Punctured Torus Groups and 2-Bridge Knot Groups (I) (Lecture Notes in Mathematics) (v. 1)

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris 1909 Hirotaka Akiyo...

Author: Hirotaka Akiyoshi | Makoto Sakuma | Masaaki Wada | Yasushi Yamashita

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Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

1909

Hirotaka Akiyoshi · Makoto Sakuma Masaaki Wada · Yasushi Yamashita

Punctured Torus Groups and 2-Bridge Knot Groups (I)

ABC

Authors Hirotaka Akiyoshi Osaka City University Advanced Mathematical Institute 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585 Japan e-mail: [email protected]

Makoto Sakuma

Masaaki Wada Yasushi Yamashita Department of Information and Computer Sciences Nara Women’s University Kita-uoya Nishimachi Nara 630-8506 Japan e-mail: [email protected] [email protected]

Department of Mathematics Hiroshima University 1-3-1, Kagamiyama Higashi-Hiroshima 739-8526 Japan e-mail: [email protected]

Library of Congress Control Number: 2007925679 Mathematics Subject Classification (2000): 57M50, 30F40, 57M25, 20H10 ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN-10 3-540-71806-0 Springer Berlin Heidelberg New York ISBN-13 978-3-540-71806-2 Springer Berlin Heidelberg New York DOI 10.1007/978-3-540-71807-9 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 ° The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting by the authors and SPi using a Springer LATEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper

SPIN: 12042029

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543210

Preface

The main purpose of this monograph is to give a full description of Jorgensen’s theory on the space QF of quasifuchsian (once) punctured torus groups with a complete proof. Our method is based on Poincare’s theorem on fundamental polyhedra. This geometric approach enabled us to extend Jorgensen’s theory beyond the quasifuchsian space and apply to knot theory.

1. History By the late 70’s Troels Jorgensen had made a series of detailed studies on the space QF of quasifuchsian (once) punctured torus groups from the view point of their Ford fundamental domains. These studies are summarized in his famous unﬁnished paper [40]. In it, he gave a complete description of the combinatorial structure of the Ford domain of every quasifuchsian punctured torus group, and showed that the space QF can be described in terms of the combinatorics of the faces of the Ford domain. This led to the description of QF in terms of the Farey triangulation, or the modular diagram. As a byproduct, the ﬁrst examples of surface bundles over the circle with complete hyperbolic structures were obtained (cf. [41] and [43]). To date, most of Jorgensen’s work has not been published, yet it became widely known, motivated various research projects, and was successfully applied. His work, together with Riley’s construction [67] of the complete hyperbolic structure on the ﬁgure-eight knot complement, has motivated Thurston’s uniformization theorem of surface bundles over the circle [77] (cf. [63]). It had also motivated the experimental study by Mumford, McMullen and Wright [60] of the limit sets of quasifuchsian punctured torus groups. This work was sublimated into the beautiful book [61] by Mumford, Series and Wright, which displays deeply hidden fractal shapes of the space QF and the limit sets of punctured torus Kleinian groups.

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2. Motivation The authors’ interest in Jorgensen’s work grew from knot theory. We are interested in hyperbolic knots, and more generally hyperbolic links, i.e., mutually disjoint circles embedded in the 3-sphere S 3 whose complements admit complete hyperbolic structures of ﬁnite volume. Recall that the Ford domain of a complete cusped hyperbolic manifold of ﬁnite volume is the geometric dual to the canonical ideal polyhedral decomposition introduced by Epstein and Penner [27] (cf. [81]). Thus, by virtue of Mostow rigidity, the combinatorial structure of the Ford domain is a complete invariant of the topological type of such a manifold. In particular, by the knot complementary theorem due to Gordon and Luecke [32], this gives a complete invariant of a hyperbolic knot. In the joint work [71] with Weeks, the second author gave certain topological decompositions of 2-bridge link complements into topological ideal tetrahedra, by imitating Jorgensen’s decomposition of punctured torus bundles over the circle (cf. [29]), and conjectured that they are combinatorially equivalent to the canonical decompositions. Here, a 2-bridge link is a link which can be drawn with only two local maxima and minima in the vertical direction (see Fig. 0.1). We had thought that if we could understand Jorgensen’s work, then we would be able to prove the conjecture.

3. Extending of Jorgensen’s theory beyond the quasifuchsian space and application to 2-bridge links Fortunately, this turned out to be the case. Namely, we found a very natural way to understand the hyperbolic structures and the canonical decompositions of the 2-bridge link complements in the context of Jorgensen’s work. To describe the idea, recall that the 2-bridge links are parametrized by pairs (p, q) of relatively prime integers (see [22, Chap. 12]) and that the complement of the 2-bridge link of type (p, q) is homeomorphic to (the interior of) the manifold obtained from S × [−1, 1], with S a 4-times punctured sphere, by attaching 2-handles along α × (−1) and β × 1, where α and β are simple loops on S of slopes 1/0 and q/p, respectively (see Sect. 2.1, p. 16, for the deﬁnition of a slope); in particular, the link group (i.e., the fundamental group of the complement of the link) is isomorphic to the quotient group π1 (S)/α, β, where · denotes the normal closure. The extended Jorgensen’s theory realizes the operation of attaching 2-handles by a continuous family of hyperbolic cone-manifolds, whose cone axes are the union of the upper and lower tunnels, i.e., the co-cores of the 2-handles (see Fig. 0.1). According to Keen-Series’ theory of pleating varieties [44, 45, 46, 47, 49], QF is foliated by the pleating varieties, P(λ− , λ+ ), where (λ− , λ+ ) runs over (ordered) pairs of distinct projective measured laminations of the punctured torus T . By extending Jorgensen’s theory beyond the quasifuchsian space (cf.

Preface

θ+ = 0 θ− = 0

VII

θ+ = 0 θ− = 2π

0 < θ+ < 2π 0 < θ− < 2π

0 < θ+ < 2π θ− = 2π

θ+ = 2π θ− = 2π Fig. 0.1. Continuous family of hyperbolic cone-manifolds M (θ− , θ+ )

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Fig. 0.2a. Riley’s pioneering exploration of groups generated by two parabolic transformations. This computer-drawn picture has been circulated among the experts and has inspired many researchers in the ﬁelds of Kleinian groups and knot theory. This speciﬁc copy of the picture was obtained directly from Prof. Riley by the third author when he visited SUNY Binghamton in February 1991.

Fig. 0.2b. Riley slice of Schottky space together with pleating rays and their extensions

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IX

[66]), we found that if (λ− , λ+ ) is rational, i.e., each of λ± are rational, then the following hold: 1. The pleating variety P(λ− , λ+ ) has a natural extension to the outside of QF in the space of type-preserving representations of the fundamental group π1 (T ). 2. Each point in the extension is the holonomy representation of a certain hyperbolic cone manifold, which is commensurable with the hyperbolic cone manifold, M (θ− , θ+ ), whose underlying space is the complement of a 2-bridge link and whose singular set is the union of the upper and lower tunnels, which have the cone angles θ+ and θ− , respectively. Moreover the 2-bridge link is of type (p, q), or of slope q/p, if (λ− , λ+ ) is equivalent to (1/0, q/p) by a modular transformation. 3. If the (edge path) distance d(1/0, q/p) in the Farey triangulation is ≥ 3, namely if q ≡ ±1 (mod p), then the hyperbolic cone manifold M (θ− , θ+ ) exists for every pair of cone angles in [0, 2π]. Thus we have a continuous family of hyperbolic cone manifolds connecting M (0, 0), the quotient hyperbolic manifold of a doubly cusped group, with M (2π, 2π), the complete hyperbolic structure of the 2-bridge link complement. 4. If 1 ≤ d(1/0, q/p) ≤ 2, namely if q ≡ ±1 (mod p) and p = 0, then the hyperbolic cone manifold M (θ− , θ+ ) exists for every pair of cone angles in [0, 2π], except the pair (2π, 2π). In addition, if p ≥ 3, M (θ− , θ+ ) collapses to the base orbifold of the Seifert ﬁbered structure of the link complement as both cone angles approach 2π. 5. The holonomy group of M (θ− , θ+ ) is discrete if and only if θ± ∈ {2π/n | n ∈ N} ∪ {0}. In particular, that of M (2π, 2π/n) is generated by two parabolic transformations, which Riley called a Heckoid group in [68]. Actually, we have constructed these hyperbolic cone manifolds by explicitly constructing “Ford fundamental polyhedra”. In other words, we have extended Jorgensen’s description of the Ford fundamental polyhedra for quasifuchsian punctured torus groups to those of the hyperbolic cone manifolds arising from the 2-bridge links. In particular, we have shown that the canonical decompositions of hyperbolic 2-bridge link complements are isotopic to the topological ideal tetrahedral decompositions constructed in [71], proving the conjecture which motivated our project. The above result also enables us to locate the 2-bridge link groups in the representation space (Fig. 0.2b). The shaded region of the ﬁgure illustrates (the ﬁrst quadrant of) the Riley slice of the Schottky space, i.e., the subspace of C consisting of those complex numbers ω such that the group 10 11 , Gω = ω1 01 is discrete and free and such that the quotient Ω(Gω )/Gω of the domain of discontinuity is homeomorphic to the 4-times punctured sphere S (Deﬁnition

X

Preface

5.3.5). Each shaded region represents groups whose Ford domains have the same combinatorics. The lines in the shaded region are pleating rays of the Riley slice ([45]) and their extensions to the outside of the Riley slice correspond to the hyperbolic cone-manifolds M (2π, θ). In particular the endpoints with positive imaginary parts represent hyperbolic 2-bridge link groups and those on the real line represent the orbifold fundamental groups of the base orbifolds of the Seifert ﬁbered structures of non-hyperbolic 2-bridge link complements. We think this realizes what Riley had in mind, for he devoted time and eﬀort to identify 2-bridge link groups in the space of non-elementary two parabolic groups, yielding the mysterious output in Fig. 0.2a ([69]). This describes a relation between the hyperbolic structure and the bridge structure of a 2-bridge link complement. Since a bridge structure is a kind of Heegaard structure, it is naturally expected that a similar relation holds between the hyperbolic structures and the Heegaard structures of hyperbolic manifolds. In particular, we conjecture that this is the case for tunnel number 1 hyperbolic knots and their unknotting tunnels. An unknotting tunnel for a knot K is an arc τ in S 3 with τ ∩ K = ∂τ such that the complement of an open regular neighborhood is homeomorphic to a genus 2 handlebody. A knot which admits an unknotting tunnel is said to have tunnel number 1. For example, a 2-bridge knot has tunnel number 1 and each of the upper and lower tunnels is an unknotting tunnel. Tunnel number 1 knots have been extensively studied, and in particular, non-hyperbolic tunnel number 1 knots were classiﬁed by [59]. An unknotting tunnel τ of a tunnel number 1 knot K gives a Heegaard structure of the knot complement S 3 − K, in the sense that S 3 − K is homeomorphic to (the interior) of the manifold obtained from the genus 2 handlebody by adding a 2-handle, where τ corresponds to the co-core of the 2-handle. We would like to propose the following conjecture. Conjecture. Let K be a tunnel number 1 hyperbolic knot and let τ be an unknotting tunnel for K. Then there is a continuous family of hyperbolic cone manifolds whose underlying space is the knot complement and whose cone axis is the unknotting tunnel τ , where the cone angle varies from 0 to 2π. In particular, τ is isotopic to a geodesic in the hyperbolic manifold S 3 −K.

4. Related results Some of these results were announced in [8, 9, 10], and our original plan was to write a single paper or a book which contains the whole story. However, we found it very diﬃcult to explain the whole theory at once, and thus decided to divide it into a few papers. This monograph is the ﬁrst part of the series, and its main purpose is to give a full description of Jorgensen’s theory on the space QF with a complete proof. For Jorgensen’s theory on the space QF, supervised by Dunﬁeld and partially inﬂuenced by [9] and [78], Schedler [72] gave a treatment based on the

Preface

XI

theory of holomorphic motions. Though the bijectivity of the side parameter map is not proved in his paper, his approach using holomorphic motions is natural and further development is expected in the future. Our approach in turn is based on Poincare’s theorem on fundamental polyhedra. This geometric approach enables us to extend Jorgensen’s theory beyond the quasifuchsian space, where we need to treat indiscrete groups. For (attempts of) expositions of Jorgensen’s theory without proof, see [75, 8, 9, 65, 70]. The ﬁrst author has extended Jorgensen’s theory to the closure of QF in [2]. In particular, a rigorous proof was given to the well-known description of the Ford domain of the punctured torus bundles over the circle (cf. [12, 64]). We note that Lackenby [52] gave a topological proof to the fact that Jorgensen’s ideal triangulations of punctured torus bundles are genuine geometric decompositions. Gueritaud [33] also gave an alternative proof to this fact by using the angle structure. In the appendix of the paper, Futer proves by modifying Gueritaud’s argument that the topological ideal triangulations of the 2-bridge link complements in [71] are also geometric. Moreover, Gueritaud [34] also proved that these geometric decompositions are canonical. In [3], the ﬁrst author has found a nice relation between Jorgensen’s parameter of QF and the conformal end invariant of elements of QF. This together with Brock’s results [21] leads to an estimate of the convex core volume in terms of Jorgensen’s parameter. He has also found interesting applications of Jorgensen’s theory to knot theory in [4]. The computer program, OPTi [78] (cf. [79]), has been developed by the third author for the project, and it has been a driving force for our work. It is our pleasure that it has now become a favorite tool for various colleagues in the world. Collaborating with Komori and Sugawa, the third and last authors launched a project to draw Bers’ slices of QF, and various mysterious pictures have been produced ([50] and [82]).

5. A quick trip through Jorgensen’s theory and its generalization Jorgensen’s theory enables us to intuitively understand how a simple fuchsian group evolves into complicated quasifuchsian punctured torus groups and boundary groups, by looking at their Ford domains (see Figs. 0.3–0.10, 0.17, 0.19–0.21 and 1.2). Jorgensen expresses this phenomenon as follows. The Ford domain records the history of how the quasifuchsian group evolved from a simple fuchsian group.

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s(P0 )

s(P2 ) ν− = ν+ s(P1 )

P−2

P−1

P0

P1

P2

P3

P4

Fig. 0.3. (1/3, 1/3, 1/3)

ν+ s(P2 )

s(P0 )

ν− s(P1 ) P0

P1

P2

P3

P4

P5

P6

Fig. 0.4. (0.421397 − 0.0483593i, 0.295605 − 0.0422088i, 0.282998 + 0.0905681i) ∗ P−2

P4∗

P1∗

s(P1∗ ) ν+ s(P0 )

s(P2 ) ν−

P0

P1

P2

P3

P4

P5

s(P1 ) P6

Fig. 0.5. (0.433791 − 0.0551654i, 0.290295 − 0.0481496i, 0.275914 + 0.103315i)

5.1. A fuchsian punctured torus group The starting point of Jorgensen’s theory is the fuchsian group illustrated in Fig. 0.11. For each integer j, let Lj be the geodesic in the upper half plane model H2 of the hyperbolic plane, represented by the Euclidean half circle with center j/3 and radius 1/3. Let Pj be the order 2 elliptic transformation whose √ ﬁxed point is equal to the highest point (j + i)/3 of Lj where i = −1. Then Pj interchanges the inside and outside of Lj and acts on Lj as a Euclidean isometry. The product Pj+2 Pj+1 Pj is equal to the parabolic transformation K(z) = z + 1. Note that Pj+3n = K n Pj K −n for every j, n ∈ Z. Let Γ be

Preface ∗ P−2

P1∗

XIII

P4∗ s(P1∗ ) ν+ s(P0 )

s(P2 ) ν−

P0

P1

P2

P4

P3

P5

s(P1 ) P6

Fig. 0.6. (0.444228 − 0.0608968i, 0.285823 − 0.0531522i, 0.269949 + 0.114049i) ∗ P−2

P1∗

P4∗ s(P1∗ ) ν+ s(P0 )

s(P2 ) ν−

P0

P1

P2

P3

P4

P5

P6

s(P1 )

Fig. 0.7. (0.496414 − 0.0895542i, 0.263465 − 0.0781648i, 0.240121 + 0.167719i)

ν+

ν−

Fig. 0.8. (0.549741 − 0.118838i, 0.240619 − 0.103725i, 0.20964 + 0.222563i)

the group generated by {Pj | j ∈ Z}. Then it is generated by three successive elements, say P0 , P1 and P2 . Consider the shaded region R in Fig. 0.11. Then the edges of R are paired by P0 , P1 , P2 and K. By applying Poincare’s theorem on fundamental polyhedra to this setting, we see that R is a fundamental domain of the group Γ and Γ ∼ = (Z/2Z) ∗ (Z/2Z) ∗ (Z/2Z). = P0 , P1 , P2 | P02 = P12 = P22 = 1 ∼

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ν+

ν−

Fig. 0.9. (0.594262 − 0.143287i, 0.221545 − 0.125063i, 0.184193 + 0.26835i)

ν+

ν−

Fig. 0.10. (0.652971 − 0.175526i, 0.196392 − 0.153203i, 0.150637 + 0.328729i)

As shown in Fig. 0.12, the quotient H2 /Γ is a hyperbolic orbifold, O, with underlying space once-punctured sphere and with three cone points of cone angle π. The subgroup Γ0 of Γ of index 2, obtained as the kernel of the homomorphism Γ → Z/2Z sending each generator Pj to the generator of Z/2Z, is a rank 2 free group generated by A := KP0 = P2 P1 and B := K −1 P2 = P0 P1 . The union R ∪ K(R) is a fundamental domain of Γ0 , and the quotient H2 /Γ0 is homeomorphic to the once-punctured torus, T , where the puncture corresponds to the commutator [A, B] = K 2 . Thus Γ0 is a fuchsian

Preface

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K

R

P0

P1

P2

Fig. 0.11. Fuchsian group Γ = P0 , P1 , P2 and its fundamental region R

K π

π

v

π

v π

π

π

Fig. 0.12. By applying the edge pairings P0 , P1 and P2 to the fundamental region R, we obtained the surface on the left hand side. By further applying the edge pairing K to this surface, we obtain the orbifold O with underlying space once-punctured sphere and with three cone points of cone angle π.

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punctured torus group, i.e., it is a discrete free group generated by two elements with parabolic commutator. It is well-known that every fuchsian punctured torus group has a Z/2Z-extension with quotient homeomorphic to O as a topological orbifold (cf. [40, Sect. 2]). Thus we abuse terminology and call the extended group a fuchsian punctured torus group. Now look at the region, F , exterior to all Lj . Then this is a “fundamental domain of Γ (resp. Γ0 ) modulo K (resp. K 2 )” (cf. Proposition 1.1.3). This region is called the Ford polygon of Γ (resp. Γ0 ). This can be regarded as the “Dirichlet domain of Γ centered at ∞”, because F = {x ∈ H2 | d(x, H∞ ) ≤ d(x, ZH∞ ) for every Z ∈ Γ }, where H∞ is a suﬃciently small horodisk centered at ∞. This implies that the image of ∂F in H2 /Γ is equal to the cut locus of H2 /Γ with respect to the cusp, i.e., the set of points of H2 /Γ which has more than two shortest geodesics to the cusp. See Proposition 5.1.3, for a description of the Ford polygons of general fuchsian punctured torus groups.

5.2. 3-dimensional picture of the fuchsian punctured torus group Figure 0.3 gives a 3-dimensional picture of the group Γ in Fig. 0.11. The elliptic transformation Pj acts on the upper half space model H3 of the hyperbolic 3-space as √ the π-rotation around the geodesic joining the two points (j ± i)/3, where i = −1. (Here we identify the complex plane C with the boundary of 3 the closure H = H3 ∪ C.) The isometric circle I(Pj ) = {z ∈ C | |Pj′ (z)| = 1} has center c(Pj ) = j/3 and radius 1/3. The hyperplane of H3 bounded by the isometric circle I(Pj ) is called the isometric hemisphere of Pj and is denoted by Ih(Pj ). Then Pj interchanges the exterior Eh(Pj ) and the interior Dh(Pj ) of the isometric hemisphere Ih(Pj ), and acts on Ih(Pj ) as a Euclidean isometry. By the argument in Subsection 5.1, we see that the common exterior ∩j Eh(Pj ), where j runs over Z, is a “fundamental domain of the action of Γ (resp. Γ0 ) on H3 modulo K (resp. K 2 )”. Thus it is equal to the Ford domain P h(Γ ) of Γ , which is deﬁned to be the common exteriors of the isometric hemispheres of all elements of Γ that do not ﬁx ∞ (see Deﬁnition 1.1.2 and Proposition 1.1.3). As in the previous subsection, the Ford domain can be regarded as the “Dirichlet domain of Γ centered at ∞”, namely P h(Γ ) = {x ∈ H3 | d(x, H∞ ) ≤ d(x, ZH∞ ) for every Z ∈ Γ }, where H∞ is a suﬃciently small horoball centered at ∞. Thus the image of ∂P h(Γ ) in H3 /Γ ∼ = O × (−1, 1) is equal to the cut locus of H3 /Γ with respect

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to the cusp, i.e., the set of points of H3 /Γ which has more than two shortest geodesics to the cusp: We call it the Ford complex of Γ . 3 Let P h(Γ ) be the closure of the Ford domain P h(Γ ) in H . Then the intersection P (Γ ) := P h(Γ ) ∩ C has precisely two connected components, the upper polygon P + (Γ ) and the lower polygon P − (Γ ). P ± (Γ ) is a fundamental domain of the action of Γ on Ω ± (Γ ) modulo K, where Ω + (Γ ) and Ω − (Γ ) are the upper and lower components of the domain of discontinuity Ω(Γ ) = C − R.

5.3. Parameters for punctured torus groups Jorgensen’s theory describes what happens to the Ford domain when we deform the group Γ keeping the condition that K = P2 P1 P0 is the parabolic transformation z → z + 1, or equivalently, deform the group Γ0 keeping the condition that [A, B] is the parabolic transformation z → z + 2. Thus we ﬁrst need to describe the space of all such groups. Let X be the space of the equivalence classes of marked subgroups Γ of P SL(2, C) generated by an elliptic generator triple (P0 , P1 , P2 ), i.e., a triple of order 2 elliptic transformations P0 , P1 and P2 such that the product K := P2 P1 P0 is a parabolic transformation (cf. Deﬁnition 2.1.1). Two such marked groups Γ and Γ ′ , endowed with elliptic generator triples (P0 , P1 , P2 ) and (P0′ , P1′ , P2′ ), respectively, are equivalent if they are conjugate in P SL(2, C) and if the conjugation maps (P0 , P1 , P2 ) to (P0′ , P1′ , P2′ ) (cf. Deﬁnition 2.2.6). We do not distinguish between an element of X and its representative. The space X is identiﬁed with a quotient of a subspace of the cartesian product P SL(2, C)3 and thus is endowed with the quotient topology. By a marked punctured torus group, we mean an element Γ of X such that Γ is discrete and isomorphic to (Z/2Z) ∗ (Z/2Z) ∗ (Z/2Z). Then X is identiﬁed with the space of the equivalence classes of the nontrivial Markoﬀ triples. Here a Markoﬀ triple is a triple (x, y, z) ∈ C3 satisfying the Markoﬀ equation x2 + y 2 + z 2 = xyz. It is non-trivial if it is diﬀerent from (0, 0, 0). Two Markoﬀ triples (x, y, z) and (x′ , y ′ , z ′ ) are equivalent if the latter is equal to (x, y, z), (x, −y, −z), (−x, y, −z) or (−x, −y, z). We associate to each marked group Γ ∈ X , endowed with an elliptic generator triple (P0 , P1 , P2 ), the equivalence class of a Markoﬀ triple (x, y, z) by the following rule. (x, y, z) = (tr(KP0 ), tr(KP1 ), tr(KP2 )) = (tr(A), tr(AB), tr(B)), where A = KP0 and B = K −1 P2 . Note that the right hand side is deﬁned only up to sign, because it depends on the lifts to SL(2, C). If we take the lifts so that the lift of AB is the product of the lift of A and that of B, then (x, y, z) satisﬁes the Markoﬀ equation and its equivalence class is uniquely

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determined by the equivalence class of the marked group Γ , and vice versa (Propositions 2.3.6 and 2.4.2). For example, the group in Fig. 0.11 corresponds to the Markoﬀ triple (3, 3, 3). More geometric parameter of (a subspace of) X is the complex probability deﬁned as follows. Let (x, y, z) be a Markoﬀ triple such that none of the entries are zero. Then its equivalence class is completely determined by the triple (a0 , a1 , a2 ) ∈ (C − {0})3 deﬁned by (a0 , a1 , a2 ) = (

x y z , , ). yz zx xy

The only constraint on this triple is the identity a0 + a1 + a2 = 1, and thus this triple is called a complex probability. This parameter has the geometric meaning that each entry gives the diﬀerence between the centers of the isometric circles of the elliptic generators. Namely, a0 = c(P2 )−c(P1 ),

a1 = c(P3 )−c(P2 ),

a2 = c(P4 )−c(P3 ) = c(P1 )−c(P0 ).

Here P3 = KP0 K −1 and P4 = KP1 K −1 . Moreover, there is a nice geometric construction of a marked group from the corresponding complex probability (see Sect. 2.4, p. 29). The complex probability for the marked group in Fig. 0.3, for example, is ( 31 , 31 , 31 ). The triples in the captions of Figs. 0.3–0.10, 0.17, 0.19–0.25 are the complex probabilities of the corresponding marked punctured torus groups.

5.4. Small deformation of the fuchsian punctured torus group Now let us study what happens to the Ford domain if we deform the group Γ in Fig. 0.3 a little in the space X , namely we perturb the complex probability from ( 13 , 31 , 31 ) a little. The answer is that nothing happens to the combinatorial structure of the Ford domain (see Fig. 0.4). Namely, the polyhedron ∩j Eh(Pj ) continues to be the Ford domain of the (deformed) group Γ . This fact can be proved by using Poincare’s theorem on fundamental polyhedra and the following facts for the (deformed) group Γ , which are consequences of the “chain rule for isometric circles” (Lemmas 4.1.2 and 4.1.3). 1. Each face Ih(Pj ) ∩ (∩j Eh(Pj )) of the polyhedron ∩j Eh(Pj ) is symmetric with respect to Pj . 2. The sum of the dihedral angles of the polyhedron ∩j Eh(Pj ) along any three successive edges Ih(Pj ) ∩ Ih(Pj+1 ) is equal to 2π.

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XIX

The above conclusion on the Ford domain corresponds to a special case of Proposition 6.2.1 (Openness), whose rigorous proof is given in Chap. 7. We note that Schedler [72] explains this phenomenon by developing a general theory of Ford domains based on the theory of holomorphic motions. A key fact behind both proofs is that P h(Γ ) has no hidden isometric hemispheres, that is, for every element A of Γ which does not ﬁx ∞, P h(Γ ) ∩ Ih(A) = ∅ only if Ih(A) support a 2-dimensional face of the polyhedron ∩j Eh(Pj ), i.e., A = Pj for some j for the group in Fig. 0.4 (see Lemma 7.1.6). Here Ih(A) 3 denotes the closure of Ih(A) in H . The deformed group Γ is a marked quasifuchsian punctured torus group, i.e., it is a quasi-conformal deformation of the fuchsian punctured torus group in Fig. 0.3, or equivalently, the limit set of Γ continues to be a Jordan circle in the Riemann sphere ∂H3 and the quotient (H3 ∪ Ω(Γ ))/Γ continues to be homeomorphic to the product O × [−1, 1]. The quasifuchsian punctured torus space QF is the subspace of X consisting of all (marked) quasifuchsian punctured torus groups. QF is an open subset of the 2-dimensional complex manifold X . Bers’ simultaneous uniformization theorem says that a quasifuchsian punctured torus group Γ is uniquely determined by the pair (Ω − (Γ0 )/Γ0 , Ω + (Γ0 )/Γ0 ) of punctured torus Riemann surfaces (see, for example, [38] or [55]). This correspondence implies a holomorphic isomorphism between the quasifuchsian space QF and the product Teich(T ) × Teich(T ) ∼ = H2 × H2 of the Teichmuller space of T . Jorgensen’s theory also gives yet another parameterization of QF in terms of H2 × H2 (see Main Theorem 1.3.5). Though the space QF itself has a simple structure, its location in X is very complicated. Jorgensen’s theory enables us to plot the shape of QF. See Fig. 0.13 illustrating a slice of QF in X . This is an output of OPTi [78], which in turn is based on Jorgensen’s theory. See also the beautiful pictures in [61].

5.5. Birth of a new face in the Ford domain We can continue the deformation in the previous subsection until some of the circular edges of the upper/lower polygons P ± (Γ ) shrinks to a point (see Fig. 0.5). Assume for simplicity that the circular edge I(P1 ) ∩ P + (Γ ) of the upper polygon P + (Γ ) shrinks to a point, v. (The existence of such a deformation is guaranteed by Jorgensen’s theory.) What happens to the Ford domain under further deformation? The answer is given by Fig. 0.6. To describe it, we note that the “chain rule for isometric circles” (Lemmas 4.1.2 and 4.1.3) implies that the isometric circle I(P1∗ ) of P1∗ := P2 P1 P2 passes through the vertex P2 (v) of the upper polygon P + (Γ ) in Fig. 0.5. Thus Ih(P1∗ ) is a hidden isometric hemisphere of P h(Γ ). Moreover this isometric hemisphere and its translates by powers of K are the only hidden isometric hemispheres of P h(Γ ) (Lemma 7.1.6). By using this fact we see that if we

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Fig. 0.13. QF in the slice of X at a0 = 1/3

deform the group further then the Ford domain undergoes the following changes (see Fig. 0.6). 1. The hidden isometric hemisphere Ih(P1∗ ) breaks out through the vertex P2 (v) and becomes to support a 2-dimensional face of P h(Γ ). 2. The vertices v and P2 (v) of the old upper polygon P + (Γ ) lying in the complex place C are lifted to vertices of the new Ford domain P h(Γ ) in the hyperbolic space H3 . 3. The new upper polygon P + (Γ ) is described by the sequence {Pj′ } deﬁned by P0′ := P0 ,

P1′ := P2 ,

′ Pj+3n := K n Pj′ K −n

P2′ := P1∗ = P2 P1 P2 , (j ∈ {0, 1, 2}, n ∈ Z).

Namely, the edges of ∂P + (Γ ) are I(Pj′ ) ∩ P + (Γ ) (j ∈ Z) in this order. Moreover, the new Ford domain P h(Γ ) is equal to the polyhedron (∩j Eh(Pj )) ∩ (∩j Eh(Pj′ )), and it is combinatorially dual to the elliptic generator complex in Fig. 0.14, which describes the relation between the two sequences {Pj } and {Pj′ } (Deﬁnition 3.2.3). The above description of the transition of the Ford domain is proved by using the following facts.

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XXI

s′2 P2′ =

=

=

P2

P3

P5

s′1 =

=

P0 P1

s′0

P4′

P3′

=

P1′

P0′

s0

s2 s1

P4 Fig. 0.14. Adjacent sequences of elliptic generators P1∗

P0

P3 P1

P2 P1∗ P0

P3 P1

P2

P1∗ P0

P3 P1

P2

Fig. 0.15. Birth of a new face

1. The local behavior of the hidden isometric hemisphere Ih(P1∗ ) under small deformation is controlled by the “side parameter”, which we explain in Subsection 5.7, and the only possible local behaviors are those illustrated in Fig. 0.15 (see Lemma 4.6.2). 2. The chain rule of the isometric circles (Lemma 4.1.3) guarantees that the above polyhedron (∩j Eh(Pj )) ∩ (∩j Eh(Pj′ )) satisﬁes the conditions of Poincare’s theorem on fundamental polyhedra. Jorgensen’s theory shows that the transition described in the above is essentially the only possible transition of the combinatorial structure of the Ford domain when the group Γ is deformed in QF and that one can deform the group so that the Ford domain becomes arbitrary complicated (see Figs. 0.3–0.10).

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5.6. Elliptic generators and the Farey triangulation ′ ′ Observe that any successive triple (Pj′ , Pj+1 , Pj+2 ) in the bi-inﬁnite sequence ′ {Pj } introduced in the previous subsection is also an elliptic generator triple, ′ ′ i.e., it forms a generator system of Γ and Pj+2 Pj+1 Pj′ is equal to the parabolic transformation K(z) = z + 1. Such a bi-inﬁnite sequence is called a sequence of elliptic generators, and an element of a sequence of elliptic generators is called an elliptic generator (cf. Deﬁnitions 2.1.1 and 2.1.13). By using the facts that the orbifold O is commensurable with the punctured torus T and that an elliptic generator corresponds to an essential simple loop in T , we can deﬁne the slope s(P ) for each elliptic generator P (Proposition 2.1.2 and Deﬁnition 2.1.3). Moreover, for any sequence {Pj } of elliptic generators, the following hold.

1. {s(Pj )} is a periodic sequence of period 3. 2. The set {s(P0 ), s(P1 ), s(P2 )} spans a triangle of the Farey triangulation, or the modular diagram, D of the hyperbolic plane H2 (see Fig. 0.16). 3. s(P2′ ) = s(P2 P1 P2 ) is the vertex opposite to the vertex s(P1 ) with respect to the edge s(P0 ), s(P2 ) = s(P0′ ), s(P1′ ) in D. This gives a bijection between the sequences of elliptic generators (modulo shifts of indices by a multiple of 3) and the triangles of D (Proposition 2.1.10).

2 1

3 2

1 1

2 3

1 2

3 1

1 3

0 1

1 0

− 13

− 31 − 21

− 32

− 11

− 23

− 12

Fig. 0.16. Farey triangulation

Jorgensen’s theory associates to each quasifuchsian punctured torus group Γ ∈ QF a pair (σ − , σ + ) of triangles of D, so that the combinatorial structure of the Ford domain P h(Γ ) is completely determined by the pair (σ − , σ + )

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XXIII

(see Theorem 1.2.2). To be precise, (the boundaries of) the upper and lower polygons P + (Γ ) and P − (Γ ), respectively, are described by the sequences of elliptic generators associated with σ + and σ − , and the Ford domain P h(Γ ) is dual to the elliptic generator complex associated with the chain of triangles in D joining σ − and σ + (Deﬁnition 3.2.3). If σ − = σ + , then the elliptic generator complex is identiﬁed with the real line R with vertex set Z, where the vertex j ∈ Z corresponds to the j-th elliptic generator Pj . If σ − and σ + are adjacent, then the elliptic generator complex is as illustrated in Fig. 0.14.

5.7. The side parameters of quasifuchsian punctured torus groups The correspondence Γ → (σ − , σ + ) is reﬁned to a (combinatorial) homeomorphism ν : QF → H2 × H2 , resembling the holomorphic isomorphism QF ∼ = H2 × H2 via the Bers’ simultaneous uniformization theorem described in Subsection 5.4. To explain this homeomorphism, recall that the edges of the upper and lower polygons P + (Γ ) and P − (Γ ), respectively, are supported by the isometric circles of the sequences of elliptic generators {Pj+ } and {Pj− } associated with σ + and σ − . For ǫ ∈ {−, +} and j ∈ {0, 1, 2}, let θjǫ be the half of the angle of the circular edge of the polygon P ǫ (Γ ) contained in the isometric circle I(Pjǫ ). Then the following identity holds (Proposition 4.2.16): θ0ǫ + θ1ǫ + θ2ǫ =

π . 2

Thus the triple (θ0ǫ , θ1ǫ , θ2ǫ ) can be regarded as π/2 times the barycentric coordinate of a point in the triangle σ ǫ = s(P0ǫ ), s(P1ǫ ), s(P2ǫ ) of (the abstract simplicial complex having the combinatorial structure of) the Farey triangulation D. We denote the point by ν ǫ (Γ ). Then ν ǫ (Γ ) is not equal to a vertex of σ ǫ and hence it is identiﬁed with a point in H2 ∼ = |D| − |D(0) |, where (0) D denotes the 0-skeleton of D and | · | denotes the underlying space of an abstract simplicial complex (cf. Sect. 1.3, p. 12). Set ν(Γ ) = (ν − (Γ ), ν + (Γ )) and call it the side parameter of Γ . Then Jorgensen’s theory asserts that ν : QF → H2 × H2 is a homeomorphism and that the combinatorial structure of the Ford domain P h(Γ ) of Γ ∈ QF is completely described in terms of ν(Γ ) (see Main Theorem 1.3.5).

5.8. Jorgensen’s theory for boundary groups. A marked punctured torus group is (a representative of) an element of X which is discrete and isomorphic to the free product (Z/2Z)∗(Z/2Z)∗(Z/2Z). It is classically known that every marked group in the closure QF of QF in X is a marked punctured torus group (see, for example, [55, Proposition 4.18]).

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A group in QF − QF is called a boundary group. Moreover it is a consequence of Minsky’s ending lamination theorem for punctured torus groups that every Kleinian punctured torus group is contained in the closure QF (see [58] ). Jorgensen’s theorem and its generalization enable us to get a visual understanding of these complicated groups (see Figs. 0.17, 0.19–0.21). v

P0+

P1+ Fig. 0.17. ( 13 ,

1 3

+

√ 5 i, 31 6

−

√

5 i) 6

P2+

– Singly cusped group

A singly cusped group, Γ ∗ , is obtained from a quasifuchsian group Γ by a deformation which shrinks two successive edges of, say the upper polygon P + (Γ ), into a single point, while ﬁxing ν − (Γ ) (see Fig. 8 in Jorgensen [40] and Fig. 0.17). If the edges of P + (Γ ) supported by the isometric circles I(P1+ ) and I(P2+ ) are shrunk into a point v, then the two ﬁxed points of the loxodromic transformation KP0+ of Γ are united into the point v, and the corresponding element KP0+ of Γ ∗ becomes a parabolic transformation with parabolic ﬁxed point v; this transformation is called an accidental parabolic transformation. The limit set of Γ ∗ is obtained from the Jordan curve Λ(Γ ) by pinching the two ﬁxed points of each conjugate of (the old) KP0+ into a single point. Thus the “upper part” Ω + (Γ ∗ ) of the domain of discontinuity is not connected anymore; it consists of inﬁnitely many (round) disks, whereas the lower component Ω − (Γ ∗ ) remains to be an open disk. Accordingly the quotient orbifold Ω + (Γ ∗ )/Γ ∗ becomes an orbifold with underlying space a twice-punctured sphere with a cone point of cone angle π, or equivalently,

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XXV

Fig. 0.18. Pinching the punctured torus Ω + (Γ0 )/Γ0 and the quotient orbifold Ω + (Γ )/Γ

, Fig. 0.19. ( 1+i 2

1−i 1−i , 4 ) 4

– Doubly cusped group

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Preface

Fig. 0.20. (0.198264+0.376931i, 0.528473+0.142584i, 0.273263−0.519515i) – Singly degenerate group

the quotient of a triply punctured sphere which in turn is obtained from a once-punctured torus by pinching an essential simple loop corresponding to KP0+ to a point (see Fig. 0.18). The component ν + (Γ ) of the side parameter ˆ of the elliptic generator P + . The (genν(Γ ) converges to the slope s(P0+ ) ∈ Q 0 eralized) Jorgensen’s theory associates to the boundary group Γ ∗ the point ν(Γ ∗ ) := lim ν(Γ ) and gives a description of the Ford domain P h(Γ ∗ ) in terms of ν(Γ ∗ ). A doubly cusped group, Γ ∗ , is obtained from a quasifuchsian group Γ by a deformation which shrinks two successive edges of the upper polygon P + (Γ ) into a single point and two successive edges of the lower polygon P − (Γ ) into another single point (see Fig. 6 in Jorgensen [40] and Fig. 0.19). Then the limit set of Γ ∗ becomes a circle packing, and the corresponding side parameter ν(Γ ∗ ) is a pair of distinct rational numbers. For the simplest case where ν(Γ ∗ ) consists of a Farey neighbor, say (0/1, 1/0), the limit set is the Apollonian packing, and as the parameter ν(Γ ∗ ) becomes complicated, to be precise, as the relative position between ν − (Γ ∗ ) and ν + (Γ ∗ ) becomes complicated, the limit set of the doubly cusped group becomes more and more intricate (see the beautiful Figs. 7.3, 9.15. 9.16 and 9.18 in [61]). A singly or doubly cusped group remains to be geometrically ﬁnite, i.e., there is a ﬁnite volume submanifold of the quotient hyperbolic manifold which contains all closed geodesics, or equivalently, the quotient of the convex hull

Preface

Fig. 0.21. ( 12 +

2

1 √ i, − √13 i, 21 3

+

2

1 √ i) 3

XXVII

– Doubly degenerate group

of the limit set has ﬁnite volume. Constructions of more complicated geometrically inﬁnite groups are presented below. A singly degenerate group, Γ ∗ , is obtained as the limit of a sequence of quasifuchsian (or geometrically ﬁnite) groups {Γn } such that ν + (Γn ) tends to an irrational boundary point, whereas ν − (Γn ) is ﬁxed (see Fig. 0.20). Then the upper component Ω + (Γn ) of the domain of discontinuity becomes “smaller and smaller” as n → ∞, and ﬁnally disappears at the limit. So, Ω(Γ ∗ )/Γ ∗ consists of only one component. A singly degenerate group is not geometrically ﬁnite anymore and hence is geometrically inﬁnite. A doubly degenerate group Γ ∗ is obtained as the limit of a sequence of quasifuchsian (or geometrically ﬁnite) groups {Γn } such that ν(Γn ) tends to a pair of mutually distinct irrational boundary points (see Fig. 0.21). The limit set Γ ∗ is the whole Riemann sphere, and it gives rise to a sphere ﬁlling Peano curve (cf. [24], [13], [56], [18]). The special case, where ν(Γ ∗ ) := lim ν(Γn ) is the pair of the ﬁxed points of the linear fractional transformation determined by a matrix M ∈ SL(2, Z) with | tr(M )| ≥ 3, is particularly interesting to topologists. Because it gives the inﬁnite cyclic cover of a hyperbolic punctured torus bundle over the circle with monodromy M . Figure 0.21 illustrates

XXVIII

Preface

21 the doubly degenerate group corresponding to the matrix M = . This 11 gives the inﬁnite cyclic cover of the ﬁgure-eight knot complement. Explicit constructions of this historically important group were given by Jorgensen and Marden [43] (cf. [40] and [41]) and Riley [67] independently and by completely diﬀerent methods. Recall that Minsky’s ending lamination theorem [58] proves that the map assigning each Γ ∈ QF with its end invariant gives a bijection from QF to H2 × H2 − diagonal(∂H2 ), extending the holomorphic isomorphism QF ∼ = H2 × H2 . The ﬁrst author proved, by using and resembling Minsky’s theorem, that the combinatorial homeomorphism ν : QF → H2 × H2 extends to a surjective map ν : QF → H2 × H2 − diagonal(∂H2 ) so that ν(Γ ) determines the combinatorial structure of the Ford domain for each Γ ∈ QF (see [2]).

5.9. Extension of Jorgensen’s theory beyond the quasifuchsian space. Consider a doubly cusped group Γ with ν(Γ ) = (s− , s+ ), where s− and s+ are distinct vertices of D. See Fig. 0.22 where (s− , s+ ) = (∞, 2/5). Assume, for simplicity, that s− and s+ are not Farey neighbors, i.e., the edge-path distance d(s− , s+ ) is greater than 1. Let Σ = (σ1 , · · · , σm ) be the chain of triangles of D intersecting the geodesic with endpoints s− and s+ in this order, and let Σ (0) be the set of the vertices of the triangles in Σ. Then, by (generalized) Jorgensen’s theory, the Ford domain P h(Γ ) is equal to the polyhedron Eh(Γ, Σ) := ∩{Eh(P ) | P is an elliptic generator with s(P ) ∈ Σ (0) }. For each ǫ ∈ {−, +}, let P ǫ be an elliptic generator with s(P ǫ ) = sǫ . Then the transformation Aǫ := KP ǫ is an accidental parabolic transformation, whose parabolic ﬁxed point is the point of tangency between I(Aǫ ) = I(P ǫ ) and I(KAǫ K −1 ) = K(I(P ǫ )). Now, perturb the group Γ in X so that each Aǫ becomes an elliptic transformation (see Fig. 0.23). Note that its axis Axis Aǫ is equal to Ih(Aǫ ) ∩ Ih(KAǫ K −1 ) and its rotation angle θǫ is equal to the exterior angle between I(Aǫ ) and I(KAǫ K −1 ). Generically, the resulting group Γ is not discrete anymore. However, we can see that the combinatorial structure of the corresponding polyhedron Eh(Γ, Σ) is unchanged, except that a new edge contained in Axis Aǫ appears. Moreover the pairing transformations for the Ford domain of the original group continue to pair the faces of the new polyhedron Eh(Γ, Σ). (To be precise, face pairings are deﬁned for the quotient of Eh(Γ, Σ) by K.) The space obtained from Eh(Γ, Σ) by pairing the faces turns out to be a hyperbolic cone manifold commensurable with the hyperbolic cone manifold M (2θ− , 2θ+ ) in Sect. 3. We can show that this remains valid as long as 0 ≤ θǫ < π (see Fig. 0.24).

