1, the plane containing these three points is elliptic, and we may choose a representative triple u, v, w sharing a common height. This

218

5 Mapping class groupoids and moduli spaces

height H depends upon c, d , e and the bound c; d; e < L gives a corresponding bound on H in terms of L. Fixing such a representative three points u, v, w, there is then a uniquely determined point  in the open positive light-cone realizing the inner products A0 2 with u and B 0 2 with v by Lemma 4.14 of Chapter 1. (Indeed, there is unique such point on either side of the plane through the origin containing u and v, and a unique one, namely,  on the other side of this plane from w.) For K >> L, the height of  is greater than H , which thus gives a positive modified simplicial coordinate to the edge e, as required. It follows that the modified simplicial coordinate of any edge is non-negative, the modified simplicial coordinate of any large edge is strictly positive, and a vanishing cycle for the modified simplicial coordinate is necessarily a vanishing cycle for the original coordinates. 5.2 Quotient arc complexes. Here we discuss aspects of bordered surfaces based on [137], [139], [141]. Definition 5.8. Construct a simplicial complex Arc.F / of F D Fgs called the arc complex of F by associating one p-dimensional simplex in Arc.F / to each isotopy class of ideal arc family with p C 1 ideal arcs, where the face relation corresponds to inclusion of ideal arc families. Define Arc# .F /  Arc.F / to be the subspace corresponding to all q.c.d’s of F , i.e., all arc families filling the surface in the sense that complementary components are either polygons or once-punctured polygons. In light of Theorem 4.1 of Chapter 4 and assuming F D Fg1 for simplicity (though the entire discussion goes through for partially decorated Teichmüller spaces based at a distinguished puncture), we have T .F / D Tz .F /=R>0 Arc# .F /  Arc.F / for F D Fg1 ; so Teichmüller space admits the natural simplicial completion Arc.F /, where we simply add faces corresponding to all ideal arc families whether or not they fill the surface. A cell in Arc.F /  Arc# .F / necessarily has infinite isotropy in MC.F /. Definition 5.9. Define the quotient arc complex of F D Fg1 to be the quotient QA.F / D Arc.F /=MC.F /: Since there are only finitely many combinatorial types of ideal arc families in the fixed surface F , QA.F / is a compactification of M.F / D T .F /=MC.F / Arc# .F /=M C.F /  Arc.F /=MC.F / D QA.F / called the combinatorial compactification of M.F /.

5 Compactifications

219

Remark 5.10. It is possible to reformulate this arc-theoretic construction in terms of dual fatgraphs. This is done in [137] in effect by coloring edges as “ghost” or “normal”, where a ghost edge corresponds to a removed arc, and taking the quotient by flips along ghost edges. Though this reformulation is possible, it is not clear that it is advantageous. We turn next to an analogous construction for bordered surfaces and then explain the connection with punctured surfaces. s Definition 5.11. For a bordered surface F D Fg; ıE with boundary distinguished points D and s  0, we define an ideal arc family in F to be (the isotopy class of) a collection of disjointly embedded essential arcs connecting points in D, no two of which are isotopic, so that no component arc is isotopic into @F  D. Again, construct the arc complex Arc.F / of F by associating one p-dimensional simplex to each isotopy class of ideal arc family in F with p C 1 ideal arcs, where the face relation corresponds to inclusion of arc families. Define the quotient arc complex QA.F / of F to be the quotient QA.F / D Arc.F /=PMC.F /:

Let us emphasize that we do not allow component arcs of an ideal arc family in a bordered surface that are isotopic into the boundary (whereas we did demand such arcs in an i.c.d. of F in order to give coordinates on the decorated Teichmüller space of a bordered surface for example). Furthermore, we take the quotient by the pure mapping class group for bordered surfaces. A key difference between punctured and bordered surfaces is that the arcs in a bordered surface come in a canonical linear ordering once the distinguished points on the boundary are so ordered. Indeed, one serially encounters the arcs by traversing small neighborhoods of the distinguished points, and these first encounters give a linear ordering to the set of arcs in an arc family as was claimed. Thus, the quotient arc complex of a bordered surface is a -complex in the sense of Hatcher [61]. Notice that the arcs in an arc family furthermore come with specified orientations from their first to last encounters. Suppose that ˛ is an ideal arc family in a punctured or bordered surface F and consider the possibly disconnected surface F  ˛ gotten by cutting Fgs along ˛. The cusps on the frontier of each component of F  ˛ may be regarded as distinguished points on its smooth boundary, so that F  ˛ inherits the natural structure of a bordered surface F˛ . It follows from the definition of the arc complexes that the link of the simplex in Arc.F / corresponding to ˛ is precisely the arc complex Arc.F˛ /. Furthermore, the link (in the second barycentric subdivision) of the corresponding simplex in QA.F / can be shown to be isomorphic to QA.F˛ / for a bordered surface F and to be finite cover of QA.F˛ / for a surface F D Fgs Thus, the global topology of arc complexes of bordered surfaces governs the local topology of arc complexes of punctured or bordered surfaces, and this motivates our study of the global topology of arc complexes of bordered surfaces.

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5 Mapping class groupoids and moduli spaces

Lemma 5.12. Suppose that F 0 arises from the bordered surface F by adding one additional distinguished point on some boundary component of F . Then QA.F 0 / S 0 QA.F /, i.e., the quotient arc complex of F 0 is piecewise-linearly homeomorphic to the suspension of the quotient arc complex of F . Proof. Imagine that a distinguished point p 2 @F splits into two distinguished points q0 ; q1 2 @F 0 as depicted in Figure 5.2, which furthermore illustrates an embedding of F into F 0 respecting the natural identification of the other distinguished points in @F with those in @F 0 . q0

q 0 q1 p0

p F

p1

F0

F0

p0 a1

q1 p1 a0

Figure 5.2. F  F 0 and arcs a0 , a1 in F 0 .

The quotient arc complex QA.F 0 / decomposes into three subcomplexes: • X0 is the subcomplex of QA.F 0 / corresponding to arc families that are not incident on q1 ; equivalently, X0 corresponding to all arc families that either contain the arc a0 or are disjoint from a0 . X0 is the star of the vertex in QA.F 0 / corresponding to a0 , i.e., the cone on the link of a0 , which is isomorphic to QA.F /; • the subcomplex X1 defined in analogy to X0 replacing a0 by a1 ; • the subcomplex Y corresponding to arc families that contain neither a0 nor a1 . The suspension S 0 QA.F / can be identified with QA.F /  Œ1; C1 where each of QA.F /  f1g and QA.F /  fC1g is collapsed to a distinct point. Given a class y of projectively weighted arc families in F 0 , normalize so that the sum of the weights of all the components of y is one and let Wi .y/ denote the sum of these normalized weights of arcs that are incident on qi , for i D 0; 1; likewise for a class x in F , let .x/ denote the sum of normalized weights of arcs that are incident on p. Define .y/ D 12 .W0 .y/ C W1 .y//; so  is linear on simplices. If y 2 Y , then let yN denote the point in QA.F / obtained from y by collapsing to a point the segment in @F 0 connecting q0 to q1 which is free from any other distinguished points, and we choose such a segment once and for all if necessary. There is then an embedding y 7! .y; N 12 .W0 .y/  W1 .y// of Y in S 0 QA.F / onto .x; t/ 2 QA.F /  Œ1; C1 with t 2 Œ.x/; .x/ . The complement of the image of Y in S 0 QA.F / consists of two pieces, each of which is the cone on a copy of QA.F /,

5 Compactifications

221

so these pieces are piecewise-linearly homeomorphic to X0 and X1 . Combining these three pieces X0 ; X1 ; Y , we indeed find a piecewise-linear homeomorphism between QA.F 0 / and S 0 QA.F /. s Example 5.13. For g D s D 0, F D Fg;ı is a polygon with r sides, PMC.F / is trivial, and it is a classical fact that QA.F / D Arc.F / in this case is piecewiselinearly homeomorphic to an (r  4)-dimensional sphere. (Giancarlo Rota told me that Hassler Whitney knew this.) For instance, the two cases of triangulating a quadrilateral correspond to the two points of a zero-dimensional sphere, and the pentagon relation shows that the arc complex of a pentagon is a circle. The general case then follows from the previous result since the suspension of sphere is a sphere.

(A number of further explicit examples, which are covered by the following theorem, are discussed separately in the next section since the complexes themselves may be interesting.) It is natural to wonder precisely which surfaces likewise have quotient arc complexes which are spheres as resolved by the following theorem, the main result in [141]: s with a total of n  r distinguished points Theorem 5.14. Consider a surface F D Fg;r on its boundary with at least one distinguished point on each boundary component. The quotient arc complex QA.F / is a piecewise-linear sphere only in the following cases: s QA.F0;1 / S nC2s4 for s  0I

0 QA.F0;2 / S n1 for n  2I

1 QA.F0;2 / S nC1 for n  2I

0 QA.F0;3 / S nC2 for n  3I

0 QA.F1;1 / S nC2 for n  1I

1 QA.F1;1 / S nC4 for n  1:

s / is a piecewise-linear manifold other than a sphere only in the Furthermore, QA.Fg;r 1 4 0 2 four cases F0;2 , F0;3 , F0;0 , and F1;2 , with exactly one distinguished point on each boundary component in each case, where these manifolds have respective dimensions s 5, 7, 9, and 7. In all other cases, QA.Fg;r / is not a piecewise-linear manifold.

There is the following immediate consequence: Corollary 5.15. The combinatorial compactification of Riemann’s moduli space of a punctured surface Fgs with a distinguished puncture is an orbifold if g D 0 or if g D 1 and s  2. A further discussion of the orbifold structure is given in [137]. Remark 5.16. We collect in this remark several further facts about quotient arc coms plexes from [141]. QA.Fg; ıE / has dimension 6g  7 C 3.r C s/ C #fdistinguished points in the boundaryg

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5 Mapping class groupoids and moduli spaces

and is simply connected provided its dimension is at least two. QA.F / admits a natural action of the r-torus .S 1 /r described geometrically by twisting around the various boundary components, and the diagonal circle action is fixed point free. We should mention that there is a large literature including [57], [58], [164], [62], [63] on the topology of the arc complex, but these are the only results we know on the quotient arc complex. It is most natural to wonder what are these four special manifolds that 2 1 4 0 arise as quotient arc complexes for the surfaces F0;2 , F0;3 , F0;0 , and F1;2 in more prosaic terms. s Definition 5.17. For any bordered surface F D Fg; ıE , define

QA# .F / D Arc# .F /=PMC.F /  Arc.F /=PMC.F / D QA.F / corresponding to arc families that quasi fill F . s with 2g 2C2.r Cs/ > 0, Corollary 5.18. For any bordered surface F D Fg;.ı 1 ;:::ır / let F{ be a surface with the same genus g and number s of punctures as F and with r smooth boundary components with no distinguished points. Then QA# .F / is homotopy equivalent to the moduli space M.F{ / of hyperbolic structures with geodesic boundary on F{ .

Proof. First, we may assume without loss of generality that there is a unique point in each boundary component. Indeed, we claim that if F 0 arises from F by the addition of one further distinguished boundary point, then QA# .F / is homotopy equivalent to QA# .F 0 /. To see this, use the homotopy identification of QA# .F 0 / with QA# .F /  I given in the proof of Lemma 5.12, i.e., the construction from before preserves the subspace corresponding to quasi filling arc families. rCs / with fiber the r-torus .S 1 /r , and the Furthermore, M.F{ / fibers over M.Fg;0 rCs z .F rCs /, which in base M.Fg;0 / is homotopy equivalent to its decorated version M g;0 rCs / by Theorem 4.1. Let us equivalently consider turn is homeomorphic to QA# .Fg;0 rCs the complex of possibly punctured fatgraphs dual to quasi filling arc families in Fg;0 . rCs { Certain of the punctures of Fg;0 are opened into boundary components of F . Adjoin a tail to each corresponding boundary component of the skinny surface associated rCs with a fatgraph spine of Fg;0 . Performing these additions of tails in all possible ways for these boundary components produces the fatgraph complex dual to QA# .F /. The natural torus action is given by moving each tail around its boundary component, and QA# .F / is thus homotopy equivalent to M.F{ /. Remark 5.19. In [139], we worked somewhat harder to produce an explicit and geometrically natural proper homotopy equivalence between these spaces. Turning finally to the relation between these results and the homology of Riemann’s moduli space, let us for simplicity restrict attention to the case s D 0. (More generally, one identifies all the interior punctures to a single point and proceeds with the

5 Compactifications

223

following definition/construction.) Let @ denote the boundary mapping of the chain complex fCp .QA.F // W p  0g of QA.F /. Suppose that ˛ is an arc family in F with corresponding cell  Œ˛ 2 Cp .QA.F //. A codimension-one face of  Œ˛ corresponds to removing one arc from ˛, and there is a dichotomy on such faces  Œˇ depending upon whether the rank of the first homology of Fˇ agrees with that of F˛ or differs by one from that of F˛ . This dichotomy decomposes @ into the sum of two operators @ D @1 C @2 ; where @2 corresponds to the latter case. The operators @1 , @2 are easily shown to be a pair of anti-commuting differentials, so there is a spectral sequence converging to H .QA.F // corresponding to the bi-grading 0 Eu;v D fchains on  Œ˛ 2 Cp .QA/ W v D rank.H1 .F˛ // and u D p  vg; 0 0 0 0 ! Eu1;v and the differential of the E 0 term is @2 W Eu;v ! Eu;v1 . where @1 W Eu;v In particular, the @1 -homology of the top row itself agrees with the homology of uncompactified Riemann’s moduli space for F by Corollary 5.18. These homology groups are thus related by this spectral sequence, which may also be of utility in their calculations.

5.3 Low-dimensional quotient arc complexes. We explicitly describe here a menagerie of low-dimensional examples of quotient arc complexes which together with Lemma 5.12 shows that any quotient arc complex of dimension at most four is necessarily a piecewise-linear sphere. This follows from Theorem 5.14 and is included here because the examples themselves may have a certain independent interest. s Throughout this section, we shall adopt the usual notation: F D Fg;.ı has 1 ;:::;ır / genus g  0 with s  0 punctures and r  1 boundary components with ıi > 0 distinguished points on the i th boundary component, for i D 1; : : : ; r. Example 5.20 (Polygons). If g D s D 0 and r D 1, then F is a polygon with ı1 vertices, N D ı1  4, PMC.F / is trivial, and we are in the setting of Example 5.13. Because it is so elegant, we shall give another constructive proof from [148] that the quotient arc complex of an n-gon Pn is an (n  4)-dimensional sphere. To this end, let 0 denote the empty arc family, so the collection of all non-negative real weights on arc families with its natural topology is the cone QA.F / f0g. To see that QA.F / is indeed a piecewise-linear sphere of the asserted dimension, we may equivalently prove that QA.F / f0g Rn3 , and to this end, let us choose a triangulation of Pn with its (n  3) interior edges. Assign a real parameter Xi 2 R to each edge of the chosen triangulation and construct separately in each triangle a partial foliation as indicated in Figure 5.3, where in effect, positive parameters are taken as transverse measures and negative as tangential measures in the sense of train tracks [144]. These foliated triangles are then combined in the natural way indicated in Figure 5.4 to produce a weighted family of arcs in Pn whose two-sided inverse is directly described.

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5 Mapping class groupoids and moduli spaces

X1  0

1 2 .X1

C X2

X3 /

X1  0

X2

X1  0

1 2 X2

X2  0 1 .X1 2

C X3

X2 /

X1

X2  0 X3  0

1 .X2 2

C X3

X1 /

all Xi are positive and satisfy all three weak triangle inequalities: Xj  Xj C Xk for fi; j; kg D f1; 2; 3g

X1  0

X3

X2

1 X 2 1

X3  0

X2 C X3

X1

X2

1 2 X3

X1  0 and X2 ; X3  0 for instance

X1  0

X1  0

X3

X2

1 X 2 1

X1  0 and X2 ; X3  0, for instance, with X2  X3

X3  0

all Xi are positive and some triangle inequality fails, say X1  X2 C X3

X2  0

X2  0

X3  0

1 X 2 2

1 X 2 1

X2  0

X3  0

X2  0

X3

X3

X1  0 and X2 ; X3  0, for instance, with X3  X2

X3  0 all Xi are negative

Figure 5.3. Partial measured foliations of triangles.

Figure 5.4. Combine foliations in triangles.

1 X 2 3

5 Compactifications

225

Example 5.21 (Once-punctured polygons). If g D 0, r D s D 1, and ı1 < 2, then Arc.F / D ;. If ı1 D 2, then F is a once-punctured bigon and as in Figure 2.1 in Chapter 2 for a quasi flip, Arc.F / S 0 . For any ı1  3, it follows by induction using Lemma 5.12 again that any once-punctured polygon has quotient arc complex a sphere. Example 5.22 (Annuli). If g D s D 0, r D 2, and ı1 D ı2 D 1, then F is an annulus with one distinguished point on each boundary component. It is elementary that any essential arc must have distinct endpoints and its isotopy class is classified by an integral “twisting number”, namely, the number of times it twists around the core of the annulus; furthermore, two non-parallel essential arcs can be disjointly embedded in F if and only if their twisting numbers differ by one. Thus, Arc.F / is in this case isomorphic to the real line, the mapping class group PMC.F / is infinite cyclic and generated by the Dehn twist along the core of the annulus, which acts by translation by one on the real line, and the quotient QA.F / is a circle. As usual, the case of an arbitrary number of distinguished points follows by induction. Definition 5.23. If an ideal arc a in the bordered surface F is separating and its endpoint coincide, then the two components of F  a are distinguished in that the one-cusped component is the component of Ffag which inherits one new cusp from a and the two-cusped component which inherits two. Example 5.24 (Twice-punctured polygons). If g D 0, s D 2, and r D 1 D ı1 , then F is a twice-punctured monogon. If a is an essential arc in F , then its one-cusped component contains a single puncture, which may be either of the two punctures in F , and the two-cusped component contains the other puncture of F . Labeling the punctures of F by 0 and 1, choose disjointly embedded arcs ai so that ai contains puncture i in its one-cusped complementary component, for i D 0; 1. The Dehn twist  along the boundary generates PMC.F /, and the 0-skeleton of Arc.F / can again be identified with the integers, where the vertex corresponding to f j .ai /g is identified with the integer 2j C i for any integer j and i D 0; 1. Applying Example 5.21 to the two-cusped component of Ffai g , for i D 0; 1, we find exactly two one-simplices incident on the vertex corresponding to fai g, namely, there are two ways to add an arc disjoint from ai corresponding to the two cusps in the two-cusped component, and these arcs are identified with the integers i ˙ 1. Thus, Arc.F / is again identified with the real line, PMC.F / acts this time as translation by two, and QA.F / is again a circle. This complex is pictured in Figure 5.5, where the equality indicates isotopic arcs rel boundary, and the two representatives of the PMC.F /-orbit of ideal arcs illustrated on the left of the figure differ by a Dehn twist around the boundary. Example 5.25 (Once-punctured annuli). If g D 0, r D 2, and s D 1 D ı1 D ı2 , then F is a once-punctured annulus. To understand QA.F /, first look at the subcomplex

5 Mapping class groupoids and moduli spaces

H

226

2 Figure 5.5. The quotient arc complex of F0;1 is a circle.

of all arc families whose component arcs have distinct endpoints. In a maximal such family, the puncture lies in a bigon, and it is then easy to see that there are exactly 2 two cells in this subcomplex of QA.F / which combine to give an embedded torus. The complement of this torus retracts onto the subcomplex corresponding to arc families each of whose component arcs have coincident endpoints. Arguing as in Example 5.22, this subcomplex is isomorphic to the disjoint union of two embedded circles (and so here is a copy of the classical Hopf fibration of S 3 ). More explicitly, the various cells are enumerated in Figure 5.6, where the subscript ˙ is explained as follows. For instance, we have depicted the arc family ˛ in the figure, and the arc family ˛C is obtained from ˛ by rotating the page about the puncture in the natural way to interchange the two boundary components. The arcs are enumerated so that Roman letters indicate arc families where the puncture does not lie in a monogon, and the Greek letters otherwise. As illustrated in Figure 5.7, the cells a, c, d , g, h combine to give the torus mentioned above, the cells with fixed subscript ˙ lie on one side of this torus, and we drop the subscript in the figure, where we draw the inside of this torus. Each 2-cell ;  is a cone with vertex ˛ and base i . The three cells C , D fill these cones, and A, B fill out the complement of these cones meeting along  and . Finally, notice that the subcomplex consisting of arc families so that the puncture does not lie in a monogon is comprised of the torus together with the two compressing disks i˙ [ f˙ [ b˙ .

227

5 Compactifications 0-cells:

2-cells:

a

˛˙

b˙

g

h

i˙

'˙

˙

˙

˙

˙

˙

1-cells:

c

d

e

f˙

ˇ˙

˙

ı˙

"˙

3-cells:

C˙

B˙

A˙

D˙

1 Figure 5.6. The arc families in F0;2 .

g ı ˇ c



˛

d



h

a '

f

b

i

e

"

 1 Figure 5.7. The Hopf fibration of QA.F0;2 /.

Example 5.26 (Torus-minus-a-disk). Suppose g D r D ı1 D 1 and s D 0. Each 0 is non-separating, and there is thus a unique vertex Œa of QA.F /. arc a in F D F1;1 Furthermore, there are two types of one-simplex depending upon whether the corre-

228

5 Mapping class groupoids and moduli spaces

sponding arc family ˛ is separating, and these two types are represented by the weighted arc families in Figure 5.8, where the nearby label denotes the weight depending upon the parameter 0  s  1. In the non-separating case, the two arcs s D 0 and s D 1 differ by a (right) Dehn twist m on the meridian, and we shall also let ` denote the (right) Dehn twist on the longitude. Thus, m and ` generate PMC.F / and satisfy the relations D m ı ` ı m D ` ı m ı ` , where 4 is represented by the (right) Dehn twist on the boundary. In the separating case, the two arcs s D 0 and s D 1 differ by the “half Dehn twist” 2 . 1

s1 1 C s2 1 s2

s1 a) Seperating case, circle C1

1 b) Non-seperating case, circle C2

0 Figure 5.8. Two circles in QA.F1;1 /.

Thus, there are two circles C1 , C2  QA.F / as illustrated in Figures 5.8a and 5.8b respectively. It is not difficult to see that C1 , C2 are embedded in Arc.F / and C1 \C2 D fŒa g. Let C be the abstract CW-complex decomposing a circle into a single zero cell and a single one cell. Take the quotient .C C /= of the join C C by collapsing to a point the join line connecting the two vertices in the two natural copies of C in C C . We leave it to the reader to confirm that QA.F / is isomorphic as a cell complex to .C C /= , which is evidently piecewise-linearly homeomorphic to the three sphere C C . 3 Example 5.27 (Thrice-punctured polygons). Label the various punctures of F D F0;1 with distinct members of the set S D f1; 2; 3g. If ˛ is an arc family in F and a is a component arc of ˛, then a has a corresponding one-cusped component containing some collection of punctures labeled by a proper non-empty subset S.˛/  S, which we regard as the “label” of a itself. More generally, define a “tableau”  labeled by S to be a rooted tree embedded in the plane where: the (not necessarily univalent) root of  is an unlabeled vertex, and the other vertices of  are labeled by proper non-empty subsets of S ; for any n  1, the vertices of  at distance n from the root are pairwise disjoint subsets of S; and, if a simple path in  from the root passes consecutively through the vertices labeled S1 and S2 , then S2 is a proper subset of S1 . Given an arc family ˛ in F , we inductively define the corresponding tableau  D  .˛/ as follows. For the basis step, choose as root some point in the component of F˛ which contains the boundary of F . For the inductive step, given a vertex of  lying

5 Compactifications

229

in a complementary region R of F˛ , enumerate the component arcs a0 ; a1 ; : : : ; am of ˛ in the frontier of R, where we assume that these arcs occur in this order in their canonical linear ordering, and a0 separates R from the root. Each arc ai separates R from another component Ri of F˛ , and we adjoin to  one vertex in each such component Ri with the label S.ai / together with a one-simplex connecting R to Ri , for each i D 1; 2; : : : ; m, where the one-simplices are disjointly embedded in F . This completes the inductive definition of  D .˛/, which evidently satisfies the conditions required of a tableaux. Furthermore, the isomorphism class of  as a labeled rooted tree in the plane is obviously well defined independent of the representative ˛ of the PMC.F /-orbit Œ˛ . It follows immediately from the topological classification of surfaces that PMC.F /orbits of cells in Arc.F / are in one-to-one correspondence with the isomorphism class of tableaux labeled by S. Furthermore, since the edges of  .˛/ are in one-to-one correspondence with the component arcs of ˛, a point in the interior of the cell corresponding to ˛ is uniquely determined by a projective positive weight on the edges of  .˛/. It follows that QA.F / itself is identified with the collection of all such projective weightings on all isomorphism classes of tableaux labeled by S . Let us adopt the convention that given an ordered pair ij , where i; j 2 f1; 2; 3g, we shall let k D k.i; j / D f1; 2; 3g  fi; j g, so k actually depends only upon the 3 are enumerated, labeled, and indexed unordered pair i , j . The various tableaux for F0;1 in Figure 5.9, where in each case, ij varies over all ordered pairs of distinct members of f1; 2; 3g, k D 1; 2; 3, and the bullet represents the root. In this notation and letting @ denote the boundary mapping in QA D QA.F /, one may directly compute incidences of cells summarized as follows. @Cij @Dij @Ek @Fk @Kij

D Ai  Aj ; @Gij D Cij  Dj i C Dij ; D Aj  Bk ; @Hij D Dij  Cj k C Fk ; D Ak  Bk ; @Iij D Dij  Ek C Ckj ; D Bk  Ak ; @Jij D Cij  Cik C Cj k ; D Gij  Hij C Hj i  Jij ; @Lij D Iij  Gij C Jki  Ij i :

We may symmetrize and define subcomplexes Xk D Xij [ Xj i , for X D K; L, and furthermore set Mk D Kk [ Lk , for k D 1; 2; 3. Inspection of the incidences of cells shows that each of Kk and Lk is a three ball embedded in QA.F /, as illustrated in Figures 5.10a and 5.10b, respectively, with Kij on the top in part a) and Lj i on the top in part b). Gluing together Lk and Kk along their common faces Gij , Gj i , we discover that Mk is almost a three ball embedded in QA.F / except that two points in its boundary are identified to the single point Ak in QA.F /, as illustrated in Figure 5.10c. Each Mk , for k D 1; 2; 3, has its boundary entirely contained in the subcomplex J of Arc spanned by fAk W k D 1; 2; 3g [ fCij ; Jij W i; j 2 f1; 2; 3g are distinctg:

230

5 Mapping class groupoids and moduli spaces ij i,j

Ak

Bk

i

j

Kij

ij

k

k k

i

j

Lij

ij i

j

j

Cij

Gij

Fk ij j

j

j

k

Ek k k

ij

ij i

i,j i,j

k

Dij

Hij

Iij

j

i

k

Jij

3 Figure 5.9. The tableaux for F0;1 .

Jij

Cij Gij

Gj i

Bk

Aj

Hij

Dij

Cj i

Cjk Ak

Ai

Aj

I ij Dij Ckj

Cik

Ek Ij i Ak

Cj i

Jj i

a) The complex Kk

Jij

Ak Cki Ckj

Ai Aj

Ai

Bk

Cki

Gij

Gj i

Cij Jki

B k Dj i

Dj i F k Hj i

Jkj

Cik Ak Cik Jkj

Cij

Jki

b) The complex Lk

Cj i

Jj i

c) The complex Mk

Figure 5.10. Mk and the balls Kk ; Lk .

In order to understand J , we again symmetrize and define Jk D Jij [ Jj i , so each Jk is isomorphic to a cone, as illustrated in Figure 5.11; we shall refer to the onedimensional simplices Cik , Cj k as the “generators” and to Cij , Cj i as the “lips” of Jk . The one-skeleton of J plus the cone Jk is illustrated in Figure 5.11. Imagine taking k D 3 in Figure 5.11 and adjoining the cone J2 so that the generator C12 of J2 is attached to the lip C12 of J3 , and the generator C13 of J3 is attached to the lip C13 of J2 . Finally, J itself is produced by symmetrically attaching J1 to J2 [ J3 in this lip-to-generator fashion. In order to finally recognize the three sphere, it is best to take a regular neighborhood of J in QA.F /, whose complement is a disjoint union of three balls. Each such three ball is naturally identified with the standard “truncated ” three simplex, where a

231

5 Compactifications Cj i Ai

Cij

Aj

Jj i Jij Cjk

Cik

Ckj

Cki Ak

Figure 5.11. The cone Jk in the one-skeleton of J .

polyhedral neighborhood of the one-skeleton of the standard three simplex has been excised. These truncated simplices are identified pairwise along pairs of faces to produce the three sphere in the natural way. Proposition 5.28. The examples in this section establish that any quotient arc complex of dimension at most four is piecewise-linearly homeomorphic to a sphere. Proof. By Lemma 5.12, we may assume that there is exactly one distinguished point on each boundary component. Enumerating the solutions to 6g  7 C 4r C 2s  4, one finds precisely the examples of this section. 5.4 Laminations and tropicalization. In this section we treat joint work [126] with Athanase Papadopoulos. The space ML0 D ML0 .F / of (possibly empty) measured laminations of compact support on F D Fgs was introduced in [158]. The quotient PL0 D PL0 .F / D .ML0 .F /  f0g/=R>0 under the homothetic action of R>0 on transverse measures is a piecewise-linear sphere of dimension 6g  7 C 2s called Thurston’s boundary, where 0 denotes the empty lamination, naturally compactifying Teichmüller space T D T .F / R6g6C2s to a closed ball Tx D Tx .F / D T .F / t PL0 .F /; upon which the mapping class group MC D MC.F / acts naturally and continuously [88]. Let ŒL 2 PL0 denote the projective class of L 2 ML0  f0g. As in [126], there is furthermore a vector space ML0 D ML0 .F /, the space of decorated measured laminations, together with a continuous map  W ML0 ! ML0 , where the fiber over a point of ML0 is isomorphic to Rs . A decorated measured lamination is regarded as a measured lamination with compact support in the usual sense except that puncture-parallel leaves, called collar curves, are now allowed and

e

e

e

232

5 Mapping class groupoids and moduli spaces

are given a R-valued “decoration”; if the decoration is positive, then this parameter is taken as the measure of a foliated band parallel to the collar curve, and if it is negative, then it is thought of as a “deficit” of such leaves. In particular, the fiber over 0 is the collection of all R-weighted collar curves. Specifically, for each ideal triangulation , there is an associated branched onesubmanifold called the dual freeway  D  as illustrated in Figure 5.12b (repeated from before for convenience), where there is a small triangle of  lying inside each complementary region to , one dual “long” edge meeting each arc of , and three “short” edges inside each complementary region. In fact,  fails to be a train track only because there are complementary once-punctured null-gons, one such normally prohibited region for each of the s punctures.

or

or

d

d

 e

e cDd

"

ı

cDd

c c

" 

ı e

e a) Fatgraph from triangulation

b) Freeway from fatgraph

Figure 5.12. Fatgraphs and freeways.

Definition 5.29. A (tangential) measure on  is an assignment of non-negative real number to each ideal arc in  satisfying the weak triangle inequalities on each complementary triangle with no condition on short edges in complementary punctured monogons. A (transverse) measure on  is the assignment of a (not necessarily nonnegative) real number .b/ to each edge of  satisfying the familiar condition from train tracks that the measure of the long edge equals the sum of the measures of the two short edges at each vertex. Let V ./ denote the vector space of all measures on  . A measure on  can dually be regarded as a non-negative measure on  ; the values on the long edges of  are those of the dual which extend uniquely to a non-negative transverse measure on all of , where in the notation of Figure 5.12 on a freeway inside a triangle and identifying an arc with its measure, we have  D d Cec , ı D cCed 2 2 " D cCd2 e , while inside a monogon, we have  D ı D 2e , " D c D d . In any case, a non-negative measure on  or  determines a partial measured foliation in F in the usual way by fattening each edge of  into a band foliated by parallel copies of the

233

5 Compactifications

edge whose width is given by the measure on the edge. This produces a well-defined (but possibly empty) measured lamination of compact support in F together with a (possibly empty) collection of foliated bands with puncture-parallel leaves.

e

Remark 5.30. In addition to these “tangential measure” coordinates on ML0 .F /, there are also “shear coordinates" [159] on laminations defined in analogy to shear coordinates on Teichmüller space, cf. Section 1.2 of Chapter 1. Namely, an ideal triangle in a surface comes equipped with a perpendicular horocyclic partial foliation determining a midpoint on each frontier edge, and the shear coordinate of an edge of  is the transverse measure of the segment contained in an ideal arc in  connecting those midpoints coming from the two ideal triangles on either side of the ideal arc (again taken with a positive sign only when the two equidistant points lie to the right of one another). More generally, suppose that 2 V ./ possibly takes negative values, and let  denote a collar curve of  . The vertices of  decompose  into a collection of sub-arcs, and the measure associates to each such arc a real number. Let fc1 ; : : : ; cn g denote the collection of real numbers associated to the small branches of  that comprise  . Define the collar weight of  to be W D minfc1 ; : : : ; cn g and make such an assignment of collar weight to each collar curve in order to determine a collar weight on F itself. Now, we modify the original measure on  by defining 0 .b/ D .b/  W if b is contained in the collar curve , for each small branch b of  . The modified 0 extends uniquely to a well-defined measure on  which is non-negative and has identically vanishing collar weights by construction, so the corresponding measured foliation is of compact support. Summarizing, we have: Proposition 5.31. For any punctured surface F D Fgs , the space of measures on  gives global coordinates on ML0 .F /. Moreover, there is a canonical fiber bundle ML0 .F / ! ML0 .F /, where the fiber over a point is given by the vector space Rs of all collar weights on F .

e

e

Remark 5.32. There is a diagonal action of the full mapping class group MC.F / on ML0 .F /, where the action on collar weights is induced by the permutation of punctures. The bundle admits an equivariant section determined by the condition that each decorated measured foliation in the image has identically vanishing collar weights. The restriction of this section may be thought of as a piecewise-linear embedding of the piecewise-linear manifold ML0 .F / into the vector space ML0 .F /

e

e

234

5 Mapping class groupoids and moduli spaces

There is the following diagram summarizing the spaces and maps thus defined: Tz

cL

RC

log

cM

R

0g

ML0

T

e

ML0

ML0

Œ

Tx

PL0

 where Tz D Tz .F / denotes decorated Teichmüller space, log W R C ! R denotes the function which in each coordinate simply takes natural logarithms, cL denotes lambda lengths and cM denotes measures on  . For convenience, we shall write xN D log x for x 2 RC .

Theorem 5.33 ([126], [131]). Fix an ideal triangulation  of F and consider a sequence i 2 R C , for i  0, whose image under  ı cL escapes each compactum in Teichmüller space. Then: • Œ ı cL .i / converges to ŒL 2 PL0 in the topology of Thurston’s boundary if and only if N i ! 2 R , where  ı cM . / D L 2 ML0 ; • the tropicalization1

eN C fN D maxfaN C c; N bN C dN g

of the Ptolemy equation ef D ac C bd describes the effect of flips on measures; • the expression 2

X

d aN ^ d bN C d bN ^ d cN C d cN ^ d a; N

where the sum is over all triangles complementary to , with coordinates a, b, c in this clockwise cyclic order compatible with the orientation of F , describes the pull-back by  ı cL of the Weil–Petersson Kähler two-form on T in lambda lengths a, b, c as well as the pull-back by  ı cM of the Thurston symplectic form N c. N b, N on ML0 in measures a, Proof. The first assertion follows directly from [126]. The second assertion is easily verified case-by-case, and the final assertion is proved for the Thurston form in [126] and for the Weil–Petersson form in Theorem 3.1 of Chapter 2. The tropicalization or “Maslov dequantization” of any function .a; b; : : : / W Rn >0 ! R is defined as the limit as t goes to infinity of 1t log .eta ; etb ; : : : / taking values in the tropical semi-ring [154]. In particular, if a formula for  is written by combining the coordinate functions of Rn using only addition and multiplication, its tropicalization takes values given by replacing each occurrence of addition by the binary maximum function and each occurrence of multiplication by addition; there is the precedence of tropical addition over maximum corresponding to the usual precedence of multiplication over addition. We can nowadays simply say that [126] showed that lambda lengths tropicalize to measures, a fact that runs yet deeper in the underlying geometry and combinatorics, cf. [143]. 1

5 Compactifications

235

5.5 Deligne–Mumford compactification. We shall here simply extend the discussion of Appendix C to partially decorated punctured surfaces in the natural way as follows. Definition 5.34. A subset A of a q.c.d.  is said to be quasi-recurrent if for each edge a 2 A, there is a quasi-efficient curve in F meeting a disjoint from   A. Let GA denote the smallest sub punctured fatgraph or cycle containing the edges dual to those in A in the punctured fatgraph G dual to , where the type (punctured or unpunctured) of a vertex of GA is inherited from that of G. In fact, A is recurrent if and only if each univalent vertex of GA is punctured, cf. Lemma 2.1 in Appendix C. Definition 5.35. Given a q.c.d.  of a punctured surface F , define a screen on  to be a subset A of the power set of  so that i)  2 A; ii) each A 2 A is quasi-recurrent; iii) if A; B 2 A with A \ B ¤ ;, then either A  B or B  A; iv) for each A 2 A, we have [fB 2 A W B ¨ Ag ¨ A. Since the arcs in  are canonically identified with the edges of its dual punctured fatgraph G, we may equivalently consider and speak of a screen on G itself. As in Appendix C, each element of A 2 A other than A D  has an immediate predecessor A0 2 A, and the inclusion A  A0 gives rise to an inclusion of corresponding fatgraphs and hence the relative boundary @A A of the associated surfaces; again, in the special case that the subsurface corresponding to A is an annulus, the core of this annulus comprises @A A by definition. The boundary of the screen A itself is then @A D [A2AfE g @A A. The proof of Theorem 5.1 of Appendix C holds without change to give: Theorem 5.36. The cell in the partially decorated Teichmüller space of a punctured surface corresponding to the possibly punctured fatgraph G is asymptotic to a stable curve with pinch curves K if and only if K is homotopic to the collection of edge-paths @A for some screen A on G. The utility of screens for providing a cell decomposition of the Deligne–Mumford compactification will be taken up elsewhere [97].

6 Further applications

This chapter treats applications of the basic theory to the Thompson group, to quantum invariants, to field theories, and to the structure of macromolecules.

1 Universal Ptolemy and Thompson groups This section covers parts of [132], [135]. We shall first define the universal Ptolemy group in three stages: groupoid, monoid, and finally group. In effect, these algebraic gadgets codify flips on edges of a tesselation of the disk with distinguished oriented edge or doe, cf. Definition 1.3 of Chapter 3. It may well be that the Lie algebra discussed in Section 4 of Chapter 3 is naturally associated to this group rather than to HomeoC .S 1 /=PSL2 .R/. Suppose that  is a tesselation with doe of D. For each geodesic  2  0 , we may perform an (enhanced) flip along  (on a tesselation with doe) to produce another tesselation  D . [ f  g/  fg as usual, where ;   are the two diagonals of an ideal quadrilateral with frontier in  \  . There is a canonical identification of   f g with  f  g, so it makes sense to stipulate that if  is not the doe of  , then  inherits the given doe of  as its doe, while if  is the doe of , then   is the distinguished edge of  taken with an orientation so that the oriented  and   in this order determine via the right-hand rule at their unique point of intersection the orientation on D. The flip on the doe thus has order 4 as illustrated in Figure 1.1a while the flip on an edge other than the doe still has order 2 by involutivity. Furthermore, the usual pentagon relation holds away from the doe, but starting from the doe, the pentagon relation becomes the decagon relation illustrated in Figure 1.1b. The usual commutativity relations hold in any case. Definition 1.1. Define the universal Ptolemy groupoid P 1 as a category, where the objects are the PSL2 .R/-orbits of any fixed tesselation of D with doe, say, the Farey tesselation  with its usual doe, and a morphism from  to  is the PSL2 .R/-orbit of a finite sequence  D 0 ; 1 ; : : : ; nC1 D  of tesselations with doe which consecutively arise one from the next via enhanced flips. In particular, if there is a morphism from  1 to  in P 1 , then they share ideal points  0 D  0  S1 . To pass to the monoid, note that for any tesselation  with doe, the characteristic 1 1 mapping f W S1 ! S1 in Definition 1.4 of Chapter 3 establishes a bijection between

1 Universal Ptolemy and Thompson groups

237

Figure 1.1. Flips on labeled oriented edges.

the edges  of the Farey tesselation with its canonical doe and the edges of  . To specify a flip along an edge of  , we might just as well specify instead the corresponding edge of  . Thus, if q 2 , then we let q   denote the tesselation with doe gotten from  by applying the flip along the edge f .q/ of  corresponding to q 2  . Define the free monoid M generated by  , and inductively extend the action in the previous paragraph by setting .qn : : : q2 q1 /   D qn  .    q2  .q1   / : : : /; thereby defining an action of M on tesselations with doe and hence on T ess as well. This replacement of groupoid by monoid is a general categorical construction applicable when the morphisms out of any object in a category are identified a priori with a fixed set. To finally define the group, consider the submonoid K of M consisting of those finite words that act identically on  as a tesselation with doe, for example, the fourth power of the doe of  lies in K . Definition 1.2. Define the three following groups: 1) The universal Ptolemy group is the quotient PG1 D M =K , of M under the equivalence relation generated by insertions or deletions of finitely many words in K .

238

6 Further applications

2) P PSL2 .Z/ is the subgroup of HomeoC .S 1 / given by those homeomorphisms A W S 1 ! S 1 so that there is a decomposition of the circle into finitely many circular intervals with endpoints among the rationals 0 so that on each such interval I  S 1 , the function A restricts to AjI 2 PSL2 .Z/. 3) The Thompson group T is the subgroup of HomeoC .S 1 / which is piecewise affine in the angular coordinate on the circle with breakpoints among the collection of all dyadic rationals. Remark 1.3. 1) To see that PG1 is actually a group, it remains only to verify that there are inverses. For example, the third power of the doe gives the inverse of the doe in PG1 as in Figure 1.1a. In the remaining case of an edge other than the doe, involutivity provides the inverse. 2) Of course, there are coherency conditions on the various restrictions AjI to ensure that they combine to give a homeomorphism, and in fact, this homeomorphism is then automatically once continuously differentiable (since the stabilizer of 01 2 0 in PSL2 .Z/ is the parabolic subgroup fixing 01 2 0 ). 3) Put another way, the Thompson group T arises as the conjugate   T D P PSL2 .Z/ 1 1 1 of P PSL2 .Z/ by the Minkowski question mark function W S1 ! S1 , discussed in Example 1.5 of Chapter 3, which maps the Farey tesselation to the dyadic tesselation.

These three groups are next identified with one another using characteristic mappings. Namely, suppose that g 2 PG1 and consider applying the flips associated to g to the Farey tesselation  with its standard doe from 10 to 10 to produce another tesselation g   with doe. There is a corresponding characteristic mapping fg D fg  , giving a representation h W PG1 ! HomeoC .S 1 /;

g 7! fg ;

which is faithful by definition of the relations in K . Theorem 1.4 (Imbert–Kontsevich, [71]). The mapping h gives an embedding of PG1 into HomeoC .S 1 / with image the subgroup P PSL2 .Z/ of HomeoC .S 1 /. Consequently, the following three groups are isomorphic: the universal Ptolemy group PG1 , Thompson’s group T , and the group P PSL2 .Z/ of all piecewise PSL2 .Z/ homeomorphisms of the circle with finitely many rational breakpoints 1 1 ! S1 of g 2 PG1 Proof. We shall first show that the characteristic mapping fg W S1 lies in P PSL2 .Z/. To this end, we identify the edges  of the Farey tesselation with the set Q  f1; C1g; 1 1 where 0 corresponds to the edge e0 connecting 01 2 S1 to 10 2 S1 ; to any other c q 2 Q  f1; C1g occurring in its Farey enumeration as q D c 0 D bbCd 0 Cd 0 , the b d 1 corresponding edge eq has endpoints b 0 ; d 0 2 S .

239

1 Universal Ptolemy and Thompson groups

Suppose q > 0 and let Q denote the quadrilateral formed by the two triangles complementary to  that contain eq in their common frontier. There is a unique geodesic eq 0 in the frontier of Q which is in the same connected component of D  Q as e0 . If q D cc0 > q 0 D bb0 , then continuing with the notation of the previous paragraph, eq 0 has endpoints aa0 , dd0 , and these four positive rationals occur in their alphabetic order 1 . in S1 1 The ideal points of Q decompose S1 into four circular intervals, and on the interval containing the endpoints of e0 , fg is the identity map by construction. On each of the three other intervals, fg is a bijection respecting the triangulation outside any compactum. Since the modular group PSL2 .Z/ is the automorphism group of  by Lemma 3.4 of Chapter 1, on each closed circular interval, fg restricts to an element 1 of the modular group. It follows that fg is a bijection of S1 which agrees with some element of PSL2 .Z/ on each of the intervals



a b ; ; a0 b 0





b ;q ; b0





q;

d ; d0





d a ; ; d 0 a0

i.e., fg 2 P PSL2 .Z/. The arguments for q < q 0 or q < 0 are analogous, so the characteristic maps indeed lie in P PSL2 .Z/. To show that h is in fact an isomorphism of groups, it is sufficient to exhibit k W P PSL2 .Z/ ! PG1 so that h ı k is the identity on P PSL2 .Z/. The elements of P PSL2 .Z/ with no breakpoints comprise the subgroup PSL2 .Z/, and there are no elements with exactly one or two breakpoints. Let g 2 P PSL2 .Z/ have breakpoints r1 ; : : : ; rp , for p  3, to which we associate the tesselation  .g 1 / D fg 1 .e/ W e 2  g and define si D g 1 .ri /, for i D 1; : : : ; p. Let P be a connected, finite-sided, ideal polygon containing e0 as well as all of the si . The frontier edges of P and any edge exterior to P occur in  as well as  .g 1 / again because PSL2 .Z/ is the automorphism group of  . On the contrary, in the interior of P , the restrictions of  and  .g 1 / do not agree. By Lemma 1.4 of Chapter 2, there is some m.g/ in the free monoid M so that the corresponding sequence of flips produces  .g 1 / from  . Define k.g/ to be the class Œm.g/ 2 PG1 of m.g/ 2 M . If g 2 PSL2 .Z/, then one writes g D .g ı h/ ı h1 , where h has several breakpoints, and defines k.g/ D Œm.g ı h/ m.h1 / . Continuing to let g 2 P PSL2 .Z/, we have that f D h ı k.g/ is the unique bijection which maps the doe of .g 1 / to e0 and the complementary triangles . .g 1 //2 to 2 . By construction g has these properties, so indeed h ı k is the identity map. Remark 1.5. One can directly calculate characteristic maps, for example: 8 z  1 for z 2 Œ 01 ; 11 ; ˆ ˆ ˆ < z1   for z 2 Œ 01 ; 11 ; z h 01 .z/ D z ˆ for z 2 Œ 10 ;  11 ; ˆ ˆ : zC11  zC1 for z 2 Œ 11 ; 01 I

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6 Further applications

and  2 

h

1

.z/ D

8 z ˆ ˆ ˆ < z

zC1

ˆ  1 ˆ ˆ : zC3 z1

for z for z for z for z

2 Π10 ; 01 ; 2 Π01 ; 11 ; 2 Π11 ; 21 ; 2 Π21 ; 10 :

Using Theorem 1.4 and a known presentation of T , Pierre Lochak and Leila Schnepps [101] gave a new essentially geometric presentation of these groups as follows. Corollary 1.6. The groups PG1 , T and P PSL2 .Z/ admit a presentation with relations ˛4;

ˇ3;

.˛ˇ/5 ;

Œˇ˛ˇ; ˛ 2 ˇ˛ˇ˛ 2 ;

Œˇ˛ˇ; ˛ 2 ˇ˛ 2 ˇ˛ˇ˛ 2 ˇ 2 ˛ 2 ;

where Œ ;  denotes the commutator and ˛ D flip along the doe; ˇ D six consecutive frames of Figure 1.1a where  is the doe. In other words, the stated relations generate the monoid K discussed before. Another consequence of Theorem 1.4 is that PG1 is simple since T is known to be so. Remark 1.7. To explain the presentation, note that ˇ cyclically permutes the doe around the triangle to its left and hence has order 3e, ˛ is the flip along the doe with order 4 as illustrated in Figure 1.1b, and ˛ˇ generates the usual pentagon relation of order 5. The further two relations respectively correspond to commutativity of flips on disjoint but contiguous quadrilaterals and quadrilaterals separated by one triangle. In fact, ˛ corresponds to Œ 01 2 PG1 and ˇ to the composition Œ 01 Œ 21 Œ 01 3 Œ 21 Œ 01 , which can be expressed explicitly in P PSL2 .Z/ as well using Remark 1.5. Moreover, there is a natural “transitive completion” of PG1 further considered in [135] which acts transitively on the fibers of the very forgetful map T ess ! Mod, but which we shall not take up here. Furthermore, [101] defines an interesting hybrid of the Ptolemy groupoid and Grothendieck’s Teichmüller group called the “Ptolemy– Teichmüller groupoid" which supports an action of the absolute Galois group.

e

2 Nilpotent theory and three-manifold invariants This section is based on [115], [13], [5], [4], which represents joint work on several projects chronologically with Shigeyuki Morita, Alex Bene, Nariya Kawazumi, Jørgen Andersen, and Jean-Baptiste Meilhan. We begin with a general discussion of Torelli– Johnson–Morita theory.

2 Nilpotent theory and three-manifold invariants

241

0 Fix a smooth, closed, oriented surface F D Fg;0 , with g  1 and fix a base point

2 F . Consider the mapping class group

MC.F; / D 0 Diff C .F; / of F relative to the base point. Let  D 1 .F; / be the fundamental group with abelianization the first integral homology group H D H1 .F I Z/. The group  has its lower central series defined recursively by 0 D  and kC1 D Œ; k for k  0, where the bracket of two groups denotes their commutator group. The kth nilpotent quotient is defined as Nk D = k , where N1 D H , and there is an exact sequence 0 ! k =kC1 ! NkC1 ! Nk ! 1;

(6.1)

which is a central extension. By [96], the quotient k = kC1 is identified with the degree k C 1 part of the free Lie algebra L.H / of H D N1 modulo the ideal generated by the symplectic class !0 2 L2 .H / D ƒ2 H . The inverse limit N1 of the tower    ! NkC1 ! Nk !    ! N1 ! 1 is the pronilpotent completion of  and may be thought of as an approximation to  itself [155], [106]. The idea is that the action of MC.F; / on  is difficult to understand, and the hopefully simpler actions on Nk might be more tractable just as happens for k D 1 which corresponds to the representation of MC.F; / in the Siegel modular group Sp2g .Z/ of 2g-by-2g integral symplectic matrices. Definition 2.1. A mapping class ' 2 MC.F; / induces actions Nk ! Nk , for each k  1, and the kernel of the corresponding homomorphism MC.F; / ! Aut.Nk / is the kth Torelli group, which is denoted by g; Œk  MC.F; /. The tower of groups g; Œ1  g; Œ2     is called the Johnson filtration of MC.F; /. For k D 1, we find the classical Torelli group 1 ! g; D g; Œ1 ! MC.F; / ! Sp2g .Z/ ! 1 of mapping classes that act trivially on homology, and for k D 2, there is the special terminology that Kg; D g; Œ2 is called the Johnson kernel. Definition 2.2. Suppose ' 2 MC.F; / acts trivially on Nk and  2 NkC1 . It follows that './ 1 lies in the kernel of NkC1 ! Nk , so by exactness of (6.1) together with elementary manipulations of commutators, there are mappings k W g; Œk ! Hom.NkC1 ; k =kC1 / D Hom.H; k = kC1 /; called the Johnson homomorphisms, where the equality follows from the fact that k = kC1 is abelian and NkC1 abelianizes to H . These homomorphisms were introduced in [74], [75] and cf. [9], [155]. See [75] for a survey of the Torelli groups and [118], [119] for further results.

242

6 Further applications

The basic point that ties this theory with decorated Teichmüller theory is the following: since the cell decomposition of Tz .F  f g/ is invariant under MC.F; / D MC.Fg1 /, it is likewise invariant under any subgroup such as g; Œk . Thus, the tower Tz .F /=R>0 D T .F /    ! TkC1 .F / ! Tk .F / !    ! M.F / of Torelli spaces Tk .F / D T .F /=g; Œk , which are manifold Eilenberg–MacLane spaces, comes equipped with a compatible tower of ideal simplicial decompositions whose codimension-one cells correspond to flips and whose codimension-two cells correspond to involutivity, commutativity, and the pentagon relation as in Section 4 of Chapter 4. Definition 2.3. Fix some trivalent fatgraph spine G of F  f g. For any abstract group z z K, define a K-marking on G to be a function W E.G/ ! K from the set E.G/ of oriented edges of G to K so that two conditions hold: reversal of orientations on z edges corresponds to group inversion, i.e., .e/ N D Œ .e/ 1 , for any pair e; eN 2 E.G/ of orientations on a common edge; and if three oriented edges a, b, c point towards a common vertex of G in this counter-clockwise cyclic order, then .a/ .b/ .c/ D 1 2 K. Remark 2.4. The first condition is common to graph connections and markings. In contrast, vertices represent ideal polygons for markings rather than points for graph connections, and we take a subset rather than a quotient. Any K-marking on G gives rise to a homomorphism  ! K in the natural way identifying the fundamental group of G as the stabilizer of an object in the fundamental path groupoid of G. Here we are treating the surface/fatgraph on par with its moduli space/fatgraph complex. Using the second condition, it is not hard to compute the evolution of K-markings under flips, and the calculation is local like lambda lengths rather than global like simplicial coordinates as in Figure 2.1. Just as elements of MC.F; / are represented by sequences of flips beginning and ending on combinatorially identical fatgraph spines, so too elements of g; Œk are represented by such sequences that moreover preserve Nk -marking. This gives infinite presentations for all the Torelli groups and finite presentations for the fundamental path groupoids of the first two Torelli spaces T1 .F /; T2 .F / as in [115]. We include here from [115] the calculation of the first Johnson homomorphism 1 for the insight it provides about the general machinery of decorated Teichmüller theory for studying cycles on quotients of Teichmüller space. Definition 2.5. Let GyT D Gy.F / denote the fatgraph complex of a bordered surface F D Fg0; 1 considered before, and define the quotient GyI D Gy=g; . Furthermore, define the combinatorial cochain j 2 C 1 .GyT ; ƒ3 H /;

243

2 Nilpotent theory and three-manifold invariants

which is sometimes called the “Morita–Penner cocycle”, by setting j.We / D a ^ b ^ c in the notation of Figure 2.1. c c

eDcCd

a b

b

b f DaCd

b c

a

d

d

a

Figure 2.1. Effect of flip on homology marking.

Theorem 2.6. The cochain j is a cocycle, i.e., j 2 Z 1 .GyT ; ƒ3 H /, and it is Mg; equivariant, i.e., j.'. // D '.j. // for any oriented one-cell  of GyT and ' 2 Mg; . Thus, j descends to a one-cocycle j 2 Z 1 .GyI ; ƒ3 H / of GyI , which is a K.g; ; 1/. The associated group homomorphism Œj 2 H 1 .GyI ; ƒ3 H / Š Hom.g; ; ƒ3 H / coincides with six times the Johnson homomorphism 1 W g; ! ƒ3 H , i.e., Œj D 61 : Proof. To begin, we check that j.We / 2 ƒ3 H is well defined, and to this end, must check that j.We / is independent of the orientation of the edge. Indeed, in the notation of Figure 2.1, a C b C c C d D 0, and so j.We / D a ^ b ^ c D c ^ d ^ a as required. According to Corollary 1.2, in order to show that j is a one-cocyle, we must check that each of the involutivity, commutativity, and pentagon relations lead to vanishing sums. Continuing in the notation of Figure 2.1, involutivity follows from j.We / C j.Wf / D a ^ b ^ c C b ^ c ^ d D 0. Commutativity follows because addition is commutative in ƒ3 H . For the pentagon relation, we compute in the notation of Figure 2.2 that a C b C c C d C e D 0, and j.Wf / C j.Wg1 / C j.Wf2 / C j.Wg3 / C j.Wf4 / is given by b ^ c ^ d C e ^ a ^ b C c ^ d ^ e C a ^ b ^ c C d ^ e ^ a D 0; as required, so that j is indeed a one-cocycle. Our one-cocyle j thus gives rise to its associated group homomorphism Œj W g; ! ƒ3 H which is Sp.2g; Z/-equivariant.

244

6 Further applications

c

b f

d

a

g

e

f1

g4

f4

g1

f2

g3 g2

f3

Figure 2.2. Pentagon relation.

aCbCd c b

aCd c a

2aCbCd c

c a b

aCd c

c

2aCbCd a b

c aCd

d

second pair of flips

aCbCd

first pair of flips

aCd aCbCd c

aCd c

b

a

first twist bCd

c

d c

2aCd c

aCbCd

second twist a b

a c

aCbCd

a

a bCd

d c

d

2aCbCd c

a c 2aCd c

a b 2aCbCd

a

a aCd c

aCbCd

aCd

Figure 2.3. The torus BP map ' 2 g; .

To determine the explicit form of Œj , we compute j.'/ for a particular ' 2 g; , where in the terminology of [76], ' W F ! F is a “torus bounding pair” mapping, i.e., ' is defined by Dehn twists in opposite directions along the boundary components of a

245

2 Nilpotent theory and three-manifold invariants

torus-minus-two-disks embedded in F . A fatgraph spine of this torus-minus-two-disks is drawn in the upper-left of Figure 2.3. There are four steps in the definition of ': first, a Dehn twist along the dotted line in the upper-left to produce the fatgraph in the upper-right; second, a pair of flips along the edges marked with crosses in the upper-right to produce the fatgraph in the lower-right; third, another Dehn twist along the dotted line in the lower-right to produce the fatgraph in the lower-left; fourth and finally, another pair of flips along the edges marked with crosses in the lower-left to produce the fatgraph in the upper-left again. The sequence of flips corresponding to the first Dehn twist is depicted on the left-hand side of Figure 2.4, and the sequence for the second Dehn twist is depicted on the right-hand side. In all, the homeomorphism ' is given by a composition of 14 flips. aCbCd c

aCbCd c aCd c

b

c

b

aCbCd a

a

a d c

c

c

b

b

aCd

d

a

c

b

a c

aCbCd

b

c

aCd aCbCd c

2aCd c 2aCbCd

b

a

aCd c

c

aCbCd aCbCd a bCd b a aCbCd

aCbCd

aCd c

b

c

a

a

a b

aCd c a d c

aCbCd

c a aCb

c

d

c a aCb

aCbCd

The first Dehn twist

aCbCd c

aCbCd c

2aCbCd c

aCbCd

2aCd c a aCd c

a c

a 2aCbCd

aCd c a

a b aCd c

a c

2aCd c

aCd

2aCbCd c

d

aCd

aCd

a c d

c a aCb

aCd c

a

a b 2aCd c

2aCbCd c

c

a b

a

c a aCb

aCbCd

aCd

2aCbCd c

2aCbCd c c

aCd c

a a c

aCd

a b

2aCbCd

d

b

c a aCb a b

bCd

aCbCd c aCbCd a aCd c 2aCbCd

2aCbCd c

2aCbCd c aCbCd aCbCd a

aCd c bCd

aCbCd c aCd c

c

d c

bCd

a

a b

a c

c a aCb

a a b

aCd c

d c a

aCbCd

a c

aCd c

d

d

The second Dehn twist

Figure 2.4. The Dehn twists as flips.

To explicitly compute the value of the crossed homomorphism at ', we calculate the contribution from the first Dehn twist on the left-hand side of Figure 2.4 to be  d ^ c ^ .a C d  c/  .b C d / ^ b ^ .a C d /  .a C b C d / ^ a ^ .a C b C d / C .a C b C d  c/ ^ c ^ .2a C b C d / C .a C d  c/ ^ b ^ .2a C b C d  c/;

246

6 Further applications

the contribution from the second Dehn twist on the right-hand side in Figure 2.4 to be  .a C b C d / ^ a ^ .a C d /  .2a C d  c/ ^ .a  c/ ^ d C .a C d  c/ ^ a ^ .a C d  c/  .2a C b C d  c/ ^ .a C b/ ^ .d  c/  .a C b C d / ^ .c  a/ ^ .a C b C d  c/; and the contributions from the two pairs of flips in Figure 2.3 to be  .2a C b C d / ^ .2a C b C d  c/ ^ .a C d  c/  .a C d / ^ .a C b C d / ^ .2a C b C d  c/ C .a C b C d  c/ ^ .d  c/ ^ d C .a C d  c/ ^ d ^ .b C d /: By direct calculation, each of these expressions is equal to 2 a^b ^c, so Œj is definitely non-zero as a cohomology class. Furthermore, the value of Johnson’s homomorphism on ' 2 g; is known [74] to be a ^ b ^ c, so 1 and 16 Œj take the same value at '; according to [74], [77], this guarantees their equality. Remark 2.7. Works of N. Kawazumi and S. Morita [86], [87] explain how to express the MMM classes of M.F / in terms of contractions of coefficients of cup product powers of j . Using the Alexander-Whitney diagonal approximation, one can explicitly calculate these powers of our expression for j , thus giving a new family of cocycles representing the MMM classes on M.F /. This construction from [115] of the MMM classes deserves further study. Remark 2.8. There is another one cochain with values in the symmetric square of ƒ2 H given in the notation of Figure 2.1 by ..a ^ c/ ˝ .b ^ d // C ..b ^ d / ˝ .a ^ c//I one checks easily and directly that this is well defined and satisfies involutivity as well as commutativity with the verification of the pentagon relation somewhat more involved. The techniques of [115] were stalled in likewise calculating a canonical cocycle representing the second Johnson homomorphism 2 , and the trick to this computation 0 turns out to be to change categories to bordered surfaces F D Fg;1 and develop an effective method of iterated integration along the boundary of the surface starting from the tail. Q ˝i denote the completed tensor algebra of H (now workNamely, let T D 1 i0 H Q ˝i ing with rational homology H however) with its natural filtration Tp D 1 , ip H for p  0. Notice that 1 C T1 D f1 C t W t 2 T1 g forms a subgroup of T .

3 Open/closed strings, TFT and CFT

247

Definition 2.9. A generalized Magnus expansion is a homomorphism  W  ! 1 C T1 so that for each element a 2 ß with underlying homology class Œa 2 H , we have .a/ D 1 C Œa C 2 .a/ C 3 .a/ C    ; where i .a/ 2 H ˝i denotes the i th graded component of the tensor .a/ 2 T . It turns out [85] that the higher Johnson homomorphisms are calculable from any generalized Magnus expansion, and we succeed in [13] in writing a canonical Magnus expansion from the fatgraph structure by computing a canonical .1 C T1 /-marking as before on any trivalent bordered fatgraph G with tail. In particular, we give a canonical cocycle representing 2 . The formula is a bit much to give casually here, but it is not so bad. Furthermore, we give explicit lifts of all the higher Johnson homomorphisms to the level of the fundamental path groupoid of the first Torelli space T1 .F / of F with explicit formulas that are readily accessible on the computer. Given any homomorphism from a subgroup I of a mapping class group MC.F / to some group K, we can ask if there is an extension to the Ptolemy groupoid Pt.F / ! K in the sense that this mapping preserves compositions and restricts to the given homomorphism K ! I on I < MC.F / < Pt.F /, where MC.F / sits inside the Ptolemy groupoid Pt.F / of F as the stabilizer of any vertex. This amounts to an assignment of elements of K to flips that satisfies involutivity, commutativity, and the pentagon relation. Not only do all the Johnson homomorphisms thus extend in this sense to the groupoid level for a bordered surface with one boundary component, so too by [5] do all of the following representations of the mapping class group: the “Nielsen” representation in the automorphism group of , the Magnus representation, and the natural representation in the symplectic group. Furthermore, Gwénaël Massuyeau [110] has used the Le–Murakami–Ohtsuki [99] construction to give Magnus expansions preserving the Hopf algebra structure and extended to the Ptolemy groupoid also Morita’s homomorphisms. Remark 2.10. In fact, these manifestations of the theory in the pronilpotent world turn out to be special cases of more general representations of the mapping class groupoids of bordered surfaces on finite type invariants of three-manifolds. See [4], where the basic combinatorics of fatgraphs are shown to determine the various choices required for the link invariant of Andersen–Mattes–Reshetikhin [6], [7]. This leads via an LMO-type construction to an invariant of suitable three-manifolds which turns out to be universal for homology cobordisms, and this leads to representations of the mapping class groupoids acting on appropriate spaces of Jacobi diagrams.

3 Open/closed strings, TFT and CFT This section is based on [82] and [84], each representing joint work with Ralph Kaufmann and the former including also Muriel Livernet.

248

6 Further applications

We model interacting strings using arc families in bordered surfaces. The idea is that as the strings move and interact, they form the leaves of a “string foliation” on their world-sheets. Dual to this foliation is another measured foliation of the world-sheets, which determines the combinatorial type of the string foliation, given by a weighted arc family. The algebra of these string interactions is then described by gluing together the string foliations along the strings, and this corresponds to appropriate gluing operations on the dual measured foliations. This algebraic structure is present already on the topological level of string interactions, and on the homology level, i.e., for integral homology groups of suitable arc complexes, it induces what we call a “c/o structure” (for closed/open strings). We shall be content here to describe the topological level, where much can already be explained, and refer the reader interested in the abstract structure to the original paper. More explicitly in Figure 3.1, each boundary component comes equipped with a non-empty collection of distinguished points that represent the branes upon which the open strings are required to have their endpoints, and the labeling will be explained presently. The physically meaningful picture arises by replacing each distinguished point in the boundary by a small distinguished arc or “window” (representing that part of the interaction that occurs within the corresponding brane) and collapsing to a point each component of the boundary disjoint from the foliation and from the windows. Thus, the distinguished points in the boundary of a bordered surface correspond to windows in the current discussion. Definition 3.1. A windowed surface F D Fgs .ı1 ; : : : ; ır / is a smooth oriented surface of genus g  0 with s  0 punctures and r  1 boundary components together with the specification of a non-empty finite subset ıi of each boundary component, for i D 1; : : : ; r. Let ı D ı1 [    [ ır denote the set of all distinguished points in the boundary @F of F and let  denote the set of all punctures. The set of components of @F  ı is called the set W of windows. According to the physics ansatz, the open string endpoints are labeled by a set of branes in the physical target. We let B denote this set of brane labels, where we assume ; … B. In order to account for all interactions, label elements of ı [  by the power set P .B/ (comprised of all subsets of B). The label ; denotes closed strings, and the label fB1 ; : : : ; Bk g  B denotes the formal intersection of the corresponding branes. This intersection in the target may be empty in a given physical circumstance. Definition 3.2. A brane-labeling on a windowed surface F is a function ˇ W ı t  ! P .B/; where t denotes the disjoint union, so that if ˇ.p/ D ; for some p 2 ı, then p is the unique point of ı in its component of @F . A brane-labeling may take the value ; at a puncture.