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XXIX

P+ θ+

θ− P− Fig. 0.22. (0.525833 + 0.110676i, 0.207579 + 0.389324i, 0.266588 − 0.5i) P+

θ+

θ− P

−

Fig. 0.23. (0.483147 + 0.115832i, 0.232288 + 0.514691i, 0.284565 − 0.630523i)

XXX

Preface θ+

P+

P−

θ− Fig. 0.24. (0.348619 + 0.115197i, 0.310165 + 1.1507i, 0.341216 − 1.265900i)

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XXXI

P+

P2+ P1+ P1− P2−

P−

Fig. 0.25. (0.136324 + 0.064967i, 0.429022 + 4.94894i, 0.434654 − 5.0139100i)

As the angle θǫ approaches π, the radius of Ih(Aǫ ) increases to ∞ and Ih(Aǫ ) converges to the vertical plane above a line in C parallel to the real line (see Fig. 0.25). So Eh(P ǫ ) = Eh(Aǫ ) converges to one of the two half spaces bounded by the limit vertical plane. When θǫ becomes π, the transformations Aǫ and P ǫ become π-rotations around vertical geodesics. Though the isometric hemisphere of P ǫ is not deﬁned, we continue to denote the above limit half space by Eh(P ǫ ). We now explain what happens to the corresponding Eh(Γ, Σ) when (θ− , θ+ ) becomes (π, π). Suppose d(s− , s+ ) ≥ 3, and assume (s− , s+ ) = (1/0, q/p) for simplicity. Then Eh(Γ, Σ) continues to be a 3-dimensional polyhedron and the pairing transformations for the Ford domain of the original group continue to pair

XXXII

Preface

the faces of Eh(Γ, Σ). The space obtained from Eh(Γ, Σ) by pairing the faces turns out to be a hyperbolic orbifold commensurable with the complement of the 2-bridge link of type (p, q). The only change in the combinatorial structure of Eh(Γ, Σ), except the above change in Eh(P ǫ ), is that the faces P1+ and P2+ (resp. P1− and P2− ) in Fig. 0.25 are united into the single face Q+ (resp. Q− ) in Fig. 0.26. The actual Ford domain of the group Γ is equal to the union of the images of Eh(Γ, Σ) by the inﬁnite dihedral group P − , P + , which acts on H3 as Euclidean isometries (see Fig. 0.26). P+

A+

Q+ Q−

P−

A− Fig. 0.26.

Suppose d(s− , s+ ) = 2, and assume (s− , s+ ) = (1/0, 1/p) with p ≥ 3 for simplicity. Then, as (θ− , θ+ ) approaches (π, π), Eh(P − ) ∩ Eh(P + ) degenerates into (a subset of) a vertical plane, and Eh(Γ, Σ) degenerates into a 2-dimensional polyhedron contained in the vertical plane. See Figs. 0.27–0.29 where p = 4. The limit group Γ preserves the vertical plane, and the polygon Eh(Γ, Σ) in the vertical plane is regarded as the Ford polygon of the action of Γ on the vertical plane. Moreover Γ is commensurable with the orbifold fundamental group of the base orbifold of the Seifert ﬁbered structure of the complement of the 2-bridge link of type (p, 1).

6. Reformulation of Jorgensen’s theory for quasifuchsian punctured torus groups For convenience, we regard a marked group Γ representing an element of X as the image of a type-preserving representation ρ : π1 (O) → P SL(2, C), and identify the space X with the space of equivalence classes of typepreserving representations (Deﬁnitions 2.2.1 and 2.2.6). Here π1 (O) is the orbifold fundamental group of the orbifold O and is isomorphic to the free product (Z/2Z) ∗ (Z/2Z) ∗ (Z/2Z). The fundamental group π1 (T ) of the oncepunctured torus T is an index 2 subgroup of π1 (O), and there is a one-to-one

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XXXIII

Fig. 0.27. (0.404256 − 0.254426i, 0.209822 − 0.303145i, 0.385922 + 0.557571i) – Doubly cusped group

Fig. 0.28. (0.273239 − 0.269885i, 0.30051 − 0.644994i, 0.426251 + 0.914879i)

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Preface

Fig. 0.29. (0.0186119 − 0.0948854i, 0.486063 − 4.97939i, 0.495325 + 5.07428i)

correspondence between type-preserving representations of π1 (T ) and those of π1 (O) (Proposition 2.2.2). Then a marked quasifuchsian punctured torus group Γ is identiﬁed with a quasifuchsian representation ρ, and we denote the Ford domain P h(Γ ) and the side parameter ν(Γ ), respectively, by P h(ρ) and ν(ρ). Jorgensen’s theory says that ν : QF → H2 × H2 is a homeomorphism and ν(ρ) encodes the combinatorial structure of the Ford domain P h(ρ). By abuse of notation, we often denote the image ν(ρ) = (ν − (ρ), ν + (ρ)) ∈ H2 × H2 by the symbol ν = (ν − , ν + ). Under this notation, let Σ(ν) be the chain of triangles of the Farey triangulation intersecting the geodesic segment [ν − , ν + ] joining ν − with ν + (Deﬁnition 3.3.3). Then Jorgensen’s theory says that the combinatorial structure of the Ford domain P h(ρ) is dual to a certain abstract simplicial

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XXXV

complex L(ν), called the elliptic generator complex associated with ν, which is constructed from Σ(ν) in Sect. 3.3 (p. 44). In particular, the faces of the Ford domain are supported by the isometric hemispheres of the images by ρ of the elliptic generators of π1 (O) whose slopes are vertices of Σ(ν) (cf. Deﬁnitions 2.1.1 and 2.1.3). By abuse of notation again, we often denote a point in H2 × H2 by the symbol ν = (ν − , ν + ), and we call it a label. Then we are led to the concept of a labeled representation, which is deﬁned to be a pair ρ = (ρ, ν) of a type-preserving representation ρ : π1 (O) → P SL(2, C) and a label ν = (ν − , ν + ) ∈ H2 × H2 (Deﬁnition 3.3.2). We say that ρ is quasifuchsian if (i) ρ is quasifuchsian, (ii) the Ford domain P h(ρ) is as described by Jorgensen, and (iii) ν is equal to the side parameter of ρ (Deﬁnition 6.1.1). Our ﬁrst task is to give a characterization of quasifuchsian labeled representations, which is suitable for the proof of Main Theorem 1.3.5. This is done by Theorem 6.1.8 (Good implies quasifuchsian), which shows that if a labeled representation is “good”, then it is quasifuchsian. Here a labeled representation is deﬁned to be good if it satisﬁes the three conditions, Nonzero, Frontier and Duality (Deﬁnition 6.1.7). Though these conditions are rather complicated, it is not diﬃcult to check if a given labeled representation satisﬁes them. Thus Theorem 6.1.8 gives us a practical method for checking if a given labeled representation is quasifuchsian. (Actually, Modiﬁed Main Theorem 6.1.11 implies that the converse to Theorem 6.1.8 is also valid, and hence a labeled representation is quasifuchsian if and only if it is good.) The proof of Theorem 6.1.8 is done by using Poincare’s theorem on fundamental polyhedra in Sects. 6.3 (p. 138), 6.4 (p. 142) and 6.5 (p. 144). Let J [QF] ⊂ QF × (H2 × H2 ) be the space of all good labeled representations. Then Jorgensen’s result is equivalent to Modiﬁed Main Theorem 6.1.11 which asserts that the projections µ1 : J [QF] → QF and µ2 : J [QF] → H2 × H2 are homeomorphisms.

7. Idea of the proof that µ1 is a homeomorphism The injectivity of µ1 is guaranteed by Proposition 6.2.5 (Uniqueness of good label), which is an easy consequence of the deﬁnition and is proved in Sect. 6.4 (p. 142). So the main task is the proof of surjectivity. Namely, we need to show that the combinatorial structure of the Ford domain of a given quasifuchsian representation is as described by Jorgensen. Roughly speaking, the proof amounts to showing that under a continuous deformation of the quasifuchsian representations, combinatorial transitions of the Ford domain happen only at their frontier in the ideal boundary, i.e., in the complex plane C, as illustrated in Fig. 4.15, and the combinatorial structure in the hyperbolic space is stable. The actual proof consists of the following two steps. 1. Proposition 6.2.1 (Openness), which guarantees the openness of the image of the projection µ1 : J [QF] → QF in QF

XXXVI

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2. Propositions 6.2.2 (SameStratum) and 6.2.4 (Closedness), which guarantee the closedness of the image of the projection µ1 : J [QF] → QF in QF Since we can easily see that the image of µ1 is non-empty, i.e., there exists a good labeled representation (Proposition 5.1.3), and since QF is connected, the above results imply that µ1 : J [QF] → QF is surjective. Step 1 is completed in Chap. 7, and a rough idea for this step is as follows. To show the openness of the image µ1 (J [QF]) in QF around ρ = µ1 (ρ, ν), we pick up a set of elliptic generators of π1 (O), which is ﬁnite modulo conjugation by the peripheral element, and study how the pattern of the corresponding isometric hemispheres change after a small perturbation of the representation ρ. By virtue of the key Lemmas 4.6.1, 4.6.2 and 4.6.7 (see Fig. 4.15) and Lemma 7.2.3 (disjointness), we see that each nearby representation ρ′ has a label ν ′ such that (ρ′ , ν ′ ) is good. We thus obtain the openness of µ1 (J [QF]). We note that Schedler [72] obtains the corresponding result, by establishing a general stability theorem of the Ford domains using the theory of holomorphic motions. However, we do not know if the idea of holomorphic motions can be applied in the proof of the generalization of Jorgensen’s theory to hyperbolic cone-manifolds. Step 2 is completed in Chap. 8. To describe a rough idea, let {(ρn , ν n )} be a sequence of good labeled representations such that lim ρn ∈ QF. Then Proposition 6.2.2 (SameStratum) guarantees that some subsequence satisﬁes the condition SameStratum (Deﬁnition 6.2.2). Roughly speaking, this conditions says that the Ford domains P h(ρn ) have the same combinatorial structure and therefore we can talk about the “behavior of a face (or an edge or a vertex) of P h(ρn ) as n → ∞”. The proof of Proposition 6.2.2 is done in Sect. 8.1 (p. 172), and it is based on the fact that the convergence ρn → ρ∞ is strong and a certain lemma (Lemma 8.1.1) due to Jorgensen, which is a prototype of Minsky’s pivot theorem in [58]. Proposition 6.2.4 (Closedness) is proved in Sects. 8.3 (p. 180) - 8.12 (p. 209) in Chap. 8, and its main ingredient is to show that no unexpected degeneration of a face of P h(ρn ) happens as n → ∞. This is the most involved part of this monograph. A reason why it is so involved is that we have to list all possible degenerations, before showing degenerations do not happen. However, as is found in Jorgensen’s original argument [40], the idea to prohibit degenerations consists of only a few simple observations (see the introduction to Chap. 8). Another reason for the complication in this step (and the previous step) lies in the treatment of the ‘thin’ case, i.e., the case when both components of ν ∞ = lim ν n belong to the interior of a single edge τ of the Farey triangulation. However, we can treat this special case by using the results established in Sect. 5.2 (p. 106).

Preface XXXVII

8. Idea of the proof that µ2 is a homeomorphism The proof consists of the following two steps. 1. Proposition 6.2.7 (Convergence), which shows that, for a sequence of good labeled representations {(ρn , ν n )}, if lim ν n ∈ H2 × H2 exists (and if it satisﬁes the condition SameStratum), then it has a subsequence such that the corresponding subsequence of {ρn } converges in QF. 2. Chapter 9, where the proof of bijectivity of the map µ2 is established by using Propositions 6.2.2 (SameStratum), 6.2.4 (Closedness) and 6.2.7 (Convergence) and an elementary intersection theory in algebraic geometry. We describe a rough idea of Step 2. In Lemma 4.2.18, we see that if (ρ, ν) is a good labeled representation which belongs to the inverse image of a label ν ∈ H2 × H2 by µ2 , then ρ (to be precise, a Markoﬀ map inducing ρ) satisﬁes a certain algebraic equation. What we need to do is to single out a unique ‘geometric’ root among numerous roots of an algebraic equation associated with ν. To this end, we introduce the concept of the geometric multiplicity dG (ν) for each ν = (ν − , ν + ) by using the elementary intersection theory in algebraic geometry (Deﬁnition 9.2.1). Then the bijectivity of the projection µ2 : J [QF] → H2 × H2 is equivalent to the assertion that dG (ν) = 1 for every label ν. After making a detailed study of the algebraic varieties associated with the labels (Sect. 9.1, p. 215), we show in Corollary 9.2.7 that dG (ν) does not depend on ν by using the “continuity of roots” (Lemmas 9.3.1 and 9.3.3) and Propositions 6.2.2 (SameStratum), 6.2.4 (Closedness) and 6.2.7 (Convergence). On the other hand, it is easy to see that dG (ν) = 1 if ν belongs to the diagonal set, i.e., ν − = ν + (Proposition 5.1.5). We thus obtain the bijectivity of µ2 : J [QF] → H2 ×H2 , and hence that of ν : QF → H2 ×H2 . We note that, though the bijectivity is claimed in [40], no indication of the proof is presented. The arguments outlined here is new in this sense. Actually, we were able to complete this step only fairly recently.

9. Organization of the monograph The monograph consists of nine chapters and an appendix. In Chap. 1, we reformulate Jorgensen’s theory from the 3-dimensional viewpoint, and give a conceptual description of Jorgensen’s theorem (Theorems 1.2.2, 1.3.2 and Main Theorem 1.3.5). We also give a description of the ideal tetrahedral complex which is a geometric dual to the Ford domain (Theorem 1.4.2). This chapter is essentially equal to the announcement in [10]. In Chap. 2, we describe the intimate relations among the Fricke surfaces, namely the punctured torus T , the four-times punctured sphere S, and the (2, 2, 2, ∞)-orbifold O. We ﬁrst give a complete description of the ‘geometric’

XXXVIII Preface

generator systems of the fundamental groups of the Fricke surfaces (Propositions 2.1.6 and 2.1.9) and describe their relation with the Farey triangulation (Proposition 2.1.10). Then we show the equivalence among the spaces of the type-preserving representations (Deﬁnition 2.2.1) of the fundamental groups of Fricke surfaces (Proposition 2.2.2). The concepts of the Markoﬀ maps and complex probabilities are introduced in Sects. 2.3 and 2.4, respectively, and explicit matrices for the type-preserving representations and its intuitive description are given (Lemma 2.3.7 and Proposition 2.4.4). Though almost all contents in this chapter seem to be known to the experts, we could not ﬁnd explicit reference for some of the contents. Thus we included full proofs for all propositions in this chapter. In Chap. 3, we introduce the deﬁnitions of a labeled representation ρ = (ρ, ν), the complex L(ν), which is a combinatorial dual to the Ford domain, and the virtual Ford domain Eh(ρ). These are used in Chap. 6 to reformulate Main Theorem 1.3.5. In Chap. 4, we ﬁrst give a detailed proof to the chain rule for isometric circles (Lemma 4.1.2), on which the whole argument is based. We then introduce Jorgensen’s side parameter (Deﬁnition 4.2.9), and prove various properties of the side parameter. In Chap. 5, we give a detailed study of some special examples. In particular, complete descriptions of real representations and isosceles representations are given in Sects. 5.1 and 5.2. Though isosceles representations themselves are simple objects, their neighborhood in QF is rich in variety. However, they are essentially controlled by side parameters (Proposition 5.2.13). These representations also form the starting point toward the proof of Jorgensen’s theory. In Sect. 5.3, we describe how two-parabolic groups arise as the images of type-preserving representations of certain kind, and explain the reason why the generalized Jorgensen’s theory is useful to the study of the 2-bridge links. In Chap. 6, we give a 2-dimensional reformulation of Main Theorem 1.3.5 (Modiﬁed Main Theorem 6.1.11). This is more similar to Jorgensen’s original description than the three dimensional picture described in Chap. 1, and is rather complicated. However, it ﬁts with the proof given in this monograph. The equivalence between the two formulations is guaranteed by Theorem 6.1.8, which is proved by using (a variation of) Poincare’s theorem on fundamental polyhedron. In Sect. 6.2, we give a route map for the proof of Modiﬁed Main Theorem 6.1.11. At the end of this chapter, we prove certain properties of the elements in QF which are useful in the actual computation of the Ford domains (see Propositions 6.7.1 and 6.7.2). These properties are also used in the Bers’ slice project in [50] (cf. [82]). The remaining Chaps. 7, 8 and 9 are devoted to the proof of Modiﬁed Main Theorem 6.1.11. To be precise, Steps 1 and 2 , respectively, presented in Sect. 7 are completed in Chaps. 7 and 8, and Steps 1 and 2 , respectively, presented in Sect. 8 are completed in Chaps. 8 and 9. (See also Sect. 6.2 for the route map.)

Preface

XXXIX

In the Appendix, we give a proof to some of the basic facts concerning the Ford domain.

Acknowledgment We would like to express our deepest thanks to Troels Jorgensen for his explanations of his results, invaluable suggestions, and for the beautiful mathematics he had produced. We were very much encouraged when he told us he had got an inspiration that there should be some relationship between his work and 2-bridge links. We also thank his collaborator, Al Marden, for his interest on our work and encouragement. It was great pleasure to explain our approach to him and Caroline Series at University of Warwick. We sincerely thank Caroline Series for her kind hospitality at University of Warwick. Comparison of our work and her joint works with Linda Keen on punctured torus groups lead us to the related works [7, 10]. We were stimulated very much by attending the wonderful workshops organized by her. We also thank members and visitors of University of Warwick, including David Epstein, Brian Bowditch, Young Eun Choi, Raquel Diaz and Mark Lackenby for their interest and encouragement. We thank Brian Bowditch for sending us his mysterious paper [17] and for fruitful discussion, which brought the ﬁrst and second authors to the related works with Hideki Miyachi [5, 6]. We thank Travis Schedler for sending us his beautiful paper [72] on Jorgensen’s work, which was partially inﬂuenced by our preliminary announcement [9]. Interplay with his paper has improved the monograph very much. In particular, Chap. 7 and Appendix A.1 were much inﬂuenced by his paper. We thank Norbert A’Campo, Michael Heusener, Eriko Hironaka and Kazuhiro Konno for helpful discussion concerning Chap. 9. We thank Stephan Hamperies, Heinz Helling, Richard Weidmann and Heiner Zieschang for their helpful suggestions concerning Sect. 2.1. Stephan Hamperies taught us a neat proof of Proposition 2.1.6, which greatly simpliﬁes our original proof. We thank Alexander Mednykh for discussion concerning Proposition 2.2.2. We thank Norbert A’Campo, Gerhard Burde, Shungbok Hong, Yoichi Imayoshi and Masaaki Yoshida for inviting the second author to series of lectures on our work at University Basel, Goethe University Frankfurt am Main, Pohang University, Osaka City University and Kyushu University. Yohei Komori encouraged us to organize a workshop to explain the ﬁrst version of the monograph. The workshop was held at Osaka University during 12-15, September, 2005. We thank the participants of the workshop, in particular to Michihiko Fujii, Yohei Komori, Hideki Miyachi, Toshihiro Nakanishi, Yoshihide Okumura, Ken’ichi Ohshika, Hiroki Sato, Kenneth Shackleton, Masahiko Taniguchi, Akira Ushijima and Xiantao Wang for their devotion, patience and suggestions. We thank Kenneth Shackleton and Yoshihide Okumura for their careful reading of the ﬁrst version. We also thank Colin Adams, Iain Aitchison, Michel Boileau, Warren Dicks, Craig Hodgson, Sadayoshi Kojima, John Parker, Joan Porti, Alan Reid, Toshiyuki

XL

Preface

Sugawa, Hyam Rubinstein and Claude Weber for stimulating conversations and encouragements. Finally, we thank the referees for their very appropriate comments on the ﬁrst version, which enabled us to improve this monograph drastically.

Contents

1

Jorgensen’s picture of quasifuchsian punctured torus groups 1 1.1 Punctured torus groups, Ford domains and EPH-decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Jorgensen’s theorem for quasifuchsian punctured torus groups (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Jorgensen’s theorem for quasifuchsian punctured torus groups (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 The topological ideal polyhedral complex Trg(ν) dual to Spine(ν) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2

Fricke surfaces and P SL(2, C)-representations . . . . . . . . . . . . . 2.1 Fricke surfaces and their fundamental groups . . . . . . . . . . . . . . . . 2.2 Type-preserving representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Markoﬀ maps and type-preserving representations . . . . . . . . . . . 2.4 Markoﬀ maps and complex probability maps . . . . . . . . . . . . . . . . 2.5 Miscellaneous properties of discrete groups . . . . . . . . . . . . . . . . . .

15 16 21 26 29 33

3

Labeled representations and associated complexes . . . . . . . . . 3.1 The complex L(ρ, σ) and upward Markoﬀ maps . . . . . . . . . . . . . 3.2 The complexes L(ρ, Σ) and L(Σ) . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Labeled representation ρ = (ρ, ν) and the complexes L(ρ) and L(ν) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Virtual Ford domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 38 41

Chain rule and side parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Chain rule for isometric circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Side parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 ǫ-terminal triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Basic properties of ǫ-terminal triangles . . . . . . . . . . . . . . . . . . . . . 4.5 Relation between side parameters at adjacent triangles . . . . . . . 4.6 Transition of terminal triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49 50 56 66 70 78 82

4

44 44

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4.7 Proof of Lemma 4.5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.8 Representations which are weakly simple at σ . . . . . . . . . . . . . . . 95 5

Special examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.1 Real representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.2 Isosceles representations and thin labels . . . . . . . . . . . . . . . . . . . . 106 5.3 Groups generated by two parabolic transformations . . . . . . . . . . 117 5.4 Imaginary representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.5 Representations with accidental parabolic/elliptic transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6

Reformulation of Main Theorem 1.3.5 and outline of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.1 Reformulation of Main Theorem 1.3.5 . . . . . . . . . . . . . . . . . . . . . . 134 6.2 Route map of the proof of Modiﬁed Main Theorem 6.1.11 . . . . 136 6.3 The cellular structure of ∂Eh(ρ) . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.4 Applying Poincare’s theorem on fundamental polyhedra . . . . . . 142 6.5 Proof of Theorem 6.1.8 (Good implies quasifuchsian) . . . . . . . . . 144 6.6 Structure of the complex ∆E and the proof of Theorem 6.1.12 . 147 6.7 Characterization of Σ(ν) for good labeled representations . . . . 151

7

Openness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.1 Hidden isometric hemispheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.2 Proof of Proposition 6.2.1 (Openness) - Thick Case - . . . . . . . . . 159 7.3 Proof of Proposition 6.2.1 (Openness) - Thin case - . . . . . . . . . . 165

8

Closedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 8.1 Proof of Proposition 6.2.3 (SameStratum) . . . . . . . . . . . . . . . . . . 172 8.2 Proof of Proposition 6.2.7 (Convergence) . . . . . . . . . . . . . . . . . . . 178 8.3 Route map of the proof of Proposition 6.2.4 (Closedness) . . . . . 180 8.4 Reduction of Proposition 8.3.5 - The condition HausdorﬀConvergence - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 8.5 Classiﬁcation of simplices of L(ν) . . . . . . . . . . . . . . . . . . . . . . . . . . 184 8.6 Proof of Proposition 8.4.4 (F ∞ (ξ) ⊂ ∂Eh(ρ∞ , L0 )) . . . . . . . . . . 185 8.7 Accidental parabolic transformation . . . . . . . . . . . . . . . . . . . . . . . 187 8.8 Proof of Proposition 8.4.5 - length 1 case - . . . . . . . . . . . . . . . . . . 189 8.9 Proof of Proposition 8.4.5 - length ≥ 2 case - (Step 1) . . . . . . . . 191 8.10 Proof of Proposition 8.4.5 - length ≥ 2 case - (Step 2) . . . . . . . . 203 8.11 Proof of Proposition 8.4.5 - length ≥ 2 case - (Step 3) . . . . . . . . 206 8.12 Proof of Proposition 8.3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

9

Algebraic roots and geometric roots . . . . . . . . . . . . . . . . . . . . . . . 215 9.1 Algebraic roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 9.2 Unique existence of the geometric root . . . . . . . . . . . . . . . . . . . . . 227 9.3 Continuity of roots and continuity of intersections . . . . . . . . . . . 229

Contents XLIII

A

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 A.1 Basic facts concerning the Ford domain . . . . . . . . . . . . . . . . . . . . 233

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

1 Jorgensen’s picture of quasifuchsian punctured torus groups

In [40, Theorems 3.1, 3.2 and 3.3], Jorgensen describes the combinatorial structure of the Ford domain of a quasifuchsian punctured torus group. It was very diﬃcult for the authors to get a conceptual understanding of the statement, because it consists of nine assertions, each of which describes some property of the Ford domain, and it does not explicitly present a topological or combinatorial model of the Ford domain. In this chapter, we construct an explicit model of the Ford domain, and reformulate Jorgensen’s theorem in terms of the model. In short, we present a 3-dimensional picture to Jorgensen’s theorem. We note that this chapter is essentially equal to the announcement [10]. In Sect. 1.1, we recall the deﬁnition of quasifuchsian punctured torus groups and their Ford domains, and then explain the geometric meaning of the Ford domain as the canonical spine of the quotient hyperbolic manifold. We also give a quick review of the duality between the Ford domains and the canonical ideal polyhedral decompositions of cusped hyperbolic manifolds introduced by Epstein and Penner [27], and recall the EPH-decomposition of a possibly inﬁnite volume cusped hyperbolic manifold, which was introduced by the ﬁrst and second authors motivated by this project. Those readers who are interested only in the Ford domains can skip this part of the section. In Sect. 1.2, we construct a family of spines of T × [−1, 1], where T is the topological once-punctured torus, and reformulate Jorgensen’s Theorems 3.1, 3.2 and 3.3 in [40] into the single Theorem 1.2.2, which essentially asserts that the Ford complex (= the canonical spine determined by the Ford domain) of the quotient quasifuchsian punctured torus hyperbolic manifold is isotopic to a spine in the family. In Sect. 1.3, we recall Jorgensen’s side parameter of a quasifuchsian punctured torus group and reformulate Jorgensen’s Theorem 4.6 in [40], which assert that the side-parameter is actually a parameter of QF, as Theorem 1.3.2.

2

1 Jorgensen’s picture of quasifuchsian punctured torus groups

In Sect. 1.4, we present a topological model of the canonical ideal polyhedral decomposition dual to the Ford domain of a quasifuchsian punctured torus manifold (Theorem 1.4.2).

1.1 Punctured torus groups, Ford domains and EPH-decompositions Let T be the topological (once) punctured torus. A marked punctured torus group is the image of a discrete faithful representation ρ : π1 (T ) → P SL(2, C) satisfying the following condition: •

If ω ∈ π1 (T ) is represented by a simple loop around the puncture, then ρ(ω) is parabolic.

Two marked punctured torus groups Γ = ρ(π1 (T )) and Γ ′ = ρ′ (π1 (T )) are equivalent if ρ is conjugate to ρ′ by an element of P SL(2, C). A marked fuchsian punctured torus group is a marked punctured torus group which is determined by a fuchsian representation, i.e., a faithful and discrete P SL(2, R)-representation. A marked quasifuchsian punctured torus group is a marked punctured torus group Γ which is a quasiconformal deformation of a fuchsian punctured torus group. (For standard terminologies and facts in Teichm¨ uller theory and Kleinian groups, see [38] and [55].) This is equivalent to the condition that the domain of discontinuity Ω(Γ ) consists of exactly two simply connected components Ω ± (Γ ), whose quotient Ω ± (Γ )/Γ are each homeomorphic to T . We employ a sign convention so that there is an orientation-preserving homeomorphism f from T × [−1, 1] to the quotient ¯ (Γ ) = (H3 ∪Ω(Γ ))/Γ such that f (T ×{±1}) = Ω ± (Γ )/Γ and that manifold M the isomorphism f∗ : π1 (T × [−1, 1]) = π1 (T ) → π1 (M (Γ )) = Γ < P SL(2, C) is identiﬁed with ρ. Since such a homeomorphism is unique up to iso¯ (Γ ), Ω − (Γ )/Γ, Ω + (Γ )/Γ ) with topy, we can identify the topological triple (M (T × [−1, 1], T × {−1}, T × {1}). The quasifuchsian punctured torus space QF is the space of the equivalence classes of marked quasifuchsian punctured torus groups. By Bers’ simultaneous uniformization theorem, QF is identiﬁed with uller the product Teich(T ) × Teich(T ) ∼ = H2 × H2 of two copies of the Teichm¨ space of T via the correspondence Γ → (Ω − (Γ )/Γ, Ω + (Γ )/Γ )). In particular, it is an open connected subset in the space of type-preserving P SL(2, C)representations of π1 (T ) modulo conjugation (Deﬁnitions 2.2.1 and 2.2.6). As a consequence of Minsky’s celebrated theorem [58], the space of marked punctured torus groups is equal to the closure QF of QF in the representation space. We now recall the deﬁnition of the isometric circles and the Ford domains. ab Deﬁnition 1.1.1. Let A = be an element of P SL(2, C) = Isom+ (H3 ) cd such that A(∞) = ∞, namely c = 0.

1.1 Punctured torus groups, Ford domains and EPH-decompositions

3

1. The isometric circle I(A) of A is deﬁned by I(A) = {z ∈ C | |A′ (z)| = 1} = {z ∈ C | |cz + d| = 1}. Thus I(A) is the circle in the complex plane with center c(A) = −d/c = A−1 (∞) and radius r(A) = 1/|c|. D(A) denotes the disk in C bounded by I(A), and E(A) denotes the closed exterior C − int D(A) of D(A). 2. The isometric hemisphere Ih(A) is the hyperplane of the upper half space H3 bounded by I(A). Dh(A) denotes the half space of H3 bounded by Ih(A) whose closure contains D(A), and Eh(A) denotes the closed exterior H3 − int Dh(A) of Dh(A). 3. Ih(A), Dh(A) and Eh(A), respectively, denote the closure of Ih(A), 3 Dh(A) and Eh(A) in the closure H = H3 ∪ C of the upper half space model of hyperbolic 3-space. Let Γ be a non-elementary Kleinian group such that the stabilizer Γ∞ of ∞ contains parabolic transformations. Deﬁnition 1.1.2. The Ford domain P h(Γ ) of Γ is the subset of the upper half space H3 which consists of all points lying exterior to each of isometric hemispheres deﬁned by Γ . The Ford polygon P (Γ ) is the subset of the complex plane which consists of all points lying exterior to each of the isometric circles 3 deﬁned by Γ . We also deﬁne P h(Γ ) to be the subset of H similarly. Namely P h(Γ ) = {Eh(A) | A ∈ Γ − Γ∞ }, P (Γ ) = {E(A) | A ∈ Γ − Γ∞ }, P h(Γ ) = {Eh(A) | A ∈ Γ − Γ∞ }

The above notations follow those of [39]: P stands for polygon and polyhedron and h stands for (3-dimensional) hyperbolic space. But they are slightly diﬀerent from those in [40], where the same sets are denoted by P(Γ ) and Ph(Γ ) respectively. We describe a geometric meaning of the Ford domain. To this end, pick a small horoball, H∞ , centered at ∞ which is precisely invariant by (Γ, Γ∞ ), that is, for any element A ∈ Γ , A(H∞ ) ∩ H∞ = ∅ if and only if A ∈ Γ∞ . The existence of such a horoball is guaranteed by the Shimizu-Leutbecher inequality (see Lemma 2.5.2(1) or [55, Lemma 0.6]). Then for each element A ∈ Γ − Γ∞ , the isometric hemisphere Ih(A) is equal to the set of points in H3 which are equidistant from H∞ and A−1 (H∞ ). This implies that P h(Γ ) can be regarded as the “Dirichlet domain of Γ centered at ∞”, because P h(Γ ) = {x ∈ H3 | d(x, H∞ ) ≤ d(x, AH∞ ) for every A ∈ Γ } = {x ∈ H3 | d(x, H∞ ) = d(x, Γ H∞ )}.

4

1 Jorgensen’s picture of quasifuchsian punctured torus groups

Just as the (usual) Dirichlet domain is a fundamental polyhedron, we have the following proposition. See Appendix (Sect. A.1) for the proof of the proposition and related facts. Proposition 1.1.3. The Ford domain P h(Γ ) is a “fundamental polyhedron of Γ modulo Γ∞ ”, in the following sense. 1. H3 = ∪{A(P h(Γ )) | A ∈ Γ }. 2. If A ∈ Γ∞ then A(P h(Γ )) = P h(Γ ), whereas if A ∈ Γ − Γ∞ then A(int P h(Γ )) ∩ int P h(Γ ) = ∅. 3. P h(Γ ) is a convex polyhedron (Deﬁnition 3.4.1(2)). 4. For any compact set K of H3 , the set {AΓ∞ ∈ Γ/Γ∞ | A(P h(Γ ))∩K = ∅} is a ﬁnite set. By the above proposition, the intersection of P h(Γ ) with a fundamental domain of Γ∞ is a fundamental domain of Γ . As is noted in [27, Sect. 4], it is more natural to work with the quotient P h(Γ )/Γ∞ in H3 /Γ∞ . In fact the hyperbolic manifold M (Γ ) is obtained from P h(Γ )/Γ∞ by identifying pairs of faces by isometries. Note that H∞ projects to a cuspidal region, C, in the quotient hyperbolic 3-manifold M (Γ ) = H3 /Γ . Then by the preceding description of the Ford domain, the image of ∂P h(Γ ) in M (Γ ) is equal to the cut locus of M (Γ ) with respect to C, which is deﬁned as follows. Deﬁnition 1.1.4. (1) The cut locus, Cut(M (Γ ), C), of M (Γ ) with respect to C is the subspace of M (Γ ) consisting of those points which have more than one shortest geodesics to a ﬁxed horospherical neighborhood. (2) The Ford complex, Ford(Γ ), of Γ is the closure of the cut locus ¯ (Γ ) = (M ∪ Ω(Γ ))/Γ . Cut(M (Γ ), C) in the Kleinian manifold M In the remainder of this section, we describe the ideal polyhedral complex, ) be the denoted by ∆E (Γ ), which is the geometric dual to Ford(Γ ). Let Ford(Γ 2-dimensional complex in the hyperbolic space obtained as the inverse image ) is the cut locus of the of Ford(Γ ) ∩ M (Γ ). It should be noted that Ford(Γ ) disjoint union, Γ H∞ , of the images of the horoball H∞ by Γ , namely, Ford(Γ 3 consists of the points in H which have more than two shortest geodesics to ). Then, generically, p is the Γ H∞ . Let p be a vertex of ∂P h(Γ ) ⊂ Ford(Γ intersection of three isometric hemispheres, Ih(Aj ) (j ∈ {1, 2, 3}), and hence it is equidistant from the four horoballs, H∞ and A−1 j (H∞ ) (j ∈ {1, 2, 3}). We regard the ideal tetrahedron spanned by the centers of these horoballs as the ), geometric dual to the vertex p. Similarly, for each edge (resp. face) of Ford(Γ we can associate an ideal triangle (an ideal edge) as its geometric dual. The family of these ideal tetrahedra, ideal triangles and ideal edges descends to an ideal polyhedral complex embedded in M (Γ ); this is the desired ideal polyhedral complex ∆E (Γ ).

1.1 Punctured torus groups, Ford domains and EPH-decompositions

5

A more precise description of ∆E (Γ ) is given as follows. For a point p ) of Γ H∞ , let Hx0 , Hx1 , · · · , Hxk be the horoballs of in the cut locus Ford(Γ Γ H∞ such that: 1. Hxi is centered at xi . 2. d(p, Hx0 ) = d(p, Hx1 ) = · · · = d(p, Hxk ) = d(p, Γ H∞ ). 3. d(p, H ′ ) > d(p, Γ H∞ ) for any horoball H ′ in Γ H∞ −{Hx0 , Hx1 , · · · , Hxk }.

p = x0 , x1 , · · · , xk be the ideal polyhedron spanned by the centers Let ∆ x0 , x1 , · · · , xk of the horoballs Hx0 , Hx1 , · · · , Hxk . If two points p and p′ belong p is regarded as p = ∆ p′ . Thus ∆ ), then ∆ to the same (open) cell of Ford(Γ ) in which p belongs to, and we the geometric dual to the cell, e, of Ford(Γ e . Then the family denote it by ∆ E (Γ ) := {∆ e | e is a cell of Ford(Γ )} ∆

determines a Γ -invariant ideal polyhedral decomposition of H3 . This descends to an ideal polyhedral complex ∆E (Γ ) embedded in M (Γ ). We note that some member of ∆E (Γ ) is not a true ideal polyhedron but a quotient of an ideal polyhedron if Γ has a nontrivial torsion. Following the argument of Epstein and Penner [27, Sect. 10] (cf. [7, Sect. 10]), we show that ∆E (Γ ) arises from the Epstein-Penner convex hull construction in the Minkowski space. Let E1,3 be the 4-dimensional Minkowski space with the Minkowski product x, y = −x0 y0 + x1 y1 + x2 y2 + x3 y3 . Then H3 = {x ∈ E1,3 | x, x = −1, x0 > 0}, together with the restriction of the Minkowski product to the tangent space, gives a hyperboloid model of the 3-dimensional hyperbolic space. Any horoball H in this model is represented by a vector, v, in the positive light cone (i.e., v, v = 0 and v0 > 0) as H = {x ∈ H3 | v, x ≥ −1}. The center of the horoball H corresponds to the ray thorough v, and as v moves away from the origin along the ray, the horoball contracts towards its center. Let v∞ denote the light-like vector representing the horoball H∞ . Then its orbit Γ v∞ is the set of light-like vectors corresponding to the horoballs in Γ H∞ . Let C be the closed convex hull of Γ v∞ in E1,3 . Now consider the p = x0 , x1 , · · · , xk in H3 which is dual to a vertex p of ideal polyhedron ∆ ). Then the horoballs Hx0 , Hx1 , · · · , Hxk in the orbit Γ H∞ centered Ford(Γ at x0 , x1 , · · · , xk attain the minimal distance from p among the members in Γ H∞ . Let vxi be the light-like vector representing the horoball Hxi . After coordinate change, we may assume the vertex p corresponds to the vector

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1 Jorgensen’s picture of quasifuchsian punctured torus groups

(1, 0, 0, 0) in the hyperboloid model. Then the points vx0 , vx1 , · · · , vxk lie in a horizontal hyperplane W : x0 = constant, because of the second condition. Moreover, by the third condition, all points in Γ v∞ − {vx0 , vx1 , · · · , vxk } lie above the hyperplane W . Hence we see that the hyperplane W is a support plane of the convex hull C (i.e., C is contained in one of the two closed half space bounded by W ), and W ∩ ∂C in the polyhedron vx0 , vx1 , · · · , vxk . In other words, vx0 , vx1 , · · · , vxk is a (top-dimensional) face of ∂C. Moreover, it is Euclidean in the sense that the restriction of the Minkowski product to the hyperplane W is positive-deﬁnite. The ideal polyhedron x0 , x1 , · · · , xk dual to p is equal to the image of the Euclidean face vx0 , vx1 , · · · , vxk of ∂C by the radial projection form the origin to H3 . In conclusion, the ideal polyhedral complex ∆E (Γ ) is obtained as follows. Consider the collection of faces of ∂C which has a Euclidean support plane, that is, the collection of the subset of E1,3 which is of the form W ∩ C for some Euclidean support plane W of C. Then their images by the radial projection compose a Γ -invariant ideal polyhedral complex embedded in H3 , and ∆E (Γ ) is equal to its image in M (Γ ) (see [7, Sect. 10]). Though ∆E (Γ ) has the nice geometric meaning that it is dual to the Ford complex , its underlying space |∆E (Γ )| looks far from nice. In general, it is not convex and is strictly smaller than the convex core M0 (Γ ). This is because we take only Euclidean faces of ∂C into account in the construction of ∆E (Γ ). If the group Γ were a ﬁnitely generated Kleinian group of coﬁnite volume with parabolic transformations, then as is proved by Epstein and Penner [27], the above construction gives a ﬁnite ideal polyhedral decomposition of the whole quotient hyperbolic manifold. Moreover, every face of ∂C is Euclidean and hence each piece of the decomposition admits a natural Euclidean structure. Thus it is called the Euclidean decomposition. However, in general, ∂C can have non-Euclidean faces. So, it is natural to try to construct an ideal polyhedral complex by taking all faces of ∂C into account. This was made explicit in [7], and we call it the EPH-decomposition and denote it by ∆(Γ ). Here the letters E, P and H, respectively, stand for Euclidean (or elliptic), parabolic and hyperbolic. At the end of this section, we note that Ford(Γ )∩M (Γ ) is a spine of M (Γ ), that is, it is a strong deformation retract of M (Γ ). Moreover, this spine is canonical in the sense that it is uniquely determined from the cusped hyperbolic manifold M (Γ ). We can apply the same construction to every cusped hyperbolic manifold, and in the special case when the manifold is of ﬁnite volume and has only one cusp (e.g. the complement of a hyperbolic knot), the combinatorial structure of the Ford complex is a complete invariant of the underlying topological 3-manifold by virtue of the Mostow rigidity theorem. This fact motivated us to study Jorgensen’s work [40], which determines the combinatorial structure of the Ford domain of punctured torus groups.

1.2 Jorgensen’s theorem for quasifuchsian punctured torus groups (I)

7

1.2 Jorgensen’s theorem for quasifuchsian punctured torus groups (I) In this section, we give a description of Jorgensen’s theorem in [40] from the view point of 3-manifolds, which describes the combinatorial structures of the Ford domains of quasifuchsian punctured torus groups. Before presenting the precise statement of Jorgensen’s theorem, we give a brief intuitive description of the idea. Let Γ be a quasifuchsian punctured torus group. Then ¯ (Γ ) is identiﬁed with T × [−1, 1]. Since P (Γ ) ⊂ C the quotient manifold M is a fundamental domain of the action of Γ on Ω(Γ ) = Ω − (Γ ) ∪ Ω + (Γ ), modulo Γ∞ , P (Γ ) is a disjoint union of P − (Γ ) and P + (Γ ) where P ± (Γ ) = P (Γ )∩Ω ± (Γ ). The ﬁrst assertion of Jorgensen’s theorem is that each of P ± (Γ ) is simply connected. This implies that the image of ∂P ± (Γ ) in T × {±1} is a spine of the punctured torus T . If Γ is fuchsian, then these two spines are identical, and the Ford complex Ford(Γ ) is equal to the product of the spine with the interval [−1, 1]. In general, these two spines are not isotopic to each other. However, they are related by a canonical sequence of Whitehead moves as described later. (This is largely why the punctured torus is so special.) The “trace” of the canonical sequence of Whitehead moves form a spine of T × [−1, 1]. The main assertion of Jorgensen’s theorem is that this spine is isotopic to the Ford complex Ford(Γ ). Thus we can say that Ford(Γ ) records the history of how the two boundary spines evolved. Now let’s give the precise statement. We begin by recalling basic topological facts on the punctured torus T . To this end, we identify T with the quotient space (R2 − Z2 )/Z2 . A simple loop in T is said to be essential , if it bounds neither a disk nor a once-punctured disk. Similarly, a simple arc in T having the puncture as endpoints is said to be essential , if it does not cut oﬀ a disk with a point on the boundary removed. Then the isotopy classes of essential simple loops (resp. essential simple arcs) in T are in one-to-one ˆ := Q∪{1/0}: A representative of the isotopy class corcorrespondence with Q ˆ is the projection of a line in R2 (the line being disjoint responding to r ∈ Q 2 from Z for the loop case, and intersecting Z2 for the arc case). The element ˆ associated to a circle or an arc is called its slope. The representative of r∈Q the isotopy class of an essential arc of slope r is denoted by βr . Consider the ideal triangle in the hyperbolic plane H2 = {z ∈ C | ℑ(z) > 0} spanned by the ideal vertices {0/1, 1/1, 1/0}. Then the translates of this ideal triangle by the action of SL(2, Z) form a tessellation of H2 . This is called the modular diagram or the Farey triangulation and is denoted by D. The abstract simplicial complex having the combinatorial structure of D is also denoted by the same symbol. The set of (ideal) vertices of D is equal to ˆ and a typical (ideal) triangle σ of D is spanned by { p1 , p1 +p2 , p2 } where Q, q1 q1 +q2 q2 p1 p 2 ∈ SL(2, Z). q1 q2

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1 Jorgensen’s picture of quasifuchsian punctured torus groups

Let σ = r0 , r1 , r2 be a triangle of D. Then the essential arcs βr0 , βr1 , βr2 are mutually disjoint, and their union determines a topological ideal triangulation trg(σ) of T , in the sense that T cut open along βr0 ∪ βr1 ∪ βr2 is the disjoint union of two 2-simplices with all vertices deleted. Let spine(σ) be a 1-dimensional cell complex embedded in T which is dual to the 1-skeleton of trg(σ). Thus spine(σ) consists of two vertices and three edges γi (i = 0, 1, 2), such that γi intersects the 1-skeleton of trg(σ) transversely precisely at a point of βri . Note that spine(σ) is a deformation retract of T and hence is a spine ˆ the slope of of T . We deﬁne the slope of an edge γi of spine(σ) to be ri ∈ Q, the ideal edge βri of trg(σ) dual to γi . Let τ = r0 , r1 be an edge of D. Then the union βr0 ∪ βr1 determines a topological ideal polygonal decomposition of T , in the sense that T cut open along it is homeomorphic to a quadrilateral with all vertices deleted. Let spine(τ ) be a 1-dimensional cell complex embedded in T which is dual to the 1-skeleton of trg(τ ). Then spine(τ ) consists of a single vertex and two edges, and it is also a spine of T . The slope of an edge of spine(τ ) is also deﬁned as explained in the preceding paragraph. Let D(i) denote the set of i-simplices of D. Then we have the following well-known fact. Lemma 1.2.1. For any spine C of T , there is a unique element δ of D(1) ∪ D(2) such that C is isotopic to trg(δ). If τ = r0 , r1 is an edge of a triangle σ = r0 , r1 , r2 of D, then spine(τ ) is obtained from spine(σ) by collapsing the edge γ2 of spine(σ) of slope r2 to a point (see Fig. 1.1). By an elementary transformation, we mean this transformation or its converse. Let (δ − , δ + ) be a pair of elements of D(1) ∪D(2) . Then, since the 1-skeleton of the dual to D is a tree, there is a unique sequence δ − = δ0 , δ1 , δ2 , · · · , δm = δ + in D(1) ∪ D(2) satisfying the following conditions. 1. For each i ∈ {0, 1, · · · , m − 1}, either δi is an edge of δi−1 or δi+1 is an edge of δi . 2. δi = δj whenever i = j.