249

3 Open/closed strings, TFT and CFT A

B

A

B a) open string propagation

;

C b) open string interaction

;

;

c) closed string propagation

A

d) closed string to open string ;

;

;

e) closed string propagation

Figure 3.1. Foliations for several string interactions, where the strings are represented by dashed lines and the dual measured foliation by solid lines.

Example 3.3. Fix some brane label A 2 B, and define the brane-labeling ˇA to be the constant function on ı [  with value fAg; ˇA corresponds to the “purely open sector with a space-filling brane label”. On the other hand, the constant function ˇ; with value ; corresponds to the “purely closed sector”, where each boundary component necessarily contains exactly one distinguished point. To explain the string phenomenology, consider a weighted arc family in a windowed surface F with brane-labeling ˇ. To each arc a in the arc family, associate a rectangle Ra of width given by the weight on a, where Ra is foliated by horizontal lines. We dissolve the distinction between a weighted arc a and the foliated rectangle Ra , thinking of Ra as a “band” of arcs parallel to a whose width is the weight. Disjointly embed each Ra in F with its vertical sides in W so that each leaf of its foliation is homotopic to

250

6 Further applications

a rel ı. These rectangles together produce a measured foliation of a closed subsurface of F , and the leaves of the corresponding unmeasured vertical foliation represent the strings. Thus, a weighted arc family in a brane-labeled windowed surface represents a string interaction. Given such surfaces Fi with weighted arc families ˛i and a choice of window wi of Fi , for i D 1; 2, suppose that the sum of the weights of the arcs in ˛1 meeting w1 agrees with the sum of the weights of the arcs in ˛2 meeting w2 . In this case as in open/closed cobordism, cf. [98], we may glue the surfaces F1 ; F2 along their windows w1 ; w2 respecting the orientations so as to produce another oriented surface F3 ; because of the condition on the weights, we can furthermore combine ˛1 and ˛2 to produce a weighted arc family ˛3 in F3 as in Figure 3.2. We use the brane-labeling to distinguish a collection of arc families in the windowed surface as follows: Definition 3.4. If ˇ is a brane-labeling on a windowed surface F D Fgs .ı1 ; : : : ; ır /, with punctures  , boundary distinguished points ı D ı1 [    [ ır and windows W , then define the sets ı.ˇ/ D fp 2 ı W ˇ.p/ ¤ ;g;  .ˇ/ D fp 2  W ˇ.p/ ¤ ;g: Define a ˇ-arc a in F to be an arc properly embedded in F with its endpoints in W so that a is not homotopic, fixing its endpoints into @F  ı.ˇ/. Given a distinguished point p 2 @F , consider the arc lying in a small neighborhood that simply connects one side of p to another in F ; a is a ˇ-arc if and only if ˇ.p/ ¤ ;. We require these arcs in order to define composition in the open sector while they are disallowed in the closed sector. The previous definition accomplishes this. Definition 3.5. Two ˇ-arcs are parallel if they are homotopic rel ı, and a ˇ-arc family is the homotopy class rel ı of a collection of ˇ-arcs, no two of which are parallel. Notice that we take homotopies rel ı rather than rel ı.ˇ/. A weighting on an arc family is the assignment of a positive real number to each of its components. An arc family is said to be exhaustive if every window has at least one incident arc with positive measure. Let Arc.F; ˇ/ denote the geometric realization of the partially ordered set of all ˇ-arc families in F , i.e., the set of all projective classes of positively weighted ˇ-arc families in F with the natural topology. Let Arcex .F; ˇ/  Arc.F; ˇ/ denote the subspace corresponding to exhaustive arc families and define the quotients QA.F; ˇ/ D Arc.F; ˇ/= PMC.F /; QAex .F; ˇ/ D Arcex .F; ˇ/= PMC.F /:

251

3 Open/closed strings, TFT and CFT

F1

F1

;

F3

;

F2

A

C

B

D A[B F2

a) closed gluing

F1 C A[B

B

F3

b) open gluing of arc with arc

F1

A

C [D

A[B [C

F3

F2

A

g

B

F2

F3

g c) open gluing of curve with curve

g

A

B

C

d) open gluing of arc with curve

g

A[C

F3

F1 D F2

e) open self-gluing of consecutive arcs not comprising a boundary component

A

F1 D F2

A F3

B

B

f) open self-gluing of consecutive arcs comprising a boundary component

Figure 3.2. The various gluing operations on weighted arc families in windowed surfaces.

0 0 Example 3.6. For the brane-labeling ˇ;  ; on F0;2 , there is a unique PMC.F0;2 /0 orbit of singleton ˇ; -arc, and there are two possible PMC.F0;2 /-orbits of ˇ; -arc 0 families with two component arcs illustrated in Figure 3.3. Thus, QA.F0;2 ; ˇ; / is 0 homeomorphic to a circle. If A ¤ ;, then QA.F0;2 ; ˇA / is homeomorphic to a three0 dimensional disk, namely, the twice iterated join with a point of the circle QA.F0;2 ; ˇ; /.

The degree zero indecomposables are illustrated in Figure 3.4, and further useful degree one indecomposables are illustrated in Figure 3.5 (whose respective parts a-e correspond to those of Figure 3.1).

252

6 Further applications ;

;

;

;

;

tm

;

m

.1 C t/m

D.t/, for 0  t  1

.1

t/m tm

D.t/, for 0  t  1

D.0/

Figure 3.3. The twist flow on QA.A; ˇ; /.

A¤; b) arc families in oncepunctured monogons

m1 > m2Cm3 m1 , m2 , m3 satisfy triangle inequalities

a) arc families in triangles

m2 > m1Cm3

A¤;

m3 > m1Cm2

;

c) arc families in annuli m1 Cm2 m3 2

d) arc families in pants

m1 Cm3 m2 2 m1 m2 m3 2

m2 Cm3 m1 2

e) arcs in pants

m2

m3

around the right leg

around the left leg

f) conventions for arcs

Figure 3.4. Geometric realizations of quotient arc complexes.

3 Open/closed strings, TFT and CFT

253

0 Figure 3.4 depicts the geometric realization of QA.F0;.3/ ; ˇ/ in part a) for some brane-labeling ˇ whose image does not contain ; and which is simply omitted from 1 the figure. There is the unique element of QA.F0;1 ; ˇ/ depicted in part b) when the 0 brane label on the boundary is non-empty. For the brane-labeling ˇ on F0;2 indicated in part c), we illustrate instead the homotopy classes of ˇ-arc families rel ı.ˇ/, rather 0 than rel ı as before. Likewise, for the brane-labeling ˇ; on F0;3 , which we omit from the figure, we consider again the homotopy classes of ˇ; -arc families rel ı.ˇ; /, where ı.ˇ; / D ; by definition, and depict the geometric realization in part d). Part e) depicts canonical models of arc families in the pair of pants under the convention of going around right cuffs as in part f). A key point is that relations are derived from decomposable elements, i.e., from the fact that a given surface admits many different decompositions into indecomposables. Thus, pictures of parametrized arc families give rise to algebraic identities. All of the known equations of open/closed string theory, including the “commutative and symmetric Frobenius algebras, Gerstenhaber–Batalin–Vilkovisky, Cardy, and center (or knowledge)” equations, hold on the chain level of weighted arc families as illustrated in Figures 3.6–3.9. In particular, this gives a new calculation of the open/closed cobordism group in dimension two [100], [98] which relies upon canonical models for measured foliations from [127] rather than on Morse theory. Higher degree relations can also be derived; see, for example, Figure 3.11. Next, we explain in more detail the indecomposables. A generalized pair of pants is a surface of genus zero with r boundary components and s punctures, where r C s D 3, with exactly one distinguished point on each 0 1 2 boundary component, that is, a surface of type F0;3 , F0;2 , or F0;1 . A generalized 0 annulus is a surface of type F0;.1;n/ . A (standard) pants decomposition … of a windowed surface F D Fgs .ı1 ; : : : ; ır / is (the homotopy class of) a collection of disjointly embedded essential curves in the interior F , no two of which are homotopic, together with the condition that complementary regions to … in F are either generalized pairs of pants or generalized annuli. If ˇ is a brane-labeling on the windowed surface F , then a generalized pants decomposition of .F; ˇ/ is (the homotopy class of) a family of disjointly embedded closed curves in the interior of F and arcs with endpoints in ı.ˇ/ [  .ˇ/, no two of which are parallel, so that each complementary region is one of the following indecomposable 0 brane-labeled surfaces: a triangle F0;.3/ with no vertex brane-labeled by ;; a gener0 1 2 alized pair of pants F0;3 , F0;2 , or F0;1 with all points ı in the boundary brane-labeled 1 by ;; a once-punctured monogon F0;1 with puncture brane-labeled by ; and boundary 0 distinguished point by A ¤ ;; an annulus F0;2 with at least one point of ı labeled by ;. For instance, if every brane label is empty, then a generalized pants decomposition is a standard pants decomposition. At the other extreme, if every brane label is nonempty, then F admits a decomposition into triangles and once-punctured monogons, namely, a quasi triangulation of F . Provided there is at least one non-empty brane label, we may collapse each boundary component with empty brane label to a puncture

254

6 Further applications B A

a

B

a) Operation

idAB a

A

a b

A

a

C

a

c) Operation ida d) Operation iaA b) Operation mABC ab for A ¤ 0 t/ a.1 at a t a

a b

b

c t

b e) Operation mab

f) Operation mabc g) Operation Bab h) Operation a

at

bt a

a.1

1

2

b.1

t/

b

t/

i) Operation ab 1

2

2

1

a

b

b

a

2

at 1

a.1

t/

b c

b

a.1

at

t/

k) Bracket f; gab

j) Operation abc

a a A

l) Operation P A for A ¤ ;

b

a

1 m) Operation P2a

2 n) Operation P1ab

Figure 3.5. Standard operations of degrees zero and one, where if there is no brane label indicated, then the label is tacitly taken to be ;.

to produce another windowed surface F 0 from F . A quasi triangulation of F 0 can be completed with brane-labeled annuli to finally produce a generalized pants decomposition of F itself. Thus, any brane-labeled windowed surface admits a generalized pants decomposition. Furthermore, any collection of disjointly embedded essential curves and arcs connecting non-empty brane labels so that no two components are parallel can be completed to a generalized pants decomposition.

255

3 Open/closed strings, TFT and CFT

C

A a

a c

b

B

;

C

A c

;

a

;

a

b D

D

B

A

a) Relation: associativity of open gluing D

D

D

a b

c

b

C

c

A

A

C

B

C c

A B

b

A b) Relation: the operator iaA is an algebra homomorphism D D

a

a

;

;

b

C

a c

A

b

B

a A

; a

B A

A

B

a

d) Relation: the center equation

C

b B

B

c) Relation: Frobenius equation in the open sector b b ; A a

b

c

b A

; a

B

B

e) Relation: the Cardy equation

Figure 3.6. Open/closed cobordism relations.

a

c

b

a

c

b

a) Relation: associativity of closed gluing

ab

c

ab

c

a

c

a

bc

a

b c

b) Relation: Frobenius equation in the closed sector

Figure 3.7. Further open/closed cobordism relations: associativity and the Frobenius equation in the closed sector.

We seek a collection of combinatorially defined transformations or “moves” on generalized pants decompositions of a fixed brane-labeled windowed surface, so that finite compositions of these moves act transitively. In particular, then any closed string

256

6 Further applications at

a.1 t/ b

b.1 t/

bt

a

at

as b

bt

at

a.1 s t/

a bs b.1 t/

bt

b a.1 t/

as

a b b.1 s t/ a.1 s/ bs a b.1 s/

Figure 3.8. Closed sector relations: compatibility of bracket and composition.

interaction (a standard pants decomposition of a windowed surface brane-labeled by the empty set) can be opened with the “opening operator” .iaA / illustrated in Figure 3.5d, say with a single brane label A; this surface can be quasi triangulated, giving thereby an equivalent description as an open string interaction. This is called “open/closed string duality”. In particular, the two moves In Figure 3.10a–b act transitively on the quasi triangulations of a fixed surface by Corollary 5.10 of Chapter 4, and likewise the two 0 0 and F0;4 of Figure 3.10c–d were shown in [59] to act “elementary moves” on F1;1 transitively on standard pants decompositions of surfaces, where we include also the s 1 generalized versions of Figure 3.10d on F0;r with r C s D 4 and Figure 3.10c on F1;0 as well (though this includes some non-windowed surfaces strictly speaking). For another example, the Cardy equation can be thought of as a move between the two generalized pants decompositions depicted in Figure 3.6e, and likewise for the four new relations in Figure 3.11b. Theorem 3.7. Consider the following set of combinatorial moves: those illustrated in Figure 3.10 together with the Cardy equation Figure 3.6e, and the four closed/open duality relations Figure 3.11b. Finite compositions of these moves act transitively on the set of all generalized pants decompositions of any surface.

257

3 Open/closed strings, TFT and CFT 1 a

1 c 1 t

1 c

a

b

c 1

1 t

b 1

s

1

c a

b

1 s

a

t

t

1

1

1

1 s

a

s

b

c 1

b 1

s

1 c

1 t

1

c 1

s

a

b c 1 s

1

st

a

1

1 a

.1 s/t

st

b

s.1 t/

b 1

.1 s/.1 t/

1 c

1 s

.1 s/t

s.1 t/

a b

c 1

.1 s/.1 t/

t 1 a

a b 1

1 t

a

c

c

b

s

1

1 1

1

.1 s/t

b 1 b 1 t

s.1 t/

c

t

a 1 s

b 1

1

a 1

b 1 1

1

a 1

b

a 1

1 t

1

t c 1

st

c

.1 s/.1 t/

c

b

t c a 1

1 s

b

c

s

a 1

Figure 3.9. BV equation: the 12 terms of the equation correspond to the 12 heavy segments in the figure.

Proof. In light of the transitivity results mentioned above by topological induction, it remains only to show that the indicated moves allow one to pass between some standard s pants decomposition and some quasi triangulation of a fixed surface F of type Fg;r . This follows from the fact that on any surface other than those in Figure 3.6e and 3.11b, one can find in F a curve  separating off one of these surfaces. Furthermore, one can complete  to a standard pants decomposition … so that there is at least one window in the same component of F  [… as  . Choose an arc a in F  [… connecting a window to  ; the boundary of a regular neighborhood of a [  corresponds to one of the enumerated moves, and the theorem follows by induction.

258

6 Further applications

a) flip

b) quasi flip

c) first elementary move

c) second elementary move

Figure 3.10. Four combinatorial moves, where absent brane labels are arbitrary.

Here is an “omnibus” theorem intended to summarize the further results from [84]: Theorem 3.8. For every brane-labeled windowed surface .F; ˇ/, the space QAex .F; ˇ/ of mapping class group orbits of exhaustive weighted arc families in F support geometrically natural operations of gluing surfaces and weighted arc families along windows. These operations descend to the level of piecewise-linear or cubical chains for example. These operations furthermore descend to the level of integral homology and induce the structure of a modular bi-operad, cf. [98]. Algebras over this bi-operad satisfy the expected equations. Furthermore, new equations can also be derived in the language of combinatorial topology: pairs of generalized pants decompositions of a common brane-labeled windowed surface give rise to families of relations. In degree zero on the homology level, we rederive the known presentation of the open/closed cobordism groups [100], [98]. Further partial algebraic results are given in higher degrees. In order to explain the “expected equations”, define QAex .m; n/ D t QAex .F; ˇ/;

for any m; n 2 Z;

where the disjoint union is over all brane-labeled windowed surfaces .F; ˇ/ with m boundary components with a single distinguished point labeled ; and with n further windows whose endpoint labels are distinct from ;. Let H denote the homology groups with integer coefficients.

3 Open/closed strings, TFT and CFT b

259

b

A

A ;

; ;

BV

a

BV

;

B

a B a) Relation: BV sandwich

b) Relations: closed/open duality

Figure 3.11. New relations.

Theorem 3.9. Suppose ; … ˇ. /. Then an algebra over the modular bi-operad ` H . n;m QA.n; m// is a pair of vector spaces .C; A/ which have the following properties: C is a commutative Frobenius BV algebra .C; m; m ; /, and A D L .A;B2P .B/P .B// AAB is a P .B/-colored Frobenius algebra (see e.g., [98] for the full list of axioms). In particular, there are multiplications mABC W AAB ˝ ABC ! AAC and a non-degenerate metric on A which makes each AAA into a Frobenius algebra. Furthermore, there are morphisms i A W C ! AAA which satisfy the following equations: letting i  denote the dual of i , 12 the morphism permuting two tensor factors, and letting A, B be arbitrary non-empty brane labels, we have 

i B ı i A D mB ı 12 ı mA .Cardy/; i A .C / is central in AA iA ı  ı i

B

D0

.Center/; .BV vanishing/:

These constitute a spanning set of operators and a complete set of independent relations in degree zero. All operations of all degrees supported on indecomposable surfaces are generated by the degree zero operators and .

260

6 Further applications

The extension from topological field theory to conformal field theory is achieved by using Theorem 5.9 of Chapter 4, namely, quasi filling arc families are necessarily exhaustive so that M.F / QA# .F /  QAex .F; ˇ; /  QA.F /: However, the operations do not preserve this subspace which is identified with moduli space M.F /. In further work, Ralph Kaufmann [80], [81] has suitably discretized and filtered the operations to resolve this issue and moreover combinatorially compute correlation functions for CFTs. Part of this work is nicely summarized in [79].

4 Computational biology Jørgen Ellegaard Andersen and I have spent several years applying and adapting to computational biology various combinatorial and geometric techniques of utility for studying not only Riemann’s moduli spaces as in this volume but also more general moduli spaces of flat connections discussed at the end of Section 1.2 of Chapter 1. Initially, our main interest was in protein geometry, but more recent work including also Christian Reidys and others has focused on RNA molecules. In either case, the basic combinatorial structure for modeling such so-called macromolecules is a familiar one as follows. Definition 4.1. A (linear) chord diagram consists of a line segment called its backbone to which are attached a number n  0 of chords with distinct endpoints. There is a natural fattening on any chord diagram where the backbone is taken to lie in the real axis and all the chords to lie in the upper half-plane. Each chord diagram thus has a natural genus of its associated surface1 . Of course, the combinatorial structure of chord diagrams is pervasive throughout pure and applied mathematics including codifying the pairings among nucleotides in RNA molecules [149] or more generally the hydrogen, Watson–Crick or other bonds or contacts among sites of any macromolecule [145], [148], [166]. In this section, we shall survey our earlier and ongoing work on RNA [150], [8], where we take the natural fattening on an appropriate chord diagram, and on protein [145], [146], where we take a different fattening which reflects the intrinsic geochemistry. The section includes new work on the geometry of hydrogen bonds and the moduli space of proteins. 1

A celebrated formula [60] computes the number cg .n/ of chord diagrams of genus g on n chords in the generating function N  X X cg .n/N nC12g 1Cz 1C2 : z nC1 D 1z .2n  1/ŠŠ n0 2gn

4 Computational biology

261

4.1 RNA. This section covers ongoing joint work with J. E. Andersen and C. M. Reidys including M. S. Waterman in [8] as well as joint work including F. W. D. Huang, P. F. Stadler, and M. Nebel in [150]. As one learns in primary school, genetic information is encoded in sequences of nucleic acids, C(ytosine), G(uanine), A(denine), T(hymine) arranged along a (sugarphospate) strand or backbone together with a complementary backbone arranged in a double helix, where the two backbones share so-called Watson–Crick basepairs C-G and A-T. One of these backbones alone is called single-stranded DNA, and a single RNA molecule is a small variation of single-stranded DNA, where T is replaced by U(racil) and the backbone is chemically altered (by removing hydroxyls OH so as to be somewhat more flexible). Definition 4.2. The primary structure of an RNA molecule is the word2 in the four-letter alphabet fC, G, A, Ug determined by the sequential nucleic acids along the backbone. This single strand of RNA can form Watson–Crick basepairs C-G andA-U with itself to fold into biologically active3 forms, but this is a simplistic view of the biophysics since the RNA molecule itself may be more dynamic and have spatial moduli. Nevertheless, there are well-accepted models of primary structure-dependent energetics, and we pose the fundamental: Problem 4.3 (RNA Folding Problem). Compute the minimum free energy configuration(s) of a folded RNA molecule from its primary structure. We may draw the backbone of an RNA molecule as a planar segment connected by chords representing the basepairs as in Figure 4.1, so the basic combinatorics is captured but the geometrical picture wildly distorted in the chord diagram compared to an associated folded RNA molecule. Sometimes, there is no crossing of chords in the chord diagram G of a folded RNA molecule, and this clearly happens if and only if the surface F associated to the natural fattening on G is planar, i.e., has genus zero. This is called a secondary structure on a folded RNA molecule. There is an algorithm for computing the minimum free energy configuration of RNA secondary structures due to Mike Waterman in the 1970s [166] requiring O.n2 /-space and O.n3 /-time computation. This dynamic programming routine depends upon two essential points: there is an efficient recursive construction of all secondary structures, and the energy is suitably adapted to this recursion. More generally, non-planar chord diagrams give examples of what are called pseudoknotted 4 RNA molecules as illustrated in Figure 4.1. A major practical difficulty 2 An inherent asymmetry in the chemistry of the backbone, which we shall not discuss, in fact determines an overall orientation on the backbone and hence a word. 3 There has been a kind of revolution in our understanding of RNA over the recent years. It was long thought that the sole function of RNA was to codify and regulate the production of proteins from RNA in the ribosome. This typically accounts for only a few percent of the genome (which for humans consists of about 3 gig nucleic acids for example), and the rest was ignominiously called “dark matter”. In fact, this dark matter has now been found to have myriad functions and structures, for instance, in HIV and certain cancers. The study of RNA has again become cutting-edge.

262

6 Further applications 87

10 40 1

30

20

50

60

70

80

(a)

1

10

20

30

40

50

60

70

80

87

(b) Figure 4.1. Chord diagram of a pseudoknotted RNA molecule.

is that the enumeration of pseudoknotted RNA structures is NP complete [3], [103]. Nevertheless, pseudoknotting does occur in nature, and in these examples, the pseudoknotting itself is typically of biophysical significance. The ad hoc methods of restricting complexity of the pseudoknotting that have been implemented to minimize free energy still lead to exponential solutions to the RNA folding problem. Definition 4.4. A chord  in a given chord diagram  is called a pimple if  connects consecutive points in the backbone, and it is called a rainbow if it connects the first and the last points in the backbone. Furthermore, two distinct chords with respective endpoints i1 < j1 and i2 < j2 are consecutively parallel if i1 D i2  1  j2 D j1 1, and consecutive parallelism generates an equivalence relation whose equivalence classes are called stacks. Notice that any planar diagram necessarily contains a pimple, whose exclusion consequently prohibits any planar sub diagrams. 4 There are rare RNA molecules with circular backbone, but the backbone is typically not closed. Thus, RNA is only rarely truly knotted in space, hence this moniker for non-planar chord diagrams.

4 Computational biology

263

Definition 4.5. An (RNA) shape is a chord diagram with a rainbow but no pimples so that every stack has cardinality one. There are only finitely many shapes with a given genus g, each of which contains at least 2g C 1 and at most 6g  1 chords (including the rainbow). For example, the four shapes of genus one are illustrated in Figure 4.2.

Figure 4.2. Shapes in genus one.

In the paper, [150], we effectively take the dynamic programming hull of those pseudoknotted RNA structures that can arise by combining ordinary secondary structures with shapes of genus one so as to produce a special class of chord diagrams of unbounded genus. This can be expressed as a multiple context free grammar in order to compute minimum free energy solutions in O.n4 /-space in O.n6 /-time computation. Our method has roughly 20 percent better positive predictive value than comparable state-of-the-art methods but, more than that, is unique in also computing with this same complexity the full partition function, so that probabilities of specific bonds can be found. The extension of these techniques to systems of interacting RNA molecules modeled as chord diagrams on several backbones represents ongoing work. There is a natural partial ordering on the collection of all shapes given by removing chords (other than the rainbow) whose endpoints are forgotten as vertices. This partial ordering further restricts to one on shapes of any fixed genus. Definition 4.6. The moduli space Rg of RNA shapes of genus g  1 is the geometric realization of the partially ordered set of all shapes of genus g. This moduli space of shapes is sufficient for enumerative purposes since there are algebraic algorithms for “inflating shapes to chord diagrams” and for “inserting vertices into chord diagrams” to allow for the possibility of un-basepaired vertices. Furthermore, one can even stipulate a minimum length along the backbone for permitted bonds (with 3 or 4 being biologically correct) and still derive algebraic enumerative schemes, cf. [8]. Shapes are the essential atoms for all of these combinatorial structures. There is a completely basic relationship between RNA shapes and Riemann’s moduli space:

264

6 Further applications

0 0 Theorem 4.7. The complex Gy.Fg;1 /=MC.Fg;1 / of fatgraphs with tail of genus g and one boundary component is isomorphic to the dual of the complex of RNA shapes of genus g on one backbone. That is, our combinatorial model for Riemann’s moduli space of a once-bordered surface of genus g, where the face relation is given by collapsing edges with unique endpoints, is naturally isomorphic to the geometric realization of the set of shapes of genus g, where the face relation is removal of chords other than the rainbow.

The proof is almost a triviality and given explicitly on the level of pairs of permutations describing fatgraphs as in Section 4.2. Namely, given a chord diagram, collapse its backbone to a point yielding a fatgraph with a single vertex that contains a unique loop which arises from the rainbow. If  and  are the permutations describing this univertex fatgraph, then the corresponding fatgraph with tail is described by the permutations  ı  and . This is essentially just the statement of Poincaré duality in a closed surface, where cycles correspond to dual vertices and conversely. Nevertheless, we call this a theorem for its amazing consequence of intimately connecting two heretofore disparate studies. For example, there is a stratification of Riemann’s moduli space corresponding to chord diagrams which are compatible with a given primary structure which would have otherwise been inconceivably unnatural. More generally, analogously defined RNA shapes on several backbones likewise correspond to surfaces with several boundary components. Remark 4.8. In fact, the existence of an isomorphism from Riemann moduli space to a partially ordered set of chord diagrams follows from Kontsevich’s famous calculation of formal Lie groups in [93] together with Morita’s description [119] of the associative version in terms of linear chord diagrams, but this isomorphism had not been explicitly described before. Furthermore, the appearance of shapes and the understanding that this ties in with RNA structures is new. 4.2 Protein. This section begins with a review of [145], which is joint work with J. E.Andersen, M. Knudsen, and C. Wiuf, and then continues discussing further ongoing collaborative work with J. E. Andersen There are 20 amino acids, 19 of which have the common basic chemical structure illustrated in Figure 4.3a, where H, C, N, O respectively denote hydrogen, carbon, nitrogen, oxygen atoms, and the residue R is one of 19 possible sub-molecules. The 20th amino acid called Proline (which is technically an imino acid) has a related chemical structure containing the ring CCCCN of atoms illustrated in Figure 4.3b. In any case, the sub-molecule COOH on the right-hand side of the amino acid is called the carboxyl group, and the NH2 on the left-hand side (or the NHC on the left-hand side for Proline) is called the amine group. The carbon atom bonded to the carboxyl and amine groups is called the alpha carbon atom of the amino acid, and it is typically denoted by C˛ .

265

4 Computational biology

˛

˛

a) Typical amino acid

b) Proline

Figure 4.3. Chemical structure of amino acids.

As depicted in Figure 4.4, a sequence of L amino acids can adjoin to form a polypeptide, where the carbon atom from the carboxyl group of i th amino acid forms a peptide bond with the nitrogen atom from the amine group of the .i C 1/st amino acid, with the resulting condensation of a water molecule (an OH from the carboxyl group of the former and an H from the amine group of the latter), for i D 1; 2; : : : ; L  1. The nature of this peptide bond, the accuracy of the implied geometry of Figure 4.4 and its further notation will be explained presently.

Oi C˛iC1

H

Ci NiC1

C˛i Oi

1

Ci

H C˛i

Ni

RiC1

i

'i

Ri

HiC1

1

Hi 1

Ri

1

Figure 4.4. A polypeptide.

Definition 4.9. The C and N atoms which participate in the peptide bonds together with the alpha carbon atoms form the backbone of the polypeptide, which is described by ˛  CL ; N1  C˛1  C1  N2  C˛2  C2      Ni  C˛i  Ci      NL  CL

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6 Further applications

indicating the standard enumeration of atoms along the backbone. The primary structure of a polypeptide is the ordered sequence of amino acids occurring in the sequential chain along the backbone, i.e., a word in the 20-letter alphabet of amino acids. The tertiary structure of a polypeptide consists of the spatial locations of each of the constituent atoms. Definition 4.10. The i th peptide unit, for i D 1; 2; : : : ; L  1, is thus composed of the consecutively bonded atoms C˛i  Ci  NiC1  C˛iC1 in the backbone plus the oxygen atom Oi from the carboxyl group bonded to Ci and one further atom, namely, the remaining hydrogen atom HiC1 of the amine group (except for Proline, for which the further atom is the carbon preceding the nitrogen of the amine group in the Proline ring). We next describe the key geometrical constraints on polypeptides, where we refer to the center of mass of the Bohr model of a nucleus as the “center” of the atom and to the line segment connecting the centers of two chemically bonded atoms as the “bond axis”. Fact 4.11. Any polypeptide satisfies the following geometric constraints: Fact A: the peptide unit is planar, i.e., the centers of the six constituent atoms lie in a plane, and the angles between the bond axes in a peptide unit are always 120 degrees; Fact B: the angles at each alpha carbon atom C˛i are tetrahedral 5 ; Fact C: the centers of the two alpha carbons of each peptide unit lie on opposite sides of the line determined by the bond axis of the peptide bond (except occasionally for the peptide unit preceding Proline). The mechanism underlying Fact C is that atoms cannot “bump into each other”, i.e., their centers cannot be closer than the sum of their van der Waals radii. This is called a steric constraint as will be pertinent to subsequent discussions. When Proline is in this anomalous configuration, it is called “cis-Proline” as opposed to the usual “trans conformation”. Fact A is fundamental to our constructions, and it arises from purely quantum effects, i.e., a complexity of shared electrons between Oi , Ci and NiC1 . Indeed, Fact A provides for the assignment of a positively oriented orthonormal three-frame, i.e., three pairwise perpendicular unit vectors in an order compatible with the orientation on three space. as follows. Facts A and B together explain the rigidity of protein as a sequence of planar peptide units meeting at tetrahedral angles at the alpha carbon atoms. However, these planes can rotate rather freely about the axes of these tetrahedral bond axes, and this accounts for the relative flexibility of polypeptides. 5 Another geometric constraint on any gene-encoded protein is that when viewed along the bond axis from hydrogen to C˛ i , the bond axes occur in the cyclic ordering Ci , residue, Ni . This imposes various chiral proclivities on proteins but plays no role in our discussion here.

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4 Computational biology

Construction 4.12. Consider a peptide unit in a polypeptide comprised of consecutive backbone atoms C˛i , Ci , NiC1 , C˛iC1 . Let uE i denote the unit displacement vector from Ci to NiC1 . Consider the displacement vector from C˛i to Ci and take the unit vector vEi which is parallel to its projection onto the orthogonal complement of uE i in the plane of the peptide unit. Define the cross product w E i D uE i  vEi , and let Fi D .E ui ; vEi ; w Ei / be the corresponding positively oriented orthonormal three-frame. Definition 4.13. For a polypeptide at equilibrium in some environment, the dihedral angle along the bond axis of Ni C˛i (and C˛i Ci ) between the bond axis of Ci1 D Ni (and Ni C˛i ) and the bond axis of C˛i Ci (and Ci D NiC1 ) is called the conformational angle 'i (and i respectively); see Figure 4.4. Illustrating the physically possible pairs .'i ; i / 2 S 1  S 1 , steric constraints for each amino acid can be plotted in a Ramachandran plot. Definition 4.14. An electronegative atom is one that attracts electrons, e.g., C, N, O in this order of increasing such tendency. When an electronegative atom approaches another electronegative atom which is chemically bonded to a hydrogen atom, the two electronegative atoms can share the electron cloud of the hydrogen atom and attract one another through a hydrogen bond. For example, the Oi or NiC1  HiC1 in one peptide unit can form a hydrogen bond with the Nj C1  Hj C1 or Oj in another peptide unit, where i ¤ j due to steric and other constraints. For another example, many of the remarkable properties of water arise from the occurrence of hydrogen bonds among HOH and OH2 molecules. The absolute potential energy of hydrogen bonds is rather large, so a polypeptide in a given environment seeks to saturate as many hydrogen bonds as possible. In the aqueous environment, the O and N atoms in the various peptide units of a polypeptide might form hydrogen bonds with one another or with the ambient water molecules of their environment, and there may also occur hydrogen bonding involving atoms comprising the residues or the alpha carbons. C˛iC1

Oi Ci C˛i

NiC1 HiC1

(or C for trans-Prolone)

Oi Ri

RiC1

HiC1 (except for trans-Prolone)

Ci C˛i

Oi

C

Oi

NiC1

for cis-Prolone

Ri

Prolone

R iC1

C˛iC1

Figure 4.5. Fatgraph building block.