Thus we obtain a canonical sequence of elementary transformations spine(δ − ) = spine(δ0 ) → spine(δ1 ) → · · · → spine(δm ) = spine(δ + ). Regard the sequence as a continuous family {Ct }t∈[−1,1] of spines of T , and set Spine(δ − , δ + ) = ∪t∈[−1,1] Ct ⊂ T × [−1, 1].

In the special case when δ − = δ + , we set

Spine(δ − , δ + ) = δ − × [−1, 1] = δ + × [−1, 1] ⊂ T × [−1, 1]. Then Spine(δ − , δ + ) is a 2-dimensional subcomplex of T × [−1, 1] satisfying the following conditions.

1.2 Jorgensen’s theorem for quasifuchsian punctured torus groups (I)

9

Fig. 1.1. Elementary transformation

1. Spine(δ − , δ + ) ∩ (T × {ǫ1}) = spine(δ ǫ ) × {ǫ1} for each ǫ = ±. 2. There is a level-preserving deformation retraction from T × [−1, 1] to Spine(δ − , δ + ). Figure 1.1 illustrates Spine(δ − , δ + ), where δ − and δ + are elements of D(2) sharing a common edge. We note that it has a natural cellular structure, consisting of a unique inner-vertex, four inner-edges and six 2-dimensional faces.

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1 Jorgensen’s picture of quasifuchsian punctured torus groups

The following theorem paraphrases Jorgensen’s results [42, Theorems 3.1-3.3], and describes the combinatorial structures of the Ford domains of quasifuchsian punctured torus groups (see [65, Sect. 3] for another exposition). Theorem 1.2.2 (Jorgensen). For any quasifuchsian punctured torus group Γ , the following hold: 1. P (Γ ) consists of two simply connected components P ± (Γ ) ⊂ Ω ± (Γ ). In particular, for each ǫ = ±, P ǫ (Γ ) is a fundamental domain for the action of Γ on Ω ǫ (Γ ) modulo Γ∞ , and the image of ∂P ǫ (Γ ) in Ω ǫ (Γ )/Γ is a spine of T , which we denote by spineǫ (Γ ). 2. Let δ ǫ be the element of D(1) ∪ D(2) such that spineǫ (Γ ) is isotopic to spine(δ ǫ ). Then the Ford complex Ford(Γ ) is isotopic to Spine(δ − , δ + ). The computer program OPTi [78] made by the third author visualizes the above theorem: we can see in real time how the Ford domain P h(Γ ) and the limit set Λ(Γ ) vary according to deformation of a quasifuchsian punctured torus group Γ . Figure 1.2(a), which was drawn using OPTi, illustrates a typical example of the Ford domain of a quasifuchsian punctured torus group Γ . We can observe the following (cf. [40], [43]). 1. Each face F of the Ford domain P h(Γ ) is preserved by an elliptic transformation, PF , of order 2. This reﬂects the fact that M (Γ ) admits an isometric involution. 2. The transformations {PF }, where F runs over the faces of the P h(Γ ), generate a Kleinian group Γ˜ which contains Γ as a normal subgroup of index 2. In fact Γ is identiﬁed with (the image of a faithful representation of) the orbifold fundamental group of the 2-dimensional orbifold which is the quotient of T by an involution with three ﬁxed points. 3. There is a parabolic transformation, K, of Γ˜ such that K(∞) = ∞ and K 2 is the element ρ(ω) ∈ Γ , where ω ∈ π1 (T ) is represented by a simple loop around the puncture. 4. If F is a face of P h(Γ ), then F ′ = K(F ) is also a face of P h(Γ ) and the transformation K ◦ PF is the element of Γ which sends F to F ′ . 5. There is a continuous family of periodic piecewise geodesic lines {Lt }t∈(−1,1) contained in ∂P h(Γ ) such that F ∩ Lt is a (possibly degenerate) geodesic segment orthogonal to F ∩ Axis(PF ) for each face F . Let St be the vertical piecewise totally geodesic plane in P h(Γ ) lying above Lt . Then St projects to a punctured torus, Tt , in M (Γ ) = T × (−1, 1) isotopic to a level surface, and the image, Ct , of ∂St forms a spine of Tt . So {Ct } gives a continuous family of spines of T . Moreover, Ct is generic or nongeneric according as Lt is disjoint from the projections of the vertices of P h(Γ ). This family {Ct } (with certain modiﬁcation near t = ±1) realizes the canonical sequence of elementary moves transforming spine− (Γ ) to spine+ (Γ ). (See Sect. 6.5 for detailed explanation.)

1.2 Jorgensen’s theorem for quasifuchsian punctured torus groups (I)

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Fig. 1.2.

Figure 1.2(b) illustrates the cross section of ∆˜E (Γ ) along a small horosphere H∞ centered at ∞, where ∆˜E (Γ ) is the Γ -invariant ideal polyhedral complex in H3 obtained as the inverse image of the dual ideal polyhedral complex ∆E (Γ ). It is also regarded as the projection of ∆˜E (Γ ) ∩ ∂H∞ to the complex plane C. Then the vertices are identiﬁed with the centers of the isometric hemispheres supporting faces of the Ford domain P h(Γ ). To be more explicit, if v is a projection of a vertex of ∆˜E (Γ ) ∩ ∂H∞ , then the vertical geodesic [v, ∞) joining v to ∞ is an edge of ∆˜E (Γ ), and v is the center of the isometric hemisphere supporting a face of P h(Γ ) dual to the edge [v, ∞) of ∆˜E (Γ ). Similarly, each triangle x0 , x1 , x2 of ∆˜E (Γ ) ∩ ∂H∞ is a horospherical cross section of an ideal tetrahedron in ∆˜E (Γ ) dual to the vertex of P h(Γ )

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1 Jorgensen’s picture of quasifuchsian punctured torus groups

obtained as the intersection of the isometric hemispheres centered at x0 , x1 and x2 . ˜ ) be the inverse image in H3 of the EPH-decomposition of ∆(Γ ). Let ∆(Γ ˜ ) ∩ ∂H∞ gives a (not necessarily locally ﬁnite) decomposition of the Then ∆(Γ inﬁnite strip which arises as the intersection of the convex hull of the limit set with ∂H∞ , because the underlying space |∆(Γ )| is equal to the convex core minus the bending laminations by [7, Corollary 1.1].

1.3 Jorgensen’s theorem for quasifuchsian punctured torus groups (II) In this section, we explain Jorgensen’s theorem [39, Theorem 4] which reﬁnes Theorem 1.2.2. A weighted spine of T is a spine of T with an assignment of a positive real number to each edge, which we call the weight on the edge, such that the sum of the weights is equal to 1. We call such an assignment a weight system on the spine. By regarding the weight on an edge as a weight on the slope of the edge, a weight system on a spine is regarded as the barycentric coordinate of a point in |D| − |D(0) |. Here |D| denotes the underlying space of the abstract simplicial complex D, and |D(0) | denotes the 0-skeleton of |D|. By ﬁxing a P SL(2, Z)-equivariant bijective continuous map from the underlying ˆ ⊂ H2 , we identify space |D| (of the abstract simplicial complex D) onto H2 ∪ Q |D| − |D(0) | with H2 . Thus each weighted spine corresponds to a unique point of H2 . For each ν ∈ H2 , we denote by spine(ν) the weighted spine of T corresponding to ν. Let Γ be a marked quasifuchsian punctured torus group. For each ǫ = ± and for each edge e of spineǫ (Γ ), let tǫ (e) be 1/π times the angle, θǫ (e), of a circular arc component of the inverse image of e in ∂P ǫ (Γ ) (see Fig. 1.2). Then we have the following (see Proposition 4.2.16): Lemma 1.3.1. The sum of tǫ (e) where e runs over the edges of spineǫ (Γ ) is equal to 1. Thus spineǫ (Γ ) has the structure of a weighted spine of T where the weight of an edge e is tǫ (e). Let ν ǫ (Γ ) be the point of D corresponding to the weighted spine spineǫ (Γ ), and put ν(Γ ) = (ν − (Γ ), ν + (Γ )). We call it the side parameter of Γ following [39]. The following reﬁnement of Theorem 1.2.2(1) justiﬁes the terminology (see [39, Theorem 4]): Theorem 1.3.2 (Jorgensen). The map ν : QF → H2 × H2 is a homeomorphism. Remark 1.3.3. See [3] for a relation between the side-parameter ν(Γ ) and the usual end invariant of Γ (see [58]), which records the conformal structure (Ω − (Γ )/Γ, Ω + (Γ )/Γ ) ∈ T eich(T ) × T eich(T ) = H2 × H2 .

1.4 The topological ideal polyhedral complex Trg(ν) dual to Spine(ν)

13

To combine Theorems 1.2.2 and 1.3.2, we introduce the following concept: Deﬁnition 1.3.4. (1) A weighted relative spine of T × [−1, 1] is a 2-dimensional subcomplex C of T × [−1, 1] satisfying the following conditions. 1. There is a level-preserving deformation retraction from T × [−1, 1] to Spine(δ − , δ + ) for some δ ± ∈ D(2) ∪ D(1) . 2. A weight system is speciﬁed on each of ∂ ± C := C ∩ T × {±1}. Two weighted relative spines are equivalent, if the underlying relative spines are isotopic and the weight systems coincide (after the isotopy). (2) For ν = (ν − , ν + ) ∈ H2 × H2 , Spine(ν) denotes the weighted relative spine, such that the underlying relative spine is Spine(δ − (ν), δ + (ν)), where δ ǫ (ν) denotes the element of D(2) ∪ D(1) whose interior contains ν ǫ , and the weight system on ∂ ± Spine(ν) is given by ν ± . Then we can summarize Theorems 1.2.2 and 1.3.2 as follows: Main Theorem 1.3.5 (Jorgensen). For each Γ ∈ QF, the Ford complex Ford(Γ ) has a structure of weighted relative spine of T × [−1, 1]. Moreover, there is a homeomorphism ν : QF → H2 × H2 , such that the weighted spine Ford(Γ ) is equivalent to Spine(ν(Γ )) for any Γ ∈ QF.

1.4 The topological ideal polyhedral complex Trg(ν) dual to Spine(ν) As explained in Sect. 1.1, the Ford complex Ford(Γ ) is a dual to the subcomplex ∆E (Γ ) of ∆(Γ ) consisting of the Euclidean faces. In this section, we describe the structure of ∆E (Γ ) following the exposition by Floyd-Hatcher [29] of Jorgensen’s ideal triangulation of punctured torus bundles over S 1 . For each element ν = (ν − , ν + ) of H2 × H2 = (|D| − |D(0) |) × (|D| − |D(0) |), we construct a topological ideal simplicial complex Trg(ν). To describe the construction, we introduce the following deﬁnition. Deﬁnition 1.4.1 (Chain). By the chain Σ(ν) spanned by ν = (ν − , ν + ) ∈ H2 × H2 , we mean the sequence (σ1 , σ2 , · · · , σn ) of triangles of D whose interiors intersect the oriented geodesic segment joining ν − with ν + in this order. Note that n = 0 if and only if ν ± is contained in a single edge τ of D. In this case we redeﬁne Σ(ν) = {τ }. Σ(ν) and ν is said to be thick or thin according as n ≥ 1 or n = 0. i ) be the ideal triangulation of R2 −Z2 obtained as Case 1. n ≥ 2. Let trg(σ i+1 ) upon trg(σ i ), we obtain an array the lift of trg(σi ). By superimposing trg(σ of ideal tetrahedra whose bottom faces compose trg(σi ) and whose top faces i , σi+1 ). i+1 ). We denote this array of ideal tetrahedra by Trg(σ compose trg(σ n−1 , σn ) up in order, we obtain a set of 1 , σ2 ), · · · , Trg(σ By stacking Trg(σ

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1 Jorgensen’s picture of quasifuchsian punctured torus groups

1 ) and whose top faces compose layers whose bottom faces compose trg(σ n ). The covering transformation group Z2 of the covering R2 − Z2 → T trg(σ freely acts on the above topological ideal simplicial complex, and we deﬁne Trg(ν) to be the quotient topological ideal simplicial complex. Case 2. n = 1. Then Trg(ν) is deﬁned to be the 2-dimensional topological ideal triangulation trg(σ1 ). Case 3. n = 0. Then δ − (ν) = δ + (ν) is an edge τ of D. We deﬁne Trg(ν) to be the 2-dimensional topological ideal cellular complex Trg(δ ± (ν)). It should be noted that Trg(ν) is not a topological ideal triangulation, but a topological ideal cellular complex. Note that the underlying space |Trg(ν)| is homeomorphic to the the quotient space of T × [−1, 1] by an equivalence relation ∼ such that (x, s) ∼ (y, t) only if x = y (cf. [71, Sect. II.2]). In particular, |Trg(ν)| is homotopy equivalent to T and has a natural embedding into T × (−1, 1). We then have the following theorem by Theorem 1.2.2. Theorem 1.4.2. For any Γ ∈ QF, ∆E (Γ ) is isotopic to Trg(ν(Γ )) in the convex core M0 (Γ ) of M (Γ ). Figure 1.2(b) illustrates the cross section of ∆E (Γ ) along a horosphere centered at ∞ for the quasifuchsian punctured torus group Γ in Fig. 1.2(a).

2 Fricke surfaces and P SL(2, C)-representations

The topological once-punctured torus T , the 4-times punctured sphere S and the (2, 2, 2, ∞)-orbifold O are commensurable, and are called Fricke surfaces (see [74]). In this chapter, we give a detailed study of the fundamental groups of Fricke surfaces and their representations to P SL(2, C). In Sect. 2.1, we classify the “geometric” generator systems of the fundamental groups of Fricke surfaces (Propositions 2.1.6 and 2.1.9), and describe the space of all geometric generators in terms of the Farey triangulation (Proposition 2.1.10). Though these results seem to be well-known, we could not ﬁnd an explicit proof in the literature. For this reason, we include a full proof, following an idea suggested to the authors by Stephan Hamperies that greatly simpliﬁed our original proof. In Sect. 2.2, we study type-preserving representations of the fundamental groups of Fricke surfaces to P SL(2, C), i.e., those irreducible representations which send peripheral elements to parabolic transformations. We show that the spaces of type-preserving P SL(2, C)-representations for Fricke surfaces are essentially identical (Proposition 2.2.2). This fact is well-known and easily proved for T and O. However, we could not ﬁnd the result for S in the literature. Thus we include the proof, because it relates the study of 2-bridge link groups to that of punctured torus groups (cf. Sect. 5.3). We also point out Proposition 2.2.8, which reduces the study of Ford domains of punctured torus groups to that of Ford domains of Kleinian groups obtained as the image of π1 (O) by type-preserving representations. In Sect. 2.3, we describe the space of type-preserving representations in terms of the aﬃne algebraic variety determined by the Markoﬀ equation x2 + y 2 + z 2 = xyz, following [17] (Proposition 2.3.4 and 2.3.6). In Sect. 2.4, we recall the complex probability parameter of type-preserving representations, and describe a conceptual geometric construction of a typepreserving representation from a given complex probability, i.e., a triple of complex numbers (a0 , a1 , a2 ) such that a0 + a1 + a2 = 1 (Proposition 2.4.4). In the last section, Sect. 2.5, we collect several well-known properties of discrete groups, which we use in this paper.

16

2 Fricke surfaces and P SL(2, C)-representations

2.1 Fricke surfaces and their fundamental groups Let T , S and O, respectively, be the once-punctured torus, the 4-times punctured sphere, and the (2, 2, 2, ∞)-orbifold (i.e., the orbifold with underlying space a punctured sphere and with three cone points of cone angle π). They ˜ respechave R2 −Z2 as the common covering space. To be precise, let G and G, tively, be the groups of transformations on R2 − Z2 generated by π-rotations about points in Z2 and ( 12 Z)2 . Then T = (R2 − Z2 )/Z2 , S = (R2 − Z2 )/G and ˜ In particular, there is a Z2 -covering T → O and a Z2 ⊕ Z2 O = (R2 − Z2 )/G. covering S → O: the pair of these coverings is called the Fricke diagram and each of T , S, and O is called a Fricke surface (see [74]). A simple loop in a Fricke surface is said to be essential if it does not bound a disk, a disk with one puncture, or a disk with one cone point. Similarly, a simple arc in a Fricke surface joining punctures is said to be essential if it does not cut oﬀ a “monogon”, i.e., a disk minus a point on the boundary. Then the isotopy classes of essential simple loops (resp. essential simple arcs with one end in a given puncture) in a Fricke surface are in one-to-one correspondence ˆ := Q ∪ {1/0}: A representative of the isotopy class corresponding to with Q ˆ r ∈ Q is the projection of a line in R2 (the line being disjoint from Z2 for the ˆ associated loop case, and intersecting Z2 for the arc case). The element r ∈ Q to a circle or an arc is called its slope. An essential loop of slope r in T or O ˜ r ). The notation reﬂects the following fact: (resp. S) is denoted by αr (resp. α After an isotopy, the restriction of the projection T → O to αr (⊂ T ) gives a homeomorphism from αr (⊂ T ) to αr (⊂ O), while the restriction of the projection S → O to α ˜ r gives a two-fold covering from α ˜ r (⊂ S) to α (⊂ O). Since T and S are ﬁnite regular coverings of the orbifold O, the fundamental groups of T and S are regarded as normal subgroups of the orbifold fundamental group of O of ﬁnite index. These groups have the following group presentations: π1 (T ) = A0 , B0 ,

(2.1)

π1 (S) = K0 , K1 , K2 , K3 | K0 K1 K2 K3 = 1, π1 (O) = P0 , Q0 , R0 | P02 = Q20 = R02 = 1,

(2.2) (2.3)

Here the generators satisfy the following conditions: Set K = (P0 Q0 R0 )−1 , then K is represented by the puncture of O and satisﬁes the relations K 2 = [A0 , B0 ], K0 = K,

A0 = KP0 = R0 Q0 ,

K1 = K P0 ,

K2 = K Q0 ,

B0 = K −1 R0 = P0 Q0 , K3 = K R0 ,

where X Y denotes Y XY −1 . Throughout this paper, we reserve the symbol K to denote the element of π1 (O) deﬁned in the above. Deﬁnition 2.1.1 ((Elliptic) generator triple). (1) An ordered pair (A, B) of elements in π1 (T ) is a generator pair of π1 (T ) if they generate π1 (T ) and

2.1 Fricke surfaces and their fundamental groups

17

satisfy [A, B] = K 2 . In this case, A and B are, respectively, called the left and right generators, and (A, AB, B) is called a generator triple. (2) An ordered triple (P, Q, R) of elements of π1 (O) is called an elliptic generator triple if they generate π1 (O) and satisfy P 2 = Q2 = R2 = 1 and (P QR)−1 = K. A member of an elliptic generator triple is called an elliptic generator. EG denotes the set of all elliptic generators. We can easily see the following proposition. Proposition 2.1.2. If (P, Q, R) is an elliptic generator triple of π1 (O), then (KP, KQ, K −1 R) is a generator triple of π1 (T ). Conversely, if (A, AB, B) is a generator triple of π1 (T ), then (K −1 A, K −1 AB, KB) is an elliptic generator triple. Deﬁnition 2.1.3. (1) If an elliptic generator triple (P, Q, R) and a generator triple (A, AB, B) are related as in Proposition 2.1.2, then we say that they are associated with each other. (2) The slope, s(A), of a generator A of π1 (T ) is deﬁned to be the slope of an essential loop in T representing the conjugacy class of the generator. (3) The slope, s(P ), of an elliptic generator P of π1 (O) is deﬁned as the slope of the generator KP of π1 (T ). Remark 2.1.4. If (P, Q, R) is an elliptic generator triple, then it follows from Proposition 2.1.6(1.1) below that (R, P K , QK ) is also an elliptic generator triple. Thus KR is a (left) generator of π1 (T ), and hence the slope of R is also well-deﬁned. Convention 2.1.5. Throughout this paper, we assume (s(A0 ), s(A0 B0 ), s(B0 )) = (s(P0 ), s(Q0 ), s(R0 )) = (0/1, 1/1, 1/0) = (0, 1, ∞). for the generators in the group presentations (2.1) and (2.3). We have the following classiﬁcation of elliptic generator triples. Proposition 2.1.6. (1) For any elliptic generator triple (P, Q, R), the following holds: (1.1) The triple of any three consecutive elements in the following biinﬁnite sequence is also an elliptic generator triple. ··· ,PK

−1

, QK

−1

, RK

−1

, P, Q, R, P K , QK , RK , · · ·

(1.2) (P, R, QR ) and (QP , P, R) are also elliptic generator triples. (2) Conversely, any elliptic generator triple is obtained from a given elliptic generator triple by successively applying the operations in (1).

18

2 Fricke surfaces and P SL(2, C)-representations

Proof. Since (1) can be proved by direct calculation, we give proof of (2). We present a proof indicated to us by Stephan Hamperies (cf. [23, Notation 2.3]), and greatly simplifying our original proof. Let (P0 , P1 , P2 ) be an elliptic generator triple, and let σ0 and σ1 be the “braid” automorphism of π1 (O) deﬁned by (σ0 (P0 ), σ0 (P1 ), σ0 (P2 )) = (P0 P1 P0 , P0 , P2 ), (σ1 (P0 ), σ1 (P1 ), σ1 (P2 )) = (P0 , P1 P2 P1 , P1 ). Then σ0 and σ1 preserve K and hence they map elliptic generator triples to elliptic generator triples. Moreover, we have the following lemma. Lemma 2.1.7. The group of automorphisms of π1 (O) preserving K is generated by σ0 and σ1 . Proof. Though the proof of this lemma is parallel to [22, Proof of Proposition 10.7], we include it the proof for completeness. Let f be an automorphism of π1 (O) which preserves K. Since f (Pj ) has order 2 and since π1 (O) is isomorphic to the free product of three cyclic groups Pj of order 2, f (Pj ) is conjugate to Pτ (j) for some τ (j) ∈ {0, 1, 2}. Since f (P2 P1 P0 ) = P2 P1 P0 , τ must be a permutation on the set {0, 1, 2}. Hence we have f (Pj ) = Wj Pτ (j) Wj−1 , for some element Wj ∈ π1 (O). Express Wj as a reduced word in {P0 , P1 .P2 }, and let l(Wj ) be the minimal word length of Wj . Set λ(f ) = l(W1 )+l(W2 )+l(W3 ). We prove the lemma by induction on λ(f ). If λ(f ) = 0, then we have f = id. Suppose λ(f ) > 0. Then there will be reduction in the product W2 Pτ (2) W2−1 W1 Pτ (1) W1−1 W0 Pτ (0) W0−1 = P2 P1 P0 −1 such that some Pτ (j) is canceled out by Pτ (j) contained in Wj+1 or Wj−1 . −1 (Otherwise all Wj+1 Wj must be reduced to 1, and hence all Wj must be trivial −1 −1 words.) Suppose Wj+1 cancels Pτ (j) . Then we see l(Wj+1 Wj Pτ (j) Wj−1 ) < −1 to the elliptic generator triple (f (P0 ), f (P1 ), f (P2 )). l(Wj+1 ). Apply f σj f Then

(f σj f −1 )(f (Pj )) = f (Pj )f (Pj+1 )f (Pj ) −1 = (Wj Pτ (j) Wj−1 Wj+1 )Pτ (j+1) (Wj+1 Wj Pτ (j) Wj−1 ).

This implies λ(f ) > λ((f σj f −1 )f ) = λ(f σj ). By using the inductive hypothesis, we see that f is contained in the group generated by σ0 and σ1 . Similarly, if Wj−1 cancels Pτ (j) , then we obtain the same conclusion by −1 −1 f . applying f σj−1 By the above lemma, we have only to show that σ0 and σ1 are contained in the group generated by the following automorphisms: f1 : (P0 , P1 , P2 ) → (P1 , P2 , P0K ),

f2 : (P0 , P1 , P2 ) → (P0 , P2 , P1P2 ).

But we can check that σ0 = f1 f2−1 f1−1 and σ1 = f2−1 . This completes the proof of Proposition 2.1.6.

2.1 Fricke surfaces and their fundamental groups

19

The following lemma can be veriﬁed by simple calculation. Lemma 2.1.8. Let (A, AB, B) = (KP, KQ, K −1 R) be the generator triple associated with an elliptic generator triple (P, Q, R). Set (s0 , s1 , s2 ) = (s(A), s(AB), s(B)) = (s(P ), s(Q), s(R)). Then the following hold: 1. (AB, ABA−1 , A−1 ) and (B −1 , A, BA), respectively, are the generator −1 triples associated with (Q, R, P K ) (resp. (P K , Q, R)), and we have: (s(AB), s(ABA−1 ), s(A−1 )) = (s(Q), s(R), s(P K )) = (s1 , s2 , s0 ), (s(B −1 ), s(A), s(BA)) = (s(P K

−1

), s(Q), s(R)) = (s2 , s0 , s1 ).

2. (A, ABA−1 , BA−1 ) and (AB −1 , A, B), respectively, are the generator triples associated with (P, R, QR ) and (QP , P, R), and we have: (s(A), s(ABA−1 ), s(BA−1 )) = (s(P ), s(R), s(QR )) = (s0 , s2 , s′1 ), (s(AB −1 ), s(A), s(B)) = (s(QP ), s(P ), s(R)) = (s′1 , s0 , s2 ), where s′1 is the vertex of D which is opposite to s1 with respect to the edge s0 , s2 . By Proposition 2.1.6 and Lemma 2.1.8, we obtain the following proposition. Proposition 2.1.9. (1) For any generator triple (A, AB, B), the following hold. (1.1) Both (AB, ABA−1 , A−1 ) and (B −1 , A, BA) are generator triples. (1.2) Both (A, ABA−1 , BA−1 ) and (AB −1 , A, B) are generator triples. (2) Conversely, any generator triple is obtained from the standard generator triple by successively applying the operations in (1). For the slopes of elliptic generator triples, we have the following proposition. Proposition 2.1.10. (1) For any elliptic generator triple (P, Q, R), the oriented triangle s(P ), s(Q), s(R) of D is coherent with the oriented triangle 0/1, 1/1, 1/0 = s(P0 ), s(Q0 ), s(R0 ). (2) For any oriented triangle s0 , s1 , s2 of D coherent with the oriented triangle 0/1, 1/1, 1/0, there is a bi-inﬁnite sequence {Pj }j∈Z of elliptic generators satisfying the following conditions: 1. For each j ∈ Z, we have s(Pj ) = s[j] , where [j] denotes the integer in {0, 1, 2} such that [j] ≡ j (mod 3). 2. The triple of any three consecutive elements Pj−1 , Pj , Pj+1 is an elliptic generator triple. m 3. PjK = Pj+3m . 4. For any elliptic generator P with s(P ) = sk ∈ {s0 , s1 , s2 }, there is a unique integer j with [j] = k such that P = Pj .

20

2 Fricke surfaces and P SL(2, C)-representations

Furthermore, such a sequence is unique modulo shift of indices by multiples of 3. Proof. (1) This follows from Proposition 2.1.6 and Lemma 2.1.8. (2) By Proposition 2.1.6(1.1) and Lemma 2.1.8, the bi-inﬁnite sequence · · · , P0K

−1

, QK 0

−1

, R0K

−1

K , P0 , Q0 , R0 , P0K , QK 0 , R0 , · · ·

obtained by expanding the standard generator triple (P0 , Q0 , R0 ) satisﬁes the conditions 1, 2 and 3 for the oriented triangle 0, 1, ∞. By Proposition 2.1.6(1) and Lemma 2.1.8, we can also construct such a bi-inﬁnite sequences for each of the three oriented triangles adjacent to 0, 1, ∞. By repeating this procedure, we can construct a bi-inﬁnite sequence {Pj } of elliptic generators satisfying the conditions 1, 2 and 3 for every oriented triangle s0 , s1 , s2 of D. To show that this sequence also satisﬁes the condition 4, pick an elliptic generator P such that s(P ) = sk ∈ {s0 , s1 , s2 }. Then by using Proposition 2.1.6(2) and Lemma 2.1.8, we can ﬁnd a bi-inﬁnite sequence {Pj′ } of elliptic generators which contains P and satisﬁes the conditions 1, 2 and 3. We have only to show that {Pj′ } coincides with {Pj } modulo shift of indices by multiples of 3. This is a consequence of the following Lemma 2.1.11, which shows that there is no ambiguity in the construction of such a sequence for a triangle from that for an adjacent triangle. This also proves the uniqueness of such a sequence.

Lemma 2.1.11. For an elliptic generator triple (P, Q, R), we have n

(QK )(R

Kn

)

n

= (QR )K ,

QP = (QR )K

−1

.

Hence the bi-inﬁnite sequences obtained by expanding the elliptic generaKn n n n tor triples (P, R, QK ), (P K , RK , (QK )(R ) ) and (QP , P, R) all coincide modulo shift of indices. Proof. This can be proved by simple calculation. Convention 2.1.12 (Ordered triangle). When we mention to a triangle σ = s0 , s1 , s2 of D, we always assume that the order of the vertices is ﬁxed so that the orientation is coherent with that of s(P0 ), s(Q0 ), s(R0 ) = 0, 1, ∞ (cf. Convention 2.1.5). Deﬁnition 2.1.13. (1) For a triangle σ = s0 , s1 , s2 of D, the bi-inﬁnite sequence {Pj } in Proposition 2.1.10(3) is called the sequence of elliptic generators associated with σ , and it is denoted by EG(σ). (2) An ordered pair (P, Q) of elliptic generators are called an elliptic generator pair if P and Q appear successively in this order in a sequence of elliptic generators. (3) More generally, for a subcomplex Σ of D, EG(Σ) denotes the set of elliptic generators deﬁned by EG(Σ) = {P ∈ EG | s(P ) ∈ Σ (0) }.

2.2 Type-preserving representations

21

By Proposition 2.1.10(1), we may assume the following when considering the sequence of elliptic generators associated with adjacent triangles of D. Assumption 2.1.14. (Adjacent triangles) The symbols σ = s0 , s1 , s2 and σ ′ = s′0 , s′1 , s′2 denote triangles of D sharing the edge τ := s0 , s2 = s′0 , s′1 . The symbols {Pj } and {Pj′ }, respectively, denote the sequences of elliptic generators associated with σ and σ ′ , such that P0′ = P0 , P1′ = P2 and P2′ = P1P2 (see Fig. 2.1)

s1

=

=

P0′

P1′

P3′

′ P−1

P3

s0

s2

=

=

′ P−2

P2

=

P−1

P0

=

P1

s′1

s′0

P2′

s′2

Fig. 2.1. Adjacent sequences of elliptic generators

2.2 Type-preserving representations In this section, we introduce the family of P SL(2, C)-representations of the fundamental groups of the Fricke surfaces which are studied in this paper. Deﬁnition 2.2.1 (Type-preserving P SL(2, C)-representation). (1) A P SL(2, C)-representation of π1 (T ) (resp. π1 (O)) is type-preserving if it is irreducible (equivalently, it does not have a common ﬁxed point in ∂H3 ) and sends peripheral elements to parabolic transformations. (2) A P SL(2, C)-representation ρ of π1 (S) is type-preserving if it is irreducible, sends peripheral elements to parabolic transformations, and satisﬁes the following condition: If Fix(ρ(Ki )) = Fix(ρ(Kj )) for some 0 ≤ i < j ≤ 3, then ρ(Ki ) = ρ(Kj−1 ). (3) For X = T , O and S, Homtp (π1 (X), P SL(2, C)) denotes the space of all type-preserving P SL(2, C)-representations of π1 (X). Proposition 2.2.2. The spaces Homtp (π1 (X), P SL(2, C)) with X = T , O and S are all identical in the following sense. 1. The restriction of any type-preserving P SL(2, C)-representation of π1 (O) to π1 (T ) (resp. π1 (S)) is type-preserving. 2. Conversely, every type-preserving P SL(2, C)-representation of π1 (T ) (resp. π1 (S)) extends to a unique type-preserving P SL(2, C)-representation of π1 (O).

22

2 Fricke surfaces and P SL(2, C)-representations

Remark 2.2.3. In this paper, we need the assertion only for T , which is wellknown. We give a proof for S for completeness and for application to 2-bridge links in the forthcoming paper [11]. Proof. (1) Since the assertion for π1 (T ) is well-known (see e.g. [76, Sect. 5.4]), we prove the assertion for π1 (S). Let ρ : π1 (O) → P SL(2, C) be a typepreserving representation. Suppose Fix(ρ(K0 )) = Fix(ρ(K1 )). Then Fix(ρ(K2 )) = ρ(Q)(Fix(ρ(K0 ))) = ρ(Q)(Fix(ρ(K1 ))) = Fix(ρ(K1Q )) = Fix(ρ(K3 )). Since ρ is irreducible, we must have Fix(ρ(K0 )) = Fix(ρ(K1 )) = Fix(ρ(K2 )) = Fix(ρ(K3 )). Suppose ρ(K0 K1 ) = 1. Then ρ(K0 K1 ) = ρ((K2 K3 )−1 ) is parabolic, and we have Fix(ρ(K0 )) = Fix(ρ(K0 K1 )) = Fix(ρ((K2 K3 )−1 )) = Fix(ρ(K3 )), This is a contradiction. Hence ρ(K0 ) = ρ(K1−1 ). Suppose, in general, that Fix(ρ(Ki )) = Fix(ρ(Kj )) for some 0 ≤ i < j ≤ 3. Then, after cyclic permutation, we may assume (i, j) = (0, 1) or (0, 2). The former case is settled by the above argument. The second case is settled by applying the same argument to the quadruple (K0 , K2 , K3 , K0 K1 K0−1 ). (2) Since the assertion for π1 (T ) is well-known (cf. [40, Sect. 2]), we prove the assertion for π1 (S) by using the arguments in [35, Proof of Proposition 1.1]. Throughout the proof, we employ the following convention: For a geodesic L in H3 , we denote the π-rotation about L by the same symbol L. Let ρ be an element of Homtp (π1 (S), P SL(2, C)), and set gi = ρ(Ki ). We show that ρ extends to a type-preserving representation of π1 (O). As in the proof of Lemma 2.2.4, after cyclic permutation of indices, we may assume Fix(g0 ) = Fix(g1 ). Let L01 , L0 and L1 be the geodesics in H3 satisfying the following conditions. 1. L01 joins Fix(g0 ) and Fix(g1 ). 2. Fix(g0 ) (resp. Fix(g1 )) is an endpoint of L0 (resp. L1 ). 3. g0 = L0 L01 and g1 = L01 L1 . Then we can ﬁnd a geodesic H01 intersecting L01 orthogonally and the πrotation H01 maps the geodesic L0 to the geodesic L01 (L1 ), the image of L1 by the π-rotation L01 . Since the π-rotation around the geodesic L01 (L1 ) is equal to L01 L1 L01 , we have L01 L1 L01 = H01 L0 H01 . Hence H01 g0 H01 = H01 L0 L01 H01 = H01 L0 H01 L01 = L01 L1 L01 L01 = L01 L1 = g1 . The assumption Fix(g0 ) = Fix(g1 ) implies Fix(g2 ) = Fix(g3 ). So there are geodesics L23 , L2 , L3 and H23 such that g2 = L2 L23 , g3 = L23 L3 and H23 g2 H23 = g3 . Set f01 = g0 H01 and f23 = g2 H23 . Then

2.2 Type-preserving representations

23

−2 2 . f01 = g0 H01 g0 H01 = g0 g1 = (g2 g3 )−1 = (g2 H23 g2 H23 )−1 = f23

If ρ is fuchsian, then both f01 and f23 are purely hyperbolic transformations −1 and hence the above equality implies f01 = f23 (see [35, Proof of Proposition 1.1]). To show that the same identity holds for the general case, we need the following lemma. Lemma 2.2.4. There is a continuous path {ρt }t∈[0,1] in Homtp (π1 (S), P SL(2, C)) satisfying the following conditions. 1. ρ1 = ρ and ρ0 is fuchsian. (t) (t) 2. Fix(ρt (K0 )) = Fix(ρt (K1 )) and hence the transformations f01 and f23 are deﬁned as in the above for every for every t ∈ [0, 1]. (t) (t) 3. (f01 )2 = (f23 )−2 is loxodromic except possibly at t = 1. Proof. Pick a fuchsian representation ρ′ ∈ Homtp (π1 (S), P SL(2, C)). We show that there is a continuous path {ρt }t∈[0,1] in Homtp (π1 (S), P SL(2, C)) satisfying the required conditions with ρ0 = ρ′ . To this end, set gi′ = ρ′ (Ki ), and let L′i , L′01 and L′23 be the geodesics in H3 determined by gi′ (0 ≤ i ≤ 3) as in the preceding paragraph. Then it is obvious that there is a continuous fam(t) (t) (t) ily of triples of mutually distinct geodesics {(L01 , L0 , L1 )}t∈[0,1] connecting ′ ′ ′ (L01 , L0 , L1 ) with (L01 , L0 , L1 ) which satisfy the following conditions. (t)

(t)

(t)

1. L01 shares an endpoint with each of L0 and L1 . ¯ (t) are disjoint, except possibly at t = 1. ¯ (t) and L 2. L 0 1 (t) (t) 3. Let H01 be the geodesic as in the preceding paragraph, and let G01 be (t) (t) (t) (t) ¯ and G ¯ are disjoint, except the axis of the π-rotation L01 H01 . Then L 0 01 possibly at t = 1. (t)

(t)

(t)

(t)

(t)

(t)

(t)

(t)

Set g0 := L0 L01 and g1 := L01 L1 . Then g0 and g1 are parabolic, (t) (t) (t) (t) and g0 g1 = L0 L1 is loxodromic except possibly at t = 1. Moreover, (t) (t) (t) (t) (t) f01 := g0 H01 = L0 G01 is loxodromic except possibly at t = 1. (t) Pick a continuous family of geodesics {L2 }t∈[0,1] connecting L′2 with L2 , (t) (t) (t) such that L2 intersects the axis of the loxodromic transformation g0 g1 orthogonally for each t ∈ [0, 1). Then there is a continuous family of geodesics (t) (t) (t) (t) (t) {L3 }t∈[0,1] connecting L′3 with L3 , such that L3 L2 = g0 g1 for every (t)

t ∈ [0, 1]. Let {L23 }t∈[0,1] be a continuous family of geodesics connecting L′23 (t) (t) (t) with L23 , such that L23 shares an endpoint with each of L2 and L3 . Set (t) (t) (t) (t) (t) (t) g2 = L2 L23 and g3 = L23 L3 . Then there is a representation ρt : π1 (S) → (t) P SL(2, C) such that ρt (Ki ) = gi (0 ≤ i ≤ 3), and {ρt }t∈[0,1] gives the desired path in Homtp (π1 (S), P SL(2, C)).

Let {ρt }t∈[0,1] be a continuous path in Lemma 2.2.4. By the last condition (t)

in the lemma, the transformation (f01 )2 , with t ∈ [0, 1), is loxodromic and

24

2 Fricke surfaces and P SL(2, C)-representations

hence it has precisely two square roots which diﬀer by the composition of (t) (t) (t) (t) π-rotation about its axis. Since f01 = f23 at t = 0, we have f01 = f23 for every t ∈ [0, 1] by continuity. Hence we have f01 = f23 . Since H01 is contained in the normalizer of the group g0 , g1 , so is f01 . −1 is Similarly f23 is contained in the normalizer of g2 , g3 . Hence f01 = f23 contained in the normalizer of g0 , g1 , g2 , g3 , and so is H01 . In fact the action of H01 on g0 , g1 , g2 , g3 by conjugation is given by (g0 , g1 , g2 , g3 ) → (g1 , g0 , g0−1 g3 g0 , g0−1 g3−1 g2 g3 g0 ). On the other hand, the action of P on π1 (S) by conjugation is given by (K0 , K1 , K2 , K3 ) → (K1 , K0 , K0−1 K3 K0 , K0−1 K3−1 K2 K3 K0 ). Hence the representation ρ of π1 (S) extends to a representation of the subgroup P, K0 , K1 , K2 , K3 which maps P to H01 . Suppose that Fix(gi ) (0 ≤ i ≤ 3) are mutually distinct. Then by the above argument, we can ﬁnd a π-rotation H02 (resp. H03 ) such that the representation ρ of π1 (S) extends to a representation of the subgroup Q, K0 , K1 , K2 , K3 (resp. R, K0 , K1 , K2 , K3 ) which maps Q to H02 (resp. R to H0,3 ). This can be done by applying the argument to the generator system (K0 , K2 , K3 , K3−1 K2−1 K1 , K2 , K3 ) (resp. (K3 , K0 , K1 , K2 )). Since π1 (O) is the free product of three cyclic groups P , Q and R, we have a representation ρ∗ : π1 (O) → P SL(2, C) such that ρ∗ maps the triple (P, Q, R) to (H01 , H02 , H03 ). To show that the restriction of ρ∗ to π1 (S) is equal to the original representation ρ, pick an element W of π1 (S). Then by the choice of H01 , H02 and H03 , ρ∗ (W ) normalizes ρ(π1 (S)) and the action of ρ∗ (W ) on ρ(π1 (S)) by conjugation is equal to that of ρ(W ). On the other hand, since ρ is irreducible there are inﬁnitely many points in ∂H3 which are ﬁxed by elements of ρ(π1 (S)). Hence we have ρ∗ (W ) = ρ(W ). Thus ρ∗ is an extension of ρ, and this completes the proof of Proposition 2.2.2(2) for the generic case. Suppose Fix(gi ) = Fix(gj ) for some 0 ≤ i < j ≤ 3. By changing the generator system of ρ(π1 (S)) if necessary, we may assume Fix(g0 ) = Fix(g1 ). Since ρ is type-preserving, we have g0 = g1−1 and hence g2 = g3−1 . Since ρ is irreducible, Fix(g0 ) = Fix(g2 ). Let H be the geodesic joining Fix(g0 ) and Fix(g2 ). Then Hg0 H = g0−1 = g1 and Hg2 H = g2−1 = g3 . Set H01 = g0−1 H. Then H01 is an elliptic transformation of order 2 and we can easily check H01 g0 H01 = g1 ,

H01 g2 H01 = g0−1 g3 g0 .

Hence the action of H01 on g0 , g1 , g2 , g3 by conjugation is compatible with that of P on π1 (S). Moreover, since Fix(g0 ) = Fix(g2 ) = Fix(g3 ), the elements H02 and H03 are deﬁned. By the argument for the generic case, we see that the representation ρ∗ : π1 (O) → P SL(2, C) which maps the triple (P, Q, R) to (H01 , H02 , H03 ) is the desired extension of ρ. This completes the proof of Proposition 2.2.2.

2.2 Type-preserving representations

25

Remark 2.2.5. In Deﬁnition 2.2.1(2) of a type-preserving P SL(2, C)-representation of π1 (S), the last condition is essential. To see this, let ρ : π1 (S) → P SL(2, C) be a representation deﬁned by 1 ω1 1 ω0 ρ(K0 ) = , , ρ(K1 ) = 0 1 0 1 1 0 1 −ω0 − ω1 ρ(K2 ) = , , ρ(K3 ) = 4/(ω0 + ω1 ) −3 −4/(ω0 + ω1 ) 1 where ω0 and ω1 are non-zero complex numbers such that ω0 + ω1 = 0. Then ρ does not satisfy the last condition, though it is irreducible and sends peripheral elements to parabolic transformations. Thus ρ does not extend to a representation of π1 (O) by Proposition 2.2.2. By Proposition 2.2.2, the following is well-deﬁned. Deﬁnition 2.2.6. The space of the equivalence classes of type-preserving P SL(2, C)-representations of π1 (X) where X = T , O or S is denoted by X . Namely, X = Homtp (π1 (X), P SL(2, C))/P SL(2, C). Throughout this paper, we employ the following convention. Convention 2.2.7. (1) We do not distinguish between an element of X and its representative: they are denoted by the same symbol so long as there is no fear of confusion. (2) When we choose a representative ρ : π1 (O) → P SL(2, C) of an element of X , we always assume that ρ is normalized so that the following identity is satisﬁed. 11 ρ(K) = 01 (3) Let {ρn } be a sequence in X which converges to ρ∞ in X . Then it is well-known that there are normalized representatives ρn : π1 (O) → P SL(2, C) which converge to a representative ρ∞ : π1 (O) → P SL(2, C) in Homtp (π1 (X), P SL(2, C)) (cf. Corollary 2.3.8). We always assume that representatives of the elements ρn and ρ∞ of X satisfy this condition. At the end of this section, we note the following basic fact, which reduces the study of Ford domains of punctured torus groups to that of Ford domains of the Kleinian groups obtained as the images of π1 (O) by type-preserving representations. Proposition 2.2.8. Let ρ : π1 (O) → P SL(2, C) be a type-preserving representation which is discrete. Then, under Convention 2.2.7, the following hold. 1. The Ford domain P h(ρ(π1 (O))) of the Kleinian group ρ(π1 (O)) is equal to the Ford domain P h(ρ(π1 (T ))) of the Kleinian subgroup ρ(π1 (T )).

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2 Fricke surfaces and P SL(2, C)-representations

2. The Ford complex Ford(ρ(π1 (O))) ⊂ M (ρ(π1 (O))) is equal to the image ¯ (ρ(π1 (T ))) in M ¯ (ρ(π1 (O))). of Ford(ρ(π1 (T ))) ⊂ M 3. ∆E (ρ(π1 (T ))) = ∆E (ρ(π1 (O))). Thus ∆E (ρ(π1 (O))) is equal to the image of ∆E (ρ(π1 (T ))) in in M (ρ(π1 (O))).