Remark 4.15. Roughly, protein secondary structure corresponds to hydrogen bonding along the backbone. More precisely, there are two efficient patterns of such hydrogen bonding called “alpha helices” and “beta strands”, and the occurrences and relationships between them is the secondary structure.

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Following [145], we may assign a fatgraph to a polypeptide at equilibrium: Construction 4.16. Given a polypeptide P, construct a corresponding fatgraph G.P/ as follows. Assign to each peptide unit a fatgraph building block in the natural way as illustrated in Figure 4.5. Concatenate these building blocks as illustrated in Figure 4.6, where the two concatenated horizontal segments combine into a single edge called an alpha carbon linkage. Each such edge connects fatgraph building blocks corresponding to two consecutive peptide units, say with corresponding three-frames Fi1 and Fi . ui ; vEi ; w E i / and a unique element There is a unique A 2 SO.3/ carrying Fi1 to Fi D .E B 2 SO.3/ carrying Fi1 to Fi0 D .E ui ; E vi ; w E i /. Twist the alpha carbon linkage, i.e., put an icon  upon it in the notation of Section 4 of Chapter 5 if and only if A is closer than B to the identity in the bi-invariant metric on SO.3/ (with the reverse construction only when exiting a Proline which violates Fact C). This completes the construction of the backbone model in which the endpoints of the vertical segments above and below the horizontal segment respectively represent the atoms Oi and HiC1 except for the vertical segments below the horizontal segment preceding a Proline alpha carbon, whose endpoint represents the non-alpha carbon atom bonded to NiC1 in the corresponding Proline ring.

alpha carbon linkage

Figure 4.6. Concatenating fatgraph building blocks.

Construction 4.17. If there is a hydrogen bond from Hi to Oj in the polypeptide, then add a segment to the backbone model connecting the corresponding vertical segment. Twist the added edge under the analogous condition that the element of SO.3/ carrying Fj to Fi is closer to the identity in SO.3/ than the element carrying it to Fi0 . Adding one such edge for each hydrogen bond among backbone C and N atoms completes the construction of the fatgraph G.P/ from P. Remark 4.18. This abstract typically twisted fatgraph G.P/ is a fattening on the usual graph representing hydrogen bonding that is drawn in biophysics so that the topological type of its associated surface depends robustly on errors. It is actually the equivalence class, not the strong equivalence class, of the fatgraph G.P/ that captures the picture usually drawn in biophysics, cf. [145]. The SO.3/ graph connection discussed in Constructions 4.16 and 4.17 has trivial SO.3/ holonomy; however, the twisted fatgraph

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269

G.P/ that arises from its Z=2-reduction may well have non-trivial Z=2 holonomy,i.e, a closed cycle in G.P/ may traverse an odd number of twisted edges. Certain polypeptides occur as the “proteins” which regulate life as we know6 it. Through experiment, it has been shown that “most” proteins (more specifically socalled globular proteins of a reasonably small size, say, less than 150–200 residues) fold spontaneously at biological conditions to a well-defined “folded” structure. This leads to the famous Problem 4.19 (Protein Folding Problem). Predict the spatial locations of the constituent atoms from the primary structure of a protein. Remark 4.20. In a folded protein, about 70 percent of the backbone C or N atoms participate in hydrogen bonds. The complexity of the basic quantum or classical physics underlying the energetics includes steric, geometric, entropic, electrostatic and so many other constraints that the formulation of a realistic potential function for the thousands to millions of constituent atoms including the ambient water boggles the mind. More generally than our constructions so far, given any two peptide units, there is a corresponding element of SO.3/ carrying the corresponding three-frames to one another, for example, when the two peptide units share a hydrogen bond. We have computed the statistics of which elements of SO.3/ arise for hydrogen bonding over the entire CATH database [125] in the following form. Consider a four-tuple WXYZ, where each of W, X, Y, Z is one of the 20 amino acids. Suppose there are two peptide units P and Q sharing a hydrogen bond from P to Q, where peptide unit P has residues W, X in order along the backbone, and likewise peptide unit Q has residues Y, Z . In this case, we deposit in a data file labeled WXYZ the element of SO.3/ mapping the 3-frame of P to that of Q. We thus produce a library with 204 D 160; 000 files. In Figure 4.7, we illustrate several examples of the distribution of points on SO.3/ for several possibilities of nearby amino acids in scatter plots. In these plots, a point of SO.3/ can be described in its angle-axis form as rotation by an angle ˛ about a unit vector .x; y; z/, and we may conveniently plot the point ˛.x; y; z/ in 3-space, where x is horizontal, y goes into the page, and z is vertical in the figure; this representation is a good one provided the absolute value of ˛ is somewhat less than . For example, the top-left plot labeled *AAL illustrates the distribution of elements of SO.3/ occurring when there is a hydrogen bond from a peptide unit with primary label *A, where * denotes a wildcard, to a peptide unit with primary label AL, i.e., the figure represents the union of all the files with primary descriptor *AAL in our library, where * varies over all 20 amino acids for any possible secondary structures. 6 The collective knowledge of protein primary structures is deposited in the manually curated data banks SWISS-PROT [11] and UNI-PROT [174], which contain about 6,000,000 entries. The collective knowledge of protein tertiary structure is deposited in the Protein Data Bank (PBD) [14], which contains roughly 65,000 proteins at this moment. All of these data are readily available in the public domain online. The atomic locations of each of the constituent atoms of each of these proteins is recorded in the PDB, so it is a vast resource. It is worth emphasizing that the quality of data in the PDB varies wildly from one entry to another.

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6 Further applications

*AAL

DL*D

AA*A

V*GV

Scatter plots for hydrogen bonding over the entire CATH database involving the amino acids A = Alanine, D = Aspartiv acid, G = Glycine, L = Leucine, V = Valine, * = wildcard

Figure 4.7. Distributions on SO.3/ for various primary structures.

These plots in Figure 4.7 are therefore the 3d analogues for hydrogen bonds of the usual Ramachandran plots of conformational angles along the backbone. One sees clearly that there is clustering of the achieved rotations in each of these examples, and this may be expected since a pair of peptide units with fixed primary structure should be able to come into spatial proximity in only several essential ways because of steric and other constraints, i.e., one might reasonably expect “peptide unit legos”. Furthermore, one sees that varying the primary structure in the examples leads to different clustering, and this is of evident relevance and value for the protein folding problem. The “moduli space of proteins” is defined as follows. Given the length L of a protein, define a collection of graphs G, where:

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271

• G contains its “primary backbone”, which is an interval of length L whose vertices correspond to peptide units; • the “secondary edges” represent hydrogen bonds oriented from donor to acceptor connecting the i th to the j th vertices if there is a hydrogen bond between the two corresponding peptide units, where ji  j j  2 and there are at most two such edges pointing towards and at most one away from each vertex, thus allowing simply bifurcated hydrogen bonds; • the “tertiary edges” represent spatial proximity and are not oriented, where a tertiary edge connects vertices i and j if the peptide units are within some distance in a specified sense, where ji  j j  3 and the number of tertiary edges incident on any vertex is at most six, reflecting the empirical fact that the maximum coordination numbers of protein residues is somewhat less than six. The edges E D E1 [ E2 [ E3 of such a graph G thus decompose naturally into types 1, 2, and 3, and we let Ez denote the oriented edges of G. A “geochemical structure” on G is the assignment of z

A  X1  X2  X3 2 .SO.3//E  Œ0; x1m E1  Œ0; x2m E2  Œ0; x3m E3 ; for some chosen huge real numbers xim , for i D 1; 2; 3, where we demand only that A.e/ N D .A.e//1 , i.e., A inverts under reversal of edges of G. Let .G/ denote the collection of all geochemical structures on G with the natural topology, and define an equivalence relation based upon the choice of three cutoffs 0  xic  xim , for i D 1; 2; 3, as follows. The “reduction” of the geochemical structure .G; A; X1 ; X2 ; X3 / arises by erasing any edge (and forgetting its corresponding rotation matrix) whenever X1 < x1c , X2 < x2c , or x3c < X3 . In effect, if the chemical or hydrogen bond is too weak for a primary or secondary edge or the distance is too great for a tertiary edge, then we simply forget the interaction altogether and erase the edge. There is an equivalence relation identifying two geochemical structures which have the same reduction. Definition 4.21. The protein moduli space of length L based on the cutoffs is PL D PL .x1c ; x2c ; x3c / D ..tG .G//= /=SO.3/; where the SO.3/ action is the standard one in gauge theory given by the diagonal action on the vertices. Notice that the topological type of PL depends only on the topological types of .Œ0; xim  fxic g/, for i D 1; 2; 3, and that restricting to x1c D x1m , for example, does not permit the backbone to break and gives the moduli space for a single protein molecule of fixed length L. Of course, the SO.3/ graph connection A on G determined by the locations of the peptide units evidently has no holonomy by construction. On the other hand, we might compute suitable averages over SO.3/ for various subsets of the PDB , and this will essentially always exhibit non-trivial holonomy. Thus, the protein moduli space PL contains the “quasi-physical subspace” where there is no holonomy. Upon

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specifying the primary structure of the protein, there are furthermore steric constraints that come into play and define the “physical subspace” of quasi-physical space where actual protein structures occur. Remark 4.22. The basic construction of PL extends the usual notion of graph connection by including also a further scalar strength of interaction to the edges of the graph. This is seemingly a new and mathematically interesting elaboration of the usual construction in gauge theory.

Epilogue

Though it is pooh-poohed by the poo-bahs, we remain boyishly optimistic that the punctured solenoid may be related to Grothendieck’s absolute Galois theory beyond the obvious adelic coincidence, perhaps involving some pronilpotent analogue. Even further out on this limb, we feel that the decorated Teichmüller space of the punctured solenoid with its locally flickering disparate genera may provide a clue as to what quantum geometry might entail. Teichmüller space has provided paradigms for celebrated properties shared by various moduli spaces of flat connections such as the Goldman bracket and cluster variety structure. We hope that the ideal triangulation of decorated Teichmüller space may likewise find such an extended incarnation. Beyond these bullet points of hope and feeling seems to lie a new frontier of applications to biology. Insofar as the combinatorics of interacting one-dimensional objects is independent of scaling, it is not so surprising that these techniques that have found application in string theory on the Planck scale might also apply at the Ångström scale of macromolecules.

Appendix A. Geometry of Gauss product

The Geometry of the Gauss Product1 R. C. Penner

1. Introduction In Disquisitiones Arithmeticae [1801], Gauss defined a law of composition of PSL.2; Z/ classes of suitable binary integral quadratic forms. Here we give a new geometric interpretation of this Gauss product in the case of definite forms; indeed, we shall find that the product is intimately connected with incidences of hypercycles (that is, loci equidistant to a geodesic) on the modular curve, and the product will be found to be analogous to addition on a non-singular cubic but using suitable hypercycles (instead of lines). We shall also elaborate briefly on the case of indefinite forms, which was actually our starting point. This entire note is based on the group  D PSL.2; Z/, which is intended as a paradigm for the general case of a finite-index subgroup  < PSL.2; Z/. Throughout our discussion, though, we shall keep in mind the more general situation, say, of torsionfree finite-index subgroups  < PSL.2; Z/. Many of our constructions generalize readily as we briefly discuss at the end; however, a suitable geometric interpretation of the Gauss product should give natural analogues of the Gauss groups for each such , and this we have not achieved. Given the very classical nature of what we describe here and the activity in this realm during the period 1940–1970, we remain surprised that this picture of the Gauss product seems to be new. On the other hand, our main result is really about an algorithm for computing Gauss products rather than about the product itself. We can furthermore imagine that nobody bothered to return to the baby quadratic case of ideal class groups in the special case of definite forms armed not only with hyperbolic geometry but also with the 1968 extension of the product described by Butts–Estes–Pall in [25] and [26]. Indeed, recent surveys have described only special cases of this extension (and in fact, we must extend their formulation a bit further still below). We have strived to keep this note entirely self-contained starting from scratch at least in the definite case. The only exceptions are that some routine calculations will be suppressed, our survey below of the contemporary number-theoretic point of view on this is, after all, just a survey without proofs, and our starting point is Dirichlet’s [1851] formulation of the product rather than Gauss’. 1

Included with the kind permission of Springer Science and Business Media ©1996.

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Appendix A. Geometry of Gauss product

We begin by gently introducing and surveying the elementary algebra from first principles and then go on to give a self-contained proof of the existence of the Gauss product in the general (i.e., either definite or indefinite) case in Theorem 6, and our new techniques are already of value here (cf. Lemma 1 below); it is worth emphasizing that we are not saying anything new about the Gauss product at this point other than that there is a well-definedness property of an algorithm for computing it. (Experts should certainly skip the first two sections which are elementary, partly expository, and included for completeness and just glance at Lemma 1 and Theorem 6 in the third section.) We then specialize to the definite case and undertake the geometric study of fundamental roots. Our main result is Theorem 13 giving an explicit geometric formulation of the Gauss product, which is in a sense insufficient for a completely geometric description as we shall discuss. Various number-theoretic and geometric points are finally described in closing remarks, but it remains to be seen whether our formulation of the product might be of real utility in number theory; in the other direction, we can say that the existing databases of class numbers and related data can be interpreted as describing various (reasonably arcane) enumerative behaviors of hypercycles in the modular curve. This note was originally composed as a letter from the author to Yuri Manin on the happy occasion of his birthday, and this explains the informal parenthetical remarks, most of which I have decided to leave in the text. I am lucky to have Dennis Estes as a colleague here at USC and want to thank him for sharing his time and insights over the last months and years. Let me also thank Francis Bonahon, Bob Guralnick, Dennis Sullivan, and especially Don Zagier for helpful and stimulating questions, comments, and corrections. I finally wanted to praise [28] (which has been my basic reference) as well as [175] and to acknowledge the support of the National Science Foundation. 2. Notation, Basics, and Context We study here integral quadratic forms defined on the two-dimensional lattice Z2 , i.e., we study expressions f .x; y/ D ax 2 C bxy C cy 2

for a; b; c 2 Z and x; y 2 Z;

and we shall typically write simply f D Œa; b; c , referring to a, b, c respectively as the “first, middle, last” coefficient of f . We say that f is primitive if gcdfa; b; cg D 1 and shall also call a lattice point .x; y/ primitive if gcdfx; yg D 1.  a b=2 Of course, the symmetric bilinear form corresponding to Œa; b; c is Bf D b=2 , c so f .x; y/ D .x; y/  Bf  .x; y/t , where t denotes the transpose. The discriminant of f is D D D.f / D b 2  4ac D 4 det Bf ;

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277

where det Bf is called the determinant of f . Notice that b  0 .mod 2/ () D  0 .mod 4/; b  1 .mod 2/ () D  1 .mod 4/; so in particular b  D .mod 2/, and D is always equivalent to either 0 or 1 mod 4. We say that f is definite if D.f / < 0, that f is indefinite if D.f / > 0, and that f is singular if D.f / D 0. Of course, if f is definite, then a and c have the same sign (which we shall often take to be positive). The natural action of  2 PSL.2; Z/ (or SL.2; Z/) is by change of basis f .x/ E 7! f .  x/ E on quadratic forms. The corresponding action on symmetric bilinear forms is Bf 7!  t  Bf   D Bf 0 , and we write :f D f 0 and f f 0 in this case. The action evidently leaves invariant the discriminant and furthermore preserves primitivity since a form f is not primitive if and only if there is some prime dividing each element of f .Z2 / (cf. Lemma 2 below). This action induces the natural equivalence relation on the set of forms, and if f D Œa; b; c is a quadratic form, then we shall write Œf D ŒŒa; b; c for the class of f . In light of the previous remarks, both primitivity and the discriminant of a class are well defined. Given a discriminant D 2 Z (i.e., an integer equivalent to either 0 or 1 mod 4), define the Gauss group G .D/ D fŒf W f is primitive and D.f / D Dg; which at this moment is to be regarded as just a set. When D is fixed, we shall write simply G D G .D/. In fact, for each D, G .D/ is a finite set. We shall not prove this here other than to say that one first proves the interesting fact (due to Hermite) that given a primitive f of discriminant D and determinant d D D=4, there is some primitive vector xE so 1 that f .x/ E ¤ 0 and jf .x/j E  jd j 2 . (This, in turn, is proved by simply completing the square in 4af .x; y/ and applying the Division Algorithm.) In Disquisitiones Arithmeticae, Gauss defined a finite abelian group structure on p each G .D/, which is “essentially” (cf. below) the ideal class group of Q. D/, and here is his original idea. Given forms fi , i D 1; 2; 3, we shall think of each with its own copy .xi ; yi / 2 Z2 of the lattice, so fi .xi yi / D ai xi2 C bi xi yi C ci yi2 : Following Gauss, we say that Œf1 Œf2 D Œf3 if f3 .x3 ; y3 / is transformed into the pointwise product f1 .x1 ; y1 /f2 .x2 ; y2 / by a transformation 1 0 x1 x2 Bx1 y2 C C .x3 ; y3 / D T  B @x2 y1 A ; y1 y2

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Appendix A. Geometry of Gauss product

where T is a four-by-two integral matrix whose two-by-two minors generate Z as an ideal over Z (plus a further technical condition on the signs). We have included (nearly) Gauss’original definition just in order to view the product in this natural pointwise way. In fact, our starting point is actually Dirichlet’s definition to be presented below. Our basic plan in the next two sections is to recall and then generalize Cassels’ version [28] of Dirichlet’s proof that the product is well defined on classes of forms, and then in subsequent sections to discuss the underlying geometry in the definite case. To close this section, let us give a quick description of some of the number-theoretic significance and context of this Gauss product. (It is problematic as I am sure you are more expert here than I, but I proceed nonetheless to record various facts mostly from [28].) There is a further equivalence relation on each G D G .D/, where we say that two forms are in the same genus if for all primes p the two forms are equivalent as forms over the p-adic integers. (A specific example of two inequivalent forms in the same genus is Œ1; 0; 82 and Œ2; 0; 41 .) As an abelian group, G .D/ has a unit which we shall denote 1 D 1D 2 G .D/, and the genus of the unit 1D is called the principal genus. An explicit form representing 1D will be given when we need it later, and we choose to write G multiplicatively for our notational convenience. A celebrated calculation of Gauss (boiling down to the pigeon-hole principle!) described in [28] proves that the class Œf 2 G of a form f lies in the principal genus if and only if Œf D Œg 2 for some Œg 2 G ; that is, the principal genus is G 2 . In fact, for any finite abelian group, we may consider the kernel K and cokernel K  of the squaring map g 7! g 2 ; since K and K  are equinumerous Z=2 vector spaces, we find an (non-canonical) isomorphism K K  , so in our case, we find G =G 2 D fgenerag ker.G ! G 2 / D fŒf 2 G W Œf 2 D 1g D fambiguous classesg; where a class Œf is said to be ambiguous if Œf 2 D 1. (The terminology is due to Gauss, and perhaps the idea is that these are the fixed points of the action of the absolute Galois group, which is by the way simply given in this quadratic context by Œa; b; c 7! Œa; b; c .) Let us next make precisepthe sense in which G .D/ is “essentially” the ideal class group of K D K.D/ D Q. D/. If D is either unity or the discriminant of a quadratic field, then it is said to be fundamental, so in the respective cases D  1 .mod 4/ and D  0 .mod 4/, we have equivalently that either D or D=4 is square-free (and in the latter case D D 4ı, where ı is square free and equivalent to either 2 or 3 mod 4). Any discriminant D can be written uniquely as D D df 2 where d is fundamental, and any fundamental discriminant is

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279

uniquely expressed as a product of prime discriminants, i.e., fundamental discriminants with one prime factor, the list of such being 4, ˙8, Cp with p  1 .mod 4/ prime, and q with q  3 .mod 4/ prime. (The fundamental discriminants are the basic ones in the sense that all class numbers are calculable in terms of those of fundamental discriminants; cf. below.) One must introduce the finer equivalence relation of “strict” equivalence on ideals where the ideal A is identified with the ideal xA for x 2 K provided the norm of x is positive (this agrees with the usual notion of equivalence in the definite case), and the corresponding group of ideal classes is the “strict” ideal class group of K. This strict ideal class group surjects onto the usual ideal class group, and the kernel is of order 1 in the definite case and of order either 1 or 2 in the indefinite case. (Indeed for fundamental discriminants, the kernel has order 1 if and only if the ring of integers of K has a unit of norm 1.) The strict ideal class group is isomorphic to the group G .D/ for fundamental discriminants D ¤ 1 in the indefinite case D > 0. In the definite case, one must specialize further and (following Gauss) consider only positive definite forms to construct a Gauss group GC .D/ (so G GC  Z=2Z), and then it is GC .D/ which is isomorphic to the strict ideal class group for fundamental discriminants D < 0. In either the indefinite or definite case, though, there are vast databases of class numbers available (as well as related data). In fact, we were surprised to learn that this formulation of class numbers in terms of quadratic forms is perhaps the most tractable approach computationally, and as a practical matter, special values of Dedekind L functions are actually estimated in terms of quadratic form data (rather than the other way around!). We shall see later how to interpret this known data in the definite case in terms of the geometry of hypercycles in the modular curve. From a contemporary point of view, then, the Gauss groups can be thought of as a sort of quadratic pre-cursor to Kummer’s ideal class groups, certainly at least in the definite case to which we shall turn our attention shortly. On the other hand, Gauss’ genus theory (together with the Hasse–Minkowski invariant) is the contemporary formalism for the local-to-global theory of binary quadratic forms over Z. 3. Dirichlet’s Definition and Its Elementary Consequences Let us first observe that if Œa; b; c is a form of discriminant D, then we may solve for 2 D 2 to conclude that aj b 4D , where we write ujv for u; v 2 Z if u divides v. c D b 4a Now suppose that f1 and f2 are primitive forms of the same discriminant D. We say that f1 and f2 are (Dirichlet) unitable if their classes Œf1 and Œf2 admit respective representatives Œa1 ; b1 ; c1 and Œa2 ; b2 ; c2 where (i) b1 D b2 D b, (ii) gcdfa1 ; a2 g D 1. We shall say that the specific forms Œa1 ; b; c1 and Œa2 ; b; c2 are themselves (Dirichlet) united in this case.

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According to our observation above, there is an integral form Œa1 a2 ; b; of disb 2 D is integral; meanwhile, we similarly conclude criminant D if and only D 4a 1 a2 that 4a1 jb 2  D and 4a2 jb 2  D, so the assumed relative primality of a1 and a2 (together with the fact that b  D .mod 2/) guarantees the existence of an integral form Œa1 a2 ; b; of discriminant D. At the same time, since b2  D c1 c2 D D ; a1 4a1 a2 a2 one can check without pain that the form Œa1 a2 ; b; is primitive if the forms f1 and f2 are primitive. In fact, Dirichlet proved that two classes of primitive forms Œf1 , Œf2 of the same discriminant are unitable, and if Œa1 ; b; c1 , Œa2 ; b; c2 and Œa10 ; b 0 ; c10 , Œa20 ; b 0 ; c20 are pairs of united representatives of Œf1 , Œf2 , respectively, then ŒŒa1 a2 ; b; D ŒŒa10 a20 ; b 0 ; 0 . Thus, the (Gauss) product Œf1 Œf2 D ŒŒa1 a2 ; b; is well defined. This is Dirichlet’s formulation of Gauss’ product, and we shall prove (a generalization of) its well-definedness in Theorem 6 below. Observe that the relative primality condition (ii) is not really so natural, for instance, it is not invariant under the action of PSL.2; Z/. A weaker (and in a sense weakest possible analogous) condition (in the notation above) is (ii)0

b 2 D 4a1 a2

2 Z, that is, a1 a2 j b

2 D

4

:

A pair of forms satisfying conditions (i) and (ii0 ) is said to be (Cassels) concordant, so a united pair is automatically concordant. Given a concordant pair as above, one defines a putative product using the same formula as before. First of all, Cassels proves that the putative product is well defined on concordance classes. We shall find that his proof generalizes handily to the more general setting of (a concordance/united type extension of) the Butts–Estes formulation to be discussed in the next section. Secondly, in the definite case, we shall prove a kind of PSL.2; Z/ invariance of concordant pairs geometrically. Our immediate goal (in the next section) is to formulate a concordance version of the Butts–Estes–Pall product and (following Cassels) prove that this product is well defined; only after these general considerations do we turn finally to definite forms and the modular curve. For the remainder of this section, let us just assume temporarily that given two elements of G , there are concordant representatives the class of whose product (as above) is well defined, and let us investigate some of the elementary group-theoretic consequences. It is convenient and traditional just now to call the equivalence relation generated by PSL.2; Z/ (proper) equivalence and denote it by or p ; we shall say that two forms f , f 0 are improperly equivalent if there is a two-by-two integral matrix  of

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determinant 1 so that :f D f 0 . (So improper equivalence is not an equivalence relation in this standard parlance.) Here are three useful calculations and tricks: • Œa; b; c  is properly equivalent to :Œa; b; c D Œc; b; a using the matrix  D 0 1 , and we call this a “flip”. 1 0 • Œa; b; c is properly to :Œa; b; c D Œa; b C 2a`; a`2 C b` C c using  equivalent  1 ` the matrix  D 0 1 , and we call this “translation by `”. • Œa; b;c is improperly equivalent to :Œa; b; c D Œa; b; c using the matrix 0  D 1 0 1 , and this is the action of Galois as was mentioned before. The unit 1D of G .D/ is evidently represented (with middle coefficient b) by 2 Œ1; b; b 4D , and we may translate to arrange that b D 0; 1 so that ´ if D  0 .mod 4/; ŒŒ1; 0;  D 4 1D D 1D ŒŒ1; 1; 4 if D  1 .mod 4/: We also have ŒŒa; b; c ŒŒc; b; a D ŒŒac; b; 1 D ŒŒ1; b; ac D 1D ; so inversion in the group G is easily described in general as ŒŒa; b; c 1 D ŒŒc; b; a : To close this section, we briefly discuss several generalities, and to begin, we claim that Œf 2 D 1D (that is, Œf is ambiguous) if and only if any representative of Œf is improperly equivalent to itself. Indeed, Œf is ambiguous by definition if and only if Œf D Œf 1 , so if Œa; b; c represents Œf , then Œa; b; c p Œc; b; a p Œa; b; c , which is in turn improperly equivalent to Œa; b; c . As to explicit representatives of ambiguous classes, we have the following two families of ambiguous forms: Œa; 0; c for D D 4ac; Œa; a; c for D D a.a  4c/: It is straight-forward to check that these forms represent ambiguous classes using the tricks and remarks above, but the proof that these represent all ambiguous classes requires a small further discussion of reduction theory. For the reduction to canonical forms, one uses the tricks above and some further calculations to prove • A definite form is equivalent to a “reduced” form Œa; b; c where jbj  a  c, and with the exception of Œa; b; a Œa; b; a and Œa; a; c Œa; a; c , no two reduced definite forms are equivalent.

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p • An indefinite form is equivalent to a “reduced” form Œa; b; c with 0 < b < D p p and Db < 2jaj < DCb, and in this case, the reduced forms are partitioned into disjoint “cycles”, where two reduced indefinite forms are equivalent if and only if they lie in a common cycle. In either case, one considers the principal root p b C D !D ; 2a so ! solves the quadric az 2 C bz C c D 0 and transforms as usual (by integral fractional linear transformations) under the action of PSL.2; Z/. In the definite case, a form is reduced if and only if the principal root lies in the usual fundamental domain for PSL.2; Z/. This is the classical geometric identification whose elaboration is really the main focus of this note. In the indefinite case, the corresponding identification is to consider the two real roots of az 2 C bz C c D 0 as the endpoints of a hyperbolic geodesic which is normalized (i.e., reduced) so that it hits the usual fundamental domain of PSL.2; Z/, and this geodesic corresponds to a cycle of equivalent reduced indefinite forms. Returning finally to ambiguous classes in either case, the respective reduction algorithms show that every ambiguous class contains an ambiguous form. (Furthermore, among reduced definite forms, only Œa; a; a and Œa; 0; a admit isotropy in PSL.2; Z/, while isotropy for indefinite forms is related to integral solutions of the (positive) Pell equation u2  Dv 2 D C4.) 4. Concordance Extension of Butts–Estes United Forms We shall say that two primitive classes Œf1 ; Œf2 are unitable if D.fi / D ti2 d for some discriminant d with ti 2 Z for i D 1; 2, i.e., if D.f1 /D.f2 / is an integral square. In this case, we may set ti0 D ti = gcdft1 ; t2 g

for i D 1; 2:

Following [25], we shall say that a pair of forms representing the classes Œf1 , Œf2 are united if there are respective representatives of the form f1 D Œa1 ; t10 b; t10 a2 c ; 2

f2 D Œa2 ; t20 b; t20 a1 c ; 2

where gcdfa1 ; a2 g D 1. We conclude from primitivity that gcdfai ; ti0 g D 1 for i D 1; 2, and furthermore gcdft10 ; t20 g D 1 always holds. Given united forms in the notation above, we may scale the forms (destroying primitivity) to produce forms of the same discriminant .t20 t1 /2 d D .t10 t2 /2 d , namely, t20 f1 D Œt20 a1 ; t10 t20 b; t10 t20 a2 c ; 2

t10 f2 D Œt10 a2 ; t10 t20 b; t10 t20 a1 c ; 2

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which thus also have the same middle coefficient. Define the product of unitable classes to be the class of Œa1 a2 ; b; c , i.e., Œf1 Œf2 D ŒŒa1 a2 ; b; c ; which we may think of as arising by applying the formula for the product of Dirichlet united forms here to the non-primitive forms t20 f1 , t10 f2 and finally scaling by .t10 t20 /1 . One finds that Œa1 a2 ; b; c is primitive and its discriminant is .gcdft1 ; t2 g/2 d . The result from [25] or [26] which is relevant to us here (which follows from our Theorem 6 and is not the main result of these papers) is Theorem ([25]). Given two unitable classes, there exist united representatives, and the class of the `product 2is well defined. Fixing a fundamental discriminant d and setting .d / D t1 G .t d /, the product above gives .d / the structure of an abelian semigroup. Though [BP] similarly describes a product in the unitable case (using moduletheoretic methods), it is the [25] notion of united representatives that is most important for us here. By the way, [BP] remarks that Gauss already knew about extensions of the product beyond the case of primitive forms with the same discriminant, and Estes and I have checked that the full semigroup is more or less already described in Disquisitiones Arithmeticae. Moreover, upon further review, Estes tells me that Theorem 6 below follows from remarks of Gauss plus remarks in [25]. To get some sense of these semigroups .d / before we continue, it seems worthwhile to pause and recall what is known from order theory: If p is a prime and D is a discriminant, then all of the primitive classes with discriminant p 2 D are represented by the primitive forms on the following list of forms: Œa; pb; p 2 c ; Œah2 C bh C c; p.b C 2ah/; p 2 a , where Œa; b; c runs over G .D/, and h runs through all integers from 0 to p  1. Of course, it follows from primitivity that then gcdfa; ng D 1. We shall not take the time to prove this here since we shall really only need the fact that if n2 divides the discriminant of a form f , for some integer n, then f is equivalent to a form so that n divides the middle coefficient and n2 divides the last coefficient. (This is easily proved directly by taking a representative of the class whose first coefficient is relatively prime to n as in Lemma 2 below, completing the square, and then translating.) In fact, there are canonical surjections G .n2 D/ ! G .D/ defined by simply taking the product with 1D (and the cardinalities of these kernels are known explaining why it suffices to compute class numbers only of fundamental discriminants). Thus, the directed system of Gauss groups has a natural inverse limit, which seems to have not been studied. There is also the following amusing and immediate consequence: Given discriminants D and n2 D and any prime q, the classes of order q k for some k  0 in G .n2 D/ map under the canonical surjection to classes in G .D/ of order q i for some i  k. In order to give an example of the Butts–Estes–Pall product and to better explain the canonical surjections, we observe that if Œf has discriminant n2 D, then by the

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discussion above we may find a representative of the form Œa; nb; n2 c with gcdfa; ng D 1. To take the product with 1D , we may translate (in each case of b even or odd) our standard representative n1D to arrange that the middle coefficient agrees with nb and the first coefficient remains n. The product Œf 1D is thus represented by Œa; b; , where we solve for so that the discriminant is D, i.e., we have ŒŒa; nb; n2 c 1D D ŒŒa; b; c . Here finally is the concordance extension of the Butts–Estes definition. We shall say that two unitable forms are concordant if they admit united representatives, but where we remove the condition that gcdfa1 ; a2 g D 1. Just as in the previous case, we use the same formula Œa1 ; t10 b; t10 a2 c Œa2 ; t20 b; t20 a1 c D Œa1 a2 ; b; c 2

2

to define a product of concordant forms. The main result of this section is simply that this product is well defined on the level of classes (and this subsumes all the various well-definedness-of-product results mentioned before). This extension may seem stupid until one realizes that Lemma 1. Suppose that Œai ; bi ; ci for i D 1; 2 are primitive forms of respective discriminants D1 , D2 , where b1 b2  0. Then the two forms are concordant if and only if the following two conditions hold: • D1 b22 D D2 b12 , and thus Di D ti2 d and bi D bti0 for i D 1; 2; • a1 a2 j b

2 d.gcdft ;t g/2 1 2

4

.

Notice that the conditions b1 p b2  0, D1 b22 D D2 b12 of the lemma are equivalent p to the condition b1 jD2 j D b2 jD1 j. The point of Lemma 1 is that the discriminant divided by the square of the middle coefficient is an “invariant” (whose geometric significance we discover in the next section) which puts concordance into proper perspective and simplifies subsequent calculations. Proof. It is immediate that the stated conditions follow from the definition of concordance. For the converse, suppose first just that f1 D Œa1 ; b1 ; c1 , f2 D Œa2 ; b2 ; c2 satisfy D1 b22 D D2 b12 . It follows that D1 and D2 have the same square-free kernel, so we may write D1 D dt12 ; D2 D dt22 (possibly in several different ways) and set ti0 D ti = gcdft1 ; t2 g as before. Thus, we find that t20 f1 and t10 f2 have the same discriminant, so D1 =b12 D D2 =b22 gives t20 b1 D t10 b2 (and it is here that we use the hypothesis of the lemma that b1 b2  0). Since gcdft10 ; t20 g D 1, we conclude that b1 D bt10 and b2 D bt20 and have proved t20 f1 D Œt20 a1 ; t10 t20 b; t20 c1 ; t10 f2 D Œt10 a2 ; t10 t20 b; t10 c2 : As to the integrality condition, just notice that (for a1 a2 ¤ 0) b 2  d.gcdft1 ; t1 g/2 1 b 2  dti2 D 02 i 4a1 a2 4a1 a2 ti

for i D 1; 2;

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and so (even if a1 a2 D 0), a2 t10 2 jc1 and a1 t20 2 jc2 . It follows directly from this that f1 ; f2 are as stated. q.e.d. In order to prove that the product of concordant forms is well defined, we continue by recalling several standard lemmas (from [28]), where we must here observe that the standard hypothesis of primitivity is a red herring. For this reason, to abide by our stated goal of remaining self-contained, and because of their geometric significance, we shall also briefly recall the proofs. Here are the three lemmas: Lemma 2. Given a primitive form f and any integer M , there is a primitive vector .x; y/ with f .x; y/ relatively prime to M . Lemma 3. If fi D Œai ; bi ; ci , for i D 1; 2 are (not necessarily primitive) forms with gcdfa1 ; a2 g D 1 and b1  b2 .mod 2/, then there are translations 1 ; 2 2 PSL.2; Z/ so that 1 :f1 and 2 :f2 have the same middle coefficient. Lemma 4. Suppose that fi D Œai ; b; ci , for i D 1; 2 are (not necessarily primitive) forms and that there is some ` 2 Z so that `jc1 ; `jc2 ; and gcdfa1 ; a2 ; `g D 1: Then

Œa1 ; b; c1 Œa2 ; b; c2 H) Œ`a1 ; b; `1 c1 Œ`a2 ; b; `1 c2 :

As to the (absolutely standard) proof of Lemma 2, consider the primes p that divide M . Define x; y by taking pjy, p  j x if p  j a (and similarly taking pjx, p  j y if p j c), while if pja, pjc, then p j b by primitivity and we take p j x, p j y. Since gcdfx; yg D 1 by construction, there is some  2 PSL.2; Z/ with first column .x; y/t , whence the first coefficient of :Œa; b; c is relatively prime to M . For Lemma 3, since gcdfa1 ; a2 g D 1, there are integers `1 , `2 with a1 `1 a2 `2 D 1. Since b1  b2 .mod 2/, we may translate fi by `i .b2  b1 /=2 for i D 1; 2 to arrange that the forms have a common middle coefficient. Lemma 4 requires a small calculation. Since we assume f1 f2 (whether or not they are primitive), there is some  D . rt us / 2 PSL.2; Z/ with :f1 D f2 . Equating coefficients (and using that f1 ; f2 have the same middle coefficient), we may eliminate r, u (this is the calculation) to get a1 s C c2 t D 0; a2 s C c1 t D 0:  1 Since `jci and gcdfa1 ; a2 ; `g D 1, we conclude that `js, and the matrix `tr ` u s does the trick. Notice that the form resulting from a primitive form via Lemma 4 is not necessarily primitive (for instance, if `jb and `2 jc). Proposition 5. Unitable classes admit united representatives.

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Proof. Given primitive classes Œf1 ; Œf2 of respective discriminants D1 D t12 d; D2 D t22 d , we may choose (as before) representatives f1 D Œa1 ; b1 ; c1 ; f2 D Œa2 ; b2 ; c2 with gcdfti ; ai g D 1 and ti0 jbi . Let us then apply Lemma 2 to arrange that gcdfa1 ; a2 g D 1, so that then t10 a2 is relatively prime to t20 a1 . Since t20 f1 has the same discriminant as t10 f2 (and discriminants and middle coefficients always have the same parity mod 2), we conclude that t20 b1  t10 b2 .mod 2/. By Lemma 3, we may translate to arrange that t20 f1 and t10 f2 have the same middle coefficient. Since they also have the same discriminant, we find t202 D1 t102 D2 D2 D1 D D D 2; 0 0 2 2 2 .t2 b1 / .t1 b2 / b1 b2 so the first condition of Lemma 1 holds, and b1 b2  0 is automatic. One checks the integrality condition as usual using that gcdfa1 ; a2 g D 1, so the forms are concordant q.e.d. by Lemma 1 and in fact united since gcdfa1 ; a2 g D 1. Theorem 6. The class of a product of concordant forms is well defined giving .d / the structure of an abelian semigroup for d fundamental and G .D/ the structure of a finite abelian group for any D. Proof. Following [28], suppose that we have two concordant pairs f10 D Œa10 ; t10 b 0 ; t102 a20 c 0 ; f20 D Œa20 ; t20 b 0 ; t102 a10 c 0 and similarly f100 , f200 of primitive forms representing a pair of unitable classes. Applying Lemma 2 twice (the first time to f10 with M D t10 t20 a10 a20 a100 a200 and the second time to f20 with M D t10 t20 a10 a20 a100 a200 a1 ), we may find united representatives, say f1 D Œa1 ; t10 b; t102 a2 c ; f2 D Œa2 ; t20 b; t202 a1 c ; respectively, where gcdfa1 ; a2 g D 1 D gcdfa1 a2 ; t10 t20 a10 a20 a100 a200 g: We shall show that

Œa1 a2 ; b; c Œa10 a20 ; b 0 ; c 0 ;

and the result then follows by symmetry (of primed and double-primed variables). To this end, by the relative primality of a1 a2 and t10 t20 a10 a20 , we may apply Lemma 3 as in the proof of Proposition 5 to conclude that there are integers B, C , C 0 with Œa1 a2 ; b; c Œa1 a2 ; B; C D fN; f1 Œa1 ; t 0 B; t 02 a2 C D fN1 ; f2

1 1 0 Œa2 ; t2 B; t202 a1 C

D fN2 ;

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287

Œa10 a20 ; b 0 ; c 0 Œa10 a20 ; B; C 0 D fN0 ; f 0 Œa0 ; t 0 B; t 02 a0 C 0 D fN0 ; 1 f20

1 1 1 2 0 0 Œa2 ; t2 B; t202 a10 C 0

1

D fN20 :

Furthermore, since f1 and f10 have the same discriminant, we find a1 a1 C D a10 a20 C 0 , and by the relative primality of a1 a2 and a10 a20 , there is some integer K with C D a10 a20 K and C 0 D a1 a2 K. Thus, in fact fN D Œa1 a2 ; B; a10 a20 K fN1 D Œa1 ; t 0 B; t 02 a2 a0 a0 K ; 1

1

1 2

fN2 D Œa2 ; t20 B; t202 a1 a10 a20 K ; fN0 D Œa10 a20 ; B; a1 a2 K fN10 D Œa10 ; t10 B; t102 a20 a1 a2 K ; fN0 D Œa0 ; t 0 B; t 02 a0 a1 a2 K ; 2

2

2

2

1

and it remains only to show that fN fN0 . Since fN1 f1 f10 fN10 , we may apply Lemma 4 with ` D t10 a20 to fN1 fN10 to conclude that Œt10 a1 a20 ; t10 B; t10 a2 a10 K Œt10 a10 a20 ; t10 B; t10 a1 a2 K ; and so

Œa1 a20 ; B; a10 a2 K Œa10 a20 ; B; a1 a2 K D fN0 :

Applying Lemma 4 in the same way to fN2 fN20 with ` D t20 a1 gives fN D Œa1 a2 ; B; a10 a20 K Œa1 a20 ; B; a10 a2 K ; so indeed fN D fN0 , completing the proof of well-definedness. Associativity follows as above using Lemmas 2 and 3, units and inverses have already been discussed, and commutativity is obvious. q.e.d. 5. The Geometry of Fundamental Roots 1=2

Let !.f / D bCD be the fundamental root of the primitive form f D Œa; b; c of 2a discriminant D. We assume in this section that f is definite, so D < 0. Thus, r r p b D r p !.f / D D Ci Ci 2 Q C i Q; 2 2a 4a q s where p; q; r; s 2 Z, and we may take gcdfp; qg D 1 D gcdfr; sg, and r; s > 0. A point in upper half space U with rational real and square imaginary parts will be called

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simply a CM point of U, since (as Zagier points out) these correspond to the elliptic curves that admit a complex multiplication. In fact, one can easily check directly that each element of PSL.2; Z/ leaves invariant this CM locus, and it therefore descends to the collection of CM points on the modular curve M D U=PSL.2; Z/ itself. A direct calculation using high-school algebra (which we omit and which is surely standard) proves q Proposition 7. Given a CM point ! D pq C i rs 2 U, the positive definite primitive form f .!/ proportional to f 0 .!/ D Œq 2 s; 2pqs; p 2 s C q 2 r inverts the formula above for fundamental roots. In our first proof of this (unaware of the direct calculation above starting from the fundamental root), we worked in Minkowski space, and it is worth briefly describing this for the insights it affords. Identify R3 with the space of all symmetric bilinear v pairings as usual by .u; v; w/ 7! . wu v wCu /. Pass to positive real projective classes of rational forms and identify a ray from the origin with its point of intersection with the unit hyperboloid; one sees the two disk components of projective definite forms (corresponding to first and last coefficients both positive or both negative) and the annulus of projective indefinite forms. Direct calculation shows that the function in Proposition 7 is given by radial projection of the upper sheet from .0; 0; 1/ followed by the usual complex fractional linear transformation mapping the Poincaré disk (the unit disk at height zero in R3 ) to U. This establishes an isomorphism between the set of projective classes of positive definite rational quadratic forms and the set of CM points in U. As to the primitive form f D f .!/ in the projective class of f 0 D f 0 .!/, observe that if  is a prime dividing the coefficients of f 0 , then jq or js (since  divides the first coefficient), and in either case jq if and only if js (since  divides the last coefficient and using the assumed relative primality of p; q and r; s). Thus, if  divides the coefficients of f 0 , then j gcdfq; sg. In particular, if gcdfq; sg D 1, then f 0 D f is itself primitive, and in this case the discriminant is D D 4q 4 rs. One sees that our map ˚ above is wildly discontinuous. One can go a bit further and show that gcdfq; sg gcd q; s= gcdfq; sg actually divides the coefficients of f 0 , but the explicit calculation of the primitive form f in the projective class of f 0 seems to be out of reach. We regard this overall scale as essentially non-geometric data and must stick to homogeneous rational functions of degree zero in the coefficients. In light of the previous discussion about unitable pairs of forms, we are led to consider the level sets of D=a2 , D=b 2 , D=c 2 , for if two primitive forms lie on a common level set of one of these functions, then they are necessarily unitable. Setting

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!D

p q

Ci

q

r s

D u C iv 2 U, one finds that

r ˛2 ˛ D () v D ; s 4 2 2 p r D=b 2 D ˇ 2 () D ˇ 2 2 () v D ˙ˇu; s q r r 2 p s r D=c 2 D  2 () 2 1 D 2 C ()  2 D u2 C .v   1 /2 ; q r s D=a2 D ˛ 2 ()

where ˛; ˇ;  > 0. These respective loci are thus horizontal lines, rays from the origin, and circles tangent to R at zero as in Figure 1. D b2

ˇ 2; b > 0

bD0

D b2

ˇ 2; b < 0

D a2

˛12

D c2

22

D a2

˛22

D c2

12

Figure 1

To understand the corresponding loci in U and M, we first recall some standard hyperbolic constructions of Poincaré. The limit of a pencil of hyperbolic circles in U passing through a common point with common tangent as the radius and center approach infinity is called a horocycle. In other words, a horocycle in U is either a horizontal line (“centered” at infinity) or a Euclidean circle in U tangent to R (“centered” at the corresponding point of tangency). An "-hypercycle to a geodesic g in U (i.e., g is a vertical half-line or a Euclidean semi-circle perpendicular to R) is a component of the locus of points at distance "  0 from g. In particular, there are two "-hypercycles for each " > 0, and g is itself the 0-hypercycle to g. (Put another way, these hypercycles are the loci of constant curvature which we think of as interpolating between geodesics and horocycles; as such, Bonahon has asked the reasonable question of whether the hypercyclic flow is also ergodic.) In particular, the hypercycles to the imaginary axis in U are simply the Euclidean rays from the origin. See Figure 1. Projections to M of horocycles centered at infinity or hypercycles to the imaginary axis in U will be called simply horocycles or hypercycles in M. See Figure 2. The horizontal foliation in U corresponds to the horocycles centered at infinity, and the leaves of this foliation descend to simple closed curves in M provided they

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lie above height one (i.e., provided they have hyperbolic length less than unity). We shall refer to this canonically foliated once-punctured open disk as the “cusp” of M and to its complement as the “body” of M. Let us take the Euclidean heights of these embedded horocycles (or equivalently their hyperbolic lengths) as a parameter, so the “-horocycle” in M is the horocycle in the cusp corresponding to the leaf at height   1 in U.

hypercycle of large slope

embedded horocycles

Figure 2

Insofar as each hypercyclic ray is asymptotic to the puncture, there is a first and last intersection of any hypercycle with any horocycle that it meets, so each hypercycle has naturally two ends. Furthermore, an end of a hypercyclic ray is standard in the cusp, namely, it is modeled (up to rotation) on a line of constant slope in U. Thus, an end of a hypercycle in M is described by one real parameter (a “slope”) plus a choice of sign (so a “signed slope”), or equivalently by a signed distance (namely, along a fixed embedded horocycle to this last intersection). In fact, the parameter " above for hypercycles is useful only in that discussion, and we shall actually use a different parameter for hypercycles, namely, the  -hypercycle to the imaginary axis in U is the one of Euclidean slope  . This is the first of several appearances of this Euclidean structure on the modular curve, and there is truly something to observe in contrast to the usual case. It is precisely because we are taking hypercycles to a bi-infinite geodesic running from puncture to puncture that this Euclidean structure is defined. (In contrast, the attempt to define an analogous structure relative to a closed geodesic is foiled by the remaining degree of freedom given by translation along the geodesic.)

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Notice that the  -hypercycle flips to the ( )-hypercycle, so we shall also speak of the j j- hypercycle in M. Usually when we uniformize a hypercycle on M, we shall take the representative in U with positive slope. It seems worth pausing to describe two figures. An easy calculation using Proposition 7 and our formulas for ambiguous forms shows that the fundamental roots of the ambiguous forms consist exactly of the CM points either lying on the geodesic in M corresponding to the imaginary axis in U or lying on the projection to M of the frontier of the usual fundamental domain for PSL.2; Z/. Furthermore, an easy verification using Proposition 7 and our formulas for units of Gauss groups, shows that 8q < CD if D  0 .mod 4/; 4q !.1D / D : 1 C CD if D  1 .mod 4/: 2

4

Thus, all units except 13 and 14 have fundamental roots lying in the cusp, and their lifts to the usual fundamental domain for PSL.2Z/ in U alternate between real part zero and real part ˙ 12 as illustrated in Figure 3.

1

1

8

1

7

1

3

4

ambiguous classes

Figure 3

Our final figure to consider is standard and rather involved. First, take the Farey tesselation  (i.e., the tesselation of U generated by reflecting the triangle spanned by 0; 1; 1 about its sides, and so on), so the orientation-preserving symmetry group of  is exactly PSL.2; Z/. Furthermore, PSL.2; Z/ acts transitively on the set of oriented edges of  , so any hypercycle to any geodesic in  admits a lift to U as a

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ray from the origin of positive slope. As usual, the ideal points of  are given their Farey enumeration by the rationals, and the PSL.2; Z/-orbit of the horocycle at height one is a circle-packing, where at each rational number of the form p=q we take the Euclidean circle of radius 1=q 2 . See Figure 4 and imagine how hypercycles to the edges of  may intersect one another and how they may intersect the horocycles in this circle-packing; these are the sorts of configurations in M we shall consider below.

 11

 23  12  13

 01 D

0 1

1 1 2 1 4 3 5 2

3 2 3 5 3 4

1 1

Figure 4

Armed with this discussion of hypercycles and horocycles in M, we return to our calculations before of level sets of D=a2 , D=b 2 , and D=c 2 , which we find correspond respectively to horocycles centered at infinity, hypercycles, and horocycles centered at zero. It is a nice picture. We shall concentrate on the hypercyclic case and relegate the discussion of the (interesting!) horocyclic case to the next section. It is time to reap some results from the discussion, and it is cleanest to formulate them on the level of M. We shall simply identify not only a form with its corresponding CM point in U but also its class with the corresponding CM point of M. Thus, if f is a definite form, then we think of f D !.f / 2 U and Œf D Œ!.f / 2 M. Furthermore, if h is a hypercycle in U to a geodesic in  , then we shall let Œh denote the corresponding hypercycle in M. Corollary 8. Œf1 ; Œf2 2 M are unitable if and only if they lie on a common hypercycle. Furthermore, if Œf1 ; Œf2 lie on a common horocycle, then they are unitable as well. The first part follows from the discussion above and Proposition 5, but notice that two CM points on a given hypercycle might not represent concordant forms, that is, we may have to choose another hypercycle to see them as concordant. Indeed, given a concordant pair f1 ; f2 , there is a corresponding hypercycle h to the imaginary axis

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293

with f1 ; f2 2 h. The second part also follows from the previous discussion, and there is the amusing (and it seems to me highly non-trivial) geometric consequence that two CM points of M which lie on a common horocycle also lie on a common hypercycle (and we wonder about the converse). Corollary 9. Suppose that f1 , f2 are concordant, say with corresponding hypercycle h. Then Œf1 Œf2 also lies on the hypercycle Œh in M. There is truly nothing to prove in light of Theorem 6. This is a remarkable “focusing” property of hypercycles on the modular curve and explains the basic connection between geometry and the Gauss product. We shall further develop this theme and give a geometric characterization of the product in Theorem 13 below. It is worth pausing, though, and inviting you to imagine products, relations, squares, and units in the Gauss groups from this geometric point of view. In preparation for our next result (which is a basic compactness property of M), we develop some generalities about the distribution of G .D/ in M. To this end, let us fix some discriminant D and consider the various b  0 of the same parity as D modulo p two. The points of U which project to G .D/ and lie on the hypercycle of slope D=b 2 are in natural one-to-one correspondence with the various divisors of .b 2  D/=4. In particular, the first and last such points evidently represent the unit 1D , and we have Proposition 10. For any discriminant D, each point of G .D/ lies either in the body of M or in the cusp of M below 1D . It might be interesting to combine Proposition 10 with estimates (which I do not know) for the “discriminant D injectivity radius” to estimate class numbers. On the p other hand, for any Œf 2 G .D/, there is some b with 0  b  D=3 and a p 2 representative f lying on the hypercycle of slope D=b , so one observes a seeming non-uniformity in the distribution of G .D/ in M which suggests that this estimate on class numbers might not be too handy. (Zagier points out that this and “all” other such elementary hyperbolic estimates on class numbers have been tried; furthermore, much more is known about the distribution of heights of G .D/ in M, for instance, all other points of G .D/ are at most half as high as 1D , and there are at most a points whose height is 1=a times the height of 1D .) Arguing as in the proof of Theorem 6, we find that given Œf ; Œg 2 G .D/, there are concordant representatives lying on a common hypercycle. Furthermore, for each fixed D, we can similarly find a b D b.D/ so thatpeach point of G .D/ is represented by a point on the “saturated” hypercycle of slope D=b 2 . Finding such a b (which I do not know how to do effectively) could also give an estimate on class numbers. At the same time, it seems like a nice combinatorics problem to try and express class numbers in terms of multiplicities on a saturated hypercycle. Turning now to the distribution of squares of G .D/ in M, suppose that Œf D Œg 2 2 G .D/, so there is a concordant pair g1 , g2 of forms representing Œg lying

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on the corresponding hypercycle h. In other words, Œg1 D Œg2 is a double point (i.e., self-intersection) of Œh in M. A double point of the hypercycle h is said to be concordant if it arises as above from a concordant pair of forms. There is a particularly simple class of double points as follows: If h is a hypercycle to the imaginary axis in U, then a cuspidal double point is a point of intersection of h with some integral translate of the flip of h. Thus, the imaginary axis itself (as a hypercycle) has no cuspidal double points, and every other hypercycle has infinitely many. Notice that any cuspidal double point lying on a hypercycle of positive slope at least two necessarily lies in the cusp, and for any slope, all but a finite number of its cuspidal double points necessarily lie in the cusp, hence the terminology. (It is easy to construct non-cuspidal double points starting from Figure 4.) Notice that hypercycles are generally dramatically far from general position. For instance, arguing as Theorem 6, we can find hypercycles with multiple points of arbitrarily high orders at any specified set of elements of the Gauss group. (The extent to which the orders of multiplicity can also be specified is an interesting question.) There is a pleasant geometric description of concordance in the case of common discriminants (and at one point I thought just homogeneity of the formula was a small miracle), as follows. Proposition 11. Suppose that two forms f1 ; f2 2 U of the same discriminant lie on a common hypercycle. Then f1 , f2 are concordant if and only if jf1 jjf2 j 2 Z, where jf j denotes the Euclidean norm of f 2 U as a vector in R2 . Proof. Let us write fi D Œai ; b; ci with fundamental root ui C i vi 2 U and discriminant D, for i D 1; 2. It follows from Proposition 7 that b=ai D 2pi =qi D 2ui and D=ai2 D 4ri =si D 4vi2 . Concordance is equivalent to integrality of b b b2  D D C 4a1 a2 2a1 2a2

s

D D 4a12 4a22

D u1 u2 C v1 v2 q q D u21 C v12 u22 C v22 D jf1 jjf2 j; where the next-to-last equality follows from the extreme case of the Cauchy–Schwarz (in)equality using that f1 and f2 are parallel as vectors in R2 since they lie on a common hypercycle. q.e.d. In fact, given two forms f1 , f2 lying on a common hypercycle (not necessarily of the same discriminant), integrality of jf1 jjf2 j is again a necessary condition for the forms to be concordant; a further sufficient condition for concordance is that t10 t20 divide jf1 jjf2 j as one can easily check. In a sense this is a perfectly suitable answer (for given two points on a common hypercycle, we can apply the Euclidean algorithm to find

Appendix A. Geometry of Gauss product

295

their primitive representatives, hence t10 t20 , and hence verify concordance); on the other hand, we would hope for a more intrinsic geometric characterization of concordance for two points on a common hypercycle. Proposition 12. Every cuspidal double point is concordant. Proof. Let h be a hypercycle, say of slope ˇ > 0. A general cuspidal double point on h is then given by f1 D n=2 C i ˇn=2 for n 2 Z. This point on h translates to n=2 C i ˇn=2 on the hypercycle of slope ˇ, which in turn flips to f2 D

2 .1 C i ˇ/ n.1 C ˇ 2 /

on h, and we find jf1 jjf2 j D

2 ˇn 2ˇ n C D 1 2 Z: 2 n.1 C ˇ 2 / 2 n.1 C ˇ 2 / q.e.d.

The result then follows from Proposition 11.

Here is the promised geometric interpretation of the product (and Sullivan points out that never mind anything else, this is a theorem about the Euclidean plane). Theorem 13. Suppose that f1 ; f2 are concordant with corresponding hypercycle h. Then Œf1 Œf2 is represented by the point f 2 h closest to the origin with the property that whenever f1 , f2 translate to concordant forms on the corresponding hypercycle h0 , then f also translates to h0 . Proof. First we show that the usual product on h has the property stated above for f , and then we show that this is actually the closest such point f to the origin. Suppose that f1 D Œa1 ; t10 b; a2 t102 c ;

and

f2 D Œa2 ; t20 b; a1 t202 c

are primitive concordant forms which translate respectively to (primitive) concordant forms f10 D Œa1 ; t10 B; a2 t102 C ; and f20 D Œa2 ; t20 B; a1 t202 C ; so that we have B1 D t10 B D t10 b C 2`1 a1 ;

and

B20 D t20 B D t20 b C 2`2 a2 :

Solving t20 B1 D t10 B2 , we find . /

t20 `1 a1 D t10 `2 a2 :

Of course 1 D gcdft10 ; t20 g, and also 1 D gcdft10 ; a1 g D gcdft20 ; a2 g by primitivity. One concludes from . / that therefore .

/

t10 a2 j`1 a1 I

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Appendix A. Geometry of Gauss product

furthermore, t10 a2 divides the last coefficient of f1 by inspection, so it also divides `1 times this last coefficient. Thus, we may show that t10 a2 j`1 .t10 b/

./

and use that f1 is primitive to conclude that therefore t10 a2 j`1 , as required. To establish that the product Œa1 a2 ; b; c has the stated property, it therefore remains only to prove the integrality condition ./, i.e., we must prove a2 jb`1 . To this end, recall that f1 , f2 and f10 , f20 are each supposed to be concordant pairs, so a1 a2 j

b 2  d.gcdft1 t2 g/2 ; 4

and

a1 a2 j

B 2  d.gcdft1 t2 g/2 ; 4

using the notation of Lemma 1. Taking the difference, we find that a1 a2 divides B 2  b2 1 .t 0 B/2  .ti0 b/2 D 02 i 4 4 ti 1 .ti0 b C 2`i ai /2  .ti0 b/2 4 ti02 1 D 02 `i ai .ti0 b C `i ai /; ti

D

for each i D 1; 2. Taking i D 1, multiplying by t102 , and dividing through by a1 , we conclude that t102 a2 j`1 .t10 b C `1 a1 /: On the other hand, we have that t10 a2 j`1 a1 from .

/, so t10 a2 divides the second term. Hence, t10 a2 also divides the first term, completing the proof of the first part (as was explained before). We must still prove that the product is closest to the origin. Specifically, we show that if a 2 Z satisfies μ Bt10  bt10 .mod 2a1 / H) B  b .mod 2a/; Bt20  bt20 .mod 2a2 / then aja1 a2 , and the desired result then follows from the formula for the real part of the fundamental root. To this end, first notice that Bti0  bti0 .mod 2ai / , B  b .mod 2ai / since gcdfti ; ai g D 1, for i D 1; 2. Thus, we may take t10 D t20 D 1 in the previous inset equation. Now, given a 2 Z, let us take B D b C 2`a1 a2 , for ` 2 Z, where gcdf`; ag D 1. Reduce modulo 2a to find that aj`a1 a2 ; and use gcdf`; ag D 1 to conclude that indeed aja1 a2 .

q.e.d.

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297

To the extent that our characterization of concordance is not completely geometric, as discussed above, so too is the characterization of the product in Theorem 13 deficient. (It may be that a converse of Theorem 13 holds and gives the desired entirely geometric description of concordance as well as the product.) To close, we offer brief remarks on several disparate topics. Throughout this discussion, we shall refer to a torsion-free finite-index subgroup  of PSL.2; Z/ as an arithmetic group. 6. Clever Euclidean Geometry Consider homothety H .z/ D z on z 2 U, so for each  > 0, H setwise preserves each hypercycle to the imaginary axis in U. We wish to analyze two cases of forms Œa; b; c , namely, in the first case: jb; 2 jc (and hence also gcdfa; g D 1); and in the second case: jc and gcdf; ag D 1. In the first case, the map H1 is simply the well-defined homomorphism multiplication by 1d=2 as we calculated before. In the second case, at least H1 is well defined as follows from Lemma 4 above. Of course, we wonder as to the further extent to which homothety is well defined on classes of points on a hypercycle, and we find it remarkable that the Euclidean geometry is somehow clever enough to detect these cases so that homothety acts in its correct well-defined way on these points on a fixed hypercycle in each case. 7. Definite Forms in the Arithmetic Case For any arithmetic group , we can consider -equivalence classes of definite forms to get a corresponding collection of CM points in the surface U= . It is of course tempting to define the notion of -concordance and -Gauss groups geometrically (perhaps by analytic continuation along hypercycles), but I am not certain of some of the details. An improvement in our geometric characterization of concordance would presumably illuminate these points and lead to a simple definition of -Gauss groups. We have also worked on this algebraically for the congruence subgroups .N /, and it looks promising (to simply mimic Lemmas 2–4 above with specified residues modulo N of certain coefficients). From the point of view of the absolute Galois group and the universal Ptolemy group (cf. [135]), the inverse limit of such -Gauss groups (over the usual subgroups of PSL.2; Z/ directed by reverse-inclusion) seems a natural construction. 8. Multiplication on Horocycles Turning to the case of two forms lying on a common horocycle in U, Estes tells me that the classical point of view gives no clue on what to expect from the Gauss product in this case, but I offer the following conjectural idea. There is a canonical involution of the divisible group Q=Z defined as follows: Given a representative p=q 2 Q with gcdfp; qg D 1, there is an essentially unique element  2 PSL.2; Z/ mapping p=q to infinity and hence the horocycle centered at p=q of Euclidean radius 1=q 2 to the

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Appendix A. Geometry of Gauss product

horocycle centered at infinity of height one. Take the image of p=q C i=q 2 under  to define a new point in the horocycle at height one and check that this image point is well defined up to integral translation, so its real part is well defined in Q=Z. More explicitly, one can compute that this involution is given by p=q 7! s=q, where ps  rq D 1. I suspect that the Gauss product on embedded horocycles is related to the twisting of Q=Z by this canonical involution. It is worth pointing out that there are analogous twistings of Q=Z for each arithmetic group ; indeed, there is one such twisting for each puncture of the corresponding surface F D F D U=. Explicitly, given a CM point on a small horocycle near a specified puncture x of F , consider the geodesic through asymptotic to x; the other end of this geodesic is asymptotic to another puncture y of F . If x ¤ y, then is taken to be a fixed point of the involution, whereas if x D y, then the involution interchanges

and , where  arises as the intersection of the small horocycle about x D y with this other end of the geodesic. My original intuition about this was that it should be some sort of “jet of Gauss groups” of definite forms about a singular form. This is a reasonable geometric analogue of considering pairs (ideal class, unit in underlying ring) as is efficaciously done in number theory. Such an extension to the singular case seems to be a new idea numbertheoretically however. 9. Indefinite Forms in the Arithmetic Case A basic difference between the definite and indefinite cases is that PSL.2; Z/ acts discontinuously on U in the former case (and the quotient is the modular curve M), but it does not act discontinuously on the one-sheeted hyperboloid H in Minkowski space in the latter case. (To see this, just take a minimal geodesic lamination on some surface F D U=, where  is arithmetic.) Our basic formulas in Lemma 1 as well as many of our calculations and arguments are not sensitive to whether the forms in question are definite or indefinite, so there is definitely some indefinite version of the theory we have described in this note. It certainly seems natural to imagine H =PSL.2; Z/ (the “indefinite modular curve”) as a non-commutative space in the sense of Alain Connes. A seemingly completely separate aspect in the indefinite case (and this was my starting point in quadratic forms) is the connection with “Markoff tuples” as follows. Fix an arithmetic group , so the quotient F D U=  comes equipped with an ideal triangulation (inherited from Farey). We have described global (“lambda length”) coordinates on the decorated Teichmüller space (see [128]) relative to a specification of ideal triangulation, and we consider the decorated surface corresponding to setting all coordinates equal to unity on this ideal triangulation. This point (the “center of the cell” in the parlance of [128]) is uniformized by ; its coordinates transform under our “Ptolemy transformations” to a collection of integral coordinates, and these tuples of coordinates are called Markoff tuples for . The reason for the terminology is that in the case of the once-punctured torus, these are exactly the classical Markoff triples as discussed in [128]. (Estes pointed this out about my formulas ten years ago!)

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299

It seems clear how to generalize Markoff’s own argument (cf. [38]) from PSL.2; Z/ to the case of arithmetic groups , and we imagine each  as a group of extreme forms, in the sense of Markoff’s theorem, indexed by the corresponding Markoff tuple. Kapranov tells me that Seminaire Rudakov studied a kind of generalized Markoff tuple, and it will thus be interesting to compare our constructions. As a starting point, it should be straight-forward to just check that our Markoff tuples satisfy Rudakov’s generalized Markoff equations.

Appendix B. Dual to the Kähler two form

The Poincaré Dual of the Weil–Petersson Kähler two form 1 by R. C. Penner 

Abstract. We consider a family of conjectures due to Witten which relate the Miller–Morita–Mumford cohomology classes to explicit cycles in a certain cell-decomposition of the moduli space of punctured Riemann surfaces. A version of the first of these conjectures is proved here, and the computational geometric proof leads directly to Rogers’ version of the dilogarithm and its Abel–Spence functional equation.