Proof. To show the ﬁrst assertion, we have only to show that, for any element γ ∈ π1 (O) such that ρ(γ) does not ﬁx ∞, Ih(ρ(γ)) = Ih(ρ(γ0 )) for some γ0 ∈ π1 (T ). If γ ∈ π1 (T ), this is obvious. So assume that γ ∈ π1 (O) − π1 (T ). Then γ = Kγ0 for some γ0 ∈ π1 (T ). Thus we have Ih(ρ(γ)) = Ih(ρ(K)ρ(γ0 )) = Ih(ρ(γ0 )) (cf. Lemma 4.1.1(2)). Hence we obtain the ﬁrst assertion. The second assertion is a consequence of the ﬁrst one. Deﬁnition 2.2.9. Under Convention 2.2.7, suppose ρ is discrete. Then P h(ρ) denotes the Ford domain P h(ρ(π1 (T ))) = P h(ρ(π1 (O))). If there is no fear of confusion, then we occasionally denote Ford(ρ(π1 (O))) or Ford(ρ(π1 (O))) by the symbol Ford(ρ). We also occasionally denote ∆E (ρ(π1 (O))) or ∆E (ρ(π1 (T ))) by the symbol ∆E (ρ). Remark 2.2.10. The Ford domain of the Kleinian group ρ(π1 (O)) is not necessarily equal to the Ford domain of ρ(π1 (S)).

2.3 Markoﬀ maps and type-preserving representations In this section, we give a description of the type-preserving P SL(2, C)representations of π1 (O) following [17] and [40]. We begin with the following lemma. Lemma 2.3.1. (1) Every type-preserving representation ρ : π1 (T ) → P SL(2, C) lifts to a representation ρ˜ : π1 (T ) → SL(2, C) such that tr(˜ ρ(K 2 )) = −2. Moreover, there are precisely four such lifts for each ρ. (2) Conversely, every representation ρ˜ : π1 (T ) → SL(2, C) such that tr(˜ ρ(K 2 )) = −2 and ρ˜(K 2 ) = −I descends to a type-preserving P SL(2, C)representation of π1 (T ). Proof. Since π1 (T ) is a rank 2 free group, every P SL(2, C)-representation of π1 (T ) has precisely four lifts. Moreover we can easily see that a representation ρ˜ : π1 (T ) → SL(2, C) is irreducible if and only if tr(˜ ρ(K 2 )) = −2 and ρ˜(K 2 ) = −I. Hence we obtain the desired results. Deﬁnition 2.3.2 (Type-preserving SL(2, C)-representation). (1) A representation ρ˜ : π1 (T ) → SL(2, C) is type-preserving if it is irreducible and sends peripheral elements to parabolic transformations. (2) The space of the equivalence classes of type-preserving SL(2, C)representations of π1 (T ) is denoted by X˜ . Namely, X˜ = Homtp (π1 (T ), SL(2, C))/SL(2, C),

2.3 Markoﬀ maps and type-preserving representations

27

where Homtp (π1 (X), SL(2, C)) denotes the space of all type-preserving SL(2, C)-representations of π1 (T ). Following Bowditch [17], we introduce the following notion. Deﬁnition 2.3.3 (Markoﬀ map associated with a type-preserving representation). For a type-preserving SL(2, C)-representation ρ˜ : π1 (T ) → ˆ to C deﬁne by φ(r) = SL(2, C), let φ = φρ˜ be the map from D(0) = Q tr(˜ ρ(αr )), where αr is an element of π1 (T ) represented by a simple loop of slope r. We call it the Markoﬀ map associated with ρ˜. The following proposition describes the genesis of the above terminology. Proposition 2.3.4. For any type-preserving SL(2, C)-representation ρ˜ of π1 (T ), the corresponding Markoﬀ map φ = φρ˜ : D(0) → C satisﬁes the following conditions: (1) For any triangle s0 , s1 , s2 of D, the triple (φ(s0 ), φ(s1 ), φ(s2 )) is a Markoﬀ triple, that is, it is a solution of the Markoﬀ equation, x2 + y 2 + z 2 = xyz. (2) For any pair of triangles s0 , s1 , s2 and s1 , s2 , s3 of D sharing a common edge s1 , s2 , we have: φ(s0 ) + φ(s3 ) = φ(s1 )φ(s2 ). Proof. This follows from the following well-known trace identities for matrices X and Y in SL(2, C): tr(X)2 + tr(Y )2 + tr(XY )2 − tr(X) tr(Y ) tr(XY ) = 2 + tr([X, Y ]), tr(XY ) + tr(XY −1 ) = tr(X) tr(Y ). Deﬁnition 2.3.5. (1) By a Markoﬀ map, we mean a map φ : D(0) → C satisfying the conclusions of the above lemma. (2) The trivial Markoﬀ map is the Markoﬀ map which sends everything to 0. (3) Φ denotes the space of all non-trivial Markoﬀ maps. Note that there is a bijective correspondence between Markoﬀ maps and Markoﬀ triples, by ﬁxing a triangle s0 , s1 , s2 of D, and by associating to a Markoﬀ map φ the triple (φ(s0 ), φ(s1 ), φ(s2 )). This gives an identiﬁcation of Φ with the variety in C3 (with 0 deleted) determined by the Markoﬀ equation x2 + y 2 + z 2 = xyz. In particular, Φ gets a topology as a subset of C3 . The following proposition is proved by [17, Sect. 4] and [40, Sect. 2]. Proposition 2.3.6. The correspondence ρ˜ → φρ˜ induces a homeomorphism from X˜ to Φ.

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2 Fricke surfaces and P SL(2, C)-representations

The following lemma, obtained by Jorgensen [40], gives a (local) crosssection of the above correspondence: Lemma 2.3.7. Let φ be a Markoﬀ map, and put (x, y, z) = (φ(0/1), φ(1/1), φ(1/0)). Assume that y = 0. (1) Let ρ˜ : π1 (T ) → SL(2, C) be a representation determined by the following formula: y −1/y x − z/y x/y 2 , ρ˜(A0 B0 ) = , ρ˜(A0 ) = x z/y y 0 z − x/y −z/y 2 . ρ˜(B0 ) = −z x/y Then ρ˜ is type-preserving and the Markoﬀ map φρ˜ associated with ρ˜ is equal to φ. (2) Let ρ be the P SL(2, C)-representation of π1 (T ) induced by the above ρ˜. Then it extends to a type-preserving P SL(2, C)-representation of π1 (O) satisfying the following identities: 0 −1/y z/y (yz − x)/y 2 , ρ(P0 ) = , ρ(Q0 ) = −x −z/y y 0 11 −x/y (xy − z)/y 2 . , ρ(K) = ρ(R0 ) = 01 −z x/y (3) The restriction of the above P SL(2, C)-representation ρ to π1 (S) satisﬁes the following identities: 1 + xz/y z 2 /y 2 11 ρ(K0 ) = , , ρ(K1 ) = −x2 1 − xz/y 01 1 − xz/y x2 /y 2 1 0 . , ρ(K ) = ρ(K2 ) = 3 −z 2 1 + xz/y −y 2 1 Proof. This lemma can be veriﬁed by direct calculation. For recipes for ﬁnding the cross section, see [70, Sect. 3] and [80, Sect. 1] By using the above lemma, we can prove the following well-known fact. Corollary 2.3.8. Let {ρn } be a sequence in X which converges to ρ∞ in X . Then we can ﬁnd normalized representatives ρn : π1 (O) → P SL(2, C) which converges to a representative ρ∞ : π1 (O) → P SL(2, C) in Homtp (π1 (X), P SL (2, C)). Proof. Let φ∞ (resp. φn ) be the Markoﬀ map inducing ρ∞ (resp. ρn ). Suppose φ∞ (1/1) = 0. Then φn (1/1) = 0 for all suﬃciently large n, and the representations constructed by Lemma 2.3.7 from φn and φ∞ satisfy the desired property. In the general case, pick an elliptic generator triple (P, Q, R)

2.4 Markoﬀ maps and complex probability maps

29

such that φ∞ (s(Q)) = 0. Then we can construct representatives as in Lemma 2.3.7, by using (P, Q, R) instead of (P0 , Q0 , R0 ), for each ρ ∈ X induced by a Markoﬀ map φ such that φ(s(Q)) = 0. By using this fact we obtain the desired result through the above argument.

2.4 Markoﬀ maps and complex probability maps Let φ be a Markoﬀ map with (x, y, z) = (φ(0/1), φ(1/1), φ(1/0)). We give a geometric description of the representation ρ = ρφ : π1 (O) → P SL(2, C) constructed from φ in Lemma 2.3.7. Case 1. xyz = 0. Note that the Markoﬀ identity x2 + y 2 + z 2 = xyz implies the identity, x y z a0 + a1 + a2 = 1, where a0 = , a1 = , a2 = . (2.4) yz zx xy Following Jorgensen, we call the triple (a0 , a1 , a2 ) the complex probability associated with the Markoﬀ triple (x, y, z). We note that the Markoﬀ triple up to sign is recovered from the complex probability, because: x2 =

1 , a1 a2

y2 =

1 , a2 a0

z2 =

1 . a0 a1

(2.5)

By the formula in Lemma 2.3.7, we have the following (see Fig. 2.2): ρ(P0 ) = the π-rotation about the geodesic with endpoints −a0 ± i/x,

ρ(Q0 ) = the π-rotation about the geodesic with endpoints 0 ± i/y, ρ(R0 ) = the π-rotation about the geodesic with endpoints a1 ± i/z.

The images by ρ of the members in the sequence of elliptic generators {Pj } associated with 0/1, 1/1, 1/0, such that (P0 , Q0 , R0 ) = (P0 , P1 , P2 ), are given as follows. By repeatedly drawing the vectors in the complex probability (a0 , a1 , a2 ) in this cyclic order, we obtain a bi-inﬁnite, possibly singular, broken line L on the complex plane C with vertices {cj } such that cj+1 − cj = a[j−1] and (c0 , c1 , c2 ) = (−a0 , 0, a1 ). Draw the vector i/x, i/y, or i/z from cj according as j is congruent to 0, 1, or 2 modulo 3. Note that the vector emanating from cj is equal to (−aj+1 )(aj−1 ), and hence it bisects the angle between the edges of L incident on cj and its length is equal to the multiplicative mean of the lengths of the two edges. Then ρ(Pj ) is the π-rotation about the geodesic with endpoints cj ± i/x, cj ± i/y, or cj ± i/z according as j ≡ 0, 1, or 2 (mod 3). In particular, the isometric circle I(ρ(Pj )) has center cj and radius 1/|x|, 1/|y| or 1/|z| accordingly. We can easily check, by using only elementary Euclidean geometry, that the product ρ(K) = ρ(Pj+1 )ρ(Pj )ρ(Pj−1 ) is the translation (z, t) → (z + 1, t). This gives a geometric description of Jorgensen’s cross section in Lemma 2.3.7. We note that ρ(Pj+3k ) is the conju1k k . Thus the isometric gate of ρ(Pj ) by the Euclidean isometry ρ(K) = 01

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circle I(ρ(Pj+3k )) is equal to the image of the isometric circle I(ρ(Pj )) by the translation z → z + k. i/y i/x

i/x

i/z Q

P

R z/xy

x/yz

y/zx

K Fig. 2.2. Isometric circles of elliptic generators for the general case xyz = 0

Case 2. xyz = 0. We assume without losing generality that x = 0 and z = iy. Then we have the following (see Fig. 2.3): ρ(P0 ) = the π-rotation about the vertical geodesic above −1/2,

ρ(Q0 ) = the π-rotation about the geodesic with endpoints ±i/y, ρ(R0 ) = the π-rotation about the geodesic with endpoints ±i/z. Here a vertical geodesic above a point z ∈ C means the geodesic {z} × R+ in H3 = C × R+ . The axes of ρ(Q0 ) and ρ(R0 ) intersect orthogonally and ρ(R0 Q0 ) is the π-rotation about the vertical geodesic above 0. We see that ρ(K) = ρ(R0 Q0 P0 ) is the parabolic transformation (z, t) → (z + 1, t). Q P

R

K Fig. 2.3. Isometric circles of elliptic generators for the special case x = 0

2.4 Markoﬀ maps and complex probability maps

31

In order to reﬁne the above geometric description, we introduce the concept of a complex probability map following [17] and [40]. Let T be a binary tree (a countably inﬁnite simplicial tree all of whose vertices have degree 3) → e , of T can be thought properly embedded in H2 dual to D. A directed edge, − of an ordered pair of adjacent vertices of T , referred to as the head and tail of − → → e . We introduce the notation − e ↔ (s1 , s2 ; s0 , s3 ) to mean that s0 , s1 , s2 and → e and that (2) s3 are the ideal vertices of D such that (1) s1 , s2 is the dual to − − → − → s0 , s1 , s2 (resp. s1 , s2 , s3 ) is dual to the head (resp. tail) of e . Let E (T ) be − → the set of directed edges of T . For a Markoﬀ map φ, let E φ (T ) be the subset − → → → of E (T ) consisting of those directed edges − e such that if − e ↔ (s1 , s2 ; s0 , s3 ) − → then φ(s1 )φ(s2 ) = 0. Then we deﬁne a map ψ = ψφ : E φ (T ) → C by → ψ(− e)=

φ(s0 ) . φ(s1 )φ(s2 )

(2.6)

We call ψ = ψφ the complex probability map associated with the Markoﬀ − → →

by setting ψ(− map φ. We extend ψ to a map ψ : E (T ) → C e ) = ∞ if φ(s1 )φ(s2 ) = 0. (Note that (i) if φ(s1 )φ(s2 ) = 0 then φ(s0 ) = 0 and that → → → e 0, − e 1, − e 2 ) of (ii) if φ(s0 ) = 0 then φ(s1 )φ(s2 ) = 0.) An ordered triple (− − → elements of E (T ) is said to be dual to an oriented simplex σ = s0 , s1 , s2 if − → e j (0 ≤ j ≤ 2) is dual to sj+1 , sj+2 (the indices are considered modulo 3) and has the vertex of T dual to σ as the head. Then we have the following:

→ → → Lemma 2.4.1. (1) Let σ be an oriented triangle of D and (− e 0, − e 1, − e 2) a − → triple of elements of E φ (T ) dual to σ. Then: → → → ψ(− e 0 ) + ψ(− e 1 ) + ψ(− e 2 ) = 1.

→ → → We call the ordered triple (ψ(− e 0 ), ψ(− e 1 ), ψ(− e 2 )) the complex probability of φ (or the value of ψ) at σ. − → − → → → (2) For each − e ∈ E φ (T ), let −− e be the element of E φ (T ) obtained from − → e by reversing the direction. Then → → ψ(− e ) + ψ(−− e ) = 1. → → → (3) Under Notation 2.1.14 (Adjacent triangles), let (− e 0, − e 1, − e 2 ) and − → − → → → ′ ′ − ′ − ( e 1 , e 2 , e 3 ), respectively, be cyclically ordered triples of elements of E φ (T ) dual to σ = s0 , s1 , s2 and σ ′ = s′0 , s′1 , s′2 . Then we have the following: → → e 1) ψ(− e 0 )ψ(− → ψ(− e ′0 ) = , − → 1 − ψ( e 1 ) → → ψ(− e ′2 ) = 1 − ψ(− e 1 ).

→ → e 2) ψ(− e 1 )ψ(− → ψ(− e ′1 ) = , − → 1 − ψ( e 1 )

Proof. This can be proved by direct calculation (cf. [17]).

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Let Ψ be the set of complex probability maps. Then we can put a topology on Ψ so that the map φ → ψ is a four-fold covering map Φ → Ψ . Indeed, the transformations (x, y, z) → (−x, −y, z) and (x, y, z) → (x, −y, −z) generate a free Z2 ⊕ Z2 -action on the space of Markoﬀ triples and that on the space Φ: the quotient of Φ by this action is identiﬁed with Ψ . Thus we have the following proposition: Proposition 2.4.2. We have the following commutative diagram of Z2 ⊕ Z2 coverings: ∼ = Φ −−−−→ X˜ ⏐ ⏐ ⏐ ⏐ ∼ =

Ψ −−−−→ X

Convention 2.4.3. (1) When we give mention to one of the elements φ ∈ Φ, ψ ∈ Ψ , ρ˜ ∈ X˜ and ρ ∈ X , the remaining symbols represent elements related to the mentioned one by the above commutative diagram so long as this does not create confusion. When we need to be precise, we use symbols ρφ , ψρ and φρ (even though φ is not uniquely determined by ρ). (2) We do not distinguish between an element of X˜ (resp. X ) and its representative: they are denoted by the same symbol so long as there is no fear of confusion. Moreover, When we choose a representative ρ˜ : π1 (T ) → SL(2, C) or ρ : π1 (O) → P SL(2, C), we always assume that it is normalized so that the following identity is satisﬁed. 11 −1 −2 2 . , ρ(K) = ρ˜(K ) = 01 0 −1 The geometric description of a (normalized) type-preserving representation in the above is summarized as follows. Proposition 2.4.4. Let φ be a nontrivial Markoﬀ map, and ρ : π1 (O) → SL(2, C) the (normalized) type-preserving representation induced by ρ˜. (1) Let P be an elliptic generator of slope s. (1.1) Suppose φ(s) = 0. Then ρ(P ) is the π-rotation about the geodesic with endpoints c(ρ(P )) ± i/φ(s), where c(ρ(P )) := ρ(P )−1 (∞) is the center of the isometric circle I(ρ(P )). In particular, the radius, r(ρ(P )), of I(ρ(P )) is equal to |1/φ(s)|. (1.2) Suppose φ(s) = 0. Then ρ(P ) is the π-rotation about a vertical geodesic. In particular, ρ(P ) stabilizes ∞ and acts on H3 as a Euclidean isometry. (2) Let σ = s0 , s1 , s2 be a triangle of D and let {Pj } be the sequence of elliptic generators associated with σ. Then the following hold: (2.1) Suppose that φ(s0 )φ(s1 )φ(s2 ) = 0. Then c(ρ(Pj+1 )) − c(ρ(Pj )) =

φ(s[j−1] ) → = ψ(− e [j−1] ), φ(s[j] )φ(s[j+1] )

2.5 Miscellaneous properties of discrete groups

33

− → → → → where (− e 0, − e 1, − e 2 ) is the triple of elements of E (T ) dual to σ. Moreover, the open line segment proj(Axis(ρ(Pj ))) bisects the angle ∠(c(ρ(Pj−1 )), c(ρ(Pj )), c(ρ(Pj+1 ))), where proj : H3 → C denotes the projection. In other words, arg

i/φ(s[j] ) c(ρ(Pj−1 )) − c(ρ(Pj )) = arg c(ρ(Pj+1 )) − c(ρ(Pj )) i/φ(s[j] )

(mod 2π).

Furthermore I(ρ(Pj+3k )) is equal to the image of I(ρ(Pj )) by the translation z → z + k. (2.2) Suppose that φ(s0 )φ(s1 )φ(s2 ) = 0, say φ(s0 ) = 0. Then ρ(P3j+1 ) and ρ(P3j+2 ) share the same isometric hemisphere and their axes intersect orthogonally. The axis of ρ(P3j ) is the vertical geodesic above the midpoint of c(ρ(P3j−2 )) = c(ρ(P3j−1 )) and c(ρ(P3j+1 )) = c(ρ(P3j+2 )). Furthermore I(ρ(Pj+3k )) is equal to the image of I(ρ(Pj )) by the translation z → z + k, for every j with j ≡ 0 (mod 3). Remark 2.4.5. Under the assumption of the above lemma, we have d(Axis(ρ(Pj )), Axis(ρ(Pj+1 ))) = 2 cosh−1 (φ(s[j−1] )/2), where Axis(·) denotes the axis and d denotes the complex distance. In this paper, we often consider the situation where the assumption of (2.1) of Proposition 2.4.4 is satisﬁed, so we summarize it as the following assumption. Assumption 2.4.6 (σ-NonZero). We assume the following: 1. σ = s0 , s1 , s2 is an ordered triangle of D (cf. Convention 2.1.5 (Ordered triangle)), {Pj } is the sequence of elliptic generators associated with σ, − → → → → e 1, − e 2 ) is the ordered triple of elements of E (T ) dual to σ. (− e 0, − 2. ρ : π1 (O) → P SL(2, C) is a type-preserving representation, and φ ∈ Φ, ψ ∈ Ψ , ρ˜ ∈ X˜ are as in Convention 2.4.3. → → e 0 ), ψ(− e 1 ), 3. We assume that φ−1 (0)∩σ (0) = ∅, and hence (a0 , a1 , a2 ) = (ψ(− − → ψ( e 2 )) is deﬁned. We call it the complex probability of ρ at σ.

2.5 Miscellaneous properties of discrete groups In this section we collect several well-known properties of discrete groups, which we use in this paper. Lemma 2.5.1. Let X, Y ∈ P SL(2, C) with c(X) = c(Y ). Then r(XY −1 ) = r(X)r(Y )/|c(X) − c(Y )|.

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2 Fricke surfaces and P SL(2, C)-representations

′ ′ ab a b Proof. Let X = and Y = ′ ′ . Then r(X) = 1/|c|, r(Y ) = 1/|c′ |, cd c d

′ ad − bc′ −ab′ + ba′ ′ ′ −1 c(X) = −d/c and c(Y ) = −d /c . Since XY = , we cd′ − dc′ −cb′ + da′ have 1 1 1 1 = = r(X)r(Y )/|c(X) − c(Y )|. |cd′ − dc′ | |c| |c′ | | − d/c + d′ /c′ |

11 Lemma 2.5.2. Let Γ be a Kleinian group containing . Then the follow01 ing hold: r(XY −1 ) =

1. For any X ∈ Γ − Γ∞ , r(X) ≤ 1. 2. For any X, Y ∈ Γ − Γ∞ with c(X) = c(Y ), we have |c(X) − c(Y )| ≥ r(X)r(Y ). 3. For any X, Y ∈ Γ − Γ∞ with c(X) = c(Y ), we have r(X) = r(Y ). Proof. The assertion (1) follows from the Shimizu-Leutbecher lemma or the Jorgensen’s inequality (see, for example, [14, Theorem 5.4.1] or [54, Sect. II.C]). To prove (2) and (3), let X, Y ∈ Γ − Γ∞ . Suppose c(X) = c(Y ). Then, by Lemma 2.5.1, r(X)r(Y ) = r(XY −1 )|c(X) − c(Y )|. Since c(X) = c(Y ), we have XY −1 ∈ Γ − Γ∞ . Thus, by the conclusion (1), r(XY −1 ) ≤ 1, and hence we obtain the assertion (2). Suppose c(X) = c(Y ). Then since X −1 KX is a parabolic transformation ﬁxing c(X) and since Y −1 X also ﬁxes c(X), Y −1 X cannot be a loxodromic transformation (cf. [14, Theorem 5.1.2]). Hence we obtain the assertion (3). Corollary 2.5.3. Let Γ and Γ∞ be as in Lemma 2.5.2, and let X and Y be elements of Γ − Γ∞ . If |c(X) − c(Y )| < r(X)r(Y ), then Ih(X) = Ih(Y ). Lemma 2.5.4. Let ρ be an element of X and φ a Markoﬀ map inducing ρ. (1) If ρ ∈ QF, then, for any elliptic generator P , ρ(KP ) is a loxodromic transformation. Thus φ−1 [−2, 2] is empty. (2) If ρ ∈ QF, then the following hold.

1. φ−1 (−2, 2) = ∅. 2. For any elliptic generator P , the radius of I(ρ(P )) is bounded above by 1. 3. For any two distinct elliptic generators P and Q, the distance between c(ρ(P )) and c(ρ(Q)) is bounded below by r(ρ(P ))r(ρ(Q)). In particular, c(ρ(P )) = c(ρ(Q)). 4. The complex probability of ρ at any triangle σ in D is well-deﬁned and is contained in (C − {0})3 .

Proof. (2.1) Let ρ be an element of QF. Then ρ˜ : π1 (T ) → SL(2, C) is faithful. ˆ and A a generator of slope s. Then ρ˜(A) is of inﬁnite Let s be an element of Q order, and hence φ(s) = tr ρ˜(A) ∈ (−2, 2).

2.5 Miscellaneous properties of discrete groups

35

(2.2) Let ρ be an element of QF and P an elliptic generator. Then, by (2.1), φ(s(P )) = 0 and hence r(ρ(P )) = 1/|φ(s(P ))| < ∞ by Proposition 2.4.4(1.1). In particular, ρ(P ) does not stabilize ∞. Thus, by Lemma 2.5.2(1), r(ρ(P )) ≤ 1. (2.3) By (2.2) both ρ(P ) and ρ(Q) do not stabilize ∞. If c(ρ(P )) = c(ρ(Q)), then r(ρ(P )) = r(ρ(Q)) by Lemma 2.5.2(3), and hence ρ(QP ) ∈ ρ(π1 (T )) stabilizes every point in the vertical geodesic connecting c(ρ(P )) and ∞. Thus ρ(QP ) is either elliptic or the identity. This contradicts the fact that ρ|π1 (T ) is faithful. Therefore c(ρ(P )) = c(ρ(Q)). By Lemma 2.5.2(2), the distance between c(ρ(P )) and c(ρ(Q)) is bounded below by r(ρ(P ))r(ρ(Q)). (2.4) is a direct consequence of (2-1).

3 Labeled representations and associated complexes

In this chapter, we introduce the notation which we use to reformulate the main theorems in Chap. 6. As explained in Sect. 2.4, the inﬁnite broken line in C obtained by successively joining the centers c(ρ(Pj )) of the isometric circles I(ρ(Pj )), where {Pj } is a sequence of elliptic generators, recovers the type-preserving representation ρ. Moreover, this broken line plays a key role in the description of the combinatorial structure of the Ford domain in the case ρ is quasifuchsian. Thus we introduce, in Sect. 3.1, the notation L(ρ, σ) to represent the broken line, where σ is the triangle of the Farey triangulation spanned by the slopes of {Pj }. Then we introduce the concept for a Markoﬀ map to be upward at σ (Deﬁnition 3.1.3), and show that precisely one Markoﬀ map among the four Markoﬀ maps inducing a given representation is upward (Lemma 3.1.4). This concept is used in Sect. 4.2 to deﬁne the side parameter. In Sect. 3.2, we generalize the deﬁnition of L(ρ, σ) to L(ρ, Σ), where Σ = (σ1 , · · · , σm ) is a chain of triangles in the Farey triangulation. It may well be a singular 2-dimensional simplicial complex in C, whose 1-skeleton is the union of L(ρ, σi )’s. The main theorems imply that if ρ is quasifuchsian, then there is a unique chain Σ such that L(ρ, Σ) is dual to the Ford domain. If L(ρ, Σ) is non-singular, its combinatorial structure depends only on Σ. Actually, it can be described by using the description of the space of elliptic generators in terms of the Farey triangulation (Sect. 2.1). This leads to the deﬁnition of the elliptic generator complex L(Σ) (Deﬁnition 3.2.3). In Sect. 3.3, we deﬁne a labeled representation as a pair ρ = (ρ, ν) of a type-preserving representation ρ and a label ν = (ν − , ν + ) ∈ H2 × H2 = (|D| − |D(0) |) × (|D| − |D(0) |). What we actually have in mind are pairs (ρ, ν) of a quasifuchsian representation ρ and its side parameter ν = ν(ρ). This concept is used in Chap. 6 to reformulate the main theorems. In Sect. 3.4, we deﬁne the virtual Ford domain Eh(ρ) for a labeled representation ρ and related notation. The main theorems imply that for every element ρ ∈ QF there is a unique label ν such that the Ford domain is equal to the virtual Ford domain Eh(ρ), where ρ = (ρ, ν), as weighted complexes.

38

3 Labeled representations and associated complexes

3.1 The complex L(ρ, σ) and upward Markoﬀ maps The geometric description of (normalized) type-preserving representations in Proposition 2.4.4 motivates the following deﬁnition. Deﬁnition 3.1.1. Under Assumption 2.4.6 (σ-NonZero): (1) L(ρ, σ) denotes the (possibly singular) bi-inﬁnite broken line in C which is obtained by successively joining the centers {c(ρ(Pj ))} of the isometric circles, where {Pj } is the sequence of elliptic generators associated with σ. We note that L(ρ, σ) is invariant by the transformation z → z + 1. → (2) − c (ρ; Pj , Pj+1 ) denotes the oriented edge with foot c(ρ(Pj )) and head c(ρ(Pj+1 )) of L(ρ, σ). It is also regarded as the complex number deﬁned by − → c (ρ; Pj , Pj+1 ) = c(ρ(Pj+1 )) − c(ρ(Pj )).

(3) L(ρ, σ) is said to be simple if the underlying space |L(ρ, σ)| is homeomorphic to the real line R and {c(ρ(Pj ))} sit on it in the order of the suﬃx j ∈ Z. (4) L(ρ, σ) is said to be weakly simple if there is a sequence {ρn } of typepreserving representations converging to ρ such that each L(ρn , σ) is simple. The following lemma describes the relationship between L(ρ, σ) and L(ρ, σ ′ ) for adjacent triangles σ and σ ′ . Lemma 3.1.2. Under Notation 2.1.14 (Adjacent triangles) and Assumption 2.4.6 (σ-NonZero), suppose that φ−1 (0) ∩ (σ (0) ∪ σ ′(0) ) = ∅. Then the Euclidean triangle ∆(c(ρ(P0 )), c(ρ(P1 )), c(ρ(P2 ))) is similar to ∆(c(ρ(P1′ )), c(ρ(P2′ )), c(ρ(P3′ ))). In particular, the union of the two inﬁnite broken lines L(ρ, σ) and L(ρ, σ ′ ) form a bi-inﬁnite sequence of mutually similar triangles (see Fig. 3.1). Here the triangles are possibly degenerate and the interiors of the triangles possibly intersect. Proof. This is a direct consequence of Lemma 2.4.1(3). Deﬁnition 3.1.3 (Upward). Under Assumption 2.4.6 (σ-NonZero), suppose that L(ρ, σ) is simple. Then we say that the Markoﬀ map φ is upward at σ, if for every j ∈ Z, the vector i/φ(s[j] ) at c(ρ(Pj )) is “upward” with respect to L(ρ, σ), that is, the following inequality holds (cf. Fig. 2.2): → i/φ(s[j] ) −− c (ρ; Pj−1 , Pj ) = arg < π. 0 < arg − → i/φ(s[j] ) c (ρ; Pj , Pj+1 )

Here arg z ∈ (−π, π] denotes the argument of a non-zero complex number z, and the identity among the two angles follows from Proposition 2.4.4(2.1). This is also equivalent to the condition: → i/φ(s[j] ) −ψ(− e [j+1] ) 0 < arg − < π. = arg → i/φ(s ) ψ( e ) [j−1]

[j]

3.1 The complex L(ρ, σ) and upward Markoﬀ maps c(ρ(P0 )) =

c(ρ(P0′ ))

c(ρ(P3 )) =

39

c(ρ(P3′ )) s1

c(ρ(P1 ))

′ c(ρ(P−1 ))

c(ρ(P2′ ))

′ c(ρ(P−1 )) = c(ρ(P−2 ))

s0

=

=

s2 s′1

s′0 s′2

c(ρ(P2 )) = c(ρ(P1′ ))

Fig. 3.1. L(ρ, σ) and L(ρ, σ ′ )

Lemma 3.1.4. Under Assumption 2.4.6 (σ-NonZero), suppose that L(ρ, σ) is simple. Then precisely one Markoﬀ map φ among the four Markoﬀ maps inducing ρ is upward. Moreover, for any other Markoﬀ map φ′ inducing ρ, there is only one element j ∈ {0, 1, 2} such that the vector i/φ′ (sj ) is upward with respect to L(ρ, σ). Proof. Let φ0 be the integral Markoﬀ map determined by φ0 (s0 ) = φ0 (s1 ) = φ0 (s2 ) = 3, and let ρ0 be the type-preserving representation determined by φ0 . Then L(ρ0 , σ) is simple and φ0 is upward with respect to L(ρ0 , σ). Let ρ be a type-preserving representation such that L(ρ, σ) is simple. Claim 3.1.5. There is a continuous family {Lt }0≤t≤1 of periodic broken lines in C satisfying the following conditions: 1. L0 = L(ρ0 , σ) and L1 = L(ρ, σ). 2. Lt is invariant by the map z → z + 1 and each period consists of three line segments. 3. Lt is simple for each t ∈ [0, 1]. Proof. Set c = min{ℑz | z ∈ L1 }, and consider the horizontal line L = {z ∈ C | ℑz = c}. Then L1 lies in the closed upper half space bounded by L. By using the periodic region bounded by L and L1 , we can deform L1 to a periodic broken line whose underlying space is equal to L. Such a periodic broken line can be deformed to L0 . Thus we obtain the desired result. By the second condition of the claim, Lt corresponds to a complex probability, and hence Lt = L(ρt , σ) for some type preserving representation ρt . By the continuity of {Lt }, {ρt } is continuous, and therefore it lifts to a continuous family of Markoﬀ maps {φt }. Since φ0 is upward at σ, the third condition of the claim implies that φt is upward at σ for every t ∈ [0, 1]. Since ρ1 = ρ by the third condition of the claim, we obtain the ﬁrst assertion of the lemma. The second assertion follows from the observation stated just before Proposition 2.4.2.

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3 Labeled representations and associated complexes

We now introduce the following deﬁnition, which we occasionally use in this paper. Deﬁnition 3.1.6. (1) Let ℓ be an oriented (topological) line in C which com Then HL (ℓ) and HR (ℓ), respectively, denote the pletes to a simple loop of C. closure of the components of C − ℓ which lies on the left and right hand side of ℓ. We say that a subset A of C lies on the left (resp. right) hand side of ℓ if A ⊂ HL (ℓ) (resp. A ⊂ HR (ℓ)). → −−→ −−→ (2) For an oriented line segment − z− 0 z1 in C, HL (z0 z1 ) and HR (z0 z1 ), respectively, denotes HL (ℓ) and HR (ℓ), where ℓ is the oriented (straight) line → in C obtained by extending − z− 0 z1 on both sides. We note that the following is obtained as a corollary to Claim 3.1.5. Lemma 3.1.7. Under Assumption 2.4.6 (σ-NonZero), suppose that L(ρ, σ) is simple. Then the region of C − |L(ρ, σ)| which lies in the above is equal to the region which lies on the left hand side of |L(ρ, σ)|, where |L(ρ, σ)| is oriented so that it is coherent with the oriented line segment with initial and terminal points c(ρ(P0 )) and c(ρ(P1 )), respectively. Proof. The assertion is obvious in the case when φ = φ0 in Claim 3.1.5. Thus we obtain the conclusion for every φ satisfying the assumption by Claim 3.1.5. Deﬁnition 3.1.8. Under Assumption 2.4.6 (σ-NonZero), suppose that L(ρ, σ) is simple. Then we say that L(ρ, σ) is convex to the above or convex to the (+)-side at c(ρ(Pj )) if ℑ

− − → → e [j−1] c (ρ; Pj , Pj+1 ) < 0. = ℑ − → − → c (ρ; Pj−1 , Pj ) e [j+1]

We say that L(ρ, σ) is convex to the below or convex to the (−)-side at c(ρ(Pj )) if − − → → e [j−1] c (ρ; Pj , Pj+1 ) ℑ − > 0. = ℑ → − → c (ρ; Pj−1 , Pj ) e [j+1] Lemma 3.1.9. Under Assumption 2.4.6 (σ-NonZero), suppose that L(ρ, σ) is simple. If ℑc(ρ(P1 )) is greater (resp. less) than both ℑc(ρ(P0 )) and ℑc(ρ(P2 )), then L(ρ, σ) is convex to the above (resp. below) at ℑc(ρ(P1 )). Proof. We prove the lemma only in the case when the inequality ℑc(ρ(P2 )) ≤ ℑc(ρ(P0 )) < ℑc(ρ(P1 )) holds. (A similar argument works for the remaining cases.) Let (a0 , a1 , a2 ) be the complex probability of ρ at σ. Then, from the assumption, it follows that − → → c (ρ; P1 , P2 ). ℑa2 > 0 and that ℑa0 < 0. Let l be the oriented line segment − Suppose to the contrary that L(ρ, σ) is not convex to the above at ℑc(ρ(P1 ))

3.2 The complexes L(ρ, Σ) and L(Σ)

41

c(ρ(P1 ))

→ − l c(ρ(P0 )) c(ρ(P−1 ))

1

c(ρ(P2 ))

Fig. 3.2. ℑc(ρ(P2 )) ≤ ℑc(ρ(P0 )) < ℑc(ρ(P1 )) and L(ρ, σ) is convex to the below at ℑc(ρ(P1 ))

(see Fig. 3.2). Since ℑ(a0 /a2 ) = 0 from the assumption, L(ρ, σ) is convex to the below at ℑc(ρ(P1 )), and hence ℑ(a0 /a2 ) > 0. Thus c(ρ(P0 )) is contained − → in int HL ( l ). On the other hand, since c(ρ(P−1 )) = c(ρ(P2 )) − 1, c(ρ(P−1 )) − → is contained in int HR ( l ). Let S be the horizontal strip {z ∈ C | ℑc(ρ(P2 )) ≤ − → ℑz ≤ ℑc(ρ(P1 ))} in C. Then c(ρ(P−1 )), c(ρ(P0 )) and l are contained in S by − → the assumption. Moreover S − l has the two components S ∩ HL and S ∩ HR . − → → Thus the edge − c (ρ; P−1 , P0 ) of L(ρ, σ) intersects the edge l of L(ρ, σ). This contradicts the assumption that L(ρ, σ) is simple.

3.2 The complexes L(ρ, Σ) and L(Σ) Deﬁnition 3.2.1 (Chain). (1) By a thick chain in the Farey triangulation D, we mean a (non-empty) ﬁnite sequence Σ = (σ1 , · · · , σm ) of mutually distinct triangles in D such that σi+1 is adjacent to σi for each i (1 ≤ i ≤ m−1). We call σi−1 (resp. σi+1 ) the predecessor (resp. successor) of σi in the chain. We also say that σi−1 (resp. σi+1 ) lies on the (−)-side (resp. (+)-side) of σi . (2) By a thin chain in D, we mean a sequence Σ = (τ ) of length 1 consisting of a single edge τ of D. (3) Σ is called a chain if it is either a thick chain or a thin chain. A chain Σ is also regarded as a subcomplex of D, and Σ (d) denotes the set of d-simplices of the simplicial complex Σ. (4) A vertex s ∈ Σ (0) is called a pivot of Σ if it is a vertex of more than two triangles of Σ. Deﬁnition 3.2.2 (Simple). Let ρ : π1 (O) → P SL(2, C) be a type-preserving representation, and let Σ = (σ1 , · · · , σm ) be a thick chain in D. Assume that φ−1 (0) ∩ Σ (0) = ∅. Then L(ρ, Σ) denotes the union of the bi-inﬁnite broken lines {L(ρ, σi )} in C. We say that L(ρ, Σ) is simple if the following conditions are satisﬁed (see Fig. 1.2(b)): 1. L(ρ, σi ) is simple for every i (1 ≤ i ≤ m).

42

3 Labeled representations and associated complexes

2. L(ρ, σi ) and L(ρ, σj ) (i = j) intersect only at the vertices {c(ρ(P )) | P ∈ EG(σi ∩ σj )} (cf. Deﬁnition 2.1.13). 3. L(ρ, σi+1 ) lies above L(ρ, σi ) for each i. Namely, |L(ρ, σi+1 )| − |L(ρ, σi )| is contained in the component of C − |L(ρ, σi )| which contains the region {z ∈ C | ℑ(z) ≥ c} for suﬃciently large number c. If L(ρ, Σ) is simple, then by adding the triangles in C spanned by edges of L(ρ, Σ), we obtain a 1- or 2-dimensional simplicial complex embedded in C, which is invariant by the translation z → z + 1. We continue to denote it by the same symbol L(ρ, Σ). If L(ρ, Σ) is simple, then the 2-dimensional complex L(ρ, Σ) is combinatorially equivalent to the abstract simplicial complex L(Σ) deﬁned as follows. Deﬁnition 3.2.3 (Elliptic generator complex). Let Σ = (σ1 , · · · , σm ) be a thick chain. (1) L(Σ) (or L(σ1 , · · · , σm )) denotes the simplicial complex constructed as follows, and we call it the elliptic generator complex associated with the chain Σ: 1. The vertex set L(Σ)(0) is identiﬁed with the set of elliptic generators whose slope is contained in Σ (0) . 2. The edge set L(Σ)(1) is identiﬁed with the set of the elliptic generator pairs (P, Q) (cf. Deﬁnition 2.1.13(2)), which appears (successively in this order) in a sequence of elliptic generators associated with a triangle of Σ. 3. The set L(Σ)(2) of the 2-simplices is identiﬁed with the set of the elliptic generator triples (P, Q, R), such that (P, Q), (Q, R) and (P, R) are edges of L(Σ).

(2) The self-map P → P K on L(Σ)(0) induces a simplicial automorphism on L(Σ), and we denote it by the symbol K. (3) L(Σ)/K and L(Σ)/K 2 , respectively, denote the abstract cell complex obtained as the quotient of L(Σ) by the group K and K 2 . The 1-skeleton of L(Σ) is obtained as the union of the 1-dimensional simplicial complex L(σi ) (σi ∈ Σ (2) ), which is obtained by joining the vertices {Pj }, the sequence of elliptic generators associated with σi , successively by edges. The following lemma is obvious from the deﬁnition. Lemma 3.2.4. (1) For an elliptic generator pair (P, Q), the following conditions are equivalent. 1. (P, Q) determines a 1-simplex of L(Σ). 2. There is a triangle σ in Σ whose (oriented) boundary contains the oriented edge s(P ), s(Q), where σ is oriented following Convention 2.1.12. 3. There is an elliptic generator R such that (P, Q, R) is an elliptic generator triple and the triangle s(P ), s(Q), s(R) belongs to Σ.

3.2 The complexes L(ρ, Σ) and L(Σ)

43

(2) For an elliptic generator triple (P, Q, R), the following conditions are equivalent. 1. (P, Q, R) determines a 2-simplex of L(Σ). 2. Both σ := s(P ), s(Q), s(R) and σ ′ := s(P ), s(R), s(QR ) belong to Σ. 3. The two triangles of D containing the edge s(P ), s(R) belong to Σ.

Let ρ and Σ be as in Deﬁnition 3.2.2, and assume that L(ρ, Σ) is simple. Then the map cρ : L(Σ)(0) → L(ρ, Σ)(0) sending P to c(ρ(P )) extends to a combinatorial isomorphism from L(Σ) to L(ρ, Σ), which is (K, ρ(K))equivariant. Moreover, if (P, Q, R), σ and σ ′ are as in Lemma 3.2.4(2), and if σ lies on the ǫ-side of σ ′ (cf. Deﬁnition 3.2.1(1)), then the Euclidean 2-simplex c(ρ(P )), c(ρ(Q)), c(ρ(R)) lies on the (−ǫ)-side of L(ρ, σ), which contains the Euclidean edges, c(ρ(P )), c(ρ(Q)) and c(ρ(Q)), c(ρ(R)), whereas it lies on the (ǫ)-side of L(ρ, σ ′ ), which contains the Euclidean edge c(ρ(P )), c(ρ(R)). Motivated by this observation, we introduce the following deﬁnition. Deﬁnition 3.2.5. In Lemma 3.2.4(2), assume that σ lies on the ǫ-side of σ ′ . Then we say that the 2-simplex (P, Q, R) lies on the (−ǫ)-side of L(σ) and that (P, Q, R) lies on the ǫ-side of L(σ ′ ). We also say that the 2-simplex (P, Q, R) lies on the (−ǫ)-side of the edges (P, Q) and (Q, R) and that (P, Q, R) lies on the ǫ-side of the edge (P, R). Deﬁnition 3.2.6. When we mention a 2-simplex of L(Σ) whose vertex set is {X, Y, Z}, and when we do not need to specify the ordering of the vertices, we denote the 2-simplex by ((X, Y, Z)). It should be noted that precisely one ordered triple among the six ordered triples consisting of the elements of {X, Y, Z} is an elliptic generator triple. Similarly, the symbol ((X, Y )) denotes the 1-simplex of L(Σ) whose vertex set is {X, Y }. We also note that precisely one of the two ordered pairs consisting of X and Y is an elliptic generator pair. For a thin chain Σ, L(Σ) is deﬁned as follows. Deﬁnition 3.2.7 (Elliptic generator complex associated with a thin chain). Let Σ = (τ ) be a thin chain. (1) L(Σ) = L(τ ) denotes the abstract simplicial complex deﬁned as follows. Set Σ ∗ = (σ ′ , σ) under Notation 2.1.14 (Adjacent triangles). Then L(τ ) is the 1-dimensional subcomplex of L(Σ ∗ ) spanned by the vertices

{P ∈ L(Σ ∗ )(0) | s(P ) ∈ τ (0) } = {Pj | j ≡ 1 (mod 3)} = {Pj′ | j ≡ 2 (mod 3)}. This complex is called the elliptic generator complex associated with the thin chain Σ = (τ ). The cellular complexes L(τ )/K and L(τ )/K 2 are deﬁned similarly. (2) Let ρ : π1 (O) → P SL(2, C) be a type-preserving representation such (0) that φ−1 = ∅. Then L(ρ, Σ) = L(ρ, τ ) denotes the image of L(τ ) by ρ (0) ∩ τ the simplicial map cρ : L(τ ) → C such that cρ (P ) = c(ρ(P )) for P ∈ L(τ )(0) . It is said to be simple if the simplicial map cρ : L(τ ) → C is an embedding.

44

3 Labeled representations and associated complexes

3.3 Labeled representation ρ = (ρ, ν) and the complexes L(ρ) and L(ν) Deﬁnition 3.3.1 (Thick/thin label). (1) By a label, we mean a point ν = (ν − , ν + ) of (|D| − |D(0) |) × (|D| − |D(0) |) = H2 × H2 (see Sect. 1.3). (2) A label ν = (ν − , ν + ) is said to be thin if there is an edge of D which contains both ν − and ν + . It is said to be thick if it is not thin. Deﬁnition 3.3.2 (Labeled representation). (1) A labeled representation is a pair ρ = (ρ, ν) of an element ρ ∈ X and a label ν = (ν − , ν + ). We call ν the label of ρ. (2) A labeled representation ρ = (ρ, ν) is said to be thick or thin according as ν is thick or thin. Deﬁnition 3.3.3 (Chain and elliptic generator complex associated with a label). (1) Let ν = (ν − , ν + ) be a thick label. Then Σ(ν) denotes the thick chain (σ1 , · · · , σm ) in D such that the interior of the oriented geodesic line segment in H2 joining ν − to ν + intersects σ1 , · · · , σm in this order. The triangles σ1 and σm , respectively, are denoted by σ − (ν) and σ + (ν). They are called the (−)-terminal triangle (resp. (+)-terminal triangle) of Σ(ν), respectively. They are often abbreviated to σ − and σ + , respectively. (2) Let ν = (ν − , ν + ) be a thin label. Then τ (ν) denotes the edge of D containing both ν − and ν + , and Σ(ν) denotes the thin chain (τ (ν)) in D. The triangles of D having τ (ν) as an edge are denoted by σ − (ν) and σ + (ν). Here the sign ± are chosen arbitrarily. (3) For a label ν = (ν − , ν + ), L(ν) denotes L(Σ(ν)), and we call it the elliptic generator complex associated with the label ν. Deﬁnition 3.3.4 (NonZero). A labeled representation ρ = (ρ, ν) is said to (0) satisfy the condition NonZero if φ−1 = ∅. ρ (0) ∩ Σ(ν) Deﬁnition 3.3.5. Let ρ = (ρ, ν) be a thick labeled representation which satisﬁes the condition NonZero. Then L(ρ) = L(ρ, ν) denotes L(ρ, Σ(ν)) in C (see Deﬁnition 3.2.2). We will show that if ρ ∈ QF and ν = ν(ρ), where ν : QF → H2 × H2 is the map in Theorem 1.3.2 (see Theorem 6.1.12), then L(ρ) is simple. Thus Fig. 1.2 gives a typical example of L(ρ) which is simple.

3.4 Virtual Ford domain We ﬁrst recall the deﬁnition of convex polyhedra following [28].

3.4 Virtual Ford domain

45

Deﬁnition 3.4.1 (Convex polyhedron). (1) A totally geodesic subspace of Hn (resp. Hn = Hn ∪ ∂Hn ) is a copy of Hd (resp. Hd ) with 0 ≤ d ≤ n (resp. 1 ≤ d ≤ n), embedded geodesically in Hn (resp. Hn ). We also regard a singleton in ∂H3 as a totally geodesic subspace of dimension 0. A hypersubspace of Hn (resp. Hn ) is a totally geodesic subspace of of Hn (resp. Hn ) of dimension n − 1. A closed half space of Hn (resp. Hn ) is the closure of a component of the complement of a hyper-subspace in Hn (resp. Hn ). (2) A subspace F of Hn (resp. Hn ) is a convex polyhedron if it is the intersection of a family H of closed half spaces with the property that each point of F has a neighborhood meeting at most a ﬁnite number of boundaries of elements of H. A singleton in ∂H3 is also regarded as a convex polyhedron of dimension 0. (3) A convex polyhedron F in Hn (resp. Hn ) is said to be spanned by a subset V of Hn (resp. Hn ), if F is the intersection of the closed half spaces of Hn (resp. Hn ) containing V . (4) Let F be a convex polyhedron in Hn (resp. Hn ). The dimension dim F is deﬁned as the the dimension of the smallest totally geodesic subspace, H, containing P . The boundary ∂F of F is the topological boundary (or the frontier) of P in H. The interior int F of F is deﬁned to be F − ∂F . A subset F ′ of ∂F is said to be a codimension-one face of F if F ′ = F ∩ H ′ for some hyperplane H ′ of H, and dim F ′ = dimF − 1. If i ≥ 2, the codimension-i faces of F are deﬁned inductively as codimension-one faces of codimension i − 1 faces of F . (5) A geodesic segment in H3 is a 1-dimensional convex polyhedron. For two (possibly identical) points a and b in H3 , we denote by [a, b] the (possibly degenerate) geodesic segment in H3 with endpoints a and b. A half geodesic (resp. complete geodesic) is a geodesic segment in H3 spanned by a point in H3 and a point in ∂H3 (resp. spanned by two points in ∂H3 .) Remark 3.4.2. It should be noted that for a given subset V of Hn (resp. Hn ), the intersection of the closed half spaces of Hn (resp. Hn ) containing V is not necessarily a convex polyhedron. Thus a ‘convex polyhedron spanned by V ’ does not necessarily exist. We now introduce the deﬁnition of a virtual Ford domain and related deﬁnitions. Deﬁnition 3.4.3 (Virtual Ford domain). Let ρ = (ρ, ν) be a labeled representation which satisﬁes the condition NonZero. (1) Eh(ρ) (resp. E(ρ)) denotes the subspace of H3 (resp. C) deﬁned as follows: Eh(ρ) := {Eh(ρ(P )) | P ∈ L(ν)(0) } E(ρ) := {E(ρ(P )) | P ∈ L(ν)(0) } We call Eh(ρ) the virtual Ford domain of ρ.

46

3 Labeled representations and associated complexes

(2) For each ǫ ∈ {−, +}, E ǫ (ρ) denotes the component of E(ρ) containing the region {z ∈ C | ǫℑ(z) ≥ L} for suﬃciently large positive number L. To describe the cellular structure of the (virtual) Ford domains, we need the following deﬁnitions. Deﬁnition 3.4.4. Let ξ be a simplex of L(ν). 1. The link lk(ξ, L(ν)) of ξ in L(ν) is the subcomplex of L(ν) spanned by the vertices X ∈ L(ν)(0) − ξ (0) such that there is a simplex of L(ν) containing both X and ξ. 2. The star st0 (ξ, L(ν)) of ξ in L(ν) is the subcomplex of L(ν) spanned by ξ and lk(ξ, L(ν)). Remark 3.4.5. If dim ξ ≥ 1, then st0 (ξ, L(ν)) is smaller than the star neighborhood of ξ in L(ν), in general. Deﬁnition 3.4.6. Let ρ = (ρ, ν) be a labeled representation which satisﬁes the condition NonZero. For each simplex ξ ∈ L(ν)(≤2) , Fρ (ξ) denotes the convex polyhedron in H3 deﬁned by Fρ (ξ) = {Ih(ρ(P )) | P ∈ ξ (0) } ∩ Eh(ρ, lk(ξ, L(ν))), where

Eh(ρ, lk(ξ, L(ν))) =

{Eh(ρ(X)) | X ∈ lk(ξ, L(ν))(0) }.

We abbreviate the image by Fρ of a 2-simplex P, Q, R (resp. a 1-cell P, Q, a 0-cell P ) as Fρ (P, Q, R) (resp. Fρ (P, Q), Fρ (P )). Thus: Fρ (P, Q, R) = Ih(ρ(P )) ∩ Ih(ρ(Q)) ∩ Ih(ρ(R)), Fρ (P, Q) = Ih(ρ(P )) ∩ Ih(ρ(Q)) ∩ Eh(ρ, lk(P, Q, L(ν))), Fρ (P ) = Ih(ρ(P )) ∩ Eh(ρ, lk(P , L(ν))).

We note that among the vertices of L(ν), only the vertices of st0 (ξ, L(ν)) are involved in the deﬁnition of Fρ (ξ). Deﬁnition 3.4.7. Let ρ = (ρ, ν) be a labeled representation which satisﬁes the condition NonZero. For each simplex ξ ∈ L(ν)(≤2) , F ρ (ξ) denotes the 3 closed convex polyhedron in H deﬁned by {Ih(ρ(P )) | P ∈ ξ (0) } ∩ Eh(ρ, lk(ξ, L(ν))), F ρ (ξ) = where

Eh(ρ, lk(ξ, L(ν))) = .

{Eh(ρ(X)) | X ∈ lk(ξ, L(ν))(0) }.

3.4 Virtual Ford domain

47

Lemma 3.4.8. Let ρ = (ρ, ν) be a labeled representation which satisﬁes the condition NonZero. Then for any simplex ξ of L(ν), we have the following. (1) F ρ (ξ) ∩ H3 = Fρ (ξ). 3 (2) F ρ (ξ) is equal to the closure Fρ (ξ) of Fρ (ξ) in H if Fρ (ξ) = ∅. Proof. (1) For any element A ∈ P SL(2, C) such that A(∞) = ∞, we have Ih(A) ∩ H3 = Ih(A) and Eh(A) ∩ H3 = Eh(A). Hence we have F ρ (ξ) ∩ H3 = Fρ (ξ). (2) Fρ (ξ) = ∩{Ih(ρ(P )) | P ∈ ξ (0) } ∩ ∩{Eh(ρ(X)) | X ∈ lk(ξ, L(ν))(0) } ⊂ ∩{Ih(ρ(P )) | P ∈ ξ (0) } ∩ ∩{Eh(ρ(X)) | X ∈ lk(ξ, L(ν))(0) } = F ρ (ξ).

To see the converse, let x be a point of F ρ (ξ). If x ∈ H3 , then x ∈ F ρ (ξ)∩H3 = Fρ (ξ) ⊂ Fρ (ξ), and so we may assume x ∈ C. Since ρ satisﬁes the condition Duality, Fρ (ξ) = ∅. Pick a point y of Fρ (ξ), and let ℓ be the geodesic in H3 joining y to x. Then ℓ ⊂ F ρ (ξ) ∩ H3 = Fρ (ξ). Hence x ∈ ℓ ⊂ Fρ (ξ). This completes the proof of Lemma 3.4.8. By an argument parallel to the above, we obtain the following lemma. Lemma 3.4.9. Let ρ = (ρ, ν) be a labeled representation which satisﬁes the condition NonZero. Then the following hold. (1) Eh(ρ) ∩ H3 = Eh(ρ) and Eh(ρ) ∩ C = E(ρ). 3 (2) Eh(ρ) is equal to the closure Eh(ρ) of Eh(ρ) in H . We need to compare the virtual Ford domains of two labeled representations sharing the same representation in Chap. 8. So, we introduce the following deﬁnition. Deﬁnition 3.4.10. Let L be an elliptic generator complex L(ν) for some label (0) ν, and let ρ be an element of X such that φ−1 = ∅. ρ (0) ∩ Σ(ν) 1. We deﬁne the virtual Ford domain Eh(ρ, L) by Eh(ρ, L) = {Eh(ρ(P )) | P ∈ L(0) }.

2. For a cell ξ ∈ L(i) , we deﬁne the virtual face F(ρ,L) (ξ) by F(ρ,L) (ξ) = {Ih(ρ(P )) | P ∈ ξ (0) } ∩ {Eh(ρ(P )) | P ∈ lk(ξ, L)(0) } .

4 Chain rule and side parameter

The essential ingredient of Jorgensen’s work in [40] is a detailed analysis of how the pattern of isometric hemispheres bounding the Ford domain change as one varies the group. This idea can be found in his preceding work [39] on the inﬁnite cyclic Kleinian groups. (See the work [25] due to Drumm and Poritz for its detailed exposition and generalization.) In this chapter we ﬁrst describe the “chain rule for isometric circles” (Lemma 4.1.2), which aﬀords a foundation on the analysis, and then we introduce the key notion of Jorgensen’s side parameter (Deﬁnition 4.2.9) and prove various of its properties. In Sect. 4.1, we give a detailed proof and an intuitive explanation of the chain rule for isometric circles (Lemma 4.1.2). In Sect. 4.2, we give the deﬁnition of Jorgensen’s side parameter θǫ (ρ, σ) for a pair (ρ, σ) and a sign ǫ ∈ {−, +} (Deﬁnition 4.2.9). If σ = s0 , s1 , s2 , it is a triple (θǫ (ρ, σ; s0 ), θǫ (ρ, σ; s1 ), θǫ (ρ, σ; s2 )) of real numbers whose sum is equal to π/2 (Proposition 4.2.16). Thus if all components of θǫ (ρ, σ) are nonnegative, then θǫ (ρ, σ) is identiﬁed with a point in σ ⊂ D (Deﬁnition 4.2.17). Moreover, if ρ ∈ QF and if the Ford complex Ford(Γ ) of Γ = ρ(π1 (T )) is isotopic to Spine(σ − , σ + ) for a pair of triangles σ − and σ + (see Theorem 1.2.2), then the point ν ǫ (Γ ) ∈ σ ǫ deﬁned in Sect. 1.3 is equal to θǫ (ρ, σ). In Lemma 4.2.18, we give an algebraic equation for a Markoﬀ map φ to “realize” a given point ν = (ν − , ν + ) ∈ H2 × H2 . This motivates the notion of an algebraic root for ν (Deﬁnition 4.2.19). The last step of the proof of the Main Theorem 1.3.5 requires detailed study of the algebraic equation (Chap. 9). In Sect. 4.3, we describe how the side parameters control relative positions of isometric circles and hemispheres (see Lemmas 4.3.1–4.3.6). Though these lemmas look obvious at a glance, their rigorous proofs require careful arguments. We then introduce the key notion of an ǫ-terminal triangle (Deﬁnition 4.3.8). If ρ is an element of QF and Σ(ν(ρ)) = (σ1 , · · · , σn ) is the chain spanned by ν(ρ) (cf. Main Theorem 1.3.5 and Deﬁnition 1.4.1), then σ1 and σn , respectively, are a (−)-terminal triangle and a (+)-terminal triangle of ρ. In Sect. 4.4, we prove a few basic properties concerning ǫ-terminal triangles. Though they also look obvious at a glance, their rigorous proofs again

50

4 Chain rule and side parameter

require careful arguments, and in particular, the proof of Proposition 4.4.4 is lengthy. However, the readers may skip the proofs, because of the following reason. Proposition 4.4.4 and its sister, Proposition 4.5.7, are used only in the proof of Lemmas 4.6.1, 4.6.2 and 4.6.7 (Proofs of Claims 4.6.3 and 4.6.5), which in turn are used in Sects. 7.2 and 7.3, in the proof of Proposition 6.2.1 (Openness). However, Propositions 4.4.4 and 4.5.7 for the quasifuchsian case are consequences of Lemma 7.1.6 (Hidden isometric hemisphere). Nevertheless, we decided to include the proofs, because the results in the section simplify the structure of the proof of the Main Theorem 1.3.5 and because they are necessary in the forthcoming paper [11], where we treat hyperbolic cone-manifolds. In Sect. 4.5, we study relation between side parameters at adjacent triangles. The results in this section are used in the next Sect. 4.6. In Sect. 4.6, we prove the key Lemmas 4.6.1, 4.6.2 and 4.6.7, which describe how the terminal triangles change according to small deformation of typepreserving representations. These lemmas hold the key to the proof of Proposition 6.2.1 (Openness in X ). Section 4.7 is devoted to the proof of the technical Lemma 4.5.5, which is used in the proof of the key lemmas in Sect. 4.5. We note that the techniques in this section plays an important role in the ﬁrst author’s study [3] which compares Jorgensen’s side parameter and the conformal invariants of quasifuchsian punctured torus groups. In Sect. 4.8, we study those representations ρ such that L(ρ, σ) is weakly simple but is not simple. We show that this happens if and only if either L(ρ, σ) is singly folded or doubly folded (Lemma 4.8.1). We also prove Lemma 4.8.7, which implies Lemma 8.9.3, which in turn plays an important role in Chap. 8.

4.1 Chain rule for isometric circles In this section, we recall fundamental facts concerning the isometric circles. Lemma 4.1.1. (1) Let A be an element of P SL(2, C) which does not ﬁx ∞. Then A(E(A) ∪ {∞}) = D(A−1 ),

A(Eh(A) ∪ {∞}) = Dh(A−1 ),

A(D(A)) = E(A−1 ) ∪ {∞},

A(Dh(A)) = Eh(A−1 ) ∪ {∞},

Moreover, if we orient circles in C counter-clockwise, then the restriction of A to I(A) is an orientation-reversing Euclidean isometry onto I(A−1 ). (2) Let W be an element of P SL(2, C) which preserves ∞ and acts on C as an Euclidean isometry. Then I(AW ) = W −1 (I(A)), Ih(AW ) = W

−1

(Ih(A)),

I(W A) = I(A), Ih(W A) = Ih(A).

4.1 Chain rule for isometric circles

51

In particular, I(W AW −1 ) = W (I(A)). (3) Let A and B elements of P SL(2, C) which do not ﬁx ∞. Then I(A) = I(B) if and only if BA−1 is a Euclidean isometry. Proof. Though these are well-known, we give proofs for completeness. (1) For a point z ∈ C, set w = A(z). Then z ∈ E(A) if and only if |A′ (z)| ≥ 1. The latter condition is equivalent to the condition that |(A−1 )′ (w)| = 1/|A′ (z)| ≤ 1, which in turn is equivalent to the condition w ∈ D(A−1 ). Hence A(E(A)) ⊂ D(A−1 ) and therefore we have A(E(A) ∪ {∞}) = D(A−1 ). The remaining assertions are consequences of this identity. (2) By the assumption, |W ′ (z)| = 1 for every z ∈ C. Hence |(AW )′ (z)| = ′ |A (W (z))W ′ (z)| = |A′ (W (z))|. Thus z ∈ I(AW ) if and only if W (z) ∈ I(A). So we have I(AW ) = W −1 (I(A)). The second identity follows from the identity |(W A)′ (z)| = |W ′ (A(z))A′ (z)| = |A′ (z)|. (3) The “if” part follows from (2). To see the converse, assume that I(A) = I(B). Let z be a point in I(A) = I(B). Then |(BA−1 )′ (A(z))| = |B ′ (A−1 (A(z))) · (A−1 )′ (A(z))| = |B ′ (z) · (A−1 )′ (A(z))|

= |(A−1 )′ (A(z))| =1

because z ∈ I(B) because A(z) ∈ I(A−1 ).

Hence the restriction of BA−1 to A(I(A)) = I(A−1 ) is a Euclidean isometry. Moreover BA−1 (D(A−1 )) = B(E(A)) = B(E(B)) = D(B). Thus we can conclude that BA−1 is a Euclidean isometry. The following lemma is called the chain rule for isometric circles because it follows from the chain rule for diﬀerentials. Though it is well-known, we cannot ﬁnd a proof in the literature. We give a proof for completeness, because it is essential to our whole argument. Lemma 4.1.2 (Chain rule for isometric circles). Let A and B be elements of P SL(2, C) which do not ﬁx ∞, such that c(A) = c(B). Then the following hold. 1. A(Ih(A) ∩ Ih(B)) = Ih(A−1 ) ∩ Ih(BA−1 ) and B(Ih(A) ∩ Ih(B)) = Ih(B −1 ) ∩ Ih(AB −1 ). 2. A(I(A) ∩ E(B)) = I(A−1 ) ∩ E(BA−1 ) and B(I(B) ∩ E(A)) = I(B −1 ) ∩ E(AB −1 ). 3. Suppose I(A) ∩ I(B) = ∅ and I(A) = I(B), namely the dihedral angle, θ(X, Y ), of E(X) ∩ E(Y ) (at I(X) ∩ I(Y )) is well-deﬁned. Then the dihedral angles of E(A−1 )∩E(BA−1 ) and E(AB −1 )∩E(B −1 ) are well-deﬁned, and the following identity holds: θ(A, B) + θ(B −1 , AB −1 ) + θ(BA−1 , A−1 ) = 2π.

52

4 Chain rule and side parameter

Proof. (1) We ﬁrst note that BA−1 and AB −1 do not ﬁx ∞ and hence I(BA−1 ) and I(AB −1 ) are deﬁned. Let z be a point in I(A) ∩ I(B). Then by the proof of Lemma 4.1.1(3), we see |(BA−1 )′ (A(z))| = 1 and hence A(z) ∈ I(BA−1 ). On the other hand, A(z) ∈ I(A−1 ) because z ∈ I(A). Hence A(z) ∈ I(A−1 ) ∩ I(BA−1 ). Thus we have proved A(I(A) ∩ I(B)) ⊂ I(A−1 ) ∩ I(BA−1 ). This implies A(I(A) ∩ I(B)) = I(A−1 ) ∩ I(BA−1 ) and A(Ih(A) ∩ Ih(B)) = Ih(A−1 ) ∩ Ih(BA−1 ). The second identity follows from the ﬁrst one by interchanging the roles of A and B. (2) Let z be a point in I(A) ∩ E(B). The above calculation also implies |(BA−1 )′ (A(z))| ≤ 1, and hence A(z) ∈ E(BA−1 ). Thus we have A(I(A) ∩ E(B)) ⊂ E(BA−1 ). Since A(I(A)) = I(A−1 ), this implies A(I(A) ∩ E(B)) ⊂ I(A−1 ) ∩ E(BA−1 ) and hence we obtain the ﬁrst identity. The second identity follows from the ﬁrst one by interchanging the roles of A and B. (3) To prove this, we need to introduce some notation and notice some elementary fact, which we present below. For a point z of the isometric circle I(X) of a M¨ obius transformation X ∈ P SL(2, C), let vz+ (X) and vz− (X), respectively, be the unit tangent vector to I(X) at z, which is coherent (resp. anti-coherent) with the counter-clockwise orientation of I(X). Let (X, Y ) be an ordered pair of M¨ obius transformations X, Y ∈ P SL(2, C) such that I(X) and I(Y ) intersect transversely. Then we can see that there is a unique point, z, in I(X)∩I(Y ) such that vz+ (X) points into E(Y ), vz− (Y ) points into E(X), and the dihedral angle θ(X, Y ) of E(X)∩E(Y ) is equal to arg(vz+ (X)/vz− (Y )), where we identify tangent vectors with complex numbers. If z ′ is the other point of I(X) ∩ I(Y ), then vz−′ (X) points into E(Y ), vz+′ (Y ) points into E(X), and θ(X, Y ) is equal to arg(vz+′ (Y )/vz−′ (X)), where we identify tangent vectors with complex numbers. We may assume that I(A) intersects I(B) transversely, because the tangential case is obtained as limit of the generic case. Let p be the point of I(A) ∩ I(B) such that vp+ (A) points into E(B), vp− (B) points into E(A), and θ(A, B) = arg(vp+ (A)/vp− (B)). By Lemma 4.1.1(1), we see dA(vp+ (A)) = vq− (A−1 ), where q = A(p). Moreover, Lemma 4.1.2(2) implies that vq− (A−1 ) points into E(BA−1 ). Thus we have θ(BA−1 , A−1 ) = arg

vq+ (BA−1 ) vq− (A−1 )

= arg

dA(w) w = arg + , + dA(vp (A)) vp (A)

where w = dA−1 (vq+ (BA−1 )). Similarly, we see that the vector dB(vp− (B)) = vr+ (B −1 ) points into E(AB −1 ), where r = B(p). This implies θ(B −1 , AB −1 ) = arg

vp− (B) dB(vp− (B)) vr+ (B −1 ) , = arg = arg dB(w′ ) w′ vr− (AB −1 )

where w′ = dB −1 (vr− (AB −1 )). On the other hand, we have vr− (AB −1 ) = d(BA−1 )(vq+ (BA−1 )) by Lemma 4.1.1(1), and hence w′ = dB −1 (vr− (AB −1 )) = dB −1 (d(BA−1 )(vq+ (BA−1 ))) = dA−1 (dA(w)) = w.

4.1 Chain rule for isometric circles

53

Hence θ(A, B)+θ(B −1 , AB −1 ) + θ(BA−1 , A−1 ) = arg

vp+ (A)

+ arg

vp− (B) ≡ arg 1 (mod 2π) ≡ 0 (mod 2π)

vp− (B) w + arg + w vp (A)

Since all dihedral angles are positive and less than π, the above congruence implies the desired identity.

vp+ (A) I(A)

vp− (B) I(B)

p

w = w′

vq+ (BA−1 ) I(BA−1 ) q

vr− (AB −1 )

vq− (A−1 ) −1

I(A

)

vr+ (B −1 )

r

I(AB −1 )

I(B −1 )

Fig. 4.1. Three pairs of circles

Since Lemma 4.1.2 (Chain rule) is so important, we also give an intuitive explanation of the lemma. To this end, consider a Kleinian group Γ such that Γ∞ consists of Euclidean isometries. Let e be an edge of its Ford complex Ford(Γ ) ⊂ M (Γ ). Then the valency of e is generically equal to 3, and we assume this condition is satisﬁed. Let Fj (j = 1, 2, 3) be the faces of Ford(Γ ) sharing the edge e. Take a base point b of M (Γ ) in the horospherical neighborhood of the main cusp, i.e., the image of a precisely (Γ, Γ∞ )-invariant horoball H∞ centered at ∞. Let αj and βj (j ∈ {1, 2, 3}) be oriented arcs in M (Γ ) satisfying the following conditions (see Fig. 4.2): 1. The endpoint of αj is equal to the initial point of αj+1 , where the indices are considered modulo 3, and the union α1 ∪ α2 ∪ α3 forms a meridian around the edge e. 2. Each αj intersects Ford(Γ ) transversely at a single point in int Fj .

54

4 Chain rule and side parameter

3. Each βj is disjoint from Ford(Γ ) and joins the base point b to the initial point of αj . β3 β2

F2 α2

α1 e

F3

b

α3

F1 β1

Fig. 4.2. Paths αj and βj

Let Aj be the element of π1 (M (Γ ), b) ∼ = Γ represented by the oriented loop −1 . Then A1 A2 A3 = 1. Since Aj is dual to Fj in the sense that it is βj · αj · βj+1 represented by a based loop intersecting Ford(Γ ) at a single point in int Fj , the following hold. 1. The isometric hemisphere Ih(Aj ) supports a face of the Ford domain P h(Γ ) which projects to Fj . 2. Ih(Aj ) ∩ Ih(A−1 j+1 ) contains an edge of P h(Γ ) which projects to e. 3. The dihedral angle θj between the faces Fj and Fj+1 along e is equal −1 to θ(Aj , A−1 j+1 ), the dihedral angle of Eh(Aj ) ∩ Eh(Aj+1 ) along Ih(Aj ) ∩ Ih(A−1 j+1 ). In the above, indices are considered modulo 3. Since θ1 + θ2 + θ3 = 2π, we have −1 −1 θ(A1 , A−1 2 ) + θ(A2 , A3 ) + θ(A3 , A1 ) = 2π. then A3 = BA−1 , because A1 A2 A3 = I. The If we set A = A1 , B = A−1 2 above identity is then nothing other than the identity in Lemma 4.1.2 (Chain rule). This gives a conceptual proof to the lemma in the special situation. As a matter of fact, this is the typical situation where we use the lemma. To be more precise, we use the chain rule in order to guarantee that the sum of the dihedral angles in a “cycle” is equal to 2π when we use Poincare’s theorem on fundamental polyhedra (Sect. 6.4). The following lemma is a consequence of Lemmas 4.1.1 and 4.1.2. Lemma 4.1.3 (Chain rule for elliptic generators). Let ρ be a typepreserving representation of π1 (O), and let {Pj } be the sequence of elliptic generators associated with a triangle σ = s0 , s1 , s2 of D. Assume that φ−1 (0)∩σ (0) = ∅ and hence the isometric hemisphere Ih(ρ(Pj )) is well-deﬁned for every j. Then the following hold.

4.1 Chain rule for isometric circles

55

(1) Ih(ρ(Pj+3n )) = ρ(K)n (Ih(ρ(Pj ))), Ih(ρ(Pj−1 Pj )) = Ih(ρ(Pj+1 )) and Ih(ρ(Pj Pj−1 )) = Ih(ρ(Pj−2 )). (2) ρ(Pj ) interchanges Ih(ρ(Pj−1 )) ∩ Ih(ρ(Pj )) and Ih(ρ(Pj )) ∩ Ih (ρ(Pj+1 )). (3) Suppose Ih(ρ(Pj )) ∩ Ih(ρ(Pj+1 )) = ∅ for every integer j. Let θj be the dihedral angle of Eh(ρ(Pj )) ∩ Eh(ρ(Pj+1 )). Then θj−1 + θj + θj+1 = 2π. (4) Suppose that φ(s′[j] ) = 0, where s′[j] is the vertex of D opposite to s[j] with respect to the edge s[j−1] , s[j+1] . Then the isometric hemispheres P P Ih(Pj j−1 ) and Ih(Pj j+1 ) exist, and the following hold (see Fig. 4.3): Pj−1

ρ(Pj−1 )(v(ρ; Pj−1 , Pj , Pj+1 )) = v(ρ; Pj−2 , Pj

, Pj−1 )

ρ(Pj )(v(ρ; Pj−1 , Pj , Pj+1 )) = v(ρ; Pj−1 , Pj , Pj+1 ) Pj+1

ρ(Pj+1 )(v(ρ; Pj−1 , Pj , Pj+1 )) = v(ρ; Pj+1 , Pj

, Pj+2 ).

In the above, v(ρ; X, Y, Z), where (X, Y, Z) is a triple of elements of π1 (O), denotes the set Ih(ρ(X)) ∩ Ih(ρ(Y )) ∩ Ih(ρ(Z)). Proof. (1) The ﬁrst identity follows from Lemma 4.1.1(2) and the facts that Pj+3n = K n Pj K −n and ρ(K) is a Euclidean translation. The remaining identities also follow from the same lemma and the following identities. Pj−1 Pj = Pj−1 Pj Pj+1 Pj+1 = K −1 Pj+1 , Pj Pj−1 = Pj Pj−1 Pj−2 Pj−2 = KPj−2 . (2) By Lemma 4.1.2(2), ρ(Pj )(Ih(ρ(Pj−1 ))∩Ih(ρ(Pj ))) = Ih(ρ(Pj−1 Pj ))∩ Ih(ρ(Pj )). Since Ih(ρ(Pj−1 Pj )) = Ih(ρ(Pj+1 )) by (1), we obtain the desired result. (3) We apply Lemma 4.1.2(3) for the pair (A, B) := (ρ(Pj ), ρ(Pj+1 )). Then θ(A, B) = θj . Since I(A−1 ) = I(ρ(Pj )) and I(BA−1 ) = I(ρ(Pj+1 Pj )) = I(ρ(Pj−1 )) by (1), we have θ(BA−1 , A−1 ) = θj−1 . Similarly, since I(B −1 ) = I(ρ(Pj+1 ))) and I(A−1 B) = I(ρ(Pj Pj+1 )) = I(ρ(Pj+2 )) by (1), we have θ(B −1 , AB −1 ) = θj+1 . Hence we obtain the desired result by Lemma 4.1.2(3). (4) By Lemma 2.1.8(2), the bi-inﬁnite sequence Pj−1

· · · , Pj−2 , Pj

Pj+1

, Pj−1 , Pj+1 , Pj

, Pj+2 , · · ·

form the sequence of elliptic generators associated with σ ′ := s[j+1] , s′[j] , s[j−1] . Hence the assumption φ(s′[j] ) = 0 implies that the isometric hemiPj−1

spheres Ih(Pj

Pj+1

) and Ih(Pj

) exist. Moreover (2) implies Pj−1

ρ(Pj−1 )(Ih(ρ(Pj−1 )) ∩ Ih(ρ(Pj+1 ))) = Ih(ρ(Pj

)) ∩ Ih(ρ(Pj−1 )), Pj+1

ρ(Pj+1 )(Ih(ρ(Pj−1 )) ∩ Ih(ρ(Pj+1 ))) = Ih(ρ(Pj+1 )) ∩ Ih(ρ(Pj Hence

)).

56

4 Chain rule and side parameter v(ρ; Pj−1 , Pj , Pj+1 )

Pj

Pj−1

Pj+1

Pj−1

v(ρ; Pj−2 , Pj

, Pj−1 )

Pj+1

v(ρ; Pj+1 , Pj

, Pj+2 )

Fig. 4.3. Chain rule

ρ(Pj−1 )(v(ρ; Pj−1 , Pj , Pj+1 )) = ρ(Pj−1 )(Ih(ρ(Pj−1 )) ∩ Ih(ρ(Pj ))) ∩ ρ(Pj−1 )(Ih(ρ(Pj )) ∩ Ih(ρ(Pj+1 ))) Pj−1

= (Ih(ρ(Pj−2 )) ∩ Ih(ρ(Pj−1 ))) ∩ (Ih(ρ(Pj−1 )) ∩ Ih(ρ(Pj =

)))

P v(ρ; Pj−2 , Pj j−1 , Pj−1 ).

Thus we have the ﬁrst identity. The remaining identities are proved similarly.

4.2 Side parameter In this section, we introduce the side parameter deﬁned by Jorgensen. Lemma 4.2.1. Under Assumption 2.4.6 (σ-NonZero), the following conditions are equivalent:

4.2 Side parameter

57

1. The positive real numbers |φ(s0 )|, |φ(s1 )|, and |φ(s2 )| satisfy the triangle inequality. 2. The positive real numbers |a0 |, |a1 |, and |a2 | satisfy the triangle inequality. 3. I(ρ(Pj )) and I(ρ(Pj+1 )) intersect in two points for some j ∈ Z. 4. I(ρ(Pj )) and I(ρ(Pj+1 )) intersect in two points for every j ∈ Z. Proof. It is obvious from the deﬁnition of the complex probability (see Sect. 2.4) that the ﬁrst condition is equivalent to the second condition. The equivalence between the third and the fourth conditions follows from Lemma 4.1.3(2) (Chain rule). We prove the equivalence between the ﬁrst and the last conditions. Recall that the radius of I(ρ(Pj )) is |1/φ(s[j] )| and that d(c(ρ(Pj )), c(ρ(Pj+1 )) = |φ(s[j−1] )/(φ(s[j] )φ(s[j+1] ))| (see Lemma 2.4.4). Hence I(ρ(Pj )) and I(ρ(Pj+1 )) intersect in two points if and only if 1 φ(s[j−1] ) 1 + > . φ(s ) φ(s φ(s )φ(s [j] [j+1] ) [j] [j+1] ) This condition is equivalent to

|φ(s[j] )| + |φ(s[j+1] )| > |φ(s[j−1] )|. Hence we see that the third condition is equivalent to the ﬁrst condition. Deﬁnition 4.2.2 (Triangle inequality). Under Assumption 2.4.6 (σ-NonZero), we say that φ (or ρ) satisﬁes the triangle inequality at σ if the (mutually equivalent) conditions of Lemma 4.2.1 are satisﬁed. Remark 4.2.3. See Lemma 4.2.11 for a geometric meaning of the similarity class of a triangle with edge lengths |φ(s0 )|, |φ(s1 )|, and |φ(s2 )|. In the following we presume the following two assumption: Assumption 4.2.4 (σ-Simple). Under Assumption 2.4.6 (σ-NonZero), we assume that the following conditions are satisﬁed: 1. φ satisﬁes the triangle inequality at σ (Deﬁnition 4.2.2). 2. L(ρ, σ) is simple (Deﬁnition 3.1.1(2)). 3. φ is upward at σ (Deﬁnition 3.1.3 and Lemma 3.1.4). Deﬁnition 4.2.5. Under Assumption 4.2.4 (σ-Simple), we use the following notation (cf. Fig. 4.5). 1. Fixǫσ (ρ(Pj )) denotes the point c(ρ(Pj )) + ǫi/φ(s[j] ) in C. 2. v ǫ (ρ; Pj , Pj+1 ) (ǫ = ±) denotes the points of I(ρ(Pj )) ∩ I(ρ(Pj+1 )), such −−−−−−−−−−−−−−−−−−−−−−→ that the vector v − (ρ; Pj , Pj+1 )v + (ρ; Pj , Pj+1 ) is upward with respect to L(ρ, σ), i.e.,

58

4 Chain rule and side parameter

arg

v + (ρ; Pj , Pj+1 ) − v − (ρ; Pj , Pj+1 ) = π/2. c(ρ(Pj+1 )) − c(ρ(Pj ))

This condition is equivalent to the following inequality. ǫ v (ρ; Pj , Pj+1 ) − c(ρ(Pj )) ǫℑ > 0. c(ρ(Pj+1 )) − c(ρ(Pj )) It should be noted that the ﬁxed point set of the M¨ obius transformation

is equal to {Fix− (ρ(Pj )), Fix+ (ρ(Pj ))}. ρ(Pj ) on C σ σ

Notation 4.2.6. If ρ, σ and {Pj } are ﬁxed under the setting in Deﬁnition 4.2.5, we employ the following abbreviations: c(j) = c(ρ(Pj )), I(j) = I(ρ(Pj )), Ih(j) = Ih(ρ(Pj )), D(j) = D(ρ(Pj )), Dh(j) = Dh(ρ(Pj )), E(j) = → E(ρ(Pj )), Eh(j) = Eh(ρ(Pj )), v ǫ (j, j + 1) = v ǫ (ρ; Pj , Pj+1 ), − c (j, j + 1) = − → ǫ ǫ − → c (ρ; Pj , Pj+1 ), Fix (j) = Fixσ (ρ(Pj )), Axis(j) = Axis(ρ(Pj )) and f (j) = −−−−−−−−−−−−−+−−−−−→ Fix− σ (ρ(Pj )) Fixσ (ρ(Pj )).

Lemma 4.2.7. Under Assumption 4.2.4 (σ-Simple), we have: ρ(Pj )(v ǫ (ρ; Pj−1 , Pj )) = v ǫ (ρ; Pj , Pj+1 ). Proof. We may assume j = 1 without loss of generality. By Lemma 4.1.3(2) (Chain rule), ρ(P1 ) sends v + (0, 1) to either v + (1, 2) or v − (1, 2). To prove the desired identity, let ℓ− be the oriented line in C obtained by extending → the oriented edge − c (0, 1). Then v + (0, 1) and v − (0, 1), respectively, lie on the left and the right of ℓ− . Similarly, v + (1, 2) and v − (1, 2), respectively, lie on the left and the right of the oriented line ℓ+ which is obtained by extending the → oriented edge − c (1, 2). On the other hand, the transformation ρ(P1 ) maps the left hand side of the oriented line ℓ− to that of the oriented line ℓ+ , because ρ(P1 ) is orientation-preserving and maps the oriented circle ℓ− ∪ {∞} to the oriented circle ℓ+ ∪ {∞}. Hence it must send v + (0, 1) to v + (1, 2). By using the above lemma, we obtain the following lemma. Lemma 4.2.8. Under Assumption 4.2.4 (σ-Simple), we have the following (see Fig. 4.4). ǫ arg

v ǫ (ρ; Pj−1 , Pj ) − c(ρ(Pj )) Fixǫσ (ρ(Pj )) − c(ρ(Pj )) Fixǫ (ρ(Pj )) − c(ρ(Pj )) ≡ π = ǫ arg ǫ σ v (ρ; Pj , Pj+1 ) − c(ρ(Pj ))

(mod 2π).

Proof. We may assume j = 1 without loss of generality. Since ρ(P1 ) acts on I(1) as an orientation-reversing Euclidean isometry, Lemma 4.2.7 implies that the above two angles are equal. In the remainder we prove that these angles are not equal to π modulo 2π. For simplicity we prove this only for the

4.2 Side parameter

59

case ǫ = +. By deﬁnition, Fix− (1) lies on the right hand side of the oriented line ℓ− in the proof of Lemma 4.2.7, and the point v + (0, 1) lies on the left hand side of ℓ− ; in fact Fix− (1) ∈ int HR (ℓ− ) and v + (0, 1) ∈ int HL (ℓ− ) (cf. Deﬁnition 3.1.6). Hence v + (0, 1) = Fix− (1). This implies that the complex number v + (0, 1) − c(1) is not a negative real multiple of Fix+ (1) − c(1). Hence we obtain the conclusion. Fix+ σ (ρ(Pj ))

θ+ (ρ, σ; s[j] )

Pj +

v (ρ; Pj−1 , Pj )

v + (ρ; Pj , Pj+1 ) Pj+1

Pj−1

Fig. 4.4. Side parameter

We now introduce the following key deﬁnition. Deﬁnition 4.2.9 (Side parameter). Under Assumption 4.2.4 (σ-Simple): (1) The angle in Lemma 4.2.8 is denoted by θǫ (ρ, σ; s[j] ) ∈ (−π, π). (See Fig. 4.4.) (2) The ǫ-side parameter of ρ at σ, denoted by θǫ (ρ, σ), is the triple deﬁned by θǫ (ρ, σ) := (θǫ (ρ, σ; s0 ), θǫ (ρ, σ; s1 ), θǫ (ρ, σ; s2 )).

We note that the terminology, “parameter”, in the above is eventually justiﬁed by Theorem 1.3.2. Notation 4.2.10. Under Assumption 4.2.4 (σ-Simple): (1) ∆ǫj (ρ, σ) , ∆ǫj for short, denotes the Euclidean triangle ∆ǫj (ρ, σ) := ∆(c(ρ(Pj )), c(ρ(Pj+1 )), v ǫ (ρ; Pj , Pj+1 )) in C spanned by the three points c(ρ(Pj )), c(ρ(Pj+1 )), and v ǫ (ρ; Pj , Pj+1 ), for each j ∈ Z. (2) ∆(ρ, σ) denotes the (similarity class) of a Euclidean triangle ∆(w0 , w1 , w2 ) with vertices w0 , w1 and w2 such that |w2 − w1 | = |φ(s0 )|,

|w0 − w2 | = |φ(s1 )|,

|w1 − w0 | = |φ(s2 )|,

60

4 Chain rule and side parameter

and α(ρ, σ; sj ) ∈ (0, π) denotes the (inner) angle of the triangle at the vertex wj . Fix+ σ (ρ(P1 )) v + (ρ; P1 , P2 )

v + (ρ; P0 , P1 ) w0 |φ(s2 )| α0 |φ(s1 )| w1

α1

α2

|φ(s0 )|

∆+ 0 P0

w2

∆− 0

P1

∆+ 1

∆+ 2 P2

∆− 1

∆+ 3 P3

∆− 2

P4

∆− 3

v − (ρ; P1 , P2 ) Fig. 4.5. ∆ǫj (ρ, σ) and ∆(ρ, σ)

The following lemma is easily proved (see Fig. 4.5). Lemma 4.2.11. Under Assumption 4.2.4 (σ-Simple), the following hold. (1) 2θǫ (ρ, σ; sj ) is equal to the “signed angle” at c(ρ(Pj )) between the two triangles ∆ǫj−1 (ρ, σ) and ∆ǫj (ρ, σ), where the sign is deﬁned so that θǫ (ρ, σ; sj ) ≥ 0 if and only if int ∆ǫj−1 (ρ, σ) ∩ int ∆ǫj (ρ, σ) = ∅. (2) ∆ǫj (ρ, σ) is similar to ∆(ρ, σ) by a similarity which maps the ordered triple (c(ρ(Pj )), c(ρ(Pj+1 )), v ǫ (ρ; Pj , Pj+1 )) to (w[j] , w[j+1] , w[j+2] ). Here is an intuitive geometric description of the sequence of the triangles {∆ǫj (ρ, σ)}j∈Z . ∆ǫj (ρ, σ) is obtained from ∆ǫj−1 (ρ, σ) by: 1. rotating about the vertex c(j) so that the edge c(j)v ǫ (j − 1, j) overlaps the edge c(j)c(j + 1), and then 2. shrinking or expanding so that the image of c(j)v ǫ (j − 1, j) coincides with c(j)c(j + 1). Lemma 4.2.12. Under Assumption 2.4.6 (σ-NonZero), suppose that L(ρ, σ) is simple. Then we have → − → ψ(− e [j−1] ) c (ρ; Pj , Pj+1 ) ∈ (−π, π). = arg − arg − → → c (ρ; P , P ) ψ( e ) [j+1]

j−1

j

Moreover, the sum of any three successive arguments is equal to 0, namely 2 j=0

arg

→ → → → ψ(− e [j−1] ) ψ(− e 1) ψ(− e 2) ψ(− e 0) = arg + arg + arg = 0. − → − → − → − → ψ( e [j+1] ) ψ( e 0 ) ψ( e 1 ) ψ( e 2 )

4.2 Side parameter

61

Proof. By Proposition 2.4.4(2.1), → − → ψ(− e [j−1] ) c (j, j + 1) =− → − → c (j − 1, j) ψ( e [j+1] ) Since L(ρ, σ) is simple, this (nonzero) complex number is not a negative real number. Hence its argument is contained in (−π, π), and we obtain the ﬁrst assertion. To prove the second assertion, pick a large positive real number, h, such that the horizontal line ℑz = h is disjoint from L(ρ, σ). Let M be the submanifold of the Euclidean annulus C/Z which is obtained as the image of the region bounded by the horizontal line and L(ρ, σ). Then the desired formula is obtained by applying the Gauss-Bonnet theorem to M . Corollary 4.2.13. Under Assumption 4.2.4 (σ-Simple), if L(ρ, σ) is convex to the above at some vertex, then it is convex to the below at some other vertex, and vice versa. In particular, if L(ρ, σ) is not ﬂat (i.e., convex to the above or below) at every vertex, then one of the following holds after a shift of indices. 1. L(ρ, σ) is convex to the above at c(ρ(P0 )) and convex to the below at c(ρ(P1 )) and c(ρ(P2 )). 2. L(ρ, σ) is convex to the below at c(ρ(P0 )) and convex to the above at c(ρ(P1 )) and c(ρ(P2 )). Proof. By the second identity in Lemma 4.2.12, it follows that if one of → → → → → → e 0 )), ℑ(ψ(− e 2 )/ψ(− e 1 )) and ℑ(ψ(− e 0 )/ψ(− e 2 )) is positive (resp. ℑ(ψ(− e 1 )/ψ(− negative) then some of them is negative (positive). The desired result follows from this fact. Lemma 4.2.14. Under Assumption 4.2.4 (σ-Simple), the following identity holds for every integer j. − → 1 c (ρ; Pj , Pj+1 ) ǫ − 2α(ρ, σ; s[j] ) θ (ρ, σ; s[j] ) = π − ǫ arg − → 2 c (ρ; Pj−1 , Pj ) → ψ(− e [j−1] ) 1 = − 2α(ρ, σ; s[j] ) , π − ǫ arg − 2 ψ(→ e [j+1] ) Proof. Throughout the proof we use the following notation. 1. arg : C − R≤0 → (−π, π) denotes the argument in (−π, π) of a complex number which does not belong to R≤0 := {z ∈ R | z ≤ 0}. 2. Let z1 , z2 and f be non-zero complex numbers such that z1 /f and z2 /f are not nonnegative real numbers and arg(z1 /f ) and arg(f /z2 ) have the same sign, that is, either they are both nonnegative or they are both nonpositive. Then we deﬁne argf

f z2 z2 ∈ (−2π, 2π). := arg + arg z1 z1 f

62

4 Chain rule and side parameter

Then argf (z2 /z1 ) is the signed length of the arc in the unit circle bounded by z1 /|z1 | and z2 /|z2 | which contains f /|f |. For simplicity, we prove the lemma only when ǫ = + and j = 1. Consider the following nonzero complex numbers (see Fig. 4.6): f = Fix+ (1) − c(1),

vL = v + (0, 1) − c(1),

vR = v + (1, 2) − c(1), → wL = −− c (0, 1), − → w = c (1, 2). R

By the deﬁnition of the side parameter,

f Pj vL vR wL

wR

Fig. 4.6. Figure of f , vL , vR , wL and wR

θ+ (ρ, σ; s1 ) = arg Hence argf

f vL ∈ (−π, π). = arg vR f

vL f vL = 2θ+ (ρ, σ; s1 ) ∈ (−2π, 2π). = arg + arg vR vR f

By the deﬁnition of Fix+ (1), arg Hence argf

f wL = arg ∈ (0, π). wR f

wL f wL ∈ (0, 2π). = arg + arg wR wR f

4.2 Side parameter

63

Since both arg(f /wR ) and arg(vR /wR ) = α(ρ, σ; s1 ) belong to (0, π), arg(f /wR ) − arg(vR /wR ) belongs to (−π, π), and hence θ+ (ρ, σ; s1 ) = arg

f vR f f = arg − arg = arg − α(ρ, σ; s1 ) ∈ (−π, π). vR wR wR wR

Similarly we have θ+ (ρ, σ; s1 ) = arg

wL wL vL wL = arg − arg − α(ρ, σ; s1 ) ∈ (−π, π). = arg f f vL f

These two equalities imply 2θ+ (ρ, σ; s1 ) = arg

wL f wL + arg − 2α(ρ, σ; s1 ) = argf − 2α(ρ, σ; s1 ). f wR wR

On the other hand, −wL /wR is not a nonpositive real by the assumption that L(ρ, σ) is simple. Therefore we have the well-deﬁned argument arg(−wL /wR ) ∈ (−π, π). Thus π − arg(−wL /wR ) is contained in (0, 2π). This implies wL argf = π − arg(−wL /wR ), wR because argf

wL ∈ (0, 2π) wR

and

argf

wL ≡ π − arg(−wL /wR ) (mod 2π). wR

Hence 2θ+ (ρ, σ; s1 ) = π − arg(−wL /wR ) − 2α(ρ, σ; s1 ). This is equivalent to the desired identity. As an immediate corollary to the above lemma, we have the following. Corollary 4.2.15. Under Assumption 4.2.4 (σ-Simple), the following holds. 1. L(ρ, σ) is convex to the above at c(ρ(Pj )), if and only if θ+ (ρ, σ; s[j] ) > θ− (ρ, σ; s[j] ). 2. L(ρ, σ) is convex to the below at c(ρ(Pj )), if and only if θ+ (ρ, σ; s[j] ) < θ− (ρ, σ; s[j] ). Proof. The desired result is a consequence of Deﬁnition 3.1.8 and the following identity, which follows from Lemma 4.2.14. θ− (ρ, σ; s[j] ) − θ+ (ρ, σ; s[j] ) = arg

→ ψ(− e [j−1] ) . → ψ(− e [j+1] )

We now obtain the following important property concerning the side parameters.