Introduction. Let M D Mgs denote the uncompactified moduli space of Riemann surfaces of genus g with s  1 punctures (so the real dimension of M is N D x denote the Deligne–Mumford compactification [37] of M, which 6g6C2s), and let M is an orbifold again of dimension N . Recall from [113], [117], [121] the (rational) x Q/, for i  1, which are Miller–Morita–Mumford cohomology classes N i 2 H 2i .MI stable classes in the sense of [57]. Of course, the class N i pulls back under the inclusion x to a well-defined class i 2 H 2i .MI Q/, for i  1, and we refer to these M M classes as the Miller–Morita–Mumford classes on M This paper is dedicated to a family of conjectures by Edward Witten [private communication] about the Poincaré duals of the Miller–Morita–Mumford classes on M. Before discussing the conjectures, we must briefly discuss some background material. Recall (see [22], [57], [128], [130]) that M comes equipped with a canonical “celldecomposition”; namely, there is an explicit decomposition of M into regions, each of which is naturally homeomorphic to the interior of a simplex together with certain of its faces. The “cell-decomposition” does not, so far as we know, extend reasonably to x Thus, we shall be forced to consider cycles of nona true cell-decomposition of M. compact support in our “cell-decomposition” of M which are expressed as rational linear combinations of interiors of simplices in this “cell-decomposition”. In fact, Witten describes for each N  2i  2 an explicit closed rational (N  2i )cycle ŒW2i of non-compact support in our “cell-decomposition” of M and makes the 1 

Included with the kind permission of International Press ©1993. Partially supported by the National Science Foundation

Appendix B. Dual to the Kähler two form

301

Conjecture (Witten). i is a constant multiple of the Poincaré dual of ŒW2i on M, for each i  1. According to [171], the Weil–Petersson Kähler two form ! is a representative x and we now let 1 D ! 2 (actually, the harmonic representative) of 2 2 N 1 on M, 2 denote this form on M. Our main result (which is a weakening of the first of the conjectures above) is that the cycle ŒW2 of non-compact support on M is a constant multiple of the “compact Poincaré dual” of ! on M, Rin the sense that for R any compactly supported closed (N  2)-form  on M, we have M ! ^  D c ŒW2  , for some constant c independent of . In fact, we have Theorem. 1 is the compact Poincaré dual of 16 ŒW2 on M. The proof is a computation using our explicit integration scheme in 7 of [129], which we use here to integrate over various cycles in M rather than just over M itself as in [131]. One ingredient of our approach is a simple lemma about any matrix-model, which allows us, in effect, to apply Stokes’ theorem one cell at a time on M. Using our coordinates on the Teichmüller space, the geometry leads naturally to exactly the Hain–MacPherson version of Rogers’ function [56] as well as to the Abel– Spence functional equation which Rogers’ function satisfies. This paper is organized as follows. 1 is dedicated to a quick review of the necessary decorated Teichmüller theory, and 2 presents our lemma about matrix-models. Various computations with our coordinates are performed in 3 in order to prove in 4 that Witten’s putative cycles are indeed cycles. The computation of the compact Poincaré dual of the Weil–Petersson Kähler two form is presented in 5. 6 is devoted to closing remarks, among which are various comments on the vagaries of compactifying M in the setting of this paper. It is a pleasure to thank Ed Witten for many stimulating conversations and to acknowledge the intellectual debt to him of some of the work presented here. JeanLouis Loday patiently explained connections between polylogarithms and K-theory and made many valuable comments. Not the least of these comments educated me about the Abel–Spence relation and thereby substantially shortened the proof given here (whose original sketch went through a painful partial derivation of the Abel– Spence relation). There were also stimulating and informative discussions with Osmo Pekonen and Dennis Sullivan. Let me finally thank Université Louis Pasteur and the Finnish Mathematical Society, especially Seppo Rickman, for support during Winter, 1990. 1 Background We begin by reviewing the decorated Teichmüller theory [128], [130] and refer the reader to [131] for a more extensive review. Let Fgs denote a fixed genus g oriented topological surface with s  1 punctures, where 2g  2 C s > 0. The (pure) mapping class group M Cgs of Fgs is the group of isotopy classes of orientation-preserving diffeomorphisms of Fgs which must pointwise

302

Appendix B. Dual to the Kähler two form

fix the punctures (whereas we consider the impure mapping class group in [128], [129], [130], [131]). The Teichmüller space Tgs of Fgs is the collection of all marked hyperbolic structures (complete with constant curvature 1) modulo push-forward by diffeomorphisms isotopic to the identity, and the quotient Mgs D Tgs =M Ggs is the moduli space of Fgs , which is the object of central interest classically. We consider the fiber-space Tzgs over Tgs , where the fiber over a point of Tgs is the collection of all s-tuples of horocycles, one horocycle about each puncture. Tzgs is called the decorated Teichmüller space of Fgs and plays a central role in [128], [129], [130], [131]. We z 2 Tzgs , so the shall typically let  2 Tgs denote a Fuchsian group underlying a point  marking is understood. The action of M Cgs on Tgs lifts to an action of M Cgs on Tzgs in the natural way. Define an ideal triangulation  of Fgs to be (the isotopy class of) a collection of arcs disjointly embedded in Fgs running between (not necessarily distinct) punctures, so that each component of Fgs   is an ideal triangle in Fgs with its vertices among the punctures. (The isotopy class of) a subset 0   of an ideal triangulation is called an ideal cell-decomposition or simply an “i.c.d.” of Fgs provided that each complementary region of 0 in Fgs is an ideal polygon. z 2 Tzgs and ˛ is (the isotopy class of) an arc, as above, connecting Suppose that  s punctures in Fg . Straighten ˛ to a geodesic for , choose a lift of ˛ to a geodesic in the hyperbolic plane, and consider the ideal points to which this lift is asymptotic. The z determines a pair of horocycles centered at these ideal points, and we decoration on  z to be define the lambda length of ˛ and  p z D 2e ı ; .˛I / where ı is the signed hyperbolic distance between these two horocycles. This quantity is clearly independent of our choices, and we have Theorem A (Theorems 3.1 of [128] and A.2 of [131]). For any fixed ideal triangulation  of Fgs , the function Tzgs ! R C;

z 7! ƒ; 

z for each ˛ 2 ; where ƒ.˛/ D .˛I /

is a homeomorphism. Furthermore, the pull-back of the Weil–Petersson Kähler two form on Tgs is given by 2

X

d log a ^ d log b C d log b ^ d log c C d log c ^ d log a;

where the sum is over the collection of triangles complementary to  in Fgs , the lambdalengths assigned to the frontier edges of this triangle are a, b, c in this order as traversed in the counter-clockwise sense (in the orientation on Fgs ), and log denotes the natural logarithm.

Appendix B. Dual to the Kähler two form

303

Another ingredient of the decorated Teichmüller theory is the M Cgs -invariant “celldecomposition” of Tzgs , whose attendant combinatorics is effectively captured by fatgraphs and the matrix-model. Recall that a fatgraph is a one-dimensional CW-complex together with a cyclic ordering on the collection of half-edges about each vertex. One associates a surface to a fatgraph in the natural way, and a marking in the usual sense on this surface is called a marking on the underlying fatgraph. A cell in the decomposition of Tgs corresponds to an i.c.d., and an i.c.d., in turn, corresponds via Poincaré duality on the surface to some marked fatgraph. We shall require both the i.c.d. and the fatgraph formalisms here, and therefore require several definitions: Let .G/ be the i.c.d. corresponding to the marked fatgraph G, let G./ be the marked fatgraph corresponding to the i.c.d. , let C ./ be the open cell in Tzgs corresponding to the i.c.d. , set C.G/ D C ..G//, and x let C.G/ denote the closure in Tzgs of C .G/. If  is an ideal triangulation of Fgs and e is an edge of , then the component of s .Fg /[feg containing e is either a once-punctured monogon or a quadrilateral. In the 0

latter case, we may replace e with the other diagonal e of the associated quadrilateral to produce a new ideal triangulation 0 D .  feg/ [ fe 0 g. We say in this case that 0 arises from  by applying an elementary move along e. By definition, the corresponding fatgraphs are related by Whitehead equivalence. The face relation in the “cell-decomposition” of Tzgs is induced by inclusion of i.c.d.’s in the natural way, and two top-dimensional cells share a common codimension-one face if and only if they differ by a elementary move. The dual of this relation gives a partial ordering on ˚ Ggs D marked isomorphism classes of fatgraphs G W .G/ is an i:c:d: of Fgs ; and we refer to the poset Ggs as the fatgraph poset. Thus, G 2 Ggs is the name of a cell C .G/ D C..G//  Tzgs , and it is in this sense that we shall employ both formalisms below. To analyze the structure of cells in the decomposition, fix some G 2 Ggs . To each z 2 C .G/  Tzgs and each edge of G, there is associated its lambda length, that  is, the lambda length of its dual ideal arc in .G/. There is furthermore associated another positive real quantity, called its “simplicial coordinate” (whose definition we shall recall when we need it in Lemma 3 below), and we have Theorem B (Theorem 3.4.1 of [131]). For any marked fatgraph G, let .Ei /N iD1 , denote the tuple of simplicial coordinates associated to the respective edges fei gN iD1 of G. This tuple .Ei /N of positive real numbers gives global coordinates on C.G/. Furthermore, iD1 if G 0 ; G 00 are marked fatgraphs so that .G 00 /  .G 0 /, then C .G 00 /  Cx .G 0 / is determined by the collection of equalities Eij D 0;

for all eij 2 .G 0 /  .G 00 /.

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Appendix B. Dual to the Kähler two form

The decorated bundle Tzgs ! Tgs admits the canonical section where each horocycle in the decoration has hyperbolic length unity, and we may thus regard Tgs  Tzgs . In fact, the “cell-decomposition” of Tzgs restricts to a “cell-decomposition” Tgs . To see this, suppose that G 2 Ggs . The fattening of G determines a marked surface whose boundary components are in natural one-to-one correspondence with the ordered set of punctures of Fgs , and each boundary component of this surface gives rise to a closed z 2 Tzgs , let `i ./ z denote the sum (with edge-path on G in the natural way. For any  unsigned multiplicity) of the simplicial coordinates of the edges traversed by this edgez is the hyperbolic length of path, for i D 1; : : : ; s. By Corollary 3.4.3 of [131], `i ./ th z It follows that the i horocycle in the decorated conformal structure . D.G/ D C .G/ \ Tgs  Tzgs is the affine slice f`i D 1 W i D 1; : : : ; sg of the positive orthant in simplicial coordinates, so D.G/ is in particular a cell, as was asserted. If  is an i.c.d. of Fgs , then we shall also write simply D./ D D.G.//. x The canonical section of Tzgs ! Tgs is everywhere transverse to @C.G/ by [131], Lemma 3.2.2, and therefore Corollary C. For any marked fatgraph G 2 Ggs , let .Ei /N iD1 denote the tuple of N simplicial coordinates associated to the respective edges fei giD1 of G. The affine slice of the simplicial coordinates on Cx .G/ determined by fi D 1 W i D 1; : : : ; sg gives x global coordinates on the closure D.G/ of D.G/ in Tgs . Furthermore, if G 0 ; G 00 are 00 x 0 / is determined by marked fatgraphs so that .G /  .G 0 /, then D.G 00 /  D.G the collection of equalities Eij D 0;

for all eij 2 .G 0 /  .G 00 /:

Turning finally to our integration scheme, given a fatgraph G, let Aut.G/ denote the unmarked fatgraph automorphism group of G, let ŒG denote the unmarked isomorphism class of G, and let # AutŒG denote the cardinality of Aut.G/ for any G 2 ŒG . Another invariant (which detects the “hyperellipticity”) of G is 8 ˆ <1 if there is a non-trivial element of Aut.G/ "ŒG D fixing each unoriented edge of GI ˆ : 0 otherwise; for any G 2 ŒG , and we say that G or ŒG is hyperelliptic if "ŒG D 1. Observe that a hyperelliptic fatgraph G must have either one or two vertices (cf. [131]); thus, there are no cells in Tgs of codimension 0,1, or 2 whose corresponding fatgraph is hyperelliptic unless Fgs D F11 ; F04 ; F12 . We shall only have occasion below to consider integrals of closed forms over certain cycles of non-compact support on Mgs . Namely, suppose that Z is a p-cycle of noncompact support in the “cell-decomposition” of Tgs which is invariant under the action

Appendix B. Dual to the Kähler two form

305

of the mapping class group M Cgs . Thus, the push-forward ŒZ D  .Z/ is a welldefined p-cycle of non-compact support on Mgs , where  W T ! M is the forgetful map, and it is cycles of this type which we shall consider in the sequel. To establish notation for such cycles, enumerate the M Cgs -orbits of p-cells in Tgs as fŒGk gK , and choose an orientation on respective representative cells fD.Gk /gK kD1 kD1 P once and for all. If ŒZ is a p-cycle on Mgs as above, then we write ŒZ D K n ŒG k k kD1 if the restriction of the corresponding cycle Z to D.G` / is nk 2 Q times the specified orientation on D.Gk /. Our integration scheme for this class of cycles on Mgs is Theorem D (Remark 1 after Theorem 2 of [131]). Given a p-cycle Z of non-compact P support on Tgs which is invariant under the action of M Cgs , write ŒZ D K kD1 nk ŒGk , as above, which depends upon the finite collection of specifications of orientations on fD.G` /gN . If is a closed p-form on Mgs , then `D1 Z

D ŒZ

K X kD1

2"ŒGk  nk # AutŒGk

Z

  . /; D.Gk /



where  . / is the pull-back of to a p-form on Tgs , and each D.Gk / is given its specified orientation. Remark 1. It is not difficult to prove that the projection Tgs ! Mgs is an embedding on the closures of cells in the first barycentric subdivision of the “cell-decomposition” of Tgs . In fact, the second barycentric subdivision of the “cell-decomposition” actually descends to a “simplicial decomposition” of Mgs (where the final quotation marks are explained as in the introduction); this point is discussed further in §6. One can easily extend the integration scheme above to one that allows arbitrary cycles of non-compact support on Mgs in this first barycentric subdivision. In fact, this modified and more general integration scheme is already suggested by Lemma 1 below. On the other hand, we shall not need this more sophisticated integration scheme in this paper. The Weil–Petersson Kähler two form ! determines a volume-form ! 3g3Cs (with non-compact support) on both the moduli space Mgs and the Teichmüller space Tgs of Fgs . ! furthermore determines a canonical orientation on the decorated Teichmüller space Tzgs , as follows. Let i denote the hyperbolic length of the horocycle about the i th puncture, for i D 1; : : : ; s. Since the punctures are ordered, !Q D ! ^ d1 ^    ^ ds is a sensible definition, and !Q is thus a canonical orientation on Tzgs . If G 2 Ggs is a trivalent fatgraph, then D.G/ inherits an orientation from that of s Tg , and the integration scheme in Theorem D becomes Z Z X 2"ŒG

D   . /; s # AutŒG Mg D.G/ ŒG

306

Appendix B. Dual to the Kähler two form

with the understood orientation on each D.G/, where the sum is over all unmarked isomorphism classes of trivalent fatgraphs whose corresponding surface is homeomorphic to Fgs . 2 Fatgraphs and Stokes’ Theorem If G 2 Ggs , then recall from 1 that ŒG denotes the unmarked equivalence class and Aut.G/ denotes the fatgraph symmetry group of G. We also let vk .G/ denote the number of k-valent vertices of G for any positive integer k and define s.G/ D s  1 to be the number of punctures of the surface Fgs corresponding to G. Notice first that the Euler characteristic of G (and hence of Fgs ) is simply X .k  2/ vk .G/; .G/ D k

and of course the pair  D .G/; s D s.G/ uniquely determines the corresponding . Furthermore, the codimension of D.G/ in Tgs (or, in other words, surface F 1s 2 fC2sg

the codimension of C .G/ in Tzgs ) is c.G/ D

X .k  3/ vk .G/: k

Notice that # Aut; vk ; s; , and c are actually well-defined functions of the equivalence class ŒG of G, and write, for instance, vk ŒG for the value of vk .G/ for any G 2 ŒG . Define . j /sg D f G 2 Ggs W cŒG D j g; Œ j sg D f ŒG W G 2 . j /sg g; so that each Œ j sg is a finite set. In the sequel, we shall typically fix the genus g and number s of punctures and uniformly drop the subscript g and superscript s. If G 2 . j /, then define @j G D f H 2 . j C1 / W .H /  .G/ g  . j C1 /; ıj G D f H 2 . j 1 / W .H / .G/ g  . j 1 /: Given a pair .G; H / of elements of G , where H 2 @j G (or, in other words, G 2 ıj 1 H ), let ŒG; H denote the unmarked isomorphism class of the pair .G; H /. Fixing ŒG; H , a marking G 2 ŒG uniquely determines a marking H 2 ŒH , and conversely. For any ŒG 2 Œ j , define ˆŒG D f ŒG; H W .H /  .G/ g ‰ŒG D f ŒH; G W .H / .G/ g:

307

Appendix B. Dual to the Kähler two form

Noticing that the elements of ˆ.ŒG / and those of ‰.ŒG / as well occur with certain well-defined multiplicities, we are led to define ŒG ŒG1 D #f H 2 @j G W ŒG; H D ŒG; G1 g ŒG1  ŒG D #f H 2 ıj C1 G1 W ŒH; G1 D ŒG; G1 g; for any ŒG1 2 Œ j C1 . A basic result which allows us to apply Stokes’ Theorem on the fatgraph complex is Lemma 1. Fix any g and s, and set Œ j D Œ j sg for each j . To each pair ŒG; G1 , assign a formal variable XŒG;G1  , where we assume that XŒG;G1  vanishes if .G1 / 6 .G/. Then X ŒG2Œj 

D

1 # AutŒG X ŒG1 2Œj C1 

X

ŒG ŒG1 XŒG;G1 

ŒG;G1 2ˆŒG

1 # AutŒG1

X

ŒG1  ŒG

XŒG;G1  :

ŒG;G1 2‰ŒG1 

Proof. Fix some ŒG; H 2 ˆŒG , and consider the coefficient of XŒG;H  on either side of the equation in Lemma 1. Let the valence tuple of ŒG be v3 D v3 ŒG ; : : : ; vK D vK ŒG , and recall the enumeration of fatgraphs with this valence tuple given in the last step of Theorem 2.1 of [130]. Thus, ŒG occurs in this enumeration ŒG times, where ŒG # AutŒG D QK vk 0 kD3 vk Š k . For each fatgraph G 2 ŒG arising in this enumeration, there are ŒG ŒH distinct edges which one may collapse to produce a fatgraph H 0 so that .G 0 ; H 0 / 2 ŒG; H . Arguing as on p. 45 of [130] (using 1-labelings), we conclude that the coefficient of XŒG;H  on the left-hand side of the formula in Lemma 1 is 1=# AutŒG; H , where Aut.G; H / is the automorphism group of the unmarked pair .G; H /. In the same way, set t3 D v3 ŒH ; : : : ; tK D vK ŒH and consider the enumeration of fatgraphs with valence tuple t3 ; : : : ; tK . Arguing exactly as in the previous paragraph, the coefficient of XŒG;H  on the righthand side of the formula in Lemma 1 is again found to be 1=# AutŒG; H , completing the proof. q.e.d.

3 Computations with Simplicial Coordinates It is more convenient here to consider the i.c.d. formalism as opposed to the fatgraph formalism. Recall that we have let .G/ denote the i.c.d. of F D Fgs corresponding to the marked isomorphism class of the fatgraph G whose associated surface is homeomorphic to F . We may add geodesics to .G/ to produce an ideal triangulation  of F , and we linearly order the edges of   .G/ D fei gM iD1 . To each ei 2 

308

Appendix B. Dual to the Kähler two form

is associated its simplicial coordinate Ei , and it follows from Corollary C above that dE1 ^    ^ dEM is a co-normal vector to D.G/ in T . Insofar as T comes equipped with its orientation derived from the two form ! (as in 1), we conclude Lemma 2. Given G 2 G , choose an ideal triangulation  containing .G/, and choose a linear ordering on   .G/. Then there is a canonically determined orientation on D.G/. This is the way in which we shall typically specify an orientation on a cell of T in the sequel. Of course, in the special case that G 2 . 1 /, the unique non-triangular region complementary to .G/ in F is an ideal quadrilateral, and an orientation on D.G/ is thus determined by a choice of diagonal of this quadrilateral. To understand how these orientations transform, suppose that , 0 are ideal triangulations of F , where 0 arises from  by applying a single elementary move, and adopt the notation of Figure 1, where the lower-case symbol next to an edge indicates

b0

b

a

e

d 

c

a0

e0

c0

d0 0

Figure 1. Notation for an elementary move.

its corresponding lambda length, and the edge labelled x 0 of 0 corresponds to the edge labelled x of . At the same time, we shall let the corresponding upper-case letter X and X 0 denote the simplicial coordinate of the respective edges. As was promised in the introduction, the formula ED

ab.c 2 C d 2  e 2 / C cd.a2 C b 2  e 2 / abcde

defines the simplicial coordinate E on the edge labelled “e” in terms of the lambda lengths a, b, c, d , e of nearby edges. The analogous formula is used to define the simplicial coordinate of any edge. Recall (see [128]) that up to the action of the Möbius group, there is a unique lift of the vertices of the quadrilateral illustrated in Figure 1 to points in the light-cone in Minkowski space so that the lambda length of each edge is the square-root of the

Appendix B. Dual to the Kähler two form

309

negative of the corresponding Minkowski inner-product. The closed convex hull of these points is a tetrahedron, and the signed volume V of this tetrahedron is related to the simplicial coordinate E on the edge labelled e by the formula V D abcde E. This gives a geometric interpretation to the simplicial coordinate. Lemma 3. Given ideal triangulations , 0 , where 0 arises from  by applying an elementary move along some edge, adopt notation as in Figure 1. Letting X denote the simplicial coordinate of an arbitrary edge of , we have X0 E0 B0 C C 0 0 A C B0 C E0 A0 C D 0 C 0 C D0 C E 0

D X; if X … fA; B; C; D; Eg D E; D B C C C E; D A C B; D A C D C E; D C C D:

The proof follows immediately from the definition of simplicial coordinates as the reader can easily verify. Actually, we shall only require the second formula E 0 D E in the sequel but include these other linear formulas here for completeness. Proposition 4. Continuing with the notation developed above for Figure 1, we have @.E 0 ; C 0 /  0 provided C D 0; @.E; C / @.A0 ; E 0 ; C 0 /  0 provided A D C D 0; .ii/ det @.A; E; C / .i/ det

where det denotes the determinant, and .X10 ; : : : ; Xn0 / with respect to .X1 ; : : : ; Xn /.

@.X10 ;:::;Xn0 / @.X1 ;:::;Xn /

Proof. We begin with part (i), and observe that  @.E 0 ; C 0 / 1 D @C 0 @.E; C / @E

0

@C 0 @C

denotes the Jacobian of

 ; 0

by the second part of Lemma 3. Thus, we must show that @C  0 if C D 0, and this @C is obvious from the geometric interpretation of simplicial coordinates: Increasing the simplicial coordinate C from zero alters the configuration of points in the light-cone in such a way that the corresponding tetrahedron has positive volume, and this in turn implies that C 0 must have increased from zero as well. Turning to part (ii), we have 0 @A0 @A0 @A0 1 0 0 0 @A @E @C @.A ; E ; C / 1 0 A D@ 0 @.A; E; C / @C 0 @C 0 @C 0 @A

@E

@C

310

Appendix B. Dual to the Kähler two form

by the second part of Lemma 3, and so @.A0 ; E 0 ; C 0 / det D  det @.A; E; C /

@A0 @A @C 0 @A

@A0 @C @C 0 @C

! :

Again resorting to the geometric interpretation of simplicial coordinates, it is obvious 0 0 0 0 that @A D 0 D @C since A D C D 0, while @A  0, @C  0 as before. q.e.d. @C @A @A @C  4 Witten’s Cycles Consider the collection Wj D f G 2 . j / W vj C3 .G/ D 1 g of cells in G D Ggs . We shall see that each cell D.G/, for G 2 W2i , admits a canonical orientation, and with these orientations, W2i becomes a cycle with non-compact support on T . In fact, these cycles are obviously equivariant for the action of the mapping class group on T , and W2i therefore descends to a cycle ŒW2i of non-compact support on M. To understand the orientation on a cell in Wj , consider the corresponding i.c.d. .G/. By definition, there is exactly one non-triangular complementary region of .G/ in F , and this region P is an ideal .j C 3/-gon. Choosing a vertex v of P , there is a canonical ideal triangulation v of P defined by the condition that each edge of v has v as an ideal endpoint. The orientation on F determines a linear ordering fek gjkD1 on v , as in Figure 2. There is thus a canonical orientation on D.G/ determined (as in Lemma 2) by each choice of vertex v of P , namely that determined by the co-normal vector dE1 ^    ^ dEj , where Ek denotes the simplicial coordinate on the edge ek , for k D 1; : : : ; j . Let v denote the orientation so determined on D.G/ for each G 2 Wj , j  1, and choice of vertex v. Lemma 5. Let P be the .j C 3/-gon complementary to .G/ in F , where G 2 Wj . Suppose that v1 and v2 are consecutive vertices of P , and let them determine the respective orientations 1 D v1 and 2 D v2 , as above, on D.G/. Then 1 D 2 if and only if j is even. That is, (i) if j  0.2/, then 1 D C2 ; (ii) if j  1.2/, then 1 D 2 . Proof. Suppose that v1 ; v2 are consecutive vertices of P occurring in this order in a counter-clockwise traversal of the frontier of P , and enumerate v1 D fek gjkD1 , where the ordering on subscripts as before reflects the canonical ordering induced by the orientation of F . Set 0 D v1 , and recursively define ` to be the ideal triangulation derived from `1 by an elementary move along e` , for ` D 1; : : : ; j . The edge e` of `1 gives rise to an edge e`0 of ` in the natural way; see Figure 3.

Appendix B. Dual to the Kähler two form

311



e1

e2

P for j D 2 Figure 2. Polygon P and its co-orientation.

As in Lemma 2, the specification of co-normal vector dE10 ^    ^ dE`0 ^ dE`C1 ^    ^ dEj ; determines an orientation on D.` /, where Ek (and Ek0 , respectively) denotes the simplicial coordinate of ek (and ek0 ). According to Proposition 4i, the first and last elementary moves in the sequence 0 ! 1 !    ! j each preserve these orientations, while each of the intermediary .j  2/  j.2/ moves reverses orientation by Proposition 4ii. Thus, the composition of moves is orientation-preserving if and only if j is even, as was asserted. q.e.d. Each cell in W2i therefore admits a canonical orientation, and we henceforth let W2i represent the corresponding chain on T of non-compact support. The main result of this section is Theorem 6. For each i, the chain W2i is a cycle of non-compact support on T . Furthermore, each such cycle descends to a rational cycle of non-compact support on M. Proof. We first show that W2i is a cycle, and suppose that G 2 W2i and G1 2 . 2iC1 / x with D.G1 /  D.G/. There are the following two possibilities for .G1 /: Either it has one complementary .2i C 4/-gon in F , or it has a complementary .2i C 3/-gon in addition to a complementary quadrilateral in F . In the first case, Lemma 5ii (together with symmetry) shows that the coefficient of D.G1 / in the boundary of W2i must vanish. In the second case, there are exactly two

312

Appendix B. Dual to the Kähler two form

1

e20

2 e1

e2

e2 e10

e10

Figure 3. Evolution of co-orientation under elementary moves.

cells in G with D.G/ in their boundary, and these contribute with opposite signs by the second part of Lemma 3. We again conclude that the coefficient of D.G1 / must vanish, so the chain W2i is indeed a cycle. Since the action of the mapping class group preserves the “cell-decomposition” and any mapping class is induced by an orientation-preserving homeomorphism of F , it follows that W2i descends to a rational cycle of non-compact support on the moduli space, as was asserted. q.e.d. Let ŒW2i denote the cycle of non-compact support on M corresponding to W2i by the previous theorem. 5 The Dual of the Weil–Petersson Kähler Two Form This entire section is dedicated to a proof of Theorem 7. Consider a surface F D Fgs and let ! denote the Weil–Petersson Kähler two form on the moduli space M of F . For any closed .6g  8 C 2s/-form  of compact support on M, we have Z Z 2 !^D : 3 ŒW2  M Our proof is computational and relies on the integration scheme. Hyperelliptic fatgraphs occur in codimension 0, 1, or 2 only for the surfaces F11 , F04 , F12 , and we assume to begin with that our surface Fgs is not among these exceptional cases. As before, we shall also typically drop the subscript g and superscript s. For this section only, we establish the conventions that for any real quantity x

Appendix B. Dual to the Kähler two form

313

defined on Tz , we set xN D log x; xQ D d log x: Thus, for any trivalent fatgraph G the Weil–Petersson Kähler two form on the corresponding moduli space is given by X ! D 2 aQ ^ bQ C bQ ^ cQ C cQ ^ a; Q where the sum is over all vertices v of G, and we assume that lambda lengths of the half-edges of G incident on v are a, b, c in this cyclic order (as determined by the fattening). Suppose now that  is a closed R .6g  8 C 2s/-form of compact support on M. Our main task here is to compute M ! ^ . According to the usual integration scheme applied to a volume-form on M, we have Z Z X 1 !^D ! ^ ; # AutŒG D.G/ M ŒG2Œ0 

where each D.G/ is given the orientation induced by ! (as in 1). For each G 2 G with ŒG 2 Œ 0 , define X 

G D aN bQ C bN cQ C cN aQ  fbN aQ C cN bQ C aN cg Q ; where the sum is again over all vertices of G and the notation is as above for !. Notice that G is not invariant under Ptolemy transformations (see [128], [129]), but G is clearly “natural” in the sense that for any f 2 Aut.G/, we have f  . G / D G . Of course, !jD.G/ D d G jD.G/ , so by Stokes’ Theorem (since  is compactly supported) Z Z !^D D.G/

x @D.G/

G ^ :

Notice that for each edge e of .G/, .G/  feg is an i.c.d. of F if F is oncepunctured, whereas in the multiply-punctured case, there are codimension-one faces of D.G/ lying “at infinity” (and, of course,  vanishes on these faces). For any G1 2 G so that ŒG; G1 2 ˆ.ŒG /, let DŒG;G1  .G1 / be the orientation on x with its specified orientation. Thus, D.G1 / induced as a face of D.G/ Z Z X 1 !^D

G ^  # AutŒG @D.G/ x M ŒG2Œ0  Z X X 1 D ŒG ŒG1

G ^  # AutŒG DŒG;G1 .G1 / ŒG2Œ0  ŒG;G1 2ˆŒG Z X X 1 D

G ^ ; ŒG1  ŒG # AutŒG1 DŒG;G1 .G1 / ŒG1 2Œ1 

ŒG;G1 2‰ŒG1 

314

Appendix B. Dual to the Kähler two form

where we have used naturality of G and the hypothesis that  is compactly supported in order to apply Lemma 1 in the last equality. Choose now an orientation on D.G1 / for each G1 2 . 1 / (that is, choose a diagonal e of the quadrilateral complementary to .G1 / in F ), and let the other possible diagonal be denoted by f , as in Figure 4. Suppose that G D G.feg [ .G1 //, H D G.ff g [ .G1 //, and define

G1 D G  H 

D aQ

bd ef





C bQ

ef ac





C cQ

bd ef



C dQ



ef ac





C eQ

b

ac bd



C fQ





ac : bd

b

e

a

f

a

c

c

d

d

Figure 4. Lambda lengths on edges near an elementary move.

Recall that ef D ac C bd by Ptolemy’s Theorem [131], whence ac.aQ C c/ Q C bd.bQ C dQ / eQ C fQ D ; ac C bd so that, in fact, 

G1 D .aQ C c/ Q

bd ac C bd



    ac C bd ac ac.aQ C c/ Q C bd.bQ C dQ / Q Q C .b C d / C : ac bd ac C bd

Notice that this previous expression for G1 is independent of e and f , is closed, and changes sign under the cyclic permutation of a, b, c, d . An elementary calculation shows that G1 D d G1 , where 

G1



bd D2' ; ac C bd

and '.x/ D 12 fLi2 .x/P  Li2 .1  x/g is the Hain–MacPherson [56] version of Rogers’ 2 n function (Li2 .x/ D x denoting Euler’s dilogarithm). Notice that G1 n1 n

Appendix B. Dual to the Kähler two form

315

changes sign under a change of orientation on D.G1 /. (The choice of integration constant here is dictated by our application of Lemma 1 below.) Invoking Stokes’ Theorem again, we find Z Z X 1 !^D G1 ^  # AutŒG1 @D.G x 1/ M ŒG1 2Œ1  Z X X 1 D ŒG1  ŒG2 G1 ^ : # AutŒG1 DŒG1 ;G2 .G2 / ŒG1 2Œ1 

ŒG1 ;G2 2ˆŒG1 

In order to proceed and apply Lemma 1, we must address the invariance of the integrals in the previous equation. There are two kinds of pairs .G1 ; G2 / here: case (a): .G2 / has two complementary quadrilaterals, and the edge in .G1 /  .G2 / triangulates one of these quadrilaterals. case (b): .G2 / has exactly one complementary pentagon, and the edge in .G1 /.G2 / decomposes this pentagon into a triangle and a quadrilateral. Pairs of the latter type obviously admit no non-trivial automorphisms. Consider a pair .G1 ; G2 / of the former type. An automorphism of the pair must fix the quadrilateral complementary to .G1 /, and we choose a diagonal e of this quadrilateral. may preserve e setwise (in which case the orientation on DŒG1 ;G2  .G2 / and G1 are both unchanged), or it may not (in which case the orientation on DŒG1 ;G2  .G2 / and G1 both change sign). Thus, we may apply Lemma 1 (again using that  is compactly supported), to the effect that Z Z X X 1 !^D ŒG G1 ^ : 1 ŒG2  # AutŒG2 M DŒG1 ;G2 .G2 / ŒG2 2Œ2 

ŒG1 ;G2 2‰ŒG2 

The terms in the previous sum are of two types as in case (a) and (b) above. In the former case, .G2 / has a pair of complementary quadrilaterals, and there are four fatgraphs Gi ; Gi i ; Gi i i ; Giv whose corresponding cells are incident on that corresponding to G2 , as illustrated in Figure 5a. Owing to the change of orientation, the contribution from ŒGi ; G2 annihilates the contribution from ŒGi i i ; G2 , and the contribution from ŒGi i ; G2 annihilates the contribution from ŒGiv ; G2 . In the remaining case (b), .G2 / has a complementary pentagon P , and we adopt the notation for the frontier edges and vertices of P as in Figure 5b. (The subscripts will be taken modulo 4 here, so v5 D v1 for instance.) There is a natural correspondence between the frontier edges and the various possible diagonals of P (as is also indicated in the figure), and we adopt the notation of Figure 5b for these diagonals. We also illustrate in Figure 5b the various i.c.d.’s m which contain .G2 / and let G m D G.m /, for  m D 0; : : : ; 4. To specify an orientation on D.G m /, we specify that the edge em is the

316

Appendix B. Dual to the Kähler two form

chosen diagonal of m so that the canonical orientation on D.G2 / (as in the previous section) actually agrees with the orientation DŒG m ;G2  .G2 / for each m D 0; : : : ; 4.

.Gi /

.Gi i /

.G2 /

.Giv /

.Gi i i /

Figure 5. The link of two disjoint quadrilaterals.

Thus, the contribution to where

R M

! ^  from ŒG2 2 Œ 2 is ˇŒG2 4 X

1 # AutŒG2 

R D.G2 /

,



 ei ei  ei ei C ei1 eiC1 iD0   4 X 1 D 2 ' e ei C1 1 C i 1 e e iD0

ˇŒG2 D 2

'

i i

4 X  D 2 ' iD0

1  ; 1  i

with i D CR.viC2 ; viC1 ; vi I vi1 / ei1 eiC1 D ei ei by [131], where CR.a; b; cI z/ is the value of z under the Möbius transformation taking a, b, c, respectively, to 0; 1; 1.

317

Appendix B. Dual to the Kähler two form

Actually, we shall employ the notation of [40] for cross ratios and set fa W b W c W d g D so

1 1i

ac ca ; bacd

D fvi1 W viC2 W vi W viC1 g; for each i D 0; : : : ; 4, and we define 0 D 1 D 2 D 3 D 4 D

1 1  2 1 1  3 1 1  4 1 1  0 1 1  1

D fv1 W v4 W v2 W v3 g; D fv2 W v0 W v3 W v4 g; D fv3 W v1 W v4 W v0 g; D fv4 W v2 W v0 W v1 g; D fv0 W v3 W v1 W v2 g:

Lemma 8. Set  D fa W b W c W d g. Then we have fb W c W d W ag D

 ; fa W c W b W d g D 1  ; 1

and fa W b W d W cg D

1 : 

We shall require these identities below but omit the proofs (which are standard computations of anharmonic ratios). Now returning to the previous computation, we have ˇŒG2 D 2

4 X

'.i /

iD0

X 5 2 L.i /; 4 3 4

D

iD0

2

where L denotes Dupont’s version [40] of Rogers’ function (so '  2L C 6 ). Recall that the functional equation on L is L.x/ C L.1  x/ D  4 X

2 ; 6

.1/i L. 1 i / D 0;

iD0

where i D fv0 W    W vyi W    W v4 g.

318

Appendix B. Dual to the Kähler two form

e2

.G 0 / 0

e3

.G 1 /

e1

e4

e0 1

4

e1

e3

2

e2

.G 4 /

3

.G2 /

e4

e0 .G 3 /

.G 2 /

Figure 6. The link of a pentagon.

On the other hand, Lemma 8 gives i D 1  1 i for i D 0; 2; 4; i D 1 i for i D 1; 3; so we conclude that o n 2 X 5 2 .1/i L. 1 4   ˇŒG2 D  i / 3 2 4

iD0

5 2 D 2  3 2 D : 3

319

Appendix B. Dual to the Kähler two form

Finally, Z

2 !^D 3 M 2 D 3

X ŒG2 2ŒW2 

Z

1 # AutŒG2

Z  D.G2 /

; ŒW2 

with D.G2 / given its canonical orientation as in 4, where the second equality follows from a final application of Theorem D. This completes the proof in case Fgs is different from F11 ; F04 ; F12 . For F11 and 4 F0 (whose moduli spaces are R of complex dimension one), we have W2 D ;, and our argument above shows that M ! ^  D 0 for any compactly supported zero-form . For F12 , one checks easily (or see Table 1 in [131]) that there is exactly one hyperelliptic fatgraph in codimension 0, 1, or 2, it occurs in codimension 2, and it is of type (a) above. The same argument therefore applies, and the proof is complete. q.e.d.

6 Closing Remarks A natural extension of Witten’s conjecture in the introduction is that the closure of x actually represents a multiple of the Poincaré dual of N i , for the cycle ŒW2i in M each i . One might also reformulate the problem and consider some other orbifold x Letting compactification M0 of M which admits a natural continuous map M 0 ! M. i0 2 H 2i .M0 ; Q/ denote the corresponding pull-back of N i for each i , one seeks a compactification M0 so that the closure ŒW2i0 of ŒW2i in M 0 is a cycle representing a multiple of the Poincaré dual of i0 for each i . In fact, we have studied a particular compactification which arises quite naturally and next briefly survey this work-in-progress to discuss certain conjectures in the current context. We shall limit the discussion here to the case of once-punctured surfaces just for simplicity. In this case, the decorated Teichmüller space Tz T R is a product, where T denotes the Teichmüller space. By definition of lambda lengths and homogeneity of our formulas, the coordinates on Tz descend to projective coordinates on T , and, what is more, the “cell-decomposition” of Tz descends to a “cell-decomposition” of T . There is thus an embedding T ! A, where A is the simplicial complex (considered in [57]) consisting of all projectively weighted families of isotopy classes of disjointly embedded essential arcs in the surface with endpoints at the puncture. A itself is therefore simply the natural simplicial completion of T  A. The mapping class group acts on A in the natural way, and we let M0 denote the quotient. Simplices in A  T have infinite isotropy. As alluded to in the Remark following Theorem D in 1 above, the second barycentric subdivision of A descends to a true simplicial complex on M0 . Conjecture 1. M0 is an orbifold.

320

Appendix B. Dual to the Kähler two form

(This conjecture is false in general, cf. Theorem 5.14 of Chapter 5.) In the setting of planar surfaces (for which the simplicial completion of T is not all of A and we do not here define M 0 ), we have proved the corresponding conjecture using a rather involved argument. It is a non-trivial geometric fact that there is actually a natural continuous mapping 0 x and we let  0 2 H 2i .M0 ; Q/ denote the pull-back of N i for each i . FurM ! M, i thermore, let W2i0 denote the collection of simplices of codimension 2i in A defined in analogy to W2i . Each simplex in W2i0 admits a canonical orientation just as before, and the associated chain W2i0 on A is again a cycle of non-compact support. This cycle is evidently invariant under the action of the mapping class group, and hence W2i0 descends to a rational cycle ŒW2i0 of compact support on M 0 . Conjecture 2. ŒW2i0 is a constant multiple of the Poincaré dual of i0 on M0 , for each i  1. We mention parenthetically that in the case of the once-punctured torus F11 , we x and the Poincaré dual of the Weil–Petersson two form is  2 times the have M0 D M, 6 class of the ideal point, where the extra factor of two here (in comparison to Theorem 7) comes from the hyperelliptic involution. Thus, both of the conjectures above hold for the surface F11 . This completes the discussion of Theorem 7 in the context of our work-in-progress on the compactification M0 of M, and we turn finally to other closing remarks. It is appealing to believe that the computational scheme employed here might be a useful tool for deriving analogues of Rogers’ function and its Abel–Spence functional equation for higher polylogarithms. Of course, a fundamental ingredient for us is an explicit hyperbolic formula for the Weil–Petersson Kähler two form (namely, the formula in Theorem A, which, in turn, depends on [169]), and the lack of such a formula for the higher Miller–Morita–Mumford classes could be a basic obstruction to such a dream. We finally mention that [92] has given explicit and simple formulas in the simplicial coordinates for Witten’s closely related “visible” cohomology classes [167] on M, and he has used the integration scheme described herein to explicitly compute their various correlations; one might also use the computational scheme of this paper to find the compact Poincaré duals of these classes.

Appendix C. Stable curves and screens

Stable curves and screens on fatgraphs1 R. C. Penner and Greg McShane

Abstract. The mapping class group invariant ideal cell decomposition of the Teichmüller space of a punctured surface times an open simplex has been used in a number of computations. This paper answers a question about the asymptotics of this decomposition, namely, in a given cell of the decomposition, which curves can be short? Screens are a new combinatorial structure which provide an answer to this question. The heart of the calculation here involves Ptolemy transformations and the triangle inequalities on lambda lengths.

Introduction Throughout this paper, F D Fgs will denote a fixed smooth oriented surface of genus g with s  1 punctures, where 2g  2 C s > 0, with mapping class group MC.F /. Let T .F / denote the Teichmüller space of F D Fgs and Tz .F / denote the trivial s (R>0 )-bundle over it. Let M.F / D T .F /=MC.F / denote Riemann’s moduli space x /. MC.F / also acts on Tz .F / by with its Deligne–Mumford compactification M.F permuting the numbers assigned to punctures. There is an MC.F /-invariant ideal cell decomposition [57], [66], [128], [153] of z T .F / which has found wide application in geometry and physics [10], [58], [60], [68], [92], [116], [115], [130]. Cells in this decomposition are in one-to-one correspondence with homotopy classes of “fatgraph spines” of F , that is, a homotopy class of embedded graph in F in the usual sense together with cyclic orderings on the half-edges about each vertex. (See the next section for further precision.) 1 Included with the kind permission of Greg McShane and Proceedings of the Research Institute for Mathematics, Kyoto University ©2007.

RCP is happy to acknowledge useful discussions with Alex Bene, Kevin Costello, and Dennis Sullivan and to thank the Laboratoire Emile Picard in Toulouse and especially the Max Planck Institute for Mathematics in Bonn for warm hospitality.

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Thus, to each fatgraph spine G of F , there is a corresponding cell C.G/  Tz .F /. x / combinatorially, it is natural to ask: In the interests of understanding M.F Question 1.1. Given G and given a collection K of non-parallel and non-punctureparallel disjointly embedded and essential simple closed curves in F , when is there zn / 2 C.G/, for n  1, so that the hyperbolic lengths of the geodesic a sequence . curves homotopic to the components of K tend to zero for large n and all other lengths remain bounded below? In other words, which multicurves can be short in C.G/? We give in this paper a complete answer to this question, as follows, where we shall concentrate in this introduction on the case that G is trivalent for simplicity. Let E denote the set of edges of G and consider any proper subset A  E. There is a smallest (not necessarily connected) subgraph GA of G containing A, and we say that A is “recurrent” if GA has no univalent vertices. (Again, see the next section for a more detailed discussion of recurrence.) Suppose A is recurrent and GA is connected, and get rid of all bivalent vertices of GA in the usual way to produce either a simple cycle in G or another trivalent fatgraph G 0 . A neighborhood of GA  G in F is a subsurface of F , an annulus in the former case and a punctured surface of negative Euler characteristic in the latter. Define the “relative boundary” of A to be the edge-path in G of the simple cycle itself in the former case and those of the boundary components of this subsurface in the latter case, where you discard any such cycles that are puncture-parallel in F itself. The new combinatorial structure which provides the answer to Question 1.1 (and was introduced in [140]), a “screen on a fatgraph G” is a subset A of the power set (i.e., the set of subsets) of the set E of edges of G with the following properties: i) E 2 A; ii) each A 2 A is recurrent; iii) if A; B 2 A with A \ B ¤ ;, then either A  B or B  A; iv) for each A 2 A, [fB 2 A W B is a proper subset of Ag is a proper subset of A. Condition i) is simply a convenient convention, conditions iii-iv) are familiar from Fulton–MacPherson [49], and here we impose the further condition ii) of recurrence. Notice that the properness condition iv) and recurrence condition ii) together imply that if GA is a simple cycle in G, then for any screen A on G with A; B 2 A and A \ B ¤ ;, we must have A  B, i.e., simple cycles are necessarily atomic in any screen. Each element A 2 A other than A D E has an immediate predecessor A0 2 A, and regarding A as a set of edges in GA0 in the natural way, has its relative S boundary @A A defined before. Finally, the “boundary” of the screen itself is @A D A2AfE g @A A. Here is the answer to Question 1.1, our main result: Theorem 1.2. For any fatgraph G, the cell C.G/ admits as short curves a family K of non-parallel and non-puncture parallel disjointly embedded and essential simple closed curves in F if and only if K D @A for some screen A on G.

Appendix C. Stable curves and screens

323

Let us immediately do several examples, where it is typically easiest to study the quotient of Tz .F / by the natural R>0 -action, the projectivized space, which we shall denote P Tz .F / D Tz .F /=R>0 T .F /  s1 ; where p denotes the open p-dimensional simplex. In particular for a once-punctured surface, we have P Tz .F / D T .F /.

Figure 1. Screens for the once-punctured torus correspond to ideal vertices in the decomposition of P Tz .F 1 /. 1

Example 1.3. For the once-punctured torus F D F11 , the ideal cell decomposition of T .F / is the Farey tesselation of the disk [128]. In Figure 1, we depict a typical top-dimensional 2-cell, which is indexed by a non-planar fatgraph G with two 3-valent vertices as is also illustrated. The codimension-one cells arise by collapsing any one of the three edges shown as darkened in the figure, and the codimension-two cells at infinity are indexed by the three possible recurrent subgraphs of G as likewise illustrated. In this example, the boundary of a screen always consists of a single curve. Example 1.4. For the four-times punctured sphere F D F04 , consider the Mercedes sign fatgraph G depicted in Figure 2. Both screens A1 D fE; fa; b; a0 ; b 0 gg and A2 D fE; fa; b; c; a0 ; b 0 g; fa; c; b 0 gg correspond to pinching to zero the closed edgepath a  b  a0  b 0 , and both screens have this same edge-path as boundary. Example 1.5. Consider the sub-fatgraph of a fatgraph G with edges E depicted in Figure 3 and the screen ˚ A D E; fa  kg; fa  gg; ff  gg; fb  eg; fi  kg

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b0

c0

a c

b

a0

Figure 2. A fatgraph for the four-punctured sphere.

i

e

f

d

h j

a k

g

c b

Figure 3. Typical example.

on G, where “ak”, for example, denotes a; b; : : : ; k. The boundary of A is comprised of the four edge-paths f  g, b  c  d  e, h  i  j  k  h  f  g, and a  b  c  d  e  a  f  g. We shall rely on “lambda length” coordinates from [128] (recalled in 3) on the decorated Teichmüller space Tz .F /, where the fiber over a point is taken to be the set of all s-tuples of horocycles, one horocycle about each puncture; one may take the hyperbolic lengths of the distinguished horocycles as a convenient coordinate on the fiber. In effect, we shall record the rates of divergence of lambda lengths regarded as projective coordinates on P Tz .F /, and the crucial point is that in the cell C.G/, the lambda length coordinates on G must satisfy all three strict triangle inequalities at each vertex of G (cf. Lemma 2.3). This is what forces the recurrence condition. The proof of Theorem 1.2 depends upon the explicit calculation of holonomies using “path-ordered products” of matrices (due to Bill Thurston and Volodya Fock [43] independently and recalled in 3). The proof further requires estimates on the absolute traces of the representing matrices. To this end, we find a condition weaker than the triangle inequalities which satisfies two properties: 1) the condition is invariant under certain “Whitehead moves” (see the next section for a definition) sufficient to simplify

Appendix C. Stable curves and screens

325

the path-ordered product; and 2) the condition guarantees the required estimates on the absolute traces. This is the heart of the paper (in 5), and the techniques involve only “Ptolemy transformations” (cf. Lemma 2.11a), path-ordered products, and the triangle inequality. Because the argument at heart only depends upon these formulae, we are optimistic that the current paper may have ramifications more generally for cluster algebras [50] and cluster ensembles [44]. Since Wolpert has recently announced [173] that lambda lengths are strictly convex along Weil–Peterssen geodesics, we are likewise optimistic about applications to the asymptotic WP geometry. There is furthermore a program to extend the cell decomposition of moduli space to the Deligne–Mumford compactification using screens, which is already well underway (as discussed in the closing remarks 7). 2. Fatgraphs and recurrence A graph is a finite one-dimensional CW-complex with no isolated vertices whose 1cells are edges and whose 0-cells are vertices. The set of edges of G will be denoted by E.G/. A half-edge of an edge e 2 E.G/ is either one of the two components of the interior of e with an interior point removed, and the valence of a vertex is the number of half-edges containing the vertex in their closures, said to be incident on the vertex. A fatgraph is a graph together with a cyclic ordering on the half-edges incident on each vertex. In particular, a finite CW-decomposition of a circle is an example of a fatgraph, as is a planar tree where the cyclic ordering is induced by the counter-clockwise orientation on the plane. A fatgraph G determines a punctured surface F 0 .G/ gotten by assigning to each kvalent vertex an oriented ideal k-gon, whose sides correspond to the incident half-edges, and finally identifying in the natural way pairs of sides of these polygons associated to pairs of half-edges contained in a common edge of G. The vertices of the ideal polygons are identified to the punctures of F 0 .G/. Each edge e 2 E.G/ gives rise to its dual ideal arc ˛.G;e/ connecting punctures in F 0 .G/. An ideal triangulation of Fgs is the homotopy class of a set of arcs connecting punctures in Fgs , called ideal arcs, which decompose the surface into a collection of triangles with vertices at the punctures. More generally, an ideal cell decomposition is the homotopy class of a subset of an ideal triangulation which decomposes the surface into polygons. Provided each vertex of G has valence at least 3, f˛.G;e/ W e 2 E.G/g is an ideal cell decomposition of F 0 .G/ said to be dual to G. Conversely, the Poincaré dual of an ideal cell decomposition of Fgs is a fatgraph embedded in Fgs each of whose vertices has valence at least 3, where the cyclic ordering in the fatgraph structure is induced by the clockwise order in the oriented surface Fgs . A fatgraph G also determines a corresponding oriented surface F .G/ with boundary constructed by assigning to each k-valent vertex an oriented .2k/-gon, whose alternat-

326

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ing sides correspond to the incident half-edges, and as before, identifying pairs of sides of these polygons corresponding to pairs of half-edges contained in a common edge of G. The alternating unpaired edges of these polygons comprise the boundary of F .G/. We may regard F .G/  F 0 .G/ as a strong deformation retraction in the natural way. In particular, G is a strong deformation retraction or spine of F .G/ or F 0 .G/. It follows that any free homotopy class of essential curve in F .G/ or F 0 .G/ gives rise to a closed edge-path in G, which is uniquely determined up to its starting point provided we demand that the edge-path is efficient in the sense that it never consecutively traverses the same edge of G with opposite orientations. A closed edge-path in G corresponding to a boundary component of F .G/ will be called a boundary component of G itself. An efficient boundary component of G must have edges of G incident on only one side. Put another way for a trivalent fatgraph, an efficient edge-path is a boundary component if and only if it consists entirely of left turns or consists entirely of right turns. Suppose that G is a fatgraph with set E D E.G/ of edges and corresponding surface F D F .G/. Any subset A  E determines a subgraph by including all vertices of G on which edges in A are incident. Furthermore by restriction, the fattening on G induces a fattening on this subgraph, which thus determines a well-defined sub-fatgraph GA . We may regard F .GA / as a subsurface embedded in the interior of F in the natural way. Define the boundary of A to be the collection @A of (unoriented) efficient closed edge-paths in G corresponding to the relative boundary of F .GA / in F D F .G/, that is, the collection of closed edge-paths corresponding to the components of the boundary @F .GA / which are not homotopic to boundary components of F itself. In particular, if GA is a circle, then @A is the closed edge-path of GA if this circle is not boundary parallel in F , and @A is empty if this circle is boundary parallel in F . We say that A  E is recurrent if for every edge a 2 A, there is an efficient closed edge-path a in G so that a traverses a and traverses only edges in A. Any subset A  E has a (possibly empty, e.g., in the case of a planar tree) maximal recurrent subset R.A/, namely, the set of edges of A traversed by an efficient closed edge-path in GA . Lemma 2.1. Suppose that G is a fatgraph and A  E D E.G/. Then the following are equivalent: i) A is recurrent; ii) there is a function W E ! Z0 whose support is A so that for each vertex of G with incident half-edges e1 ; : : : ; ek and extending the function to be defined P on half-edges in the natural way, we have that kiD1 .ei / is even, and the generalized weak triangle inequalities hold, i.e., for each j D 1; : : : ; k, X

.ei /I

.ej /  i¤j

iii) every vertex of GA has valence at least 2.

Appendix C. Stable curves and screens

327

Proof. First suppose that A is recurrent, and let a .e/ be the number of times that a chosen a traverses e for each a 2 A and e 2 E. Each a W EP ! Z0 satisfies the restrictions of condition (ii), hence so too does their sum D a2A a , which has full support on A. Thus, (i) implies (ii). Conversely, suppose that is a function supported on A satisfying the properties of condition (ii). For each k-valent vertex of G, there is a dual ideal k-gon in the corresponding punctured surface F 0 .G/, and we shall construct a family of arcs properly embedded in this k-gon realizing the values of on the dual edges of G as the geometric intersection numbers. These arc families in the k-gons then combine uniquely to produce disjointly embedded curves in the natural way, whose component simple closed curves in F have corresponding edge-paths which satisfy the required properties. The construction in each k-gon proceeds by induction on k  2 with notation for incident edges as in condition (ii). In case k D 2, simply take a collection of

.e1 / D .e2 / arcs crossing the bigon. For the case k D 3, take 1 1 Œ .ei1 / C .ei2 /  .ei3 / D Œ .ei1 / C .ei2 / C .ei3 /  2 .ei3 / 2 Z0 2 2 parallel copies of the arc joining edges ei1 to ei2 , where fi1 ; i2 ; i3 g D f1; 2; 3g. For the induction step, take a consecutive pair of edges ei ; eiC1 so that .ei / C .eiC1 / is least among all consecutive pairs of edges, here taking the indices modulo n so that enC1 D e1 . Cutting along the diagonal separating ei and eiC1 from the rest decomposes the k-gon into a .k  1/-gon and a triangle. Extend to a function defined on the edges of these regions by taking value .ei / C .eiC1 / on the diagonal, so the generalized triangle inequalities hold on each region by our choice of consecutive edges, and the parity condition holds by construction. By the inductive hypothesis, appropriate arc families exist in each region, and they combine in the natural way to give the required arc family in the k-gon itself. It follows that (i) is equivalent to (ii). If GA has a univalent vertex, say with incident edge a 2 A, then there can be no efficient edge-path in GA traversing a, so (i) implies (iii). To see that (iii) implies (ii), define to take value 2 on the edges in A and vanish otherwise, and note that satisfies condition (ii) provided GA has no univalent vertices. Suppose that e is an edge of a fatgraph G with distinct endpoints. We may collapse e to a vertex to produce a new fatgraph G 0 , where the cyclic ordering at the resulting vertex arises by combining the cyclic orderings on the half-edges incident on e in the natural way. Dually, one removes the ideal arc ˛.G;e/ from the dual ideal cell decomposition. If G is a trivalent fatgraph and e is an edge of G with distinct endpoints, then a Whitehead move on e is the fatgraph that results by collapsing e and then un-collapsing the resulting four-valent vertex in the unique distinct manner. A Whitehead move along an edge e is depicted in Figure 4, which furthermore indicates the notation near an edge e which we shall adopt in many of the calculations of this paper.

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Whitehead

d

a 



ı c

ˇ " ˛ e

move

f

b

Figure 4. Standard notation for Whitehead moves.

Using the characterization Lemma 2.1iii), it follows directly that recurrence is invariant under Whitehead moves on trivalent fatgraphs and is furthermore in any case invariant under collapse of edges e with distinct endpoints neither of which is univalent. 3. Coordinates The reader is referred to [128] or the more recent treatment in this volume for proofs and further details on the material which is recalled in this section. We begin with several formulae on horocycles in the hyperbolic plane. If h, h0 are horocycles in the hyperbolic plane with distinct centers in the circle at infinity, then consider the unique geodesic .h; h0 / connecting their centers. The horocycles h; h0 truncate .h; h0 / to a geodesic segment of some finite signed length taken to be positive ifp and only if h and h0 are disjoint. Define the lambda length of 0 0 h; h to be .h; h / D e ı . (This is a different normalization p for lambda lengths than in [128], for instance, where the lambda length is taken as 2 e ı , cf. [140].) Lemma 3.1. Suppose h1 ; h2 ; h3 ; h4 are horocycles with distinct centers occurring in this clockwise order in the circle at infinity, and let ij D .hi ; hj / for distinct i; j 2 f1; 2; 3; 4g. Then: a) Ptolemy’s equation: 13 24 D 12 34 C 14 23 ; b) Cross ratios: the Möbius transformation that takes the centers of h3 , h2 , h1 14 respectively to 0; 1; 1 also maps the center of h4 to  23 ; 12 34 c) h-lengths: the hyperbolic length of the horocyclic segment in hi with endpoints  hi \ .hi ; hj / and hi \ .hi ; hk / is given by ijjki k , where fi; j; kg D f1; 2; 3g; d) Affine duality: taking the upper sheet H of the hyperboloid in Minkowski 3space as the model for the hyperbolic plane, there is a unique isotropic vector 1 ui with positive z-coordinate so that hi D fw 2 H W w  ui D 2 2 g, for i D 1; 2; 3; 4 where  denotes the pairing with quadratic form x 2 C y 2  z 2 , and

Appendix C. Stable curves and screens

ij D

329

p ui  uj for distinct i; j 2 f1; 2; 3; 4g;

e) Simplicial coordinates: in the notation of part d), the signed volume of the Euclidean tetrahedron in Minkowski 3-space spanned by u1 , u2 , u3 , u4 is given p by 2 2 12 23 34 14 times 212 C 223  213 2 C 234  213 C 14 ; 12 23 13 14 34 13 where the sign is positive if and only if the edge of the tetrahedron connecting u1 , u3 lies below the edge connecting u2 , u4 . f) Ellipticity: in the notation of part d), the affine plane containing u1 , u2 , u3 determines an elliptic conic section if and only if 12 , 13 , 23 satisfy the three strict triangle inequalities. z 2 Tz .F / and given the homotopy class of an ideal arc ˛ in F , we Given a point  may straighten ˛ to the geodesic for the underlying hyperbolic structure and truncate this geodesic by cutting it at the horocycles centered at its endpoints coming from the decoration. This geodesic segment has a signed hyperbolic length ı taken with a positive sign if and only if the horocycles are disjoint. The basic coordinate of an ideal arc in a decorated hyperbolic surface is the lambda length (also sometimes called the p z D eı. “Penner coordinate”) defined by .˛I / Theorem 3.2. Fix any trivalent fatgraph G. Then the assignment of lambda lengths Tz .F 0 .G// ! RE.G/ >0 ;

z 7! .e 7! .˛.G;e/ I //; z 

is a real-analytic homeomorphism onto. For convenience when the fatgraph G is fixed or understood, we shall refer to the lambda length of an edge e of G rather than that of its dual arc ˛.G;e/ . We shall also often identify an arc with its lambda length for convenience. Suppose that G is a trivalent fatgraph. Consider an edge e of G and adopt the notation of Figure 4, where e has distinct endpoints with incident half-edges a, b and c, d occurring in the alphabetic counter-clockwise order about e. (If e does not have distinct endpoints or if a, b, c, d are not distinct, then adopt the corresponding notation for nearby edges in the universal cover.) Dual to each vertex of e is an ideal triangle, and each such triangle has three vertices, denoted by Greek letters in Figure 4. To each such triangle/vertex pair is naturally associated a sector of G, that is, a pair of consecutive half-edges of G incident on a common vertex, namely, the pair of half-edges adjacent to the given vertex in the given triangle. In fact, one can conveniently calculate the holonomies of based closed curves in F 0 .G/ as follows. Define the matrices     1 1 0 1 RD ; LD 2 PSL2 .R/: 1 0 1 1

330

Appendix C. Stable curves and screens

According to Lemma 3.1b), the cross ratio of the ideal quadrilateral with edges ˛.G;a/ , ˛.G;b/ , ˛.G;c/ , ˛.G;d / is given by bd=ac, where we have identified an edge of G with its lambda length for convenience, and we further define the matrix p   ac=bd 0 p Xe D :  bd =ac 0 Choosing a vertex of G as basepoint, consider a closed edge-path  in G representing an essential based closed curve in F . We may as well assume that  is efficient (though this is not necessary since RL D R3 D L3 D Xe2 D 1 2 PSL2 .R/), so that it alternately traverses edges and sectors of G and makes turns, right or left, at each sector. Suppose that  serially makes turns ti at the sectors, then traverses edges ei , for i D 1; : : : ; n, and associate the path-ordered product M D T1 Xe1 T2 : : : Xen of matrices, where Ti D R or L if ti is a right or left turn respectively. The matrix M 2 PSL2 .R/ gives the holonomy of the based curve . Of course by conjugacy invariance of trace, the absolute value of the trace of M is independent of the basepoint. We shall use these path-ordered products to detect the short curves that occur on a path in C.G/  Tz .F 0 .G//. The quadrilateral in Figure 4 is realized as a geodesic ideal quadrilateral with horocycles centered at each vertex. We define the h-length of a sector of G to be the hyperbolic length of the corresponding horocyclic segment. According to Lemma 3.1c), the h-length of a sector is the opposite lambda length divided by the product of adjacent lambda lengths. Furthermore in the notation of Figure 4, we define the simplicial coordinate of the edge e to be the quantity c 2 C d 2  e2 a b e c d e a2 C b 2  e 2 C D C  C C  : abe cde be ae ab de ce cd According to Lemma 3.1e), the simplicial coordinate is a multiple of the signed volume of the corresponding tetrahedron, and by inspection, it is a linear combination of the nearby h-lengths. From the definition, the simplicial coordinate is the sum of two terms each of which is associated to a vertex of the corresponding edge. Consider a trivalent fatgraph G with set E of edges for the surface Fgs together with an assignment of lambda lengths  W E ! R>0 . We say that  satisfies the no vanishing cycle condition provided that all the corresponding simplicial coordinates are non-negative and there is no cycle in G all of whose simplicial coordinates vanish. Theorem 3.3. For any surface F D Fgs with s  1, there is an MC.F /-invariant ideal cell decomposition of Tz .F /, where the cells in this decomposition are in one-to-one correspondence with homotopy classes of embeddings of fatgraph spines of F each of

Appendix C. Stable curves and screens

331

whose vertices has valence at least 3. The face relation in this cell decomposition is generated by Whitehead collapse. In particular, if G is a trivalent fatgraph spine of F , then the closed cell C.G/  z T .F / corresponding to it is described in lambda length coordinates with respect to G by the no vanishing cycle condition. Furthermore, suppose G 0 arises from G by collapsing to a point each component of a forest in G. Then the corresponding closed cell C.G 0 /  C.G/  Tz .F / is described by taking all simplicial coordinates on edges in the forest to vanish. Finally, any assignment of non-negative real numbers to the edges of G 0 with no vanishing cycles is realized as the simplicial coordinates of a uniquely determined collection of positive lambda lengths on G. Corollary 3.4. There is an MC.F /-invariant ideal cell decomposition of P Tz .F / for any surface F D Fgs with s  1, where the cells in this decomposition are in one-to-one correspondence with homotopy classes of embeddings of fatgraph spines of F whose vertices have valence at least three. Proof. This follows immediately from the previous theorem and homogeneity of the formula for simplicial coordinates. Lemma 3.5. Suppose that  is an efficient edge-path in G serially traversing edges ei alternating with sectors ti , for i D 1; : : : ; n. Let PnEi denote the Pnsimplicial coordinate of ei and ˛i the h-length of the sector ti . Then iD1 Ei D 2 iD1 ˛i . Proof. The proof follows from the definition of simplicial coordinates in terms of h-lengths. Lemma 3.6 ([128]). The no vanishing cycle condition implies that the lambda lengths at any vertex of G satisfy the three strict triangle inequalities. Proof. Adopt the notation of Figure 4 for the half-edges near an edge e (again, in the universal cover if the edges a, b, c, d are not distinct or if e does not have distinct endpoints). If c C d  e, then c 2 C d 2  e 2  2cd , so the non-negativity of the simplicial coordinate E of e gives 0  cd Œ.a  b/2  e 2 , and we find a second vertex so that the triangle inequality fails. This is a basic algebraic fact about simplicial coordinates. It follows that if there is any such vertex so that the triangle inequalities do fail for the lambda lengths of incident half-edges, then there must be an efficient closed edge-path  passing through such triangles. Letting ei denote the consecutive edges of G serially traversed by  and bi denote the half-edge of G incident on the common endpoint of ei and eiC1 , we find ejP C1  bj C ej , for j D 1; : : : n. Upon summing and canceling like terms, we find 0  jnD1 bj , which is absurd since lambda lengths are positive.