64

4 Chain rule and side parameter

Proposition 4.2.16. Under Assumption 4.2.4 (σ-Simple), the following hold. θǫ (ρ, σ; s0 ) + θǫ (ρ, σ; s1 ) + θǫ (ρ, σ; s2 ) =

π . 2

Proof. By Lemmas 4.2.12 and 4.2.14 2 → ψ(− e [j−1] ) 1 − 2α(ρ, σ; s[j] ) θ (ρ, σ; sj ) = π − ǫ arg − 2 j=0 ψ(→ e [j+1] ) j=0 ⎛ ⎞ 2 2 → ψ(− e [j−1] ) 1⎝ −2 = 3π − ǫ α(ρ, σ; s[j] )⎠ arg − → 2 ψ( e ) [j+1] j=0 j=0

2

ǫ

=

1 π (3π − 0 − 2π) = . 2 2

Deﬁnition 4.2.17. When all components of θǫ (ρ, σ) are non-negative, it is regarded as π/2 times the barycentric coordinate of a point in the triangle σ of the Farey triangulation D. We denote the point by the same symbol θǫ (ρ, σ). When at most one component of θǫ (ρ, σ) is zero and the other components are positive, θǫ (ρ, σ) is also identiﬁed with a point in H2 , via its identiﬁcation with H2 ∼ = |D| − |D(0) | described in Sect. 1.3. The following lemma gives an algebraic characterization of the side parameter, which is used in Chap. 9. Lemma 4.2.18. Under Assumption 4.2.4 (σ-Simple), the following identity holds for every integer j. ǫ

ǫ

φ(s[j−1] ) + ǫieǫiθ[j+1] φ(s[j] ) − ǫie−ǫiθ[j] φ(s[j+1] ) = 0, where θjǫ = θǫ (ρ, σ; sj ) and φ is the upward Markoﬀ map inducing ρ. Moreover, each identity is equivalent to the identity φ(s0 ) + αǫ φ(s1 ) + β ǫ φ(s2 ) = 0, where αǫ = ǫi exp(ǫiθ2ǫ ) and β ǫ = −ǫi exp(−ǫiθ1ǫ ). Proof. Since v ǫ (j, j + 1) and Fixǫ (j) are contained in I(j), we have |v ǫ (j, j + 1) − c(j)| = | Fixǫ (j) − c(j)|. Hence, by the deﬁnition of the side parameter, we have ǫ

v ǫ (j, j + 1) − c(j) = e−ǫiθ[j] (Fixǫ (j) − c(j)) ǫ ǫi −ǫiθ[j] =e φ(s[j] ) Thus we have

4.2 Side parameter

v ǫ (j, j + 1) = c(j) +

65

ǫ ǫi e−ǫiθ[j] . φ(s[j] )

Similarly, we also have v ǫ (j, j + 1) = c(j + 1) + Hence, c(j) +

ǫi φ(s[j+1] )

ǫ

eǫiθ[j+1] .

ǫ ǫ ǫi ǫi e−ǫiθ[j] = c(j + 1) + eǫiθ[j+1] . φ(s[j] ) φ(s[j+1] )

So, by using Lemma 2.4.4(2.1), we have ǫ ǫ φ(s[j−1] ) ǫi ǫi = c(j + 1) − c(j) = e−ǫiθ[j] − eǫiθ[j+1] . φ(s[j] )φ(s[j+1] ) φ(s[j] ) φ(s[j+1] )

This is equivalent to the following identity: ǫ

ǫ

φ(s[j−1] ) = ǫie−ǫiθ[j] φ(s[j+1] ) − ǫieǫiθ[j+1] φ(s[j] ). Thus we obtain the ﬁrst identity. The second identity is obtained by putting j = 1, and we can easily check that the identity for any integer j is equivalent to this identity. The above lemma motivates the following deﬁnition. Deﬁnition 4.2.19. (1) Let σ = s0 , s1 , s2 be a triangle of D and ν = ǫ : Φ → C denotes the map deﬁned (θ0 , θ1 , θ2 ) be a point in σ ∩ H2 . Then ζν,σ by: ǫ ζν,σ (φ) = φ(s0 ) + αǫ φ(s1 ) + β ǫ φ(s2 ), where αǫ = ǫi exp (ǫiθ2 ) and β ǫ = −ǫi exp (−ǫiθ1 ). (2) Let σ ǫ be a triangle of D and ν ǫ be a point in σ ǫ ∩ H2 for each ǫ ∈ {−, +}. Then a Markoﬀ map φ is called an algebraic root for ((ν − , σ − ), (ν + , ǫ σ + )) if ζν,σ ǫ (φ) = 0 for each ǫ ∈ {−, +}. (3) For a given ν = (ν − , ν + ) ∈ H2 × H2 , a Markoﬀ map φ is called an algebraic root for ν if it is an algebraic root for ((ν − , σ − (ν)), (ν + , σ + (ν))), where σ ǫ (ν) is the ǫ-terminal triangle of Σ(ν) (Deﬁnition 3.3.3). Remark 4.2.20. Suppose a Markoﬀ map φ realizes ν = (ν − , ν + ) ∈ H2 × H2 , namely, φ satisﬁes Assumption 4.2.4 (σ-Simple) for the two triangles σ − (ν) and σ + (ν) and θǫ (ρ, σ ǫ (ν)) is identiﬁed with ν ǫ for each ǫ ∈ {−, +}. Then φ is an algebraic root for ν. However, the converse does not hold, that is, not every algebraic root for ν does not necessarily realize ν, because it does not necessarily satisfy Assumption 4.2.4 (σ-Simple) for the two triangles σ − (ν) and σ + (ν).

66

4 Chain rule and side parameter

4.3 ǫ-terminal triangles Lemma 4.3.1. Under Assumption 4.2.4 (σ-Simple), the following conditions are equivalent for each j ∈ Z and ǫ ∈ {−, +} (see Figs. 4.7 and 4.8):

1. θǫ (ρ, σ; s[j] ) > 0 (resp. θǫ (ρ, σ; s[j] ) = 0). 2. |φ(s[j+1] )+ǫiφ(s[j−1] )| > |φ(s[j] )| (resp. |φ(s[j+1] )+ǫiφ(s[j−1] )| = |φ(s[j] )|). 3. Fixǫσ (ρ(Pj )) ∈ / D(ρ(Pj−1 )) (resp. Fixǫσ (ρ(Pj )) ∈ I(ρ(Pj−1 ))). ǫ / D(ρ(Pj+1 )) (resp. Fixǫσ (ρ(Pj )) ∈ I(ρ(Pj+1 ))). 4. Fixσ (ρ(Pj )) ∈ ǫ / D(ρ(Pj−1 )) (resp. v ǫ (ρ; Pj−1 , Pj ) = Fixǫσ (ρ(Pj ))). 5. v (ρ; Pj , Pj+1 ) ∈ ǫ / D(ρ(Pj+1 )) (resp. v ǫ (ρ; Pj , Pj+1 ) = Fixǫσ (ρ(Pj ))). 6. v (ρ; Pj−1 , Pj ) ∈ Fix+ σ (ρ(Pj ))

θ+ (ρ, σ; s[j] )

Pj +

v (ρ; Pj−1 , Pj )

v + (ρ; Pj , Pj+1 ) Pj+1

Pj−1

Fig. 4.7.

Pj+1 Pj−1

v ǫ (ρ; Pj−1 , Pj ) = Fixǫσ (ρ(Pj )) = v ǫ (ρ; Pj , Pj+1 ) Pj Fig. 4.8.

Proof. For simplicity, we prove the assertion for the case ǫ = + and j = 1. We ﬁrst show that Condition 1 is equivalent to Conditions 4 and 6. Set v ± = −−−→ −−−→ v ± (1, 2), IL = I(1) ∩ HL (v − v + ) and IR = I(1) ∩ HR (v − v + ) (cf. Deﬁnition 3.1.6).

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67

Claim 4.3.2. IR = I(1) ∩ D(2).

Proof. Since I(1) ∩ I(2) = {v − , v + }, I(1) ∩ D(2) is equal to either IL or IR . To show that it is actually equal to IR , let xL and xR , respectively, be the → c (1, 2). By using intersection points of IL and IR with the line containing − the fact that both c(2) and xR lie on the right hand side of the oriented line −−−−−−−−−−−→ − → f (1) = Fix− (1) Fix+ (1), we see d(c(2), xR ) < d(c(2), xL ). Since precisely one of xL and xR is contained in D(2), we see xR ∈ int D(2). Hence we obtain the desired result. − → We apply a rotational coordinate change of C so that arg( f (1)) = π2 . Then we have the following claim. Claim 4.3.3. ℑ(v − ) < ℑ(v + ).

Proof. By Deﬁnitions 3.1.3 (Upward) and 4.2.5(1), we have − → f (1) < π. 0 < arg − → c (1, 2) − → → c (1, 2)) ∈ (− π2 , π2 ). Since arg( f (1)) = π2 in the new coordinate, we see arg(− −− − → → On the other hand, arg(v − v + ) = arg(− c (1, 2)) + π2 by Deﬁnition 4.2.5(2). −− − → Hence arg(v − v + ) ∈ (0, π) and therefore ℑ(v − ) < ℑ(v + ). Let p : R → I(1) be the covering projection deﬁned by p(t) = c(1) + reit − → where r is the radius of I(1). By the normalization arg( f (1)) = π2 , we have Fix+ (1) = p( π2 ). Set θ+ (1) = θ+ (ρ, σ; s1 ) and v˜+ = π2 − θ+ (1). Then v + = p(˜ v + ). Let v˜− be the point of R such that v˜− < v˜+ and (˜ v − , v˜+ )∩p−1 (v − ) = ∅. − + Then IR = p([˜ v , v˜ ]). Moreover, by Claim 4.3.3, we see the following. 3π + 1. If θ+ (1) > 0, then − 3π ˜− < v˜+ = π2 − θ+ (1) < π2 . 2 < − 2 + θ (1) < v 3π π + + − 2. If θ (1) < 0, then − 2 − θ (1) < v˜ < 2 + θ+ (1) < π2 < π2 − θ+ (1) = v˜+ .

+ In fact, Claim 4.3.3 implies that − 3π ˜− < π2 − θ+ (1) or − 3π 2 + θ (1) < v 2 − π θ+ (1) < v˜− < 2 + θ+ (1) according as θ+ (1) > 0 or θ+ (1) < 0. By using these observations and Claim 4.3.2, we obtain the following claim.

Claim 4.3.4. 1. If θ+ (1) > 0, then neither Fix+ (1) nor v + (0, 1) belong to IR = I(1) ∩ D(2). 2. If θ+ (1) < 0, then both Fix+ (1) and v + (0, 1) belong to the interior of IR = I(1) ∩ D(2). 3. If θ+ (1) = 0, then Fix+ (1) = v + (0, 1) ∈ I(1) ∩ I(2).

Proof. (1) Suppose θ+ (1) > 0. Then π2 +2kπ and π2 +θ+ (1)+2kπ do not belong to [˜ v − , v˜+ ] for every k ∈ Z by the preceding observation. Since Fix+ (1) = p( π2 ) and v + (0, 1) = p( π2 + θ+ (1)), we obtain the desired result. (2) Suppose θ+ (1) < 0. Then both π2 and π2 + θ+ (1) are contained in (˜ v − , v˜+ ) by the preceding observation. So we obtain the conclusion. (3) follows from the deﬁnition of θ+ (1) = θ+ (ρ, σ; s1 ).

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The above claim implies that the Condition 1 is equivalent to the Conditions 4 and 6. By a parallel argument, we can also see that Condition 1 is equivalent to the Conditions 3 and 5. Finally, we show that Condition 2 is equivalent to Condition 3. Recall that the radius of I(0) is 1/|φ(s0 )|, c(1) − c(0) = φ(s1 )/(φ(s0 )φ(s1 )), and that / D(0) if and Fix+ (1) = c(1) + i/φ(s1 ) (see Lemma 2.4.4). Hence, Fix+ (1) ∈ only if φ(s1 ) i 1 φ(s0 )φ(s1 ) + φ(s1 ) > φ(s0 ) . This is equivalent to the inequality in Condition 2. Similarly, we can see that the condition Fix+ (1) ∈ I(0) is equivalent to the equality in the parenthesis in Condition 2. This completes the proof of Lemma 4.3.1

Lemma 4.3.5. Under Assumption 4.2.4 (σ-Simple), I(ρ(Pj )) ⊂ D(ρ(Pj−1 ))∪ D(ρ(Pj+1 )) if and only if θǫ (ρ, σ; s[j] ) ≤ 0 for both ǫ = − and +. Moreover, the following hold. (1) If θǫ (ρ, σ; s[j] ) ≤ 0 and θ−ǫ (ρ, σ; s[j] ) < 0, then Ih(ρ(Pj )) ∩ (Eh(ρ(Pj−1 )) ∩ Eh(ρ(Pj+1 ))) = ∅. (2) If θǫ (ρ, σ; s[j] ) = 0 and θ−ǫ (ρ, σ; s[j] ) < 0, then I(ρ(Pj )) ∩ (E(ρ(Pj−1 )) ∩ E(ρ(Pj+1 ))) = {v ǫ (ρ; Pj−1 , Pj )} = {v ǫ (ρ; Pj , Pj+1 )} = {Fixǫσ (ρ(Pj ))}, (3) If θ− (ρ, σ; s[j] ) = θ+ (ρ, σ; s[j] ) = 0, then Ih(ρ(Pj )) ∩ (Eh(ρ(Pj−1 )) ∩ Eh(ρ(Pj+1 ))) = Axis(ρ(Pj )). Proof. We may assume j = 1 without loss of generality. Suppose I(1) ⊂ D(0) ∪ D(2). Then Fix± (1) ⊂ D(0) ∪ D(2). Hence θ± (ρ, σ; s1 ) ≤ 0 by Lemma 4.3.1. To prove the converse, let ℓ be the oriented line in C containing proj(Axis(1)) oriented so that ℓ is upward with respect to L(ρ, σ) (cf. Deﬁnition 3.1.3). Let DL (1) (resp. DR (1)) be the closure of the component of D(1) − ℓ which lies on the left (resp. right) of ℓ. Now suppose θ± (ρ, σ; s1 ) ≤ 0. Then, by Lemma 4.3.1, both D(0) and D(2) contain Fix± (1) and hence contain the diameter, proj(Axis(1)), of D(1). This implies DL (1) ⊂ D(0) and DR (1) ⊂ D(2), because the center c(0) of D(0) lies on the left of ℓ and the center c(2) of D(2) lies on the right of ℓ. Hence I(1) ⊂ D(0) ∪ D(2). This completes the main assertion of the lemma. The remaining assertions are consequences of the following fact, which follows from the above argument and Lemma 4.3.1: If θǫ (ρ, σ; s1 ) ≤ 0 for both ǫ = − and +, then

4.3 ǫ-terminal triangles

69

I(1) ∩ ∂ (D(0) ∩ D(2)) ⊂ {Fix− (1), Fix+ (1)}. Moreover, Fixǫ (1) is contained in the set on the left hand side if and only if θǫ (ρ, σ; s1 ) = 0. Lemma 4.3.6. Under Assumption 4.2.4 (σ-Simple), Ih(ρ(Pj−1 ))∩Ih(ρ(Pj ))∩ Ih(ρ(Pj+1 )) is a singleton if and only if θǫ (ρ, σ; s[j] ) > 0 and θ−ǫ (ρ, σ; s[j] ) < 0 for some ǫ ∈ {−, +}. Moreover, in this case, the following hold. 1. Ih(ρ(Pj−1 )) ∩ Ih(ρ(Pj )) ∩ Ih(ρ(Pj+1 )) ∈ Axis(ρ(Pj )). / D(ρ(Pj−1 )) ∩ D(ρ(Pj+1 )). 2. Fixǫσ (ρ(Pj )) ∈ 3. Fix−ǫ σ (ρ(Pj )) ∈ int(D(ρ(Pj−1 )) ∪ D(ρ(Pj+1 ))). Proof. We may assume j = 1 without loss of generality. Since ρ satisﬁes the triangle inequality at σ by the assumption, we have Ih(0) ∩ Ih(1) = ∅ and Ih(1) ∩ Ih(2) = ∅ by Lemma 4.2.1. Since ρ(P1 ) interchanges Ih(0) ∩ Ih(1) and Ih(1) ∩ Ih(2) (see Lemma 4.1.3(2) (Chain rule)), Ih(0) ∩ Ih(1) ∩ Ih(2) is a singleton if and only if Axis(1) intersects Ih(0) ∩ Ih(1) transversely and nonorthogonally. On the other hand, we can see that if Axis(1) intersects Ih(0) ∩ Ih(1) orthogonally, then L(ρ, σ) is not simple, which contradicts Assumption 4.2.4 (σ-Simple) (cf. Lemma 4.8.6). Hence Ih(0)∩Ih(1)∩Ih(2) is a singleton if and only if Axis(1) intersects Ih(0) ∩ Ih(1) transversely. The latter condition is satisﬁed if and only if one of Fix± (1) is contained in int D(0) and the other is contained in int E(0). By Lemma 4.3.1, this is equivalent to the condition that one of θ± (ρ, σ; s1 ) is positive and the other is negative. Thus we have obtained the main assertion and the assertion 1. The remaining assertions follow from Lemma 4.3.1. Notation 4.3.7. Under Assumption 4.2.4 (σ-Simple), suppose θǫ (ρ, σ; s[j] ) ≥ 0. Then eǫ (ρ, σ; Pj ) , eǫ (j) for short, denotes the component of I(ρ(Pj )) − int(D(ρ(Pj−1 )) ∪ D(ρ(Pj+1 ))) = I(ρ(Pj )) ∩ (E(ρ(Pj−1 ))∩ E(ρ(Pj+1 ))) containing Fixǫσ (ρ(Pj )). Note that eǫ (ρ, σ; Pj ) is a (possibly degenerate) circular arc of angle 2θǫ (ρ, σ; s[j] ). Thus: 1. If θǫ (ρ, σ; s[j] ) > 0, then eǫ (ρ, σ; Pj ) is a non-degenerate circular arc, and ∂eǫ (ρ, σ; Pj ) = {v ǫ (ρ; Pj−1 , Pj ), v ǫ (ρ; Pj , Pj+1 )}. 2. If θǫ (ρ, σ; s[j] ) = 0, then eǫ (ρ, σ; Pj ) is a singleton consisting of the point: v ǫ (ρ; Pj−1 , Pj ) = v ǫ (ρ; Pj , Pj+1 ) = Fixǫσ (ρ(Pj )). Deﬁnition 4.3.8 (ǫ-Terminal triangle). Under Assumption 4.2.4 (σ-Simple), σ is called an ǫ-terminal triangle of ρ if the following conditions are satisﬁed:

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1. All components of θǫ (ρ, σ) are non-negative, and at most one component of θǫ (ρ, σ) can be 0. In particular, eǫ (ρ, σ; Pj ) and v ǫ (ρ; Pj , Pj+1 ) are deﬁned for every j ∈ Z. 2. For each j ∈ {0, 1, 2}, if θǫ (ρ, σ; sj ) = 0, then θ−ǫ (ρ, σ; sj ) ≥ 0. 3. Let E ǫ (ρ, σ) be the component of E(ρ, σ) containing the region {z ∈ C | ǫℑ(z) ≥ L} for suﬃciently large positive number L. Then fr E ǫ (ρ, σ) = ∪j eǫ (ρ, σ; Pj ) and it is homeomorphic to R. 4. For each j ∈ Z, some neighborhood of v ǫ (ρ; Pj , Pj+1 ) in Ih(ρ(Pj )) ∩ Ih(ρ(Pj+1 )) is contained in Eh(ρ, σ). When σ is an ǫ-terminal triangle, we often identify θǫ (ρ, σ) with a point of H2 as in Deﬁnition 4.2.17. Remark 4.3.9. We describe relations among conditions in Deﬁnition 4.3.8. (1) By Lemma 4.3.5(2), the second condition is a consequence of the last condition. But we include this condition because it is useful in the proof of the key Lemma 4.6.2 and because it is useful in determining whether a given representation is quasifuchsian (see Proposition 6.7.1). (2) The ﬁrst two conditions do not imply the third condition. Put (a0 , a1 , a2 ) = (−0.02+0.06i, 0.26−0.41i, 0.76+0.35i), and let φ be the upward Markoﬀ map having (a0 , a1 , a2 ) as the complex probability at σ, i.e., φ is the Markoﬀ √ √ √ map determined by (φ(s0 ), φ(s1 ), φ(s2 )) = (1/ a1 a2 , 1/ a2 a0 , 1/ a0 a1 ). Then L(ρ, σ) is simple and all components of θ+ (ρ, σ) are positive. However, the point v + (ρ; P0 , P1 ) is contained in the interior of D(ρ(P3 )). One can check this by putting z1 = a2 = 0.76 + 0.35i and z2 = a2 + a0 = 0.74 + 0.41i in OPTi [78] (see Fig. 4.9). (3) The ﬁrst three conditions do not necessarily imply the last condition. In fact, by perturbing the complex probability in (1), we can ﬁnd an example which satisﬁes the ﬁrst three conditions, such that v + (ρ; P0 , P1 ) is contained in I(ρ(P3 )). Such an example is obtained by putting z1 = 0.74073902362583166+ 0.33679617661012978i and z2 = 0.74 + 0.41i in OPTi. In this example, we see that Ih(ρ(P0 )) ∩ Ih(ρ(P1 )) is contained in int(Dh(ρ(P0 )) ∪ Dh(ρ(P3 ))). So, the last condition is not satisﬁed (see Fig. 4.10). (4) In the third condition, though it is natural to expect that if fr E ǫ (ρ, σ) = ∪j eǫ (ρ, σ; Pj ) then it is homeomorphic to R. But we have not been able to exclude the possibility that ∪j eǫ (ρ, σ; Pj ) has a self-tangency.

4.4 Basic properties of ǫ-terminal triangles Lemma 4.4.1. Suppose σ is an ǫ-terminal triangle of ρ. Then the following holds for every integer j and k. 1. D(ρ(Pj )) ∩ fr E ǫ (ρ, σ) = I(ρ(Pj )) ∩ fr E ǫ (ρ, σ). 2. eǫ (ρ, σ; Pj ) ∩ D(ρ(Pk )) = eǫ (ρ, σ; Pj ) ∩ I(ρ(Pk )). 3. v ǫ (ρ; Pj , Pj+1 ) ∩ D(ρ(Pk )) = v ǫ (ρ; Pj , Pj+1 ) ∩ I(ρ(Pk )).

4.4 Basic properties of ǫ-terminal triangles

71

+

I(ρ(P0 )) v (ρ; P0 , P1 ) I(ρ(P2 )) I(ρ(P3 ))

I(ρ(P1 )) Fig. 4.9. Though all components of θǫ (ρ, σ) are positive, part of int eǫ (ρ, σ; P1 ) is covered by D(ρ(P3 )). I(ρ(P0 ))

v + (ρ; P0 , P1 )

I(ρ(P2 ))

I(ρ(P3 ))

I(ρ(P1 )) ǫ

Fig. 4.10. Though fr E (ρ, σ) = ∪j eǫ (ρ, σ; Pj ), Ih(ρ(P0 )) ∩ Ih(ρ(P1 )) is covered by int Dh(ρ(P3 )), and hence the last condition is not satisﬁed.

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4 Chain rule and side parameter

Proof. (1) Since int D(j) is contained in C − E(j) ⊂ C − E(ρ, σ) ⊂ C − E ǫ (ρ, σ), it is disjoint from E ǫ (ρ, σ). This implies that int D(j) ∩ fr E ǫ (ρ, σ) = ∅, because E ǫ (ρ, σ) is closed and therefore fr E ǫ (ρ, σ) ⊂ E ǫ (ρ, σ). Hence D(j)∩ fr E ǫ (ρ, σ) = I(j) ∩ fr E ǫ (ρ, σ). (2) By using the ﬁrst assertion and the fact that eǫ (j) ∩ fr E ǫ (ρ, σ), we see eǫ (j)∩D(k) = eǫ (j)∩fr E ǫ (ρ, σ)∩D(k) = eǫ (j)∩fr E ǫ (ρ, σ)∩I(k) = eǫ (j)∩I(k). (3) Parallel to the proof of (2). Lemma 4.4.2. Suppose σ is an ǫ-terminal triangle of ρ. Then the following holds for every j ∈ Z. 1. D(ρ(Pj ))∩fr E ǫ (ρ, σ) is contained in I(ρ(Pj ))∩Aj , where Aj is the vertical strip Aj := {z ∈ C | |ℜ(z − c(ρ(Pj )))| ≤ 1/2}.

2. Suppose that I(ρ(Pj )) ∩ I(ρ(Pj+3 )) = ∅. Then I(ρ(Pj )) − (D(ρ(Pj−3 )) ∪ D(ρ(Pj+3 ))) = I(ρ(Pj )) ∩ int Aj consists of two open arcs. Moreover, I(ρ(Pj ))∩fr E ǫ (ρ, σ) lies in the closure, aǫj , of the component of I(ρ(Pj ))− (D(ρ(Pj−3 )) ∪ D(ρ(Pj+3 ))) which lies on the ǫ-side of the horizontal line ℑ(z) = ℑ(c(ρ(Pj ))). Proof. We may assume j = 0 without loss of generality. (1) Recall that I(±3) is the image of I(0) by the parallel translation ρ(K)±1 : z → z ± 1. Hence D(0) ∩ (E(−3) ∩ E(3)) ⊂ D(0) ∩ A0 . Since D(0) ∩ fr E ǫ (ρ, σ) is contained in D(0) ∩ (E(−3) ∩ E(3)), it is contained in D(0) ∩ A0 , Hence, by Lemma 4.4.1(1), we have D(0)∩fr E ǫ (ρ, σ) ⊂ D(0)∩fr E ǫ (ρ, σ)∩A0 ⊂ I(0)∩fr E ǫ (ρ, σ)∩A0 ⊂ I(0)∩A0 . This implies the assertion 1, because int D(0) is disjoint from E ǫ (ρ, σ). (2) Assume that I(0) ∩ I(3) = ∅. Then ∪m D(3m) is connected and ˜ ǫ be the closure of its component C − ∪m D(3m) has two components. Let E which lies on the ǫ-side of the horizontal line ℑ(z) = ℑc(0). Then E ǫ (ρ, σ) is ˜ ǫ . Thus contained in E ˜ ǫ ⊂ aǫ0 . I(0) ∩ fr E ǫ (ρ, σ) ⊂ I(0) ∩ E ˜ ǫ and E ˜ −ǫ , where Lemma 4.4.3. C−fr E ǫ (ρ, σ) consists of two components, E ǫ ǫ −ǫ ˜ = int E (ρ, σ) and E ˜ ⊃ ∪j int D(ρ(Pj )). E Proof. Since fr E ǫ (ρ, σ) is homeomorphic to R and is invariant under the parallel translation ρ(K), the complementary region C − fr E ǫ (ρ, σ) consists of two components. Since int E ǫ (ρ, σ) ∩ fr E ǫ (ρ, σ) = ∅, int E ǫ (ρ, σ) is con˜ ǫ . Since tained in a component of C − fr E ǫ (ρ, σ), which we denote by E ˜ ǫ = fr E ǫ (ρ, σ), we have E ˜ ǫ = int E ǫ (ρ, σ). Since the set of isometric cirfr E cles {I(j) | j ∈ Z} is locally ﬁnite in C, we see fr E(ρ, σ) = fr {∪j D(j)} and

4.4 Basic properties of ǫ-terminal triangles

73

∪j int D(j) = int {∪j D(j)}. Thus the subset fr E ǫ (ρ, σ) of fr E(ρ, σ) is disjoint from ∪j int D(j). Moreover, the union ∪j int D(j) is connected, because ρ satisﬁes the triangle inequality at σ (cf. Lemma 4.2.1 and Deﬁnition 4.2.2). Hence ∪j int D(j) is contained in one of the components of C − fr E ǫ (ρ, σ). Since ˜ ǫ , it is contained in E ˜ −ǫ . ∪j int D(j) is disjoint from int E ǫ (ρ, σ) = E Proposition 4.4.4. Suppose σ is an ǫ-terminal triangle of ρ. Then for every j ∈ Z, we have: D(ρ(Pj )) ∩ fr E ǫ (ρ, σ) = eǫ (ρ, σ; Pj ). Proof. We may assume j = 0 without loss of generality. By the deﬁnition of eǫ (0), we have D(0) ∩ fr E ǫ (ρ, σ) ⊃ eǫ (0). Thus we show that D(0) ∩ fr E ǫ (ρ, σ) ⊂ eǫ (0). Since fr E ǫ (ρ, σ) is the union of the sets int eǫ (k) with θǫ (ρ, σ; s[k] ) > 0 and the singletons v ǫ (k, k + 1), the desired inclusion is reduced to the following Claims 4.4.5–4.4.9. Claim 4.4.5. Let k be an integer such that θǫ (ρ, σ; s[k] ) > 0. Then int eǫ (0), if k = 0, ǫ D(0) ∩ int e (k) = ∅, otherwise. Claim 4.4.6. 1. If θǫ (ρ, σ; s1 ) > 0, then D(0)∩v ǫ (1, 2) = ∅. If θǫ (ρ, σ; s1 ) = 0, then v ǫ (1, 2) ⊂ eǫ (0). 2. If θǫ (ρ, σ; s2 ) > 0, then D(0) ∩ v ǫ (−2, −1) = ∅. If θǫ (ρ, σ; s2 ) = 0, then v ǫ (−2, −1) ⊂ eǫ (0). Claim 4.4.7. For any integer l with l ∈ {−1, 0}, the following holds. D(0) ∩ v ǫ (3l + 1, 3l + 2) = ∅. Claim 4.4.8. For any integer l with l = 0, the following holds. D(0) ∩ v ǫ (3l − 1, 3l) = ∅. Claim 4.4.9. For any integer l with l = 0, the following holds. D(0) ∩ v ǫ (3l, 3l + 1) = ∅. In what follows, we prove the claims in the following order: Claim 4.4.5, Claim 4.4.6, Claim 4.4.8 for the case when l = 1, Claim 4.4.9 for the case when l = −1, Claims 4.4.8 and 4.4.9 for the general case, and Claim 4.4.7. Proof (Proof of Claim 4.4.5). If k = 0, then the claim is obvious. So we may assume k = 0. By Lemma 4.4.1(2), D(0) ∩ int eǫ (k) = I(0) ∩ int eǫ (k). Suppose to the contrary that this set is non-empty, and let w be the intersection point. Then, by the above identity, I(0) and I(k) (⊃ eǫ (k)) has the common tangent line at w. Since L(ρ, σ) is simple by assumption and hence I(0) = I(k), we have the following three possibilities:

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4 Chain rule and side parameter

1. I(0) and I(k) lie on the same side of ℓ and int D(0) ⊃ I(k) − {w}. 2. I(0) and I(k) lie on the same side of ℓ and int D(k) ⊃ I(0) − {w}. 3. I(0) and I(k) lie on the diﬀerent sides of ℓ. The possibility 1 cannot happen, because it implies that eǫ (k) = eǫ (k)∩E(0) = {w}, which contradicts the assumption that θǫ (ρ, σ; s[k] ) > 0. Suppose that the possibility 2 happens. Then eǫ (0) = eǫ (0) ∩ E(k) = {w}. Thus eǫ (0) intersects the open arc int eǫ (k). This contradicts the assumption that fr E ǫ (ρ, σ) = ∪j eǫ (j) is homeomorphic to R (cf. Deﬁnition 4.3.8(3)). Finally, suppose that the possibility 3 happens. Since the set of isometric circles {I(j) | j ∈ Z} is locally ﬁnite in C and since fr E ǫ (ρ, σ) = ∪j eǫ (j) ∼ = R, there is a disk, B, centered at the point of tangency such that B ∩ fr E ǫ (ρ, σ) is a properly embedded arc in B. Then one of the two components of B − fr E ǫ (ρ, σ) has a nontrivial intersection with int D(0) and the other has a nontrivial intersection with int D(k). This contradicts the fact that one of the two sides of fr E ǫ (ρ, σ) is equal to E ǫ (ρ, σ). Proof (Proof of Claim 4.4.6). We prove only the ﬁrst assertion, because the second one is proved similarly. If θǫ (ρ, σ; s1 ) = 0, then the assertion is obvious. So assume that θǫ (ρ, σ; s1 ) > 0. Suppose to the contrary that D(0)∩v ǫ (1, 2) = ∅. Then v ǫ (1, 2) is contained in I(0) by Lemma 4.4.1(2). Thus I(0) contains the two (distinct) boundary points v ǫ (0, 1) and v ǫ (1, 2) of the non-degenerate circular arc eǫ (1). Since eǫ (1) is contained in I(1) and since I(0) = I(1), this implies that I(0) ∩ I(1) consists of the two points v ǫ (0, 1) and v ǫ (1, 2). Since ρ(P1 ) maps Ih(1) to itself and interchanges v ǫ (0, 1) and v ǫ (1, 2) (see Lemma 4.2.7), this in tern implies that ρ(P1 ) induces an involution of the complete geodesic Ih(0) ∩ Ih(1). Therefore L(ρ, σ) is folded at c(1) by Lemma 4.8.6, a contradiction. Proof (Proof of Claim 4.4.8 when l = 1). Suppose to the contrary that D(0)∩ v ǫ (2, 3) = ∅. If θǫ (ρ, σ; s2 ) = 0, then v ǫ (2, 3) = v ǫ (1, 2) and hence we have D(0) ∩ v ǫ (1, 2) = ∅. This contradicts Claim 4.4.6(1), because the condition θǫ (ρ, σ; s2 ) = 0 implies θǫ (ρ, σ; s1 ) > 0 by Deﬁnition 4.3.8(1). So we may assume θǫ (ρ, σ; s2 ) > 0. By Lemma 4.4.1(3), v ǫ (2, 3) is contained in I(0). Since v ǫ (2, 3) ∈ I(3) by deﬁnition, it follows that v ǫ (2, 3) ∈ I(0) ∩ Ih(3). This in tern implies v ǫ (−1, 0) = ρ(K)−1 (v ǫ (2, 3)) ⊂ ρ(K)−1 (I(0) ∩ I(3)) = I(−3) ∩ I(0). Hence, by Lemma 4.4.2(2), v ǫ (−1, 0) and v ǫ (2, 3), respectively, are the left and right boundary points of the closure of the component of I(0) − D(−3) ∪ D(3) which lies on the ǫ-side of the horizontal line ℑ(z) = ℑ(c(0)). Let ℓ be the line tangent to I(ρ(P2 )) at v ǫ (2, 3). Since θǫ (ρ, σ; s2 ) > 0, eǫ (2) is a non-degenerate arc of I(2), which contains v ǫ (2, 3) as an endpoint. Since

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75

eǫ (2) ⊂ fr E ǫ (ρ, σ) ⊂ E(0) ∩ E(3), this implies that a germ of ℓ at v ǫ (2, 3) intersects E(0) ∩ E(3). In particular, ℓ is not horizontal, and the left/right hand sides of ℓ are well-deﬁned. The circle I(2) is entirely contained on the left or right hand side of ℓ. Suppose ﬁrst that I(2) is contained on the right hand side of ℓ. Consider the half spaces H ǫ = {z ∈ C | ǫℑ(z) ≥ ǫℑ(v ǫ (2, 3))}

H −ǫ = {z ∈ C | ǫℑ(z) ≤ ǫℑ(v ǫ (2, 3))}. Then the horizontal line ∂H ǫ = ∂H −ǫ is contained in ∪m D(3m) ⊂ ∪j D(j), and hence int H −ǫ ∩ E ǫ (ρ, σ) = ∅. Since θǫ (ρ, σ; s2 ) = ǫ arg

Fixǫ (2) − c(2) ∈ (0, π/2], v ǫ (2, 3) − c(2)

a germ of eǫ (2) at v ǫ (2, 3) lies in H −ǫ and intersects with int H −ǫ (see Fig. 4.11) This contradicts the fact that eǫ (2) ⊂ fr E ǫ (ρ, σ).

ℓ

v ǫ (2, 3) I(ρ(P−1 ))

I(2)

eǫ (2)

I(−3)

I(0)

I(ρ(P3 ))

Fig. 4.11. The ﬁgure is for the case where I(2) lies in the right hand side of ℓ and ǫ = +.

Suppose ﬁnally that I(2) is contained on the left hand side of ℓ. Since ρ satisﬁes the triangle inequality at σ (cf. Deﬁnition 4.2.2), both I(2) and ℓ intersect I(3) transversely. This also implies that I(0) and I(3) intersect transversely. (Otherwise, ℓ is tangent to both I(0) and I(3) at v ǫ (2, 3), because a germ of ℓ at v ǫ (2, 3) intersects E(0)∩E(3).) Since v ǫ (2, 3) is also a transversal intersection point of I(0) and I(3), we also denote it by v ǫ (0, 3). Let v −ǫ (2, 3) (resp. v −ǫ (0, 3))) be the point of I(2) ∩ I(3) (resp. I(0) ∩ I(3)) diﬀerent from v ǫ (2, 3) = v ǫ (0, 3). If v −ǫ (2, 3) is contained in I(3) ∩ int D(0), then it follows that Ih(2) ∩ Ih(3) ⊂ int Dh(0). This contradicts Condition 4 of Deﬁnition

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4 Chain rule and side parameter

4.3.8 (ǫ-Terminal triangle). Hence v −ǫ (2, 3) must be contained in I(3) ∩ E(0). Let D2′ be the disk bounded by the circle which is tangent to ℓ at v ǫ (2, 3) and passes through the point v −ǫ (0, 3) (see Fig. 4.12). Then D2′ ⊂ D(2), because: 1. Both I(2) and ∂D2′ are tangent to ℓ at v ǫ (2, 3). 2. The point v −ǫ (0, 3) of ∂D2′ is contained in D(2).

Moreover, I(0) − D(3) ⊂ D2′ , because:

1. I(0) and ∂D2′ share the two points v ± (0, 3). 2. The tangent line ℓ to ∂D2′ at v ǫ (2, 3) intersects E(0) ∩ E(3) near v ǫ (2, 3).

Hence I(0) − D(3) ⊂ D2′ ⊂ D(2). Since I(0) = I(2), this implies I(0) − D(3) ⊂ int D(2), and hence v ǫ (0, 1) ⊂ int D(2). This contradicts the fact that v ǫ (0, 1) ⊂ fr E ǫ (ρ, σ). I(0)

ℓ v ǫ (2, 3) = v ǫ (2, 3)

I(3)

v ǫ (2, 3) I(2)

v −ǫ (0, 3) ∂D2′

Fig. 4.12. The ﬁgure is for the case where I(2) lies in the left hand side of ℓ, v ǫ (2, 3) ⊂ E(0) and ǫ = +.

Proof (Proof of Claim 4.4.9 when l = −1). Parallel to the proof of Claim 4.4.8 when l = 1. Proof (Proof of Claims 4.4.8 and 4.4.9 for the general case). Since v ǫ (3l−1, 3l) and v ǫ (3l, 3l + 1) are contained in I(3l) ∩ fr E ǫ (ρ, σ), they are contained in the vertical strip A3l by Lemma 4.4.2(2). If |l| ≥ 2, then A3l = ρ(K)l (A0 ) is disjoint from A0 , and hence v ǫ (3l − 1, 3l) and v ǫ (3l, 3l + 1) are disjoint from I(0) ∩ A0 ⊃ D(0) ∩ fr E ǫ (ρ, σ). Hence the claims hold in this case. So we may assume l = ±1. We prove the claims only for the case when l = 1, because a similar argument works for the case when l = −1. Since Claim 4.4.8 for the case when l = 1 is already proved, it suﬃces to show that D(0) ∩ v ǫ (3, 4) = ∅. Suppose this is not the case, namely, v ǫ (3, 4) is contained in D(0). Then it follows that v ǫ (3, 4) is contained in the vertical line A0 ∩ A3 , because v ǫ (3, 4) is also contained in A3 as observed in the above. Hence v ǫ (0, 1) = ρ(K)−1 (v ǫ (3, 4)) is contained in the vertical line A−3 ∩A0 . By Lemma 4.4.2(2),

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77

both v ǫ (0, 1) and Fixǫ (j) are contained in the closure, aǫj , of the component I(0) ∩ int A0 which lies on the ǫ-side of the horizontal line ℜ(z) = ℜ(c(0)). The above observation implies that v ǫ (0, 1) forms the left boundary point of the arc aǫj . Since θǫ (ρ, σ; s0 ) = ǫ arg

Fixǫ (0) − c(0) ∈ [0, π/2], v ǫ (0, 1) − c(0)

the above observation implies that Fixǫ (0) is also equal to the left boundary point of aǫj and therefore θǫ (ρ, σ; s0 ) = 0. Hence v ǫ (−1, 0) = v ǫ (0, 1) ∈ D(−3)), and so v ǫ (2, 3)) ∈ D(ρ(P0 )). This contradicts Claim 4.4.8 for the case when l = 1, which we have already proved. Proof (Proof of Claim 4.4.7). We prove the claim for l > 0. (A similar argument works for the case when l < 0.) Suppose to the contrary that D(0) ∩ v ǫ (3l + 1, 3l + 2) = ∅ for some l > 0. Set w1 = v ǫ (3l + 1, 3l + 2), w2 = v ǫ (3l + 2, 3l + 3). Then eǫ (3l + 2) is a circular arc joining the point w1 ∈ I(0) with the point w2 ∈ I(3l + 3). Set w2′ = ρ(K)−(l+1) (w2 ). Then w2′ is contained in I(0) ∩ fr E ǫ (ρ, σ), because w2 ∈ I(3l + 3) ∩ fr E ǫ (ρ, σ)

= ρ(K)l+1 (I(0)) ∩ fr E ǫ (ρ, σ)

= ρ(K)l+1 (I(0) ∩ fr E ǫ (ρ, σ)).

Thus both w1 and w2′ are contained in I(0)∩fr E ǫ (ρ, σ) and hence in I(0)∩A0 by Lemma 4.4.2(1). Thus there is a simple arc, τ0 , joining w2′ with w1 such that int τ0 ⊂ int D(ρ(P0 )) ∩ A0 . Consider the union τ := τ0 ∪ eǫ (3l + 2). Since int τ0 ⊂ int D(0) and eǫ (3l + 2) ⊂ fr E ǫ (ρ, σ), we see that τ0 and eǫ (3l + 2) intersects only at w1 . Hence τ is a simple arc in C with endpoints w2′ and w2 . Then τ projects onto a closed curve in the open annulus C/ρ(K). Since the closed curve represents l + 1 times the generator of the ﬁrst integral homology group of the open annulus and since l + 1 ≥ 2 by assumption, it cannot be simple. On the other hand, since int τ0 is contained in the vertical strip A0 , which forms a fundamental domain for the action of ρ(K) on C, the image of int τ0 in C/ρ(K) is simple. Similarly, we see, by using Lemma 4.4.2(1), that the image of int eǫ (3l + 2) in C/ρ(K) is also simple. Moreover the image of int τ0 is disjoint from the image of eǫ (3l + 2), because int τ0 ⊂ int D(0) and eǫ (3l + 2) ⊂ fr E ǫ (ρ, σ). Since the image of τ is a non-simple closed curve, the above observations imply that the image of w1 is equal to the image of w2 . ′ Hence, there is an integer l′ such that w2 = ρ(K)l (w1 ), namely, ′

v ǫ (3l + 2, 3l + 3) = ρ(K)l v ǫ (3l + 1, 3l + 2) = v ǫ (3(l + l′ ) + 1, 3(l + l′ ) + 2). By Condition 3 in Deﬁnition 4.3.8 (ǫ-Terminal triangle), this happens only when θǫ (ρ, σ; s2 ) = 0 and l′ = 0. Then v ǫ (3l + 2, 3l + 3) is equal to

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4 Chain rule and side parameter

v ǫ (3l + 1, 3l + 2), and hence is contained in D(0) by the primary assumption. Since l > 0, this contradicts Claim 4.4.8, which we have already proved. We have proved Claims 4.4.5–4.4.9. Thus the proof of Proposition 4.4.4 is complete.

4.5 Relation between side parameters at adjacent triangles In this section, we study relation between side parameters at adjacent triangles. To explain the results, we need the following deﬁnition. Deﬁnition 4.5.1. Let ρ be a type-preserving representation, and let σ1 and σ2 be mutually adjacent triangles of D. (1) L(ρ, {σ1 , σ2 }) is said to be simple if L(ρ, (σ1 , σ2 )) or L(ρ, (σ2 , σ1 )) is simple (cf. Deﬁnition 3.2.2). (2) L(ρ, {σ1 , σ2 }) is said to be ﬂat if both L(ρ, σ1 ) and L(ρ, σ2 ) are simple and |L(ρ, σ1 )| = |L(ρ, σ2 )|. The following lemma shows that the property that φ is upward at a triangle is inherited by adjacent triangles: Lemma 4.5.2. Under Notation 2.1.14 (Adjacent triangles) and Assumption 2.4.6 (σ-NonZero), suppose that φ−1 (0) ∩ (σ ∪ σ ′ )(0) = ∅ and that L(ρ, {σ, σ ′ }) is simple or ﬂat. Then φ is upward at σ if and only if φ is upward at σ ′ . Moreover, we have: Fixǫσ (ρ(P0 )) = Fixǫσ′ (ρ(P0′ )),

Fixǫσ (ρ(P2 )) = Fixǫσ′ (ρ(P1′ )).

Proof. We prove the lemma only when L(ρ, {σ, σ ′ }) is simple, because the proof for the case when L(ρ, {σ, σ ′ }) is ﬂat is essentially the same. Suppose that φ is upward at σ. We ﬁrst show that i/φ(s′1 ) = i/φ(s2 ) is upward at c(ρ(P1′ )) = c(ρ(P2 )) with respect to L(ρ, σ ′ ). To this end, let ℓ be the oriented line in C containing π(Axis(ρ(P1′ ))) = π(Axis(ρ(P2 ))), and orient ℓ so that it is upward with respect to L(ρ, σ). Note that ℓ bisects the angle ∠(c(ρ(P0′ )), c(ρ(P1′ )), c(ρ(P2′ ))) (see Proposition 2.4.4(2.1)). Moreover, by Lemma 3.1.2, ∠(c(ρ(P1 )), c(ρ(P2 )), c(ρ(P0 ))) = ∠(c(ρ(P3′ )), c(ρ(P1′ )), c(ρ(P2′ ))). On the other hand, since L(ρ, {σ, σ ′ }) is simple, the triangle ∆(c(ρ(P1 )), c(ρ(P2 )), c(ρ(P0 ))) intersects the triangle ∆(c(ρ(P3′ )), c(ρ(P1′ )), c(ρ(P2′ ))) only at the common vertex c(ρ(P2 )) = c(ρ(P1′ )). Hence c(ρ(P0′ )) = c(ρ(P0 )) must lie on the left of the oriented line ℓ, and c(ρ(P2′ )) must lie on the right of ℓ (see Fig. 4.13). Hence i/φ(s′1 ) = i/φ(s2 ) is upward at c(ρ(P1′ )) with respect to

4.5 Relation between side parameters at adjacent triangles

79

L(ρ, σ ′ ). By a parallel argument, we can also see that i/φ(s′0 ) = i/φ(s0 ) is upward at c(ρ(P0′ )) = c(ρ(P0 )) with respect to L(ρ, σ ′ ). These imply that i/φ(s′2 ) is also upward at c(ρ(P2′ )) with respect to L(ρ, σ ′ ), by the last assertion of Lemma 3.1.4. Hence φ is upward at σ ′ . Since the argument is symmetric, we obtain the ﬁrst assertion of the lemma. To see the second assertion, let φ be the Markoﬀ map inducing ρ which is upward at σ and σ ′ . Then Fixǫσ (ρ(P0 )) = c(ρ(P0 )) +

ǫi ǫi = Fixǫσ′ (ρ(P0′ )). = c(ρ(P0′ )) + φ(s0 ) φ(s′0 )

The remaining identity is proved similarly.

l

c1 c0 c′0

c3 c′3

c2 c′1 c′2

Fig. 4.13. Figure for Lemma 4.5.2

Lemma 4.5.3. Under Notation 2.1.14 (Adjacent triangles) and Assumption 4.2.4 (σ-Simple), suppose φ−1 (0) ∩ (σ ∪ σ ′ )(0) = ∅, L(ρ, {σ, σ ′ }) is simple or ﬂat, and ρ satisﬁes the triangle inequality at σ and σ ′ . Then the following hold. (1) θǫ (ρ, σ ′ ; s′2 ) is positive, 0, or negative according as θǫ (ρ, σ; s1 ) is negative, 0, or positive. (2) Suppose θǫ (ρ, σ; s1 ) = 0, or equivalently θǫ (ρ, σ ′ ; s′2 ) = 0. Then ǫ θ (ρ, σ; s0 ) = θǫ (ρ, σ ′ ; s′0 ) and θǫ (ρ, σ; s2 ) = θǫ (ρ, σ ′ ; s′1 ). In particular, when these two numbers are positive, θǫ (ρ, σ) and θǫ (ρ, σ ′ ) determine the same point in int τ ⊂ H2 (cf. Deﬁnition 4.2.17). Proof. (1) By Lemmas 4.5.2, the Markoﬀ map φ is also upward at σ ′ . Hence we have the following by Lemma 4.3.1: 1. θǫ (ρ, σ; s1 ) < 0 if and only if |φ(s2 ) + ǫiφ(s0 )| < |φ(s1 )|. 2. θǫ (ρ, σ ′ ; s′2 ) > 0 if and only if |φ(s′0 ) + ǫiφ(s′1 )| > |φ(s′2 )|.

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4 Chain rule and side parameter

Put (x, y, z) = (φ(s0 ), φ(s1 ), φ(s2 )). Then (φ(s′0 ), φ(s′1 ), φ(s′2 )) = (x, z, xz −y), and hence we have |φ(s2 ) + ǫiφ(s0 )||φ(s′0 ) + ǫiφ(s′1 )| = |z + ǫix| · |x + ǫiz| = |x2 + z 2 |

= |xyz − y 2 |

= |φ(s1 )φ(s′2 )|. By this identity and the above facts, we obtain the ﬁrst assertion. (2) Since θǫ (ρ, σ; s1 ) = 0, we have v ǫ (ρ; P0 , P1 ) = v ǫ (ρ; P1 , P2 ) = Fixǫσ (ρ(P1 )). By using the facts that P0′ = P0 and P1′ = P2 , we see that this point is also equal to v ǫ (ρ; P0′ , P1′ ) ∈ I(ρ(P0′ )) ∩ I(ρ(P1′ )). Moreover, Fixǫσ (ρ(P0 )) = Fixǫσ′ (ρ(P0′ )) by Lemma 4.5.2. Hence θǫ (ρ, σ; s0 ) = θǫ (ρ, σ ′ ; s′0 ). Similarly, we have θǫ (ρ, σ; s2 ) = θǫ (ρ, σ ′ ; s′1 ). Thus we have obtained the desired result. Lemma 4.5.4. Under Assumption 4.2.4 (σ-Simple), suppose θǫ (ρ, σ) = (+, 0, +) and θ−ǫ (ρ, σ; s1 ) > 0. Then the following hold under Notation 2.1.14 (Adjacent triangles): 1. L(ρ, σ ′ ) lies on the ǫ-side of L(ρ, σ), L(ρ, {σ, σ ′ }) is simple, and ρ satisﬁes the triangle inequality at σ ′ . (In particular, θǫ (ρ, σ ′ ) is well-deﬁned.) 2. θǫ (ρ, σ ′ ; s′2 ) = 0. Moreover, θǫ (ρ, σ) and θǫ (ρ, σ ′ ) determine the same point of int τ . 3. D(ρ(P2′ )) ⊂ D(ρ(P2 ))∪D(ρ(P3 )) and D(ρ(P2′ ))∩∂(D(ρ(P2 ))∪D(ρ(P3 ))) = {v ǫ (ρ; P2 , P3 )}. 4. Dh(ρ(P2′ )) ∩ (Eh(ρ(P2 )) ∩ Eh(ρ(P3 ))) = ∅. In the above lemma, θǫ (ρ, σ) = (+, 0, +) means θǫ (ρ, σ; s0 ) > 0,

θǫ (ρ, σ; s1 ) = 0,

θǫ (ρ, σ; s2 ) > 0.

To prove this lemma, we need the following lemma, whose proof is referred to Sect. 4.7. Lemma 4.5.5. Under Assumption 4.2.4 (σ-Simple), suppose that all components of θǫ (ρ, σ) are non-negative. Then any two of {∆ǫk (ρ, σ) | k ∈ Z} (see Notation 4.2.10) intersect only at a common edge or a common vertex. Proof (Proof of Lemma 4.5.4). Recall Notation 4.2.10, and set αj = α(ρ, σ; sj ). By Lemma 4.2.11, the angles of ∆ǫ0 (ρ, σ) at the vertices c(ρ(P0 )) and c(ρ(P1 )) are equal to α0 and α1 , respectively, and the angles of ∆ǫ1 (ρ, σ) at the vertices c(ρ(P1 )) and c(ρ(P2 )) are equal to α1 and α2 , respectively (see Fig. 4.5). By the assumption and Sublemma 4.5.5, ∆ǫ0 (ρ, σ) and ∆ǫ1 (ρ, σ) intersect only in the common edge c(ρ(P1 ))v ǫ (ρ; P0 , P1 ) = c(ρ(P1 ))v ǫ (ρ; P1 , P2 ), and hence ∆ǫ0 (ρ, σ) ∪ ∆ǫ1 (ρ, σ) forms a quadrangle. Note that the (inner) angles of the quadrangle are α0 , α2 , 2α1 and α0 + α1 = π − α2 . Since θǫ (ρ, σ; s1 ) = 0 and θ−ǫ (ρ, σ; s1 ) > 0, Lemma 4.2.11(3) implies:

4.5 Relation between side parameters at adjacent triangles

81

2α1 = π − θǫ (ρ, σ; s1 ) − θ−ǫ (ρ, σ; s1 ) < π. Thus every corner of the quadrangle has angle < π, and hence the quadrangle is convex. Hence the edge c(ρ(P0 ))c(ρ(P2 )) of L(ρ, σ ′ ) is contained in ∆ǫ0 (ρ, σ)∪∆ǫ1 (ρ, σ), and therefore it lies on the ǫ-side of L(ρ, σ) (see Fig. 4.14). By Lemma 3.1.2, we have the following inequalities among angles. ∠(c(ρ(P3′ )), c(ρ(P1′ )), c(ρ(P2′ ))) = ∠(c(ρ(P0 )), c(ρ(P2 )), c(ρ(P1 ))) < α2 , ∠(c(ρ(P2′ )), c(ρ(P3′ )), c(ρ(P1′ ))) = ∠(c(ρ(P1 )), c(ρ(P0 )), c(ρ(P2 ))) < α0 . Hence the vertex c(ρ(P2′ )) lies in the interior of ∆ǫ2 (ρ, σ). Therefore the edges c(ρ(P1′ ))c(ρ(P2′ )) and c(ρ(P2′ ))c(ρ(P3′ )) of L(ρ, σ ′ ) lie in ∆ǫ2 (ρ, σ). Since ∆ǫ2 (ρ, σ) and ∆ǫ0 (ρ, σ)∪∆ǫ1 (ρ, σ) intersect only at the common vertex c(ρ(P2 )) by Sublemma 4.5.5, we see that L(ρ, σ ′ ) lies on the ǫ-side of L(ρ, σ) and that L(ρ, {σ, σ ′ }) is simple. Since θǫ (ρ, σ; s1 ) = 0, I(ρ(P0′ )) = I(ρ(P0 )) and I(ρ(P1′ )) = I(ρ(P2 )) intersect at v ǫ (ρ; P0 , P1 ) = v ǫ (ρ; P1 , P2 ). Since L(ρ, σ ′ ) is simple, this intersection point must be a transversal intersection point by Lemma 4.8.6. Hence ρ satisﬁes the triangle inequality at σ ′ by Lemma 4.2.1. This completes the proof of the ﬁrst assertion. The second assertion follows from the assumption θǫ (ρ, σ; s1 ) = 0 and Lemma 4.5.3. To see the third assertion, we note that θǫ (ρ, σ ′ ; s′2 ) = 0 and θ−ǫ (ρ, σ ′ ; s′2 ) < 0, where the latter inequality follows from the assumption θ−ǫ (ρ, σ; s1 ) > 0 and Lemma 4.5.3 (1). Hence we obtain the desired result by Lemma 4.3.5(2). The last assertion is a consequence of the third assertion.

c′3 c3

c′0 c0 c′2 c1 c′1 c2 Fig. 4.14. L(ρ, σ) and ∆ǫk (ρ, σ)

By a similar argument, we obtain the following lemma. Lemma 4.5.6. Under Assumption 4.2.4 (σ-Simple), suppose θǫ (ρ, σ) = (+, 0, +) and θ−ǫ (ρ, σ; s1 ) = 0. Then the following hold under Notation 2.1.14 (Adjacent triangles):

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4 Chain rule and side parameter

1. L(ρ, {σ, σ ′ }) is ﬂat, and ρ satisﬁes the triangle inequality at σ ′ . (In particular, θǫ (ρ, σ ′ ) is well-deﬁned.) 2. θǫ (ρ, σ ′ ; s′2 ) = 0. Moreover, θǫ (ρ, σ) and θǫ (ρ, σ ′ ) determine the same point of int τ . 3. I(ρ(P2′ )) ⊂ D(ρ(P2 )) ∪ D(ρ(P3 )) and I(ρ(P2′ )) ∩ ∂(D(ρ(P2 )) ∪ D(ρ(P3 ))) = {v − (ρ; P2 , P3 ), v + (ρ; P2 , P3 )}.

4. Dh(ρ(P2′ )) ∩ (Eh(ρ(P2 )) ∩ Eh(ρ(P3 ))) = Ih(ρ(P2′ )) ∩ (Eh(ρ(P2 )) ∩ Eh(ρ(P3 ))) is a geodesic with endpoints v ± (ρ; P2 , P3 ). By combining Proposition 4.4.4 with Lemmas 4.5.4 and 4.5.6, we obtain the following analogy of Proposition 4.4.4. Proposition 4.5.7. Under Notation 2.1.14 (Adjacent triangles) and Assumption 2.4.6 (σ-NonZero), let σ = s0 , s1 , s2 be an ǫ-terminal triangle of ρ and assume that θǫ (ρ, σ; s1 ) = 0. Then the following hold. 1. θǫ (ρ, σ ′ ) is well-deﬁned, and θǫ (ρ, σ ′ ; s′2 ) = 0. Moreover, θǫ (ρ, σ) and θǫ (ρ, σ ′ ) determine the same point of int τ . 2. fr E ǫ (ρ, σ) = fr E ǫ (ρ, σ ′ ) and ′ eǫ (ρ, σ ′ ; P3j ) = eǫ (ρ, σ; P3j ), ′ eǫ (ρ, σ ′ ; P3j+1 ) = eǫ (ρ, σ; P3j+2 ), ′ ) = v ǫ (ρ; P3j+2 , P3j+3 ). eǫ (ρ, σ ′ ; P3j+2

3. For every j ∈ Z, D(ρ(Pj′ )) ∩ fr E ǫ (ρ, σ ′ ) = eǫ (ρ, σ ′ ; Pj′ ).

4.6 Transition of terminal triangles We are now ready to prove the following key Lemmas 4.6.1, 4.6.2 and 4.6.7, which describes how the terminal triangles changes according to small deformation of type-preserving representations. These lemmas hold the key to the proof of Proposition 6.2.1 (Openness in X ). Lemma 4.6.1. Under Assumption 2.4.6 (σ-NonZero), let σ be an ǫ-terminal triangle of ρ and assume that θǫ (ρ, σ) ∈ int σ. Then there is a neighborhood U of ρ in X , such that for any element ρ′ of U , σ is an ǫ-terminal triangle of ρ′ . Proof. This is almost obvious and all arguments necessary for the proof are contained in the proof of the next Lemma 4.6.2. So we omit the proof. Lemma 4.6.2. Under Notation 2.1.14 (Adjacent triangles) and Assumption 2.4.6 (σ-NonZero), let σ = s0 , s1 , s2 be an ǫ-terminal triangle of ρ and assume that θǫ (ρ, σ; s1 ) = 0 and θ−ǫ (ρ, σ; s1 ) > 0. (In particular, θǫ (ρ, σ) ∈ int τ , where τ = s0 , s2 .) For notational convenience, we set σ ∗ := σ ′ . Then there is a neighborhood U of ρ in X such that the following conditions hold for every ρ′ ∈ U (see Fig. 4.15):

4.6 Transition of terminal triangles

83

1. If θǫ (ρ′ , σ; s1 ) ≥ 0, then σ is an ǫ-terminal triangle of ρ′ . 2. If θǫ (ρ′ , σ; s1 ) < 0, then σ ∗ is an ǫ-terminal triangle of ρ′ . Moreover, D(ρ′ (P1 )) ∩ E ǫ (ρ′ , σ ∗ ) = ∅, and both Ih(ρ′ (P0 )) ∩ Ih(ρ′ (P1 )) ∩ Ih(ρ′ (P2 )) and Ih(ρ′ (P1′ )) ∩ Ih(ρ′ (P2′ )) ∩ Ih(ρ′ (P3′ )) are singletons. P2′ P0

P3 P1

P2′ P0

P2 σ∗

τ P3

σ

P2 P1 τ

σ∗

P2′

σ

P0

P3 P1

τ

P2 σ∗ σ

Fig. 4.15. Transition of terminal triangles

Proof. By the assumption and Lemmas 4.5.4, L(ρ, {σ, σ ∗ }) is simple and ρ satisﬁes the triangle inequality at σ and σ ∗ . This implies that there is a neighborhood U1 of ρ in X such that for every ρ′ ∈ U1 , L(ρ′ , {σ, σ ∗ }) is simple, and ρ′ satisﬁes the triangle inequality at σ and σ ∗ . In particular, θǫ (ρ′ , σ) and θǫ (ρ′ , σ ∗ ) are deﬁned. Since θǫ (ρ, σ) = (+, 0, +), we have θǫ (ρ, σ ∗ ) = (+, +, 0) by Lemma 4.5.3. So we can ﬁnd a neighborhood U2 of ρ in U1 such that every ρ′ ∈ U2 satisﬁes the following conditions:

1. The components of the side parameter θǫ (ρ′ , σ), except possibly θǫ (ρ′ , σ; s1 ), are positive. 2. θ−ǫ (ρ′ , σ; s1 ) > 0. 3. The components of the side parameter θǫ (ρ′ , σ ∗ ) are positive, except possibly θǫ (ρ′ , σ ∗ ; s′2 ).

Here the second condition follows from the assumption that θ−ǫ (ρ, σ; s1 ) > 0. Thus, by using Lemma 4.5.3, we see that one of the following holds for each ρ′ ∈ U2 .

84

4 Chain rule and side parameter

1. θǫ (ρ′ , σ; s1 ) > 0 and hence all components of θǫ (ρ′ , σ) are positive. 2. θǫ (ρ′ , σ; s1 ) < 0 and hence all components of θǫ (ρ′ , σ ∗ ) are positive. 3. θǫ (ρ′ , σ; s1 ) = 0 and hence all components of θǫ (ρ′ , σ) are non-negative In the following, we show, through case-by-case argument, that every ρ′ ∈ U2 satisﬁes the conclusion of Lemma 4.6.2 provided that U2 is chosen to be small enough in advance. Case 1. θǫ (ρ′ , σ; s1 ) > 0 and hence all components of θǫ (ρ′ , σ) are positive. Then the ﬁrst two conditions for σ to be an ǫ-terminal triangle (Deﬁnition 4.3.8) are satisﬁed. In particular, eǫ (ρ′ , σ; Pj ) and v ǫ (ρ′ ; Pj , Pj+1 ) are deﬁned for every j ∈ Z. To show that the remaining conditions are satisﬁed, we need the following claim. Claim 4.6.3. By choosing U2 small enough in advance, the following hold. For each j ∈ Z, eǫ (ρ′ , σ; Pj ) is a non-degenerate circular arc, and satisﬁes the following condition (cf. the upper-right ﬁgure in Fig. 4.15). ⎧ ǫ ′ e (ρ , σ; Pj ) ⎪ ⎪ ⎪ ⎨v ǫ (ρ′ ; P , P ) j j+1 eǫ (ρ′ , σ; Pj ) ∩ D(ρ′ (Pk )) = ǫ ′ ⎪ v (ρ ; P , Pj ) j−1 ⎪ ⎪ ⎩ ∅

if k = j, if k = j + 1, if k = j − 1, otherwise.

Proof. Since all components of θǫ (ρ′ , σ) are positive by the assumption, each eǫ (ρ′ , σ; Pj ) is a non-degenerate circular arc. Thus we have the ﬁrst assertion. To see the second assertion, note that, by virtue of Proposition 4.4.4 and the assumption θǫ (ρ, σ; s1 ) = 0, D(ρ(Pk )) has nonempty intersection with eǫ (ρ, σ; Pj ) if and only if one of the following conditions holds. 1. k = j or j ± 1. 2. k = j + 2 and j ≡ 0 (mod 3). 3. k = j − 2 and j ≡ 2 (mod 3). Suppose k does not satisfy any of the above conditions. Then D(ρ(Pk )) is disjoint from eǫ (ρ, σ; Pj ), and hence we may assume, by choosing U2 sufﬁciently small in advance, that D(ρ′ (Pk )) is disjoint from eǫ (ρ′ , σ; Pj ). Since the Euclidean distance between D(ρ(Pk )) and eǫ (ρ, σ; Pj ) is large when |k − j| is large, and since we have the parallel translation ρ(K), we can choose U2 in advance, so that the same conclusion holds for every such k. Suppose k = j. Then it is obvious that eǫ (ρ′ , σ; Pj ) ∩ D(ρ′ (Pj )) = ǫ ′ e (ρ , σ; Pj ). Suppose k = j ± 1. Since I(ρ′ (Pj )) intersect I(ρ′ (Pj+1 )) transversely and since eǫ (ρ, σ; Pj ) ∩ I(ρ(Pj+1 )) = v ǫ (ρ; Pj , Pj+1 ), we see eǫ (ρ′ , σ; Pj ) ∩ I(ρ(Pj+1 )) = v ǫ (ρ′ ; Pj , Pj+1 ) by choosing U2 small enough in advance. Suppose j ≡ 0 (mod 3) and k = j + 2. Then we may assume j = 0 and k = 2. By Proposition 4.4.4, eǫ (ρ, σ; P0 ) ∩ D(ρ(P2 )) is equal to the degenerate arc eǫ (ρ, σ; P1 ) = v ǫ (ρ; P0 , P1 ) = v ǫ (ρ; P1 , P2 ). By the preceding argument

4.6 Transition of terminal triangles

85

v ǫ (ρ′ ; P0 , P1 ) ∈ D(ρ′ (P2 )). Moreover, by Lemma 4.2.1, I(ρ(P0 )) = I(ρ(P0′ )) and I(ρ(P2 )) = I(ρ(P1′ )) intersect transversely. Hence we can choose U2 small enough in advance so that eǫ (ρ′ , σ; P0 ) ∩ D(ρ′ (P2 )) = ∅ (see Fig. 4.16). In fact this is guaranteed by the following observations. 1. Since I(ρ(P0 )) and I(ρ(P2 )) intersect transversely and since v ǫ (ρ′ ; P0 , P1 ) ∈ D(ρ′ (P2 )), we may assume a small neighborhood of v ǫ (ρ′ ; P0 , P1 ) in eǫ (ρ′ , σ; P0 ) is disjoint from D(ρ′ (P2 )), by choosing U2 small enough in advance. 2. Since the complement of a small neighborhood of v ǫ (ρ;P0 ,P1 ) in eǫ (ρ, σ; P0 ) is disjoint from D(ρ(P2 )), we may assume the complement of the above small neighborhood of v ǫ (ρ′ ; P0 , P1 ) in eǫ (ρ′ , σ; P0 ) is also disjoint from D(ρ′ (P2 )), by choosing U2 small enough in advance. v ǫ (ρ; P0 , P1 ) = v ǫ (ρ; P1 , P2 )

v ǫ (ρ′ ; P0 , P1 ) v ǫ (ρ′ ; P1 , P2 ) eǫ (ρ′ , σ; P )

eǫ (ρ, σ; P0 )

0

→ D(ρ(P2 ))

D(ρ′ (P2 ))

Fig. 4.16. Deformation from θǫ (ρ, σ; P1 ) = 0 to θǫ (ρ′ , σ; P1 ) > 0

Suppose j ≡ 2 (mod 3) and k = j − 2. Then by an argument similar to the above, we see eǫ (ρ′ , σ; Pj ) is disjoint from D(ρ′ (Pk )). This completes the proof of the claim. Claim 4.6.4. Let U2 be as in Claim 4.6.3. Then fr E ǫ (ρ′ , σ) = ∪j eǫ (ρ′ , σ; Pj ) ∼ = R. Proof. By Claim 4.6.3, ∪j eǫ (ρ′ , σ; Pj ) is homeomorphic to R and is contained in fr E(ρ′ , σ). Since it is invariant under the parallel translation ρ(K), the ˜ + (resp. complement C − ∪j eǫ (ρ′ , σ; Pj ) consists of two components. Let E − ǫ ′ ˜ E ) be the component of C − ∪j e (ρ , σ; Pj ) which contains the region {z ∈ C | ℑ(z) ≥ L} (resp. {z ∈ C | − ℑ(z) ≥ L}) for suﬃciently large number L. Since int E ǫ (ρ′ , σ) is disjoint from ∪j eǫ (ρ′ , σ; Pj ) ⊂ fr E(ρ′ , σ) and contains the region {z ∈ C | ǫℑ(z) ≥ L} for suﬃciently large number L, it is contained ˜ǫ. in E On the other hand ∪j int D(ρ′ (Pj )) is disjoint from ∪j eǫ (ρ′ , σ; Pj ) ⊂ ˜ − or E ˜ + . But ∪j int D(ρ′ (Pj )) fr E(ρ′ , σ) and hence it is contained in E

86

4 Chain rule and side parameter

˜ −ǫ . Thus ∪j int D(ρ′ (Pj )) contains L(ρ′ , σ), which in tern is contained in E −ǫ ǫ ˜ ˜ is contained in E . Now every point in E is joined to the region {z ∈ ˜ ǫ . The path is disjoint from ∪j int D(ρ′ (Pj )) C | ǫℑ(z) ≥ L} by a path in E ˜ ǫ ⊂ int E ǫ (ρ′ , σ). Hence E ˜ǫ = by the above observation. Thus we have E ǫ ′ ǫ ′ ǫ ǫ ′ ˜ int E (ρ , σ), and therefore fr E (ρ , σ) = fr E = ∪j e (ρ , σ; Pj ). By Claim 4.6.4, the pair (ρ′ , σ) satisﬁes the third condition in Deﬁnition 4.3.8. By Claim 4.6.3, Dh(ρ′ (Pk )) intersects (a small neighborhood of) v ǫ (ρ′ ; Pj , Pj+1 ) if and only if k = j or j + 1. So, the last condition in Deﬁnition 4.3.8 is also satisﬁed. Hence σ is an ǫ-terminal triangle of ρ′ . Case 2. θǫ (ρ′ , σ; s1 ) < 0 and hence all components of θǫ (ρ′ , σ ∗ ) are positive. Then the ﬁrst two conditions for σ ∗ to be an ǫ-terminal triangle (Deﬁ′ ) are nition 4.3.8) are satisﬁed. In particular, eǫ (ρ′ , σ ∗ ; Pj′ ) and v ǫ (ρ′ ; Pj′ , Pj+1 deﬁned for every j ∈ Z. To show that the remaining conditions are satisﬁed, we need the following claim. Claim 4.6.5. By choosing U2 small enough in advance, the following hold. For each j ∈ Z, eǫ (ρ′ , σ; Pj′ ) is a non-degenerate circular arc, and satisﬁes the following condition (cf. the lower-right ﬁgure in Fig. 4.15). ⎧ ǫ ′ ∗ ′ e (ρ , σ ; Pj ) ⎪ ⎪ ⎪ ⎨v ǫ (ρ′ ; P ′ , P ′ ) j j+1 eǫ (ρ′ , σ ∗ ; Pj′ ) ∩ D(ρ′ (Pk′ )) = ǫ ′ ′ ′ ⎪ v (ρ ; P , ⎪ j−1 Pj ) ⎪ ⎩ ∅

if k = j, if k = j + 1, if k = j − 1, otherwise.

Proof. By Proposition 4.5.7, D(ρ(Pk′ )) has nonempty intersection with eǫ (ρ, σ; Pj ) if and only if one of the following conditions holds. 1. k = j or j ± 1. 2. k = j + 2 and j ≡ 1 (mod 3). 3. k = j − 2 and j ≡ 3 (mod 3). As in the proof of Claim 4.6.3, we may assume the conclusion holds for every k which does not satisfy any of the above conditions. For each k satisfying one of the above conditions, the intersection is controlled by the side parameter as shown in the proof of Claim 4.6.3. Thus we obtain the claim. By using Claim 4.6.5 instead of Claim 4.6.3 in the proof of Claim 4.6.4, we see that fr E ǫ (ρ′ , σ ∗ ) = ∪j eǫ (ρ′ , σ ∗ ; Pj′ ) ∼ = R whenever ρ′ ∈ U2 , provided that U2 is chosen to be small enough in advance. This implies that the pair (ρ′ , σ ∗ ) satisﬁes the third condition in Deﬁnition 4.3.8. By Claim 4.6.5 and by an argument parallel to that in Case 1, the last condition in Deﬁnition 4.3.8 is also satisﬁed. Hence σ ∗ is an ǫ-terminal triangle of ρ′ . Next, we show that D(ρ′ (P1 )) ∩ E ǫ (ρ′ , σ ∗ ) = ∅. Note that E ǫ (ρ′ , σ ∗ ) ′ (ρ ∈ U2 ) moves continuously with respect to the Chabauty topology. Since

4.6 Transition of terminal triangles

87

D(ρ(P1 ))∩E ǫ (ρ, σ ∗ ) = {Fixǫσ (ρ(P1 ))} and since I(ρ(Pj )) (j = 0, 1, 2) intersect transversely at Fixǫσ (ρ(P1 )), we can ﬁnd, by using the above continuity, a round disk V with center Fixǫσ (ρ(P1 )) such that the following conditions are satisﬁed for all ρ′ ∈ U2 . 1. D(ρ′ (P1 )) ∩ E ǫ (ρ′ , σ ∗ ) ⊂ V . 2. D(ρ′ (Pj )) ∩ V is a bigon, i.e., a convex region bounded by an arc in I(ρ′ (Pj )) and an arc in ∂D, for j = 0, 1, 2. 3. (I(ρ′ (Pj )) ∩ I(ρ′ (Pj+1 ))) ∩ V = {v ǫ (ρ′ ; Pj , Pj+1 )} for j = 0, 1.

Since θǫ (ρ′ , σ; s1 ) < 0 by the assumption, we see by Lemma 4.3.1 that v ǫ (ρ′ ; P0 , P1 ) ∈ int D(ρ′ (P2 )) and v ǫ (ρ′ ; P1 , P2 ) ∈ int D(ρ′ (P0 )). These imply that the D(ρ′ (P1 ))∩V is contained in the interior of (D(ρ′ (P0 ))∩D(ρ′ (P2 )))∩ V in V . Hence we have D(ρ′ (P1 )) ∩ E ǫ (ρ′ , σ ∗ ) = ∅. Since θǫ (ρ′ , σ; s1 ) < 0 and θ−ǫ (ρ′ , σ; s1 ) > 0, Ih(ρ′ (P0 )) ∩ Ih(ρ′ (P1 )) ∩ Ih(ρ′ (P2 )) is a singleton by Lemma 4.3.6. Hence Ih(ρ′ (P1′ )) ∩ Ih(ρ′ (P2′ )) ∩ Ih(ρ′ (P3′ )) is also a singleton by Lemma 4.1.3 (Chain rule). Case 3. θǫ (ρ′ , σ; s1 ) = 0 and hence all components of θǫ (ρ′ , σ) are nonnegative. Then the ﬁrst two conditions for σ to be an ǫ-terminal triangle are satisﬁed, because θ−ǫ (ρ′ , σ; s1 ) > 0. Moreover, we have the following analogy of Claim 4.6.3.

Claim 4.6.6. By choosing U2 small enough in advance, the following hold. For each j ∈ Z, eǫ (ρ′ , σ; Pj ) is a degenerate or non-degenerate circular arc according as j ≡ 1 (mod 3) or not, and satisﬁes the following condition.

eǫ (ρ′ , σ; Pj ) ∩ D(ρ′ (Pk )) =

⎧ ǫ ′ e (ρ , σ; Pj ) if k = j, ⎪ ⎪ ⎪ ǫ ′ ⎪ ⎪ v (ρ ; P , P ) if k = j + 1, j j+1 ⎪ ⎪ ⎨ ǫ ′ v (ρ ; Pj−1 , Pj )

⎪ eǫ (ρ′ , σ; Pj+1 ) ⎪ ⎪ ⎪ ⎪ ⎪eǫ (ρ′ , σ; Pj−1 ) ⎪ ⎩ ∅

if k = j − 1, if k = j + 2 and j ≡ 0

(mod 3)

if k = j − 2 and j ≡ 0

(mod 3)

otherwise.

Proof. The proof is parallel to that of Claim 4.6.3. The following are the only diﬀerence. • •

Suppose k = j + 1 and j ≡ 1 (mod 3), say k = 1 and j = 0, then θǫ (ρ′ , σ; s1 ) = 0 and hence eǫ (ρ′ , σ; P1 ) is a singleton. So it is obvious that eǫ (ρ′ , σ; P1 ) ∩ D(ρ′ (P0 )) is equal to the degenerate arc eǫ (ρ′ , σ; P1 ). Suppose k = j + 2 and j ≡ 0 (mod 3), say k = 2 and j = 0. Then eǫ (ρ′ , σ; P1 ) is a degenerate arc and is equal to v ǫ (ρ′ ; P0 , P1 ) = v ǫ (ρ′ ; P1 , P2 ). Thus eǫ (ρ′ , σ; P1 ) is contained in the intersection eǫ (ρ′ , σ; P0 ) ∩ D(ρ′ (P2 )). On the other hand, since I(ρ′ (P0 )) and I(ρ′ (P2 )) intersect transversely and since eǫ (ρ, σ; P0 ) ∩ D(ρ(P2 )) is equal to the singleton eǫ (ρ, σ; P1 ), we see that eǫ (ρ′ , σ; P0 ) ∩ D(ρ′ (P2 )) = eǫ (ρ′ , σ; P1 ) by choosing U2 small enough in advance.

88

•

4 Chain rule and side parameter

Suppose k = j − 2 and j ≡ 0 (mod 3). Then we can apply the above argument to this case.

By using Claim 4.6.6 instead of Claim 4.6.3 in the proof of Claim 4.6.4, we see that fr E ǫ (ρ′ , σ) = ∪j eǫ (ρ′ , σ; Pj ) ∼ = R whenever ρ′ ∈ U2 , provided that U2 is chosen to be small enough in advance. This implies that the pair (ρ′ , σ) satisﬁes the third condition in Deﬁnition 4.3.8. By Claim 4.6.6 and the fact that θ−ǫ (ρ′ , σ; s1 ) > 0, the last condition in Deﬁnition 4.3.8 is also satisﬁed. Hence σ is an ǫ-terminal triangle of ρ′ . Thus we have proved that if we choose U2 small enough in advance, then every ρ′ ∈ U2 satisﬁes the conclusion of Lemma 4.6.2. This completes the proof of Lemma 4.6.2. Finally, we prove the following analogy to Lemmas 4.6.1 and 4.6.2. Lemma 4.6.7. Under Notation 2.1.14 (Adjacent triangles) and Assumption 2.4.6 (σ-NonZero), let σ = s0 , s1 , s2 be an ǫ-terminal triangle of ρ and assume that θǫ (ρ, σ; s1 ) = θ−ǫ (ρ, σ; s1 ) = 0. For notational convenience, we set σ ∗ := σ ′ . Then there is a neighborhood U of ρ in X such that the following conditions hold for every ρ′ ∈ U :

1. If θǫ (ρ′ , σ; s1 ) > 0, then σ is an ǫ-terminal triangle of ρ′ . Moreover D(ρ′ (P2′ )) is disjoint from E ǫ (ρ′ , σ). 2. If θǫ (ρ′ , σ; s1 ) < 0, then σ ∗ is an ǫ-terminal triangle of ρ′ . Moreover D(ρ′ (P1 )) is disjoint from E ǫ (ρ′ , σ ∗ ). 3. If θǫ (ρ′ , σ; s1 ) = 0 and θ−ǫ (ρ′ , σ; s1 ) > 0, then σ is an ǫ-terminal triangle of ρ′ . 4. If θǫ (ρ′ , σ; s1 ) = 0 and θ−ǫ (ρ′ , σ; s1 ) < 0, then σ ∗ is an ǫ-terminal triangle of ρ′ . 5. If θǫ (ρ′ , σ; s1 ) = 0 and θ−ǫ (ρ′ , σ; s1 ) = 0, then both σ and σ ∗ are ǫ-terminal triangles of ρ′ .

Proof. The proof of Lemma 4.6.7 is parallel to that of Lemma 4.6.2. The only diﬀerence are the following. •

•

L(ρ, {σ, σ ∗ }) is not simple but is ﬂat (Deﬁnition 4.5.1(2)). Thus for each nearby representation ρ′ , L(ρ′ , {σ, σ ∗ }) is either simple or ﬂat. However, the results in Sect. 4.5, which were used in the proof of Lemma 4.6.2, are proved under this weaker condition. The condition θ−ǫ (ρ, σ; s1 ) > 0 in Lemma 4.6.2 was used only at the following two points. 1. In the treatment of ρ′ such that θǫ (ρ, σ; s1 ) < 0, it was used to prove that both Ih(ρ′ (P0 )) ∩ Ih(ρ′ (P1 )) ∩ Ih(ρ′ (P2 )) and Ih(ρ′ (P1′ )) ∩ Ih(ρ′ (P2′ )) ∩ Ih(ρ′ (P3′ )) are singletons. However, Lemma 4.6.7 does not contain corresponding assertion. (Such a problem is studied in Sect. 7.3.)

4.7 Proof of Lemma 4.5.5

89

2. In the treatment of ρ′ such that θǫ (ρ′ , σ; s1 ) = 0. However, in Lemma 4.6.7, the corresponding case is divided into three cases according as θ−ǫ (ρ′ , σ; s1 ) is > 0, = 0 or < 0. If θǫ (ρ′ , σ; s1 ) = 0 and θ−ǫ (ρ′ , σ; s1 ) ≤ 0, then θǫ (ρ′ , σ ∗ ; s′2 ) = 0 and θ−ǫ (ρ′ , σ ∗ ; s′2 ) ≥ 0 by Lemma 4.5.3. Thus these cases are essentially equivalent to Case 3 in the proof of Lemma 4.6.2, where σ may be replaced with σ ∗ , except when θǫ (ρ′ , σ; s1 ) = θ−ǫ (ρ′ , σ; s1 ) = 0. In this exceptional case, (ρ, τ ) is an isosceles representation (see the forthcoming Deﬁnition 5.2.2) by virtue of the forthcoming Proposition 5.2.3. Thus the conclusion also holds by the forthcoming Proposition 5.2.8.

4.7 Proof of Lemma 4.5.5 Throughout this section, we use Notation 4.2.6. We shall reduce the proof of Lemma 4.5.5 to the generic case, where all components of θǫ (ρ, σ) are positive for some ǫ ∈ {−, +} and L(ρ, σ) is not ﬂat at every vertex. By Corollary 4.2.13, we may assume without loss of generality that L(ρ, σ) is convex to the above at c(0) and convex to the below at c(1) and c(2), because the argument for the other case is parallel. The following lemma describes the shape of E(ρ, σ) and L(ρ, σ) in this generic case, and it holds the key to the proof of Lemma 4.5.5. Lemma 4.7.1. Under Assumption 4.2.4 (σ-Simple), suppose that all components of θǫ (ρ, σ) are positive for some ǫ ∈ {−, +} and that L(ρ, σ) is convex to the above at c(0) and convex to the below at c(1) and c(2). Apply a rotational − → coordinate change of C so that the oriented line segment f (0) is parallel to − → the imaginary axis, i.e., arg( f (0)) = π2 . (1) If ǫ = −, then the following inequalities hold in the new coordinate (see Fig. 4.17). (i) ℜ(c(0)) < ℜ(v − (0, 1)) < ℜ(c(1)) < ℜ(c(2)) < ℜ(v − (2, 3)) < ℜ(c(3)). (ii) ℜ(c(0)) < ℜ(v − (0, 1)) < ℜ(v − (1, 2)) < ℜ(v − (2, 3)) < ℜ(c(3)). − → − → (iii) 0 < arg( f (1)) ≤ π/2 ≤ arg( f (2)) < π. → → → (iv) arg(− c (0, 1)) ∈ (−π/2, 0), arg(− c (1, 2)) ∈ (−π/2, π/2), arg(− c (2, 3)) ∈ (0, π/2). (2) If ǫ = +, then the following inequalities hold in the new coordinate (see Fig. 4.18). (i) ℜ(c(0)) < ℜ(c(1)) < ℜ(v + (1, 2)) < ℜ(c(2)) < ℜ(c(3)). (ii) ℜ(c(0)) < ℜ(v + (0, 1)) < ℜ(v + (1, 2)) < ℜ(v + (2, 3)) < ℜ(c(3)). − → − → (iii) 0 < arg( f (1)) ≤ π/2 ≤ arg( f (2)) < π. − → → → (iv) arg( c (0, 1)) ∈ (−π/2, 0), arg(− c (1, 2)) ∈ (−π/2, π/2), arg(− c (2, 3)) ∈ (0, π/2).

90

4 Chain rule and side parameter → − f (0)

c(0)

c(1)

→ − f (1)

→ − f (2)

→ − f (3) c(3)

c(2)

P0

P3

→ − c (0, 1) → − c (1, 2) → − c (2, 3) P2

−

v (0, 1)

v − (2, 3)

P1

v − (1, 2) ℜ

Fig. 4.17. Isometric circles after a coordinate change (a) v + (1, 2)

v + (0, 1) → − f (0)

v + (2, 3) P3

P0

→ − f (3)

c(0) c(3)

P1

→ − c (1, 2)

→ − c (2, 3) P2

→ − − c (0, 1) → f (1)

c(1)

→ − f (2)

c(2) ℜ

Fig. 4.18. Isometric circles after a coordinate change (b)

4.7 Proof of Lemma 4.5.5

91

Remark 4.7.2. Though one might expect the stronger inequality ℜ(c(0)) < ℜ(v + (0, 1)) < ℜ(c(1)) < ℜ(v + (1, 2)) < ℜ(c(2)) < ℜ(v + (2, 3)) < ℜ(c(3)),

it does not necessarily hold. In fact, it can happen that ℜ(c(1)) > ℜ(v + (1, 2)) or ℜ(c(2)) > ℜ(v + (2, 3)). Such a phenomena can be observed in the example in Remark 4.3.9(2). Proof. We begin by the following observation. Claim 4.7.3. Under the setting of Lemma 4.7.1, the following hold. − → − → (1) c(j − 1) ∈ int(HL ( f (j))) and c(j + 1) ∈ int(HR ( f (j))). − → → + − c (j, j + 1))). (2) v (j, j + 1) ∈ int(HL ( c (j, j + 1))) and v (j, j + 1) ∈ int(HR (− − → − → ǫ ǫ (3) v (j − 1, j) ∈ int(HL ( f (j))) and v (j, j + 1) ∈ int(HR ( f (j))). − − → − → (4) Fix+ σ (ρ(Pj )) ∈ int(HL ( c (j −1, j)))∩int(HL ( c (j, j +1))) and Fixσ (ρ(Pj )) − → − → ∈ int(HR ( c (j − 1, j))) ∩ int(HR ( c (j, j + 1))). → → c (−1, 0)) = arg(− c (0, 1)) ∈ (−π/2, 0). (5) − arg(− Proof. By Deﬁnitions 3.1.3 (Upward) and 4.2.5(1), we have 0 < arg

− → → f (j) −− c (j − 1, j) = arg < π. − → − → c (j, j + 1) f (j)

This implies the assertion (1). The assertions (2) and (4) follow from Deﬁnition 4.2.5. By the assumption that all components of θǫ (ρ, σ) are positive, we have − → f (j) v ǫ (j − 1, j) − c(j) = arg = θǫ (ρ, σ; s[j] ) > 0. arg − → ǫ (j, j + 1) − c(j) v f (j) (cf. Lemma 4.2.8.) This implies the assertion (3). − → By Deﬁnition 3.1.3 and the assumption that f (0) is parallel to the imaginary axis (in the new coordinate), we have → → − arg(− c (−1, 0)) = arg(− c (0, 1)) ∈ (−π/2, π/2). Since L(ρ, σ) is convex to the above at c(0) by the assumption, this implies that these angles lie in (−π/2, 0). Thus we obtain the assertion (5). Claim 4.7.4. ℜ(c(0)) < ℜ(c(1)) < ℜ(c(2)) < ℜ(c(3)). Proof. By Claim 4.7.3(5), we have ℜ(c(−1)) < ℜ(c(0)) < ℜ(c(1)). Since c(3)− c(2) = c(0) − c(−1), we also have ℜ(c(2)) < ℜ(c(3)). Next, we prove ℜ(c(0)) < ℜ(c(3)). Suppose to the contrary that ℜ(c(3)) ≤ ℜ(c(0)). Then we have ℜ(c(2)) < ℜ(c(3)) ≤ ℜ(c(0)) and therefore c(2) ∈

92

4 Chain rule and side parameter

− → HL ( f (0)). Moreover, since L(ρ, σ) is convex to the below at c(1) by the → → c (0, 1))). Thus c(2) ∈ int(HL (− c (0, 1)))∩ assumption, we have c(2) ∈ int(HL (− − → − → − → HL ( f (0)). Since arg( c (0, 1)) ∈ (−π/2, 0) by Claim 4.7.3(5) and since f (0) is parallel to the imaginary axis, this implies ℑ(c(2)) > ℑ(c(1)) (see Fig. 4.19). − → → On the other hand, we can also see c(1) ∈ int(HL (− c (2, 3))) ∩ HR ( f (3)), through a parallel argument, by using the fact that ℜ(c(2)) < ℜ(c(3)) ≤ ℜ(c(0)) < ℜ(c(1)) and the assumption that L(ρ, σ) is convex to the below − → → at c(2). Since arg(− c (2, 3)) ∈ (0, π/2) by Claim 4.7.3(5) and since f (3) is parallel to the imaginary axis, this implies ℑ(c(1)) > ℑ(c(2)), a contradiction. Hence ℜ(c(0)) < ℜ(c(3)). c(2)

→ − c (1, 2) c(0) − HL (→ c (0, 1)) → − f (0)

→ − HL ( f (0))

→ − c (0, 1)

c(1)

Fig. 4.19. ℑ(c(2)) > ℑ(c(0))

Finally, we prove ℜ(c(1)) < ℜ(c(2)). Suppose to the contrary that ℜ(c(2)) → ≤ ℜ(c(1)). Then c(2) ∈ int(HL (− c (0, 1))) ∩ {z ∈ C | ℜ(z) ≤ ℜ(c(1))}, because → L(ρ, σ) is convex to the below at c(1). Since arg(− c (0, 1)) ∈ (−π/2, 0), this implies ℑ(c(2)) > ℑ(c(1)). However, we can also conclude ℑ(c(1)) > ℑ(c(2)) → by using arg(− c (2, 3)) ∈ (0, π/2) and the assumption that L(ρ, σ) is convex to the below at c(2). This is a contradiction. Hence we have ℜ(c(1)) < ℜ(c(2)). − → → Claim 4.7.5. −π/2 < arg(− c (1, 2)) < π/2 and 0 < arg( f (1)) < π/2 < − → arg( f (2)) < π. Proof. Since ℜ(c(1)) < ℜ(c(2)) by Claim 4.7.4, it follows that −π/2 < → arg(− c (1, 2)) < π/2. Since L(ρ, σ) is convex to the below at c(2), we have − → → → → arg(→ c (1, 2)) < arg(− c (2, 3)) = − arg(− c (0, 1)). Thus arg(− c (0, 1)) + arg(− c (1, 2)) < 0. Hence − → π 1 → → c (0, 1)) + arg(− c (1, 2))} + ∈ (0, π/2). arg( f (1)) = {arg(− 2 2

4.7 Proof of Lemma 4.5.5

93

− → By a parallel argument, we can also prove that arg( f (2)) ∈ (π/2, π). Claim 4.7.6. ℜ(c(0)) < ℜ(v ǫ (0, 1)) < ℜ(v ǫ (1, 2)) < ℜ(v ǫ (2, 3)) < ℜ(c(3)). Proof. By Claim 4.7.3(3), we have ℜ(c(0)) < ℜ(v ǫ (0, 1)). To prove ℜ(v ǫ (0, 1)) < ℜ(v ǫ (1, 2)), note that v ǫ (0, 1) = c(1) + ǫrei(α+ǫβ) ,

v ǫ (1, 2) = c(1) + ǫrei(α−ǫβ) ,

− → where α = arg( f (1)), β = θǫ (ρ, σ; s1 ) and r = r(ρ(P1 )). Thus ℜ(v ǫ (1, 2)) − ℜ(v ǫ (0, 1)) = ℜ(rei(α−β) − rei(α+β) )

= r(cos(α − β) − cos(α + β)) = 2r sin α sin β.

Since α ∈ (0, π/2) by Claim 4.7.5 and since β ∈ (0, π/2) by the assumption that all components of θǫ (ρ, σ) are positive, the above number is positive. Hence ℜ(c(0)) < ℜ(v ǫ (0, 1)) < ℜ(v ǫ (1, 2)). The remaining inequalities are proved by a parallel argument. Claim 4.7.7. If ǫ = −, then ℜ(v − (0, 1)) < ℜ(c(1)) and ℜ(c(2)) < ℜ(v − (2, 3)). Proof. By the assertions (2) and (3) of Claim 4.7.3, v − (0, 1) is contained in − → − → → c (0, 1)) ∩ int HL ( f (1)) ∩ int HR ( f (0)). By Claims the open region int HR (− 4.7.3(5) and 4.7.5, the open region is contained in the open strip {z ∈ C | ℜ(c(0)) < ℜ(z) < ℜ(c(1))} (see Fig. 4.20). Hence we have ℜ(v − (0, 1)) < ℜ(c(1)). By a parallel argument, we also have ℜ(c(2)) < ℜ(v − (2, 3)). → − f (0)

c(0) → − c (0, 1)

→ − f (1) v − (0, 1) c(1)

→ − → − − int HR (→ c (0, 1)) ∩ int HL ( f (1)) ∩ int HR ( f (0)) Fig. 4.20. ℜ(c(0)) < ℜ(v − (0, 1)) < ℜ(c(1))

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4 Chain rule and side parameter

Claim 4.7.8. If ǫ = +, then ℜ(c(1)) < ℜ(v + (1, 2)) < ℜ(c(2)). Proof. By the assertions (2) and (3) of Claim 4.7.3, v + (1, 2) is contained in the open region − → − → → int(HL (− c (1, 2))) ∩ int(HR ( f (1))) ∩ int(HL ( f (2))). By Claim 4.7.5, this open region is contained in the following region (see Fig. 4.21). → HL (− c (1, 2)) ∩ {z ∈ C | ℜ(c(1)) ≤ ℜ(z) ≤ ℜ(c(2))} Hence we obtain the desired result.

v + (1, 2) → − f (2)

→ − f (1) c(1)

c(2) → − c (1, 2)

Fig. 4.21. ℜ(c(1)) < ℜ(v + (1, 2)) < ℜ(c(2))

This completes the proof of Lemma 4.7.1. In fact, the assertion (1-i) follows from Claims 4.7.4, 4.7.6 and 4.7.7. The assertion (2-i) follows from Claims 4.7.4 and 4.7.8. The assertions (1-ii) and (2-ii) are equivalent to Claim 4.7.6. The assertion (1-iii) and (2-iii) are contained in Claim 4.7.5. Finally, the assertions (1-iv) and (2-iv) follow from Claims 4.7.3(5) and 4.7.5. Proof (Proof of Lemma 4.5.5). Suppose that the lemma does not hold. Then there exists a pair (ρ, σ) satisfying the assumption of Lemma 4.5.5 and a couple of distinct integers j1 and j2 , such that int(∆ǫj1 ) ∩ int(∆ǫj2 ) = ∅. Since this is an open condition, we may assume that all components of θǫ (ρ, σ) are positive and that L(ρ, σ) is not ﬂat at every vertex. According to Corollary 4.2.13, we may assume that L(ρ, σ) satisﬁes the assumptions of Lemma 4.7.1. By Lemma 4.2.11(1), int(∆ǫ0 ) ∩ int(∆ǫ1 ) = int(∆ǫ1 ) ∩ int(∆ǫ2 ) = ∅. In what follows, we see that int(∆ǫ0 ) ∩ int(∆ǫ2 ) = ∅. If ǫ = −, then, by Lemma 4.7.1, the vertices c(0), c(1) and v − (0, 1) of ∆− 0 are contained in {z ∈ C | ℜ(z) ≤ ℜ(c(1))}, and the vertices c(2), c(3) and v − (2, 3) of ∆− 2 are contained in {z ∈ C | ℜ(z) ≥ ℜ(c(2))} in the coordinate in Lemma 4.7.1. Since these two

4.8 Representations which are weakly simple at σ

95

− regions are disjoint by Lemma 4.7.1(1), we have int(∆− 0 ) ∩ int(∆2 ) = ∅. If ǫ = +, then, by Lemma 4.7.1, the vertices c(0), c(1) and v ǫ (0, 1) of ∆ǫ0 are contained in {z ∈ C | ℜ(z) ≤ ℜ(v + (1, 2))}, and the vertices c(2), c(3) and + v + (2, 3) of ∆+ 2 are contained in {z ∈ C | ℜ(z) ≥ ℜ(v (1, 2))} in the coordinate + in Lemma 4.7.1. Hence int(∆+ ) ∩ int(∆ ) = ∅. Thus we have proved that the 0 2 three successive triangles ∆ǫ0 , ∆ǫ1 and ∆ǫ2 have mutually disjoint interiors. On the other hand, int(∆ǫ0 ), int(∆ǫ1 ) and int(∆ǫ2 ) are contained in the open strip {z ∈ C | ℜ(c(0)) < ℜ(z) < ℜ(c(3))}, because the vertices of these triangles are contained in the closure of the strip by Lemma 4.7.1. Since the Euclidean transformation ρ(K) maps ∆ǫj to ∆ǫj+3 , and since the images of the above open strip by the powers of ρ(K) are disjoint, we see that the triangles ∆ǫj (j ∈ Z) have mutually disjoint interiors. This completes the proof of Lemma 4.5.5.

4.8 Representations which are weakly simple at σ In this section, we collect basic facts concerning representations ρ such that L(ρ, σ) is weakly simple but is not simple (Deﬁnition 3.1.1). To this end we introduce the following deﬁnition. Deﬁnition 4.8.1. Under Assumption 2.4.6 (σ-NonZero): → 1. L(ρ, σ) is said to be folded at c(ρ(Pj )) if ψ(− e [j−1] ) = c(ρ(Pj+1 ))−c(ρ(Pj )) − → is a real negative multiple of ψ( e ) = c(ρ(P )) − c(ρ(P )). [j+1]

j

j−1

2. L(ρ, σ) is said to be singly folded if L(ρ, σ) is weakly simple and there is a unique element k ∈ {0, 1, 2} such that L(ρ, σ) is folded at c(ρ(Pj )) if and only if [j] = k. 3. L(ρ, σ) is said to be doubly folded if L(ρ, σ) is weakly simple and there are two elements k1 and k2 ∈ {0, 1, 2} such that L(ρ, σ) is folded at c(ρ(Pj )) if and only if [j] = k1 or k2 .

Lemma 4.8.2. Under Assumption 2.4.6 (σ-NonZero), suppose that L(ρ, σ) is weakly simple but is not simple. Then L(ρ, σ) is either singly folded or doubly folded. Moreover the following hold. 1. If L(ρ, σ) is singly folded, then there is a unique Markoﬀ map inducing ρ which is upward at σ. 2. If L(ρ, σ) is doubly folded, then there are precisely two Markoﬀ maps inducing ρ which are upward at σ. Proof. Since L(ρ, σ) is weakly simple, there is a sequence {ρn } of typepreserving representations converging to ρ, such that L(ρn , σ) is simple. Fol→ → lowing Notation 4.2.6, let − cn (j, j + 1) be the oriented line segment − c (ρn ; Pj , Pj+1 ) and let cn (j, j + 1) be the edge of L(ρn , σ) with endpoints cn (j) and cn (j + 1). The assumption that L(ρ, σ) is not simple implies the following claim.

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4 Chain rule and side parameter

Claim 4.8.3. There are integers k and j such that k = j, j + 1 and that the vertex c(k) is contained in the edge c(j, j + 1). Proof. Since L(ρ, σ) is not simple, one of the following condition holds. 1. c(j − 1, j) ∩ c(j, j + 1) is strictly larger than {c(j)} for some j. 2. c(j, j + 1) ∩ c(k, k + 1) = ∅ for some integers j and k such that |j − k| ≥ 2. If the ﬁrst condition holds, then L(ρ, σ) is folded at c(j) and we have either c(j − 1) ∈ c(j, j + 1) or c(j + 1) ∈ c(j − 1, j). Suppose the second conditions holds. Suppose further that int c(j, j + 1) ∩ int c(k, k + 1) = ∅. Since L(ρ, σ) is weakly simple, this intersection cannot be a transversal intersection. Hence we see that c(j, j+1) and c(k, k+1) are contained in a single line and the boundary of one of the edges is contained in the other, which implies the desired result. Suppose int c(j, j + 1) ∩ int c(k, k + 1) = ∅. Since c(j, j + 1) ∩ c(k, k + 1) = ∅, we again come to the same conclusion. Claim 4.8.4. L(ρ, σ) is folded at some vertex. Proof. By Claim 4.8.3, some vertex c(k) is contained in the edge c(0, 1) for some k = 0, 1, after a shift of indices. We may assume k > 1, because the case k < 0 is treated by a parallel argument. We also assume that k is the minimal integer greater than 1 such that c(k) ∈ c(0, 1). Case 1. k = 3j for some j ∈ N. Since c(3j) = c(0) + j, c(0, 1) is horizontal (i.e., parallel to the real axis), and ℜ(c(0)) < ℜ(c(3j)) ≤ ℜ(c(1)). Moreover we have j = 1 and k = 3 by the minimality of k. Suppose ℑ(c(2)) = ℑ(c(0)). Then c(2) and c(3) lie in the horizontal line containing c(0, 1). Hence L(ρ, σ) is folded at c(1) or c(2) according as ℜ(c(2)) < ℜ(c(1)) or ℜ(c(2)) > ℜ(c(1)). Suppose ℑ(c(2)) > ℑ(c(0)) (= ℑ(c(1))). If c(3) = c(1) then L(ρ, σ) is folded at c(2). So we may assume c(3) ∈ c(0, 1) − c(1) and hence ℜ(c(3)) < ℜ(c(1)). Thus ℑ(cn (2)) > ℑ(cn (0)), ℑ(cn (1)) and ℜ(cn (3)) < ℜ(cn (1)) for all suﬃciently → → large n. This implies that arg(− cn (2, 3)/− cn (1, 2)) ∈ (0, π). Hence L(ρn , σ) is convex to the below at cn (2). On the other hand, since ℑ(cn (2)) > ℑ(cn (0)), ℑ(cn (1)), Lemma 3.1.9 implies that L(ρn , σ) is convex to the above at cn (2). This is a contradiction. Hence c(3) = c(1) and L(ρ, σ) is folded at c(2). If ℑ(c(2)) < ℑ(c(0)), then we obtain the same conclusion by a parallel argument. Thus the desired result holds in Case 1. Case 2. k = 3j + 1 for some j ∈ N. Since c(3j + 1) = c(1) + j, c(0, 1) is horizontal, and ℜ(c(1)) < ℜ(c(3j + 1)) ≤ ℜ(c(0)). Moreover we have j = 1 and k = 4 by the minimality of k. Suppose ℑ(c(2)) = ℑ(c(0)). Then c(2) and c(3) lie in the horizontal line containing c(0, 1). Hence L(ρ, σ) is folded at c(1) or c(2) according as ℜ(c(2)) > ℜ(c(1)) or ℜ(c(2)) < ℜ(c(1)). Suppose ℑ(c(2)) = ℑ(c(0)). Let ℓ0 and ℓ2 , respectively, be the horizontal line containing c(0) and c(2). Then c(4) and c(3) = c(0)+1 lie in ℓ0 in this order, and c(2) and c(5) = c(2) + 3 lie in ℓ2 in this order. Since ℓ0 = ℓ2 , this implies that int c(2, 3) and int c(4, 5) intersect transversely. This contradicts the assumption that L(ρ, σ) is weakly simple. Thus the desired result holds in Case 2.

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97

Case 3. k = 3j + 2 for some j ∈ N ∪ {0}. If j = 0, then L(ρ, σ) is folded at c(1). Suppose j > 0. Suppose further that c(0, 1) is horizontal. Then all vertices of L(ρ, σ) lies in a single horizontal line. Hence, by using the assumption c(3j + 2) ∈ c(0, 1), we see that L(ρ, σ) is folded at some vertex. Suppose that c(0, 1) is not horizontal. Then the line ℓ containing c(3, 4) separates c(0, 1) from c(3j + 3) = c(0) + (j + 1). Since c(3j + 2) ∈ c(0, 1), int c(3j + 2, 3j + 3) intersects ℓ transversely. Since c(3j + 3) = c(0) + j + 1 and c(3, 4) = c(0, 1) + 1, the intersection point is contained in int c(3, 4). Hence int c(3, 4) and int c(3j + 2, 3j + 3) intersect transversely, a contradiction. Thus the desired result holds in Case 3. Since ρ(K)(c(j +3)) = c(j)+1, L(ρ, σ) cannot be folded at three successive vertices. Hence L(ρ, σ) is either singly folded or doubly folded. To see the remaining assertion, let φ be a Markoﬀ map inducing ρ. Suppose ﬁrst that L(ρ, σ) is singly folded. We may assume L(ρ, σ) is folded at c(0). Then we can see that φ is upward at σ if and only if the angle → i/φ(s[j] ) −− c (j − 1, j) = arg arg − → i/φ(s ) c (j, j + 1) [j]

is equal to π if j = 0 and belongs to (0, π) if j = 1 or 2. Hence there is a unique Markoﬀ map inducing ρ which is upward at σ. Suppose next that L(ρ, σ) is doubly folded. We may assume L(ρ, σ) is folded at c(1) and c(2). Then φ is upward if and only if the above angle is equal to π for j = 1 or 2. The angle for j = 0 is equal to π or −π and both possibilities actually happen. Hence there are precisely two Markoﬀ maps inducing ρ which are upward at σ. This completes the proof of Lemma 4.8.2. The following Lemmas 4.8.5 ad 4.8.6 describe characteristic properties of those (ρ, σ)’s such that L(ρ, σ) is folded. Lemma 4.8.5. Under Assumption 2.4.6 (σ-NonZero), suppose that L(ρ, σ) is folded at c(ρ(P1 )). Then Ih(ρ(P0 )) ∩ Ih(ρ(P1 )) = Ih(ρ(P1 )) ∩ Ih(ρ(P2 )), and ρ(P1 ) preserves this set. Moreover, if Ih(ρ(P0 )) ∩ Ih(ρ(P1 )) is a complete geodesic, then ρ(P1 ) interchanges the endpoints of the complete geodesic. Proof. Since L(ρ, σ) is folded at c(1), the centers c(j) (j ∈ {0, 1, 2}) lie in the line, ℓ, containing proj(Axis(1)) by Lemma 2.4.4(2.1). Suppose ﬁrst that Ih(0) ∩ Ih(1) is a complete geodesic. Then proj(Axis(1)) is orthogonal to proj(Ih(0) ∩ Ih(1)) by the above observation. Thus Axis(1) is orthogonal to Ih(0)∩Ih(1) and hence ρ(P1 ) maps Ih(0)∩Ih(1) to itself by interchanging the two endpoints. Since ρ(P1 ) maps Ih(0) ∩ Ih(1) to Ih(1) ∩ Ih(2) by Lemma 4.1.3 (Chain rule), we have Ih(0) ∩ Ih(1) = Ih(1) ∩ Ih(2), and obtain the conclusion. Next, suppose that Ih(0) ∩ Ih(1) is a singleton, then we see that the singleton is contained in proj(Axis(1)) and hence in Axis(1). Thus we obtain the conclusion as in the ﬁrst case. Finally, if Ih(0) ∩ Ih(1) is neither a complete geodesic nor a singleton, then it is the empty set and the conclusion obviously holds.

98

4 Chain rule and side parameter

Lemma 4.8.6. Under Assumption 2.4.6 (σ-NonZero), L(ρ, σ) is folded at c(ρ(P1 )), if one of the following conditions holds. 1. Ih(ρ(P0 )) ∩ Ih(ρ(P1 )) = Ih(ρ(P1 )) ∩ Ih(ρ(P2 )), and it is a complete geodesic, and ρ(P1 ) interchanges the two endpoints of the geodesic. 2. Ih(ρ(P0 )) ∩ Ih(ρ(P1 )) = Ih(ρ(P1 )) ∩ Ih(ρ(P2 )), and it is a singleton. Proof. Suppose Condition 1 holds. Then the assumption that Ih(0) ∩ Ih(1) = Ih(1) ∩ Ih(2) is a complete geodesic implies that the centers c(ρ(Pj )) (j ∈ {0, 1, 2}) are contained in the perpendicular bisector, ℓ, of the line segment proj(Ih(0) ∩ Ih(1)). The condition that ρ(P1 ) interchanges the two endpoints of the geodesic implies that proj(Axis(1)) is orthogonal to proj(Ih(0) ∩ Ih(1)) and therefore contained in ℓ. Hence by using Lemma 2.4.4(2.1) we see that L(ρ, σ) is folded at c(1). Suppose Condition 2 holds. Then by a similar argument we again see that the centers c(ρ(Pj )) (j ∈ {0, 1, 2}) are contained in a single line, ℓ, and that proj(Axis(1)) is contained in ℓ. Hence we obtain the conclusion by using Lemma 2.4.4(2.1). At the end of this section, we prove the following Lemma 4.8.7, which implies the useful Lemma 8.9.3 that in tern is used in Sect. 8.9. Lemma 4.8.7. Under Notation 2.1.14 (Adjacent triangles), let Σ be the chain (σ, σ ′ ) or (σ ′ , σ). Let ρ be an element of X , satisfying the following conditions. 1. φ−1 (0) ∩ Σ (0) = ∅. 2. L(ρ, σ) is folded at c(ρ(P1 )). 3. L(ρ, σ) is not doubly folded.

Then there is a neighborhood U of ρ in X such that L(ρ′ , Σ) is not weakly simple for any element ρ′ of U . Proof. Let (a0 , a1 , a2 ) and (a′0 , a′1 , a′2 ), respectively, be the complex probability of ρ at σ and σ ′ , which are well-deﬁned by the ﬁrst assumption. By the second assumption, a0 = λa2 for some negative real number λ. By Lemma 2.4.1(3), this implies a′0 =

λ a1 , 1+λ

a′1 =

1 a1 , 1+λ

a′2 = (1 + λ)a2 .

In particular, λ = −1 and a′0 = λa′1 . Thus we have c(ρ(P2 )) − c(ρ(P1 )) = λ(c(ρ(P1 )) − c(ρ(P0 ))),

c(ρ(P2′ )) − c(ρ(P1′ )) = λ(c(ρ(P3′ )) − c(ρ(P2′ ))).

Suppose ﬁrst that −1 < λ < 0. Then the above identity implies that c(ρ(P2 )) lies in the interior of the edge c(ρ(P0 ))c(ρ(P1 )) and that c(ρ(P1′ )) lies in the interior of the edge c(ρ(P2′ ))c(ρ(P3′ )). Hence the interiors of the

4.8 Representations which are weakly simple at σ

99

edges c(ρ(P0 ))c(ρ(P1 )) and c(ρ(P2′ ))c(ρ(P3′ )) have a common point c(ρ(P2 )) = c(ρ(P1′ )). These two edges are not parallel by the third condition, and hence their interiors intersect transversely. Therefore, for any representation ρ′ which is suﬃciently close to ρ, L(ρ′ , Σ) is weakly singular. Next, suppose that λ < −1. Then the identity implies c(ρ(P0 )) ∈ int c(ρ(P1 ))c(ρ(P2 )) and c(ρ(P3′ )) ∈ int c(ρ(P1′ ))c(ρ(P2′ )). Hence the interiors ′ ))c(ρ(P ′ )) intersect transversely of the edges c(ρ(P1 ))c(ρ(P2 )) and c(ρ(P−2 −1 ′ at the point c(ρ(P0 )) = c(ρ(P0 )). Hence we also obtain the desired result in this case.

5 Special examples

In this chapter, we give a detailed study of special examples. In Sect. 5.1, we study the real representations, i.e., the type-preserving P SL(2, R) representations. We give a direct proof to the well-known fact that every such representation is fuchsian, by explicitly constructing the Ford domains (Proposition 5.1.3). This gives the starting point to the proof of the main Theorem 1.2.2. We also show that the restriction of the side parameter map ν : QF → H2 × H2 to the subspace F of fuchsian representations is a bijection (actually a homeomorphism) to the diagonal set (Proposition 5.1.5). This proves that Theorem 1.3.2 is valid for the subspace F. In Sect. 5.2, we study the isosceles representations, i.e., those representations ρ such that L(ρ, σ) together with a horizontal line forms periodic pattern of isosceles triangles (see Deﬁnition 5.2.2). We give a complete characterization of those isosceles representations which are quasifuchsian, and determine the Ford domains (Proposition 5.2.8). These representations are important by the following reasons. 1. Various geometric quantities, including the Ford domain, the bending laminations, the convex core volume, and the width of the limit sets, of these representations can be explicitly calculated (cf. [5, Example 3.3], [57]). 2. The isosceles representations play special roles in the proof of the main theorems, because their Ford domains have “hidden isometric hemispheres” (see Deﬁnition 7.1.1 and Lemma 7.1.6). In fact we treat these representations separately in the proof Proposition 6.2.1 (Openness) in Sect. 7.3. 3. We can easily see how Jorgensen’s theory can be extended to the outside of QF for isosceles representations (Remark 5.2.9). Proposition 5.2.12, which describes the side parameters of those representations near to isosceles representations, plays an important role in the proof of the main theorems (cf. Proof of Proposition 8.3.2). In Sect. 5.3. we study the groups generated by two parabolic transformations. After describing their relation with the Markoﬀ maps which vanish at

102

5 Special examples

some vertex (Lemma 5.3.2), we describe basic properties of these groups. We also calculate the degrees of the polynomials which arise from a Markoﬀ map φ such that φ(1/0) = 0 and φ(0/1) = x (Lemmas 5.3.12 and 5.3.14). These results are used in the proof of Proposition 9.1.13 in Chap. 9. In Sect. 5.4, we recall a result of Bowditch [17] concerning the “imaginary representations” which are induced by Markoﬀ maps which take value in R∪iR (Proposition 5.4.3). In Sect. 5.5, we characterize those type-preserving representations which send a non-peripheral element of π1 (T ) to a parabolic or elliptic transformation (Lemmas 5.5.1, 5.5.3 and 5.5.4). We also present a basic Lemma 5.5.6, which is used in Sect. 8.7 in Chap. 8.

5.1 Real representations Deﬁnition 5.1.1 (Real representation). A type-preserving representation ρ : π1 (O) → P SL(2, C) is called a real representation if it is conjugate to a representation into P SL(2, R). By using the identities 2.4 and 2.5, we can easily obtain the following characterization of the real representations. Lemma 5.1.2. For a type-preserving representation ρ, the following conditions are equivalent. 1. ρ is a real representation. 2. The Markoﬀ map φ = φρ takes only real numbers. 3. The complex probability map ψ = ψρ takes only positive real numbers. As is noted by [17, Proposition 4.11]), every real representation ρ is fuchsian, i.e., discrete and faithful. The following proposition describes the Ford domains of real representations. Proposition 5.1.3. Let ρ : π1 (O) → P SL(2, R) be a type-preserving real representation. Then it is faithful and discrete. Moreover, there is a unique triangle or edge, δ(ρ), of D such that the Ford domain P h(ρ) is equal to Eh(ρ, δ(ρ)) := {Eh(ρ(P )) | P is an elliptic generator with slope s(P ) ∈ δ(ρ)(0) }

and has the structure as illustrated in Fig. 5.1. Furthermore, δ(ρ) is characterized by the following properties: 1. Suppose δ(ρ) is a triangle σ = s0 , s1 , s2 of D. Then |φρ (si )| ≤ |φρ (s′i )| for each i ∈ {0, 1, 2}, where s′i is the vertex of D opposite to si with respect to the edge of σ which does not contain si .

5.1 Real representations Fρ (Pj , Pj+1 )

Fρ (Pj )

103

Fρ (Pj+1 , Pj+2 )

Fρ (Pj+1 ) Fρ (Pj+2 )

Fig. 5.1. Eh(ρ, σ) for a real representation Fρ (P3j−1 , P3j+1 )Fρ (P3j−1 , P3j+1 )

Fρ (P3j+1 )

Fρ (P3j−1 )

I(ρ(P3j ))

Fρ (P3j+2 )

Fig. 5.2. Eh(ρ, σ) for a “thin” real representation

2. Suppose δ(ρ) is an edge, say τ . Then the two triangles having τ as an edge satisﬁes the condition in 1. Proof. By [17, Lemma 3.17], we may assume φρ takes only positive real values. Moreover, we have φρ (s) > 2 for every s ∈ D(0) by [17, Proposition 3.18]. Hence, by [17, Corollary 3.7], there is a triangle σ = s0 , s1 , s2 of D such that 0 < φρ (si ) ≤ φρ (s′i ) for each i ∈ {0, 1, 2}, where s′i is the vertex of D opposite to si with respect to the edge of σ which does not contain si . Set (x, y, z) = (φρ (s0 ), φρ (s1 ), φρ (s2 )). Then we have x ≤ yz − x,

y ≤ zx − y,

z ≤ xy − z.

This implies |x| ≤ |y ± iz|,

|y| ≤ |z ± ix|,

|z| ≤ |x ± iy|.

104

5 Special examples

In fact, x ≤ yz −x implies x2 ≤ xyz −x2 = y 2 +z 2 and therefore |x| ≤ |y ±iz|. Hence, by Lemma 4.3.1, θ+ (ρ, σ; sj ) = θ− (ρ, σ; sj ) ≥ 0 for every j ∈ {0, 1, 2}. Let {Pj } be the sequence of elliptic generators associated with σ, and set Eh(ρ, σ) := ∩j∈Z Eh(ρ(Pj )), Fρ (Pj ) := Ih(ρ(Pj )) ∩ (Eh(ρ(Pj−1 )) ∩ Eh(ρ(Pj+1 ))) ,

Fρ (Pj , Pj+1 ) := Ih(ρ(Pj )) ∩ Ih(ρ(Pj+1 ))

Suppose ﬁrst that θ+ (ρ, σ; sj ) = θ− (ρ, σ; sj ) > 0 for every j ∈ {0, 1, 2}. Then Fρ (Pj , Pj+1 ) is an edge of Eh(ρ, σ) and Fρ (Pj ) is a face of Eh(ρ, σ) which is bounded by the two edges Fρ (Pj−1 , Pj ) and Fρ (Pj , Pj+1 ) (see Fig. 5.1(1)). The involution ρ(Pj ) preserves the face Fρ (Pj ) and interchanges the edges Fρ (Pj−1 , Pj ) and Fρ (Pj , Pj+1 ). Thus the set {Fρ (Pj−1 , Pj ), Fρ (Pj , Pj+1 ), Fρ (Pj+1 , Pj+2 )} forms an “edge cycle modulo ρ(K)” for each integer j. The sum of the dihedral angles of Eh(ρ, σ) along the edges in the edge cycle is equal to 2π by Lemma 4.1.3 (chain rule). Hence we can see that Eh(ρ, σ) together with the gluing isometries {ρ(Pj )} satisﬁes the conditions for the “Poincare’s theorem on fundamental polyhedra modulo ρ(K)”. (See Sect. 6.4 for details.) From this fact, we see that ρ is discrete and faithful and Eh(ρ, σ) is a fundamental domain of ρ(π1 (O)) modulo ρ(K). Since Eh(ρ, σ) is obtained as the intersection of the exteriors of isometric hemispheres, this implies that Eh(ρ, σ) is equal to the Ford domain P h(ρ). Suppose that some component of the side parameter vanishes, say θ+ (ρ, σ; s0 ) = θ− (ρ, σ; s0 ) = 0. Then the edges Fρ (P3j−1 , P3j ) and Fρ (P3j , P3j+1 ) become an identical edge, which we denote by Fρ (P3j−1 , P3j+1 ) (see Fig. 5.2). Moreover, the face Fρ (P3j ) degenerates to the edge Fρ (P3j−1 , P3j+1 ), and Eh(ρ, σ) becomes identical with Eh(ρ, τ ) := {Eh(ρ(P )) | P is an elliptic generator with slope s(P ) ∈ τ (0) } = {Eh(ρ(Pj )) | j ≡ 0 (mod 3)}, where τ = s1 , s2 (cf. Fig. 5.5). Then ∂Eh(ρ, τ ) consists of the faces Fρ (P3j−1 ), Fρ (P3j+1 ) and the edges Fρ (P3j−1 , P3j+1 ), Fρ (P3j+1 , P3j+2 ) where j runs over Z. The involution ρ(P3j−1 ) preserves the face Fρ (P3j−1 ) and interchanges Fρ (P3j−2 , P3j−1 ) with Fρ (P3j−1 , P3j+1 ), whereas the involution ρ(P3j+1 ) preserves the face Fρ (P3j+1 ) and interchanges Fρ (P3j−1 , P3j+1 ) with Fρ (P3j+1 , P3j+2 ). Thus the set {Fρ (P3j−1 , P3j+1 ), Fρ (P3j+1 , P3j+2 )} forms an “edge cycle modulo ρ(K)” for each integer j. On the other hand, the dihedral angle of Eh(ρ, τ ) along Fρ (P3j−1 , P3j+1 ) is equal to θ3j−1 + θ3j − π,

5.1 Real representations

105

where θj be the dihedral angle of Eh(ρ(Pj )) ∩ Eh(ρ(Pj+1 )). Hence the sum of the dihedral angles of Eh(ρ, τ ) along the edges in the edge cycle is equal to π by Lemma 4.1.3 (chain rule). Hence we can see that Eh(ρ, τ ) together with the gluing isometries satisﬁes the conditions for the “Poincare’s theorem on fundamental polyhedra modulo ρ(K)”. (See Sect. 6.4 for details.) Here the above angle sum π corresponds to the fact that Fρ (P3j−1 , P3j+1 ) is equal to the axis of the involution ρ(P3j ). From this fact, we see that ρ is discrete and faithful, and Eh(ρ, τ ) is equal to the Ford domain P h(ρ) as in the previous case. Moreover, by the argument for the previous case, the condition θ+ (ρ, σ; s0 ) = θ− (ρ, σ; s0 ) = 0 implies x = yz − x. Hence the condition in 1 is also satisﬁed for the triangle sharing the edge τ with σ. This completes the proof of Proposition 5.1.3. Remark 5.1.4. Set ν = (θ− (ρ, σ), θ+ (ρ, σ)) and consider the labeled representation ρ = (ρ, ν). Then, in the generic case when δ(ρ) = σ, the convex polyhedra Eh(ρ, τ ), Fρ (Pj ) and Fρ (Pj , Pj+1 ) introduced in the proof, are equal to Eh(ρ), Fρ (Pj ), Fρ (Pj , Pj+1 ) respectively (see Deﬁnitions 3.4.3 and 3.4.6). Similar identiﬁcations are also valid for the non-generic case, as explained in the forthcoming Remark 5.2.10. Proposition 5.1.5. For any point ν ∈ H2 , there is a type-preserving real representation ρ : π1 (O) → P SL(2, R) which realizes (ν, ν), that is, P h(ρ) = Eh(ρ, σ) and θ± (ρ, σ) = ν, where σ is a triangle of D containing ν. Moreover, the element ρ in X is uniquely determined by ν. Proof. Set ν = (θ0 , θ1 , θ2 ) ∈ σ. For each positive real h, we can ﬁnd points cj (j ∈ {0, 1, 2, 3}), v01 , v12 and v23 in the complex plane which satisfy the following conditions (see Fig. 5.3). 1. cj ∈ R and c0 < c1 < c2 < c3 . 2. v01 , v12 and v23 are contained in the horizontal line ℑ(z) = h. 3. arg(v01 − c0 ) = π2 − θ0 , arg(v01 − c1 ) = π2 + θ1 , arg(v12 − c1 ) = π2 − θ1 , arg(v12 − c2 ) = π2 + θ2 , arg(v23 − c2 ) = π2 − θ2 , arg(v23 − c3 ) = π2 + θ0 . By choosing a suitable h, we may assume c3 − c0 = 1. Put (a0 , a1 , a2 ) = (c2 − c1 , c3 − c2 , c1 − c0 ). Then a0 + a1 + a2 = 1, and therefore there is a → → → complex probability map ψ such that (ψ(− e 0 ), ψ(− e 1 ), ψ(− e 2 )) = (a0 , a1 , a2 ), − → − → − → where ( e 0 , e 1 , e 2 ) is dual to the triangle σ = s0 , s1 , s2 (cf. Sect. 2.4). Let ρ be the type-preserving representation determined by ψ. Then ρ is (equivalent to) a real representation, because the complex probability (a0 , a1 , a2 ) consists of real positive numbers. To show that ρ satisﬁes the desired properties, we normalize ρ so that c(ρ(P0 )) = c0 . Observe that the triangles ∆(c0 , c1 , v01 ), ∆(v12 , c1 , c2 ) and ∆(c3 , v23 , c2 ) are mutually similar, by comparing the angles. By comparing the edge lengths, we see that these triangles are similar to a |a |, |v − v | = |a | triangle ∆(v , v , v ) such that |v − v | = 0 0 2 1 and 0 1 2 2 1 |v1 − v0 | = |a2 |, which in tern is similar to a triangle ∆(w0 , w1 , w2 ) such that |w2 − w1 | = |φρ (s0 )|, |w0 − w2 | = |φρ (s1 )| and |w1 − w0 | = |φρ (s2 )|.

106

5 Special examples

Thus, by Lemma 4.2.11(2), the above three triangles are identical with the + + triangles ∆+ 0 (ρ, σ), ∆1 (ρ, σ) and ∆2 (ρ, σ) in Notation 4.2.10(1), respectively. Thus we have θ± (ρ, σ) = ν by Lemma 4.2.11(1). The above fact also implies that vj,j+1 ∈ I(ρ(Pj )) ∩ I(ρ(Pj+1 )) for each j ∈ {0, 1, 2}. Moreover, we see that Eh(ρ, σ) has the structure as that in the proof of Proposition 5.1.3. Hence we have P h(ρ) = Eh(ρ, σ). The uniqueness of ρ follows from the fact that the conﬁguration in Fig. 5.3 is uniquely determined by ν (cf. Lemma 5.2.12).

v01

v12

v23

θ2 θ2 θ0

a2

θ0

θ1 θ1 θ1

θ0 c0

θ1

c1

θ0

θ2 θ2 a0

c2

a1

c3

Fig. 5.3. angles

5.2 Isosceles representations and thin labels By using Lemma 3.1.2, we can easily prove the following lemma. Lemma 5.2.1. Let ρ : π1 (O) → P SL(2, C) be a type-preserving represen′ (0) tation such that φ−1 = ∅ under Notation 2.1.14 (Adjacent ρ (0) ∩ (σ ∪ σ ) triangles). Then the following conditions are equivalent. 1. The complex probability (a0 , a1 , a2 ) of ρ at σ satisﬁes the following conditions. a2 = ta0 for some t > 0, a0 + a2 = a ¯1 . That is, the point c1 lies in the interior of the edge c0 c2 and the triangle ∆(c0 , c2 , c3 ) is a (possibly degenerate) isosceles triangle with base c0 c3 , where cj = c(ρ(Pj )) (see Fig. 5.4). 2. The complex probability (a′0 , a′1 , a′2 ) of ρ at σ ′ satisﬁes the following conditions. a′1 = ta′0 for some t > 0, a′0 + a′1 = a ¯′2 . That is, the point c′2 lies in the interior of the edge c′1 c′3 and the triangle ∆(c′0 , c′1 , c′3 ) is a (possibly degenerate) isosceles triangle with base c′0 c′3 , where c′j = c(ρ(Pj′ )).

5.2 Isosceles representations and thin labels v+

w+

c2

a0 a2

107

a1

c1

c3

c0 v−

w−

Fig. 5.4. Complex probabilities of an isosceles representation

Deﬁnition 5.2.2 (Isosceles representation). Let ρ : π1 (O) → P SL(2, C) be a type-preserving representation. Under Notation 2.1.14 (Adjacent triangles), the pair (ρ, τ ), where τ = σ ∩ σ ′ , is called an isosceles representation if the mutually equivalent conditions in Lemma 5.2.1 are satisﬁed. We have the following characterization of isosceles representations. Proposition 5.2.3. Let ρ : π1 (O) → P SL(2, C) be a type-preserving repre′ (0) = ∅ under Notation 2.1.14 (Adjacent sentation such that φ−1 ρ (0) ∩ (σ ∪ σ ) triangles). Then the following conditions are equivalent. 1. (ρ, τ ) is an isosceles representation, where τ = σ ∩ σ ′ . 2. The side parameters θ± (ρ, σ) are deﬁned and θ− (ρ, σ; s1 ) = θ+ (ρ, σ; s1 ) = 0. 3. The side parameters θ± (ρ, σ ′ ) are deﬁned and θ− (ρ, σ ′ ; s′2 ) = θ+ (ρ, σ ′ ; s′2 ) = 0. 4. Ih(ρ(P0 )) ∩ Ih(ρ(P2 )) = Axis(ρ(P1 )). 5. Ih(ρ(P1′ )) ∩ Ih(ρ(P3′ )) = Axis(ρ(P2′ )). Proof. (1) → (2). Suppose (ρ, τ ) is an isosceles representation. Let v + and v − be the points in C satisfying the following conditions (see Fig. 5.4). 1. v + and v − lie on the perpendicular to the edge c0 c2 passing through c1 . → 2. v + (resp. v − ) lies on the left (resp. right) to the directed line c−− 0 c2 3. ∠(c0 , v ǫ , c2 ) is the right angle for each ǫ ∈ {−, +}.

We show that the triangles ∆(c0 , c1 , v ǫ ) and ∆(c1 , c2 , v ǫ ) coincide with the triangles ∆ǫ0 (ρ, σ) and ∆ǫ1 (ρ, σ) in Notation 4.2.10, respectively. To this end, let wǫ be the image of v ǫ by the reﬂection in the perpendicular bisector of the edge c0 c3 , which is equal to the perpendicular to c0 c3 passing through c2 . Then we can see, by using the fact that ∠(c0 , v ǫ , c2 ) = π/2, that the right triangles ∆(c0 , c1 , v ǫ ), ∆(v ǫ , c1 , c2 ) and ∆(c3 , wǫ , c2 ) are mutually similar. By using this fact we have the following claim.

108

5 Special examples

Claim 5.2.4. Set (x, y, z) = (φρ (s0 ), φρ (s1 ), φρ (s2 )). Then we have: |v ǫ − c0 | =

1 , |x|

|v ǫ − c1 | =

1 , |y|

|v ǫ − c2 | = |wǫ − c2 | =

1 . |z|

Proof. We prove only the last equality, because the remaining equalities can be proved similarly. |v ǫ − c2 |2 = |v ǫ − c2 | · |wǫ − c2 |

= |c1 − c2 | · |c3 − c2 | because ∆(v ǫ , c1 , c2 ) is similar to ∆(c3 , wǫ , c2 ) x y 1 = · = 2 . yz zx |z |

The above claim implies v ǫ ∈ I(ρ(P0 ))∩I(ρ(P1 ))∩I(ρ(P2 )), wǫ ∈ I(ρ(P2 )) and that the triangles ∆(c0 , c1 , v ǫ ), ∆(v ǫ , c1 , c2 ) and ∆(c3 , wǫ , c2 ) are similar to the triangle ∆(ρ, σ) in Notation 4.2.10(2). Hence the triangles ∆(c0 , c1 , v ǫ ) and ∆(c1 , c2 , v ǫ ) coincide with the triangles ∆ǫ0 (ρ, σ) and ∆ǫ1 (ρ, σ) in Notation 4.2.10, respectively. Since L(ρ, σ) is obviously simple, this implies that θ± (ρ, σ) are deﬁned. Hence we obtain θ− (ρ, σ; s1 ) = θ+ (ρ, σ; s1 ) = 0 by Lemma 4.2.11(1). (2) → (4). Though this is a consequence of Lemma 4.3.5(3), we give a direct proof. If θ− (ρ, σ; s1 ) = θ+ (ρ, σ; s1 ) = 0, then v ± (ρ; P0 , P1 ) = v ± (ρ; P1 , P2 ) = ± Fix± σ (ρ(P1 )) by the deﬁnition of θ (ρ, σ; s1 ). Hence Axis(ρ(P1 )) = Ih(ρ(P0 ))∩ Ih(ρ(P2 )). (4) → (1). Suppose Axis(ρ(P1 )) = Ih(ρ(P0 )) ∩ Ih(ρ(P2 )). Then c0 , c1 and c2 lie on the perpendicular bisector ℓ of the open interval π(Axis(ρ(P1 ))). Since π(Axis(ρ(P1 ))) bisects the angle ∠(c0 , c1 , c2 ) (