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4. Screens Suppose that G is a trivalent fatgraph with set E of edges and corresponding surface F , and suppose that  t W E ! R>0 , i.e.,  t 2 RE >0 , is a continuous one-parameter family of lambda lengths for t  0. We shall typically apply Theorem 3.2 to regard such a one-parameter family as a path in Tz .F 0 .G// itself. There is an induced N t 2 P .RE >0 /, where P denote projectivization, and by compactness, there is an accumulation point of lim t!1 N t in P .RE 0 /. Say that  t stable if there is a unique such limit point denoted by N 1 2 P .RE 0 /. If  t is any path, then any accumulation point of N t is also the limit of some stable path since decorated Teichmüller space is path connected. E 0 N Suppose that  t 2 P .RE >0 / is stable with limit 1 2 P .R0 /. Set E D E k1 and make the following recursive definition for k  1. If E ¤ ;, then there is e 2 E k1 so that for all f 2 E k1 , we have N 1 .e/=N 1 .f / < 1, and we set E k D ff 2 E k1 W N 1 .e/=N 1 .f / D 0g; with Ek D ; if Ek D Ek1 . Notice that if two edges e1 ; e2 2 E k1 satisfy the condition that for all f 2 E k1 , we have N 1 .ei /=N 1 .f / < 1, for i D 1; 2, then we have 0 < N 1 .e1 /=N 1 .e2 / < 1, so e1 and e2 determine the same subset E k . Thus, E D E 0  E 1      E N is a well-defined nested sequence of finite length N of proper subsets, and we set E N C1 D ; for convenience. Now, suppose that  t stays for all finite t  0 in the closed cell C.G/ corresponding to G, i.e., the lambda lengths satisfy the no vanishing cycle condition by Theorem 3.3. Define A. t / D fA  E W A is the set of edges of a component of some E k g; a subset of the power set of E. Proposition 4.1. For any connected trivalent fatgraph G with set E of edges and any continuous stable one-parameter family  t 2 RE >0 , for t  0, which stays for all finite t in the cell C.G/  Tz .F .G// corresponding to G, the collection A D A. t / satisfies the following properties: i) E 2 A; ii) each A 2 A is recurrent; iii) if A; B 2 A with A \ B ¤ ;, then either A  B or B  A; iv) for each A 2 A, [fB 2 A W B is a proper subset of Ag is a proper subset of A. A subset of the power set of E satisfying properties i)–iv) is called a screen on G for any (not necessarily trivalent) recurrent fatgraph G with set E of edges.

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333

Proof. The first condition holds since G is connected and E D E 0 . Recursively applying Lemmas 2.1 and 3.6, we conclude that E k is a proper recurrent set in the possibly disconnected fatgraph GE k1 , for k D 1; : : : ; N , so the second condition holds as well. The third condition holds since two components of a topological space either coincide or are disjoint, and the fourth follows since each inclusion E k  E k1 is proper. If A is a screen, then each A 2 A  fEg has an immediate predecessor A0 , i.e., A  A0 and if B 2 E and A  B  A0 , thenB D A or B D A0 . The maximum length of a chain A  A0      E of immediate predecessors in A is called the depth of A in A, and the depth of e 2 E in A is the maximum depth of A 2 A with e 2 A. Lemma 4.2. Every screen A on every trivalent fatgraph G arises as A D A.N t / for some stable  t 2 RE >0 lying in C.G/. Proof. For any screen A on any trivalent fatgraph G, define a one-parameter family of lambda lengths by taking  t .e/ D t de , where de is the depth of e in A. For any vertex v of G, the maximum degree of the incident (half-)edges is achieved either twice or thrice by recurrence of elements of A. Thus, the contribution from v to each of the three possible simplicial coordinates of edges incident on v is positive, and so the simplicial coordinate of each edge of G for  t is also positive;  t thus lies in C.G/ by Theorem 3.3, and A.N t / D A by construction. Let @A A denote the relative boundary of F .GA / in F .GA0 /, where A0 is the immediate predecessor of A in A, and define the boundary of A itself to be [ @A D @A A: A2AfE g

Lemma 4.3. For any trivalent fatgraph G with set E of edges and stable  t 2 RE >0 lying in C.G/, each edge-path in @A.N t / is homotopic to a curve in F 0 .G/ whose hyperbolic length tends to zero as t tends to infinity. Furthermore, these are only such asymptotically short curves for  t . Proof. Let K be a component of @A, so K D @A A for some A 2 A  fEg with immediate predecessor A0 . Orient K with the subsurface F .GA / on its left. Consider zA0 respectively denote the full z G zA , and G the universal cover Fz of F D F .G/, let G, z pre-images of G, GA , and GA0 in F , and choose a lift Kz of K to Fz . We shall refer to z by which we mean the value of  t on the projection of lambda lengths of edges of G, z with the edge to F , and we will as usual identify the lambda length of an edge of G the edge itself for convenience. On the right of Kz since K is homotopic to a boundary component of F .GA /, there zA , and since K is not homotopic to a boundary component of F .GA0 /, are no edges of G zA0 not in G zA on the right. Furthermore on the left of there is at least one edge of G

334

Appendix C. Stable curves and screens

z there is at least one edge of G zA0 again since K is not homotopic to a boundary K, component of F .GA0 /. Since  t corresponds to points in C.G/, it follows that the triangle inequalities hold z by Lemma 3.6. We claim that the following on lambda lengths at each vertex of G further properties of lambda lengths follow from these facts, where all limits are taken as t ! 1: z then we have x ! 0; 1) if x is an edge on the right of Kz and y is an edge of K, y

z and y1 ; y2 z y0 is an edge on the left of K, 2) if x is an edge on the right of K, z z then are edges of K so that y0 ; y1 ; y2 are all incident at a common vertex in K, xy0 ! 0; y1 y2 3) if y0 ; y1 are consecutive edges of Kz and x is an edge on the right of Kz incident on their common endpoint, then yy01 ! 1. The first property follows from the definition of K as a relative boundary component of F .GA / in F .GA0 / and the definition of the screen A. t /. For property 2, y0 , y1 , y2 satisfy the triangle inequality y0 < y1 C y2 , so dividing by y1 y2 and multiplying by 0 x, we find yxy < yx1 C yx2 ; the right-hand side tends to zero by property 1. Finally for 1 y2 property 3 again by the triangle inequalities, we have y0 < y1 C x and y1 < y0 C x. Upon dividing the first by y1 and the second by y0 and applying property 1, we conclude 1  lim yy01  1, as required. The first key point about properties 1)–3) is that they are invariant under certain Whitehead moves. In each case, we shall perform a Whitehead move along an edge e 2 K, where one vertex of e has incident half-edges a, b and the other vertex has incident half-edges c, d , and where the edges a, b, c, d occur in this counter-clockwise order about e. We shall refer to properties 1)–3) for the fatgraph before the Whitehead move and the corresponding properties 10 )–30 ) for the resulting fatgraph, and we shall denote the edge and lambda length of the edge resulting from e under let f D acCbd e the Whitehead move. z The The first case of utility is when b, c, e lie in Kz and a; d lie on the right of K. x z properties for this fatgraph imply that: 1) y ! 0 for x 2 fa; d g and y 2 K; 2) does not involve the vertices of e; and 3) be ! 1 and ce ! 1. Property 30 ) requires which follows from property 3). Furthermore by the Ptolemy equation, f ac C bd c bd D D C !1C a ae e ea f b ca ac C bd D C !1C D de e ed d

b c

! 1,

d ; a a ; d

since ce ! 1 and be ! 1. Thus, at least one of fa , fd has a finite limit, hence f D fa ya D fd dy ! 0 for any y 2 Kz by property 1) proving property 10 ) and likewise y for property 20 ).

335

Appendix C. Stable curves and screens

The second case of utility is when b, d , e lie in Kz with a on the right and c on the z The properties for this fatgraph imply that: 1) a ! 0 for any y in K; z 2) left of K. y ! 0 for any x on the right; and 3) be ! 1. Property 30 ), namely, fd ! 1, follows from ac C bd ac b f D D C !1 d de de e 0 using properties 2)–3). Property 1 ) follows from this and property 1). Finally, since xc de

.ac C bd /b ba b 2 de bf D D C 2 !1 xc xce ex e xc using the Ptolemy equation and properties 2)–3), property 20 ) holds as well. Applying these two types of Whitehead moves along edges in K, we may alter G 0 to arrange that the edge-path for K in the resulting graph makes exactly one left turn. Furthermore as we have just proved, properties 1)–3) continue to hold for the resulting graph. We shall complete the proof by calculating that the absolute value of the trace of the holonomy of the edge-path K is asymptotic to 2, and the second key point about properties 1)–3) is that they are sufficient to guarantee this. To this end, let us adopt the notation that K traverses the consecutive edges y1 ; : : : ; ynC1 , the unique half-edge on the right is x0 , which is incident on the common endpoint of ynC1 ; y1 , and the consecutive half-edges on the left are x1 ; : : : ; xn , where xk has common endpoint in K with yk ; ykC1 , for k D 1; : : : ; n. As usual identifying an edge or a half-edge with its lambda length, which depends upon the parameter t , let us define 12 D

y2 ynC1 x0 xn ykC1 xk1 2 ; nC1 D ; and k2 D x1 x0 y1 yn xk yk1

for k D 2; : : : ; n;

so the cross ratio of edge yk is k2 , for k D 1; : : : ; n C 1. The path-ordered product of matrices to compute the holonomy of K beginning from the unique left turn is given by       0 2 0 nC1 0 1 R :::R L 1 11 0 21 0 nC1 0  1   1   1  1 0 2 2 nC1 nC1 D ::: 0 2 0 nC1 11 1 !  1  P Q 1 0 .2 : : : nC1 /1 2 : : : nC1 nkD1 jkD2 j2 D ; 11 1 0 2 : : : nC1 so the trace is found to be .1 2 : : : nC1 / C .1 2 : : : nC1 /1  12 .1 2 : : : nC1 /

k n Y X kD1 j D2

j2 :

336

Appendix C. Stable curves and screens

Finally, direct calculation shows that the product telescopes, and .1 2 : : : nC1 / D since

y1 ynC1

ynC1 !1 y1

! 1 by property 3). Furthermore, 12 D

x0 x1 x0 x1

!0 y2 ynC1 y1 y2

by properties 2)–3), and indeed, the general term in the sum also telescopes 12 22 : : : k2

x0 xk !0 yk ykC1

for k D 2; : : : ; n;

again by properties 2-3). The absolute value of the trace is thus indeed asymptotic to 2. Since the absolute value of the trace is twice the hyperbolic cosine of half the hyperbolic length, the curve K is asymptotically short as t ! 1. For the final assertion, we must show that the edge-path of an essential short curve K for  t lies in @A. We shall use the Collar Lemma [24] that an essential simple closed curve of hyperbolic length ` has an embedded collar of width at least the logarithm of the hyperbolic cotangent of 2` . Since the dual of G is an ideal triangulation and K is essential, it thus follows that if K traverses an edge e of G, then the lambda length of (the ideal arc dual to) e has divergent lambda length. It follows that K shares an edge with E k , for some k  1. Since two essential short curves cannot intersect, again by the Collar Lemma, we conclude that K cannot meet @A. t / for large t , so in fact the edge-path for K is contained in E k  E kC1 . If K is not homotopic to a boundary component of GE k , then its edge-path must make both right and left turns in GE k . Without loss, we may assume that there is a left turn followed by a right turn and adopt the following notation. Suppose that the edge-path for K traverses edges y0 ; : : : ; ynC1 in K with half-edge xj incident on the common endpoint of yj ; yj C1 for j D 1; : : : ; n, where x0 2 E k lies on the right and xn lies on the left of K. In the extreme case that n D 1, the dual arcs to x0 ; y1 ; x1 ; y0 2 E k are the consecutive edges of an ideal quadrilateral whose cross ratio is bounded near one by Lemma 3.1 since the lambda lengths x0 , x1 , y0 , y1 are comparable, i.e., the limit of the ratio of any pair is finite and non-zero. The arcs dual to y0 and y1 are therefore a bounded distance apart, contradicting that K is short. This extreme case gives a lower bound to the distance between the arcs dual to y0 and yn , so in any case, K cannot be short. This contradiction establishes the final assertion and completes the proof. 5. Proof of main result Theorem 5.1. The cell C.G/ in decorated Teichmüller space corresponding to the fatgraph G is asymptotic to a stable curve with pinch curves K if and only if K is homotopic to the collection of edge-paths @A for some screen A on G.

Appendix C. Stable curves and screens

337

Proof. Suppose that  t 2 R0>0 is a path of lambda lengths in C.G/ whose projectivization N t accumulates at some point of P .R00 /. Since C.G/ is path connected, there is a stable path, still denoted by  t , whose limit point is this accumulation point. By Lemma 4.3, the short curves for this limit point are the components represented by edge-paths in @A. t /. Conversely for any trivalent fatgraph G and any screen A on G, Lemma 4.2 shows that @A is realized as the set of short curves for a stable path in C.G/. For a general not necessarily trivalent fatgraph, we require a further ingredient for the converse, namely: Theorem 5.2 ([128]). For any cyclically ordered tuple x1 ; : : : ; xP n of positive real numbers satisfying the generalized strict triangle inequalities xj < i¤j xi , for j D 1; : : : ; n, there is a cyclic Euclidean planar polygon (i.e., the polygon inscribes in a circle) unique up to orientation-preserving isometry of the plane which realizes these numbers as its consecutive edge lengths. To apply this result, let LC denote the collection of isotropic vectors in Minkowski space with positive z-coordinate. Given a collection of coplanar points in LC lying in an affine plane determining an elliptic conic section, we may apply a Minkowski isometry to arrange that the plane containing these points is horizontal. The restriction of the Minkowski pairing to this horizontal plane is a multiple of the Euclidean metric induced on the plane, so the projectivized lambda lengths of pairs of these points agree with the projectivized Euclidean lengths in the horizontal plane. Furthermore, the intersection of the horizontal plane with LC is a round circle in this Euclidean structure. Now, given any fatgraph G 0 with vertices at least trivalent and any screen A0 on G 0 , again define lambda lengths on the edges of G 0 by 0t .e/ D t de , where de is the depth of e in A0 . According to Theorem 5.2, the previous paragraph, and Theorem 3.2, this does indeed determine a path in C.G 0 /. Choose any trivalent fatgraph G which collapses to G 0 . The lambda lengths on the edges of G  G 0 are thus given by the Euclidean lengths of the corresponding cyclic polygon again according to Theorem 5.2, the lambda lengths on the edges of G 0 have already been specified, so  t is now determined on G. It follows that there is a unique screen A on G which restricts to A0 in the natural sense with corresponding lambda lengths  t on G, and satisfying @A0 D @A as homotopy classes. The proof of Lemma 4.3 applies to  t to conclude that the components of @A are precisely the short curves for  t . 6. Closing remarks In sketch, we have seen that simplicial coordinates are given explicitly in terms of lambda lengths, and the no vanishing cycle condition is necessary and sufficient to

338

Appendix C. Stable curves and screens

guarantee that these formulae are uniquely invertible. Writing the inverse explicitly is the basic “arithmetic problem" in decorated Teichmüller theory [134]. One ingredient, which is related to the asymptotics of this arithmetic problem, towards describing the Deligne–Mumford compactification is: Theorem 6.1 ([137]). Suppose that  t is a stable one-parameter family of lambda lengths on the fatgraph G with no vanishing cycles of corresponding simplicial coordinates X t . Define I D fe 2 E W  t .e/ ! 1g and J D fe 2 E W X t .e/ ! 0g. Then I  J and R.GJ / D GI , where R.X / denotes the maximal recurrent subset of X . This result in tandem with Theorem 1.2 has interesting consequences: Take a straight-line path in the natural affine structure of simplicial coordinates on C.G/ for some fatgraph G which limits to a point that fails to satisfy the no vanishing cycle condition. Let E1  E.G/ denote the subset of edges of G whose simplicial coordinates vanish, and let R1  E1 denote its maximal recurrent subset. Depending upon the affine path, certain lambda lengths of edges in R1 diverge at various rates, i.e., a screen magically pops out as determined by the arithmetic problem. The set of all screens on all fatgraphs of a fixed topological type g, s forms a partially ordered set in the natural way, where the face relation is generated as usual by collapsing edges and now also by inclusion of screens, and the mapping class group M C.Fgs / naturally acts on the geometric realization of this screen poset. See [97] for the further elaboration of these ideas to produce a cell decomposition of a space N s /. homotopy equivalent to M.F g

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List of Notation

Œa; b; c

quadratic form a 2 C b  C c2 , 27

ŒŒa; b; c

class of quadratic form Œa; b; c , 28

A

screen, 235

Arc.F /

arc complex of F , 218

Arc.F; ˇ/

ˇ-arc complex of F , 250

Arc# .F /

subspace of Arc.F / corresponding to filling arc families, 218

Aut.G/

automorphism group of G, 182

Be ; Ce

basis vectors associated to e, 141

B

points in LC corresponding to decoration, 126

B2 .R/

real symmetric bilinear forms on two indeterminates, 19

ˇ

brane-labeling, 248

C

closed convex hull of B, 126

C ./

cell associated to i.c.d. , 124

C.G/

cell corresponding to G, 142

y PSL2 .R// Cont.G; y R / ContG .G;

y to the Möbius group, 102 group of continuous maps from G y R /, 103 G-equivariant subgroup of Cont.G;



ideal triangulation or cell decomposition, 9

.G/ z .)

ideal cell decomposition dual to fatgraph G, 10 z 124 i.c.d. associated to ,

ı

boundary distinguished points, 248

D

Poincaré disk, 16

doe

distinguished oriented edge, 89

E.G/ z E.G/

edges of G, 140 oriented edges of graph G, 133

e.G/

number of edges of G, 194

F .G/

surface associated to fatgraph G, 10

F˛

F cut along the arc family ˛, 219

Fgs .ı1 ; : : : ; ır /

windowed surface, 248

>0

>0

350

List of Notation

s Fg; ıE

surface of genus g with s punctures and vector ı of numbers of distinguished boundary points, 67

s Fg;r

a surface of genus g with r boundary components and s punctures, 1

f 0

characteristic mapping of  0 , 89

F

orthonormal three-frame, 267

G./

fatgraph dual to ideal cell decomposition , 10

G.D/ y G

Gauss group of discriminant D, 28

G .F /

ideal simplicial decomposition of Tz .F /=R>0 , 154

Gy.F /

dual fatgraph complex for F , 154



hyperbolic structure, 57

k z 

lower central series, 241 decorated hyperbolic structure, 57

H

first integral homology group, 241

H

one sheeted hyperboloid in Minkowski space, 18

h.u/

horocycle affine dual to u 2 LC , 21

H

punctured solenoid based on modular curve, 100

HG

punctured solenoid determined by group G

HM

punctured solenoid based on orbifold M , 100

H

solenoid with marked hyperbolic structure , 102 , 101

profinite completion of G, 100

H

upper sheet of hyperboloid in Minkowski space, 17

HomeoC .S /

group of orientation-preserving homeomorphisms of S 1 , 90

Homeon .S 1 /

decorated normalized homeomorphisms of S 1 , 110

g; Œk

kth Torelli group, 241

i.c.d.

ideal cell decomposition, 9

K

energy function, 142

n

nth MMM class, 6

ƒ.s/

basic deformation of lambda lengths, 94

ƒA .s/ z .˛I /

lambda length deformation corresponding to A, 111

B

1

lambda length of ideal arc ˛ with respect to decorated hyperbolic z 58 structure ,

List of Notation

351

.h1 ; h2 /

lambda length of horocycles h1 , h2 , 32

M.F / x .F / M

Riemann moduli space of F , 6

MG .F /

moduli space of flat G connections on F , 7

M .F /

mapping class groupoid of F , 176

MC.F /

mapping class group of F , 1

MC.F; /

mapping class group of F rel basepoint, 241

MC.H /

mapping class group of H , 106

M CBLP .H /

baseleaf-preserving mapping class group of H , 106

ML0 .F /

e

measured laminations of compact support in F , 231

ML0 .F /

decorated measured laminations of compact support in F , 232

MMM class

Miller–Morita–Mumford class, 6

Mod

universal moduli space, 92

Nk

kth nilpotent quotient, 241

N1

pronilpotent completion, 241

LC

open positive light-cone in Minkowski space, 18

`i

i th perimeter function, 66

sg

Weil–Petersson volume for Fgs , 185

!Œa;b;c

primitive root of Œa; b; c , 29

!

Weil–Petersson type two form with respect to ideal triangulation , 50

PG1

universal Ptolemy group, 237

P 1

universal Ptolemy groupoid, 236

PL0 .F /

space of projective measured laminations of compact support on F , 213

PMC.F /

pure mapping class group of F , 1

P PSL2 .Z/

piecewise modular group, 238

PSL2 .R/

Möbius group, 5

PSL2 .Z/

modular group, 24

Pt.F /

Ptolemy groupoid of F , 174

p q

1 Farey point in S1 , 24

Deligne–Mumford compactification of M.F /, 213

352

List of Notation

psl2

algebra of piecewise sl 2 vector fields, 112

A

fan vector field associated to A, 114

…

projection to X C , 142



fundamental group, 241 A

hyperfan vector field associated to A, 114

QA.F /

quotient arc complex of F , 218

QA.F; ˇ/

quotient ˇ-arc complex of F , 250

QAex .F; ˇ/

exhaustive subspace of QA.F; ˇ/, 250

QAex .m; n/

union of exhaustive quotient arc complexes m closed and n open windows, 258

Rg

moduli space of RNA shapes of genus g, 263

r.G/

number of boundary components of F .G/, 195



sum of all perimeter functions, 66

1 S1 C

circle at infinity, 16

SO .2; 1/

component of identity in SO.2; 1/, 19

sŒG

number of boundary components of F .G/, 202

.G/

sectors of G, 140



punctures, 248

.G/; u .G/;  t .G/ permutations associated to G, 199 T

Thompson group, 238

T .F / Tz .F /

Teichmüller space of F , 4

Tz .F / Tz .H /

partially decorated Teichmüller space of F , 67

Tz .Q/ TzP .F /

decorated Teichmüller space of polygon Q, 48

T .H /

Teichmüller space of H , 102

Tk .F / Ty .F /

kth Torelli space of F , 242 holed Teichmüller space of F , 75

T

completed tensor algebra of H over Q, 246

T ess

space of tesselations of D, 91

decorated Teichmüller space of F , 7 decorated Teichmüller space of H , 103 P -decorated Teichmüller space of F , 63

List of Notation

T ess0

space of tesselations with doe of D, 89

T ess0

space of decorated tesselations of D, 92

A TA ess 0

 

space of decorated tesselations with doe of D, 92 tesselation, 88

0

0 

353

2

tesselation with doe, 89 ideal vertices of  , 88 triangles complementary to  , 88



Farey tesselation, 24



freeway associated to , 232

c

right Dehn twist on the curve c, 1

k

kth Johnson homomorphism, 241

#

elementary vector field, 112

#e

elementary vector field associated to edge e of Farey tesselation, 113

hu; vi

Minkowski inner product of u; v, 17

u, v, w

standard light-cone basis for R3 , 32

U

upper half-plane, 14

vN

1 projection to S1 of v 2 LC , 19

v.G/

number of vertices of G, 194

vk ŒG

number of k-valent vertices of G, 202

V.G/

quadric variety determined by coupling equations on G, 133

Wi

flip on edge labeled i, 176

Œw; x; y; z x XC X; Y; X;

cross ratio, 16

Xe

putative simplicial coordinate of e, 133

xe

putative lambda length of e, 133

.G/

Euler characteristic of G, 195

YzE

affine subspace determined by zE, 143

Z .e; /

shear coordinate of ideal arc e in ideal triangulation  with respect to hyperbolic structure , 74

zE

vector in R , 141

spaces of weights on edges, 142

Index

of Poisson algebra, 83, 87 chain relation, 3 characteristic mapping, 89 circle at infinity, 14, 16 cis conformation , 266 cis-Proline, 266 class numbers, 28 class of a flip, 176 cluster algebra of type An , 50 of type Dn , 50 combinatorial classes, 192 compactification, 218 combinatorial compactification, 218 commutativity relation, 2, 175, 177 backbone, 260 compact-open topology, 47 baseleaf, 106 compactifications of M.F /, 213 baseleaf-preserving homeomorphism, 106 complete hyperbolic structure, 75 Basic Calculation, 40 complex multiplication, 29 Bers’ universal Teichmüller space, 98 complex structure of Teichmüller space, ˇ-arc, 250 122 ˇ-arc family, 250 concordant forms, 28 beta strand, 268 conformal factor, 21 bordered continued fraction, 26 fatgraph, 161 convex hull, 48 surface, 67, 160 convex hull construction, 124 braid group, 65 coupling equation, 133 braiding relation, 3 cross ratio, 16, 41 brane-labeling, 248 cyclic Euclidean polygon, 51 BV equation, 257 decagon relation, 236 canonical decomposition, 124 decorated Cardy equation, 255, 256 hyperbolic n-gon, 48 Cayley transforms, 17 hyperbolic structure, 56 center hyperbolic structure on n-gon, 47 of cell, 150 ideal quadrilateral, 39 of horocycle in a surface, 56 measured laminations, 232 of horocycle in hyperbolic space, 20 tesselation, 92

Abel–Spence equation, 192 Alexander’s trick, 3, 13 alpha carbon, 265 alpha helix, 268 amino acid, 264 anharmonic ratios, 16 arc complex of bordered surface, 219 of punctured surface, 218 arithmetic problem for bordered surfaces, 163 for decorated surfaces, 130 for partially decorated surfaces, 159 arithmetic surface, 150 augmented Teichmüller space, 213

356

Index

decorated normalized homeomorphisms, 110 decorated Teichmüller space of an n-gon, 47 of bordered surface, 67 of punctured surface, 7 of the solenoid, 103 decoration, 33, 36 on bordered surface, 67 on ideal triangle, 37 definite form, 28 degenerate form, 28 Dehn twist, 1 Deligne–Mumford compactification, 213, 235 discrete, 5 discriminant, 27 distinguished oriented edge, 89 doe, 89 dual bordered fatgraph, 161 fatgraph, 10 ideal cell decomposition, 10 punctured fatgraph, 157 q.c.d., 157, 161 dyadic tesselation, 88 earthquake, 115 elementary vector field, 113 elliptic curve, 26 involution, 13 plane, 18 transformation, 15 embedded case, 141 energy function, 142 enhanced flip, 236 Epstein-Penner decomposition, 124 equidistant points to horocycles, 37 essential arc, 57, 161 curve, 1 Euclid’s Fifth Axiom, 14

exhaustive arc family, 250 faithful, 5 fan vector field, 114 Farey sampling, 123 tesselation, 24 tree, 154 fatgraph, 10 automorphism group, 182 complex, 154 fatgraph marking, 242 Fenchel–Nielsen coordinates, 8 Theorem, 8 finite face, 142 finite isotropy, 6 flip, 11, 49, 157, 161 flip homeomorphism, 107 flip on the doe, 236 forgetting the puncture, 6 formal h-length, 141 simplicial coordinate, 141 fractional linear transformation, 14 freeway, 76, 232 Frobenius equation, 255 Fuchsian group, 5 of the first kind, 160 of the second kind, 161 fundamental path groupoid, 174 Garside relation, 3 word, 3 Gauss product, 28 generalized annulus, 253 Magnus expansion, 247 pair of pants, 253 pants decomposition, 253 G-graph connection, 7

Index

global sl2 vector field, 111 graph, 9 graph connection, 7 half edge, 9 harmonic function on surface, 21 hexagon relation, 181 Hilbert transform, 122 h-length, 37 holonomy, 61, 79 hormonica, 34 horoball, 99 horocycle, 20 horocycle in a surface, 56 hydrogen bond, 267 hyperbolic geometry, 14 plane, 18 structure, 4 transformation, 15 translation length, 15 hyperboloid of one sheet, 18 hypercycle, 30 hyperelliptic involution, 12, 13 hyperfan vector field, 114 i.c.d., 9 ideal arc, 9, 57, 161 cell decomposition, 9 hyperbolic n-gon, 48 simplicial decomposition, 12 triangle, 24 triangulation, 8, 9 ideal arc family in bordered surface, 219 in punctured surface, 218 ideal cell decomposition, 8 IJ Lemma, 215 imaginary form, 28 indecomposable brane-labeled surface, 253 inessential curve, 1

357

integration scheme, 182 involutivity relation, 175, 177 Johnson filtration, 241 homomorphism, 241 kernel, 241 Kirillov–Kostant form, 93 Kontsevich form, 192 labeled fatgraph, 176 flip, 176 pentagon relation, 177 Whitehead move, 176 ladder fatgraph, 194 lambda length, 32, 57 lantern relation, 3 lattice, 26 Lie algebra psl2 , 112 light-cone, 18 linear chord diagram, 260 long edge, 77 Möbius group, 5 transformation, 14 Madsen–Weiss–Tillman Theorem, 6 Madsen-Weiss-Tillman Theorem, 180 Magnus representation, 247 mapping class group of solenoid, 106 of surface, 1 mapping class groupoid of bordered surface, 180 of punctured surface, 176 marking on fatgraph, 242 Markoff equation, 154 triple, 154 Maslov dequantization, 234 matrix model, 183 maximal recurrent set, 215

358 measured laminations, 231 Miller–Morita–Mumford class, 6 Minkowski question mark function, 89 three-space, 17 MMM classes, 6, 191–193, 246 modular curve, 26 group, 24 moduli space of flat connections, 7 of proteins, 271 of Riemann surfaces, 6 of RNA shapes, 263 Morita–Penner cocycle, 243 Mumford’s Conjecture, 6 natural fattening, 260 naturality relation, 2, 177 n-gon, 47 Nielsen representation, 247 no quasi vanishing cycles, 162 no quasi-vanishing arcs, 162 cycles, 159 no vanishing arcs, 163 cycles, 132, 213 non-embedded case, 141, 145 non-separating relation, 3 normalized tesselation, 91 trigonometric field, 118 vector field, 113, 118 off-center polygon, 52 on-center polygon, 52 open/closed string duality, 256 orbifold, 26 ordinary fatgraph, 156 orientation double cover, 193 P-decorated Teichmüller space, 63

Index

pair of pants, 8 pants decomposition, 8 parabolic plane, 18 transformation, 15 parallel ˇ-arcs, 250 partial decoration, 63 partially decorated bordered surface, 164 decorated surface, 62, 155 decorated Teichmüller space, 66 of punctured surface, 157 path-ordered product expansion, 79 Penner compactification, 213 coordinate, 11 matrix model, 207 pentagon relation, 175, 177 peptide, 265 peptide unit, 266 perimeter function, 66, 182 section, 182 peripherals, 5, 75 piecewise parabolic, 115 pin, 114 pinch curve, 213 pinched, 95 Poincaré disk, 16 Poisson structure, 81, 82 polypeptide, 265 backbone, 265 primary structure, 265 secondary structure, 268 tertiary structure, 265 primitive form, 27 root, 29 profinite completion, 100 pronilpotent completion, 241 protein backbone, 265

Index

folding problem, 269 primary structure, 265 secondary structure, 268 tertiary structure, 265 pseudoknot, 261 Ptolemy equation, 41 groupoid, 174 Theorem, 43 Ptolemy–Teichmüller groupoid, 240 punctured arithmetic surface, 61 fatgraph, 156 vertex, 156 punctured solenoid, 100 pure mapping class group, 1 q.c.d., 155, 161 quadric variety V , 133 quasi cell decomposition, 155, 161 efficient, 159, 162 flip, 63, 157, 161 Ptolemy transformation, 63 recurrent, 235 triangulation, 63, 67, 161 quasi-conformal, 97 quasi-symmetric, 98 quotient arc complex of bordered surface, 219 r-cyclic polygon, 51 Ramachandran plot, 267 real form, 28 recurrent, 215 Riemann’s moduli space, 6 RNA folding problem, 261 primary structure, 261 secondary structure, 261 shape, 263 shape moduli space, 263 Rogers’ function, 192

screen, 235 sector, 36, 131 shape, 263 shear coordinate, 41, 73, 91 of hyperbolic structure, 74 of ideal quadrilateral, 41 of measured foliation, 233 of tesselation, 90 Shinkansen Lemma, 138 short edge, 77 side-side-side axiom, 52 Siegel modular group, 241 simplicial completion, 218 coordinate, 45, 51 spine, 10 stable cohomology of moduli space, 6 curve, 213 stack, 262 standard light-cone basis, 32 steric, 266 Stokes’ Theorem for fatgraphs, 192 superposition, 141 support plane, 126 surface with holes, 75 tail edges, 161 tangential measure, 232 tautological classes, 6 Teichmüller space of bordered surface, 67 of punctured solenoid, 102 of punctured surface, 5 of surface with holes, 75 telescoping property, 132 tesselation, 88 Thompson group, 89, 112, 238 Thompson group T , 237 Torelli group, 241 space, 242

359

360 trans conformation, 266 transitive completion of PG1 , 240 translate of form, 30 triangulation of a polygon, 48 tropicalization, 234 twisted fatgraph, 193 two-cusped component, 225 uni-trivalent fatgraph, 156 unitable forms, 28 universal moduli space, 92 Ptolemy group, 237 Ptolemy groupoid, 236 unpunctured vertex, 156 untwisted fatgraph, 193 upper half-plane, 14 sheet of the hyperboloid, 17

Index

valence, 9 very forgetful mapping, 92 virtual automorphism, 107 virtual Euler characteristic of moduli space, 206 Weil–Petersson Kähler two form, 62, 192 volume form, 185 Whitehead Classical Fact, 11 collapse, 157, 161 homeomorphism, 107 move, 11, 49, 161 Wick’s Lemma, 204 windowed surface, 248 windows, 248 Witten’s Conjecture, 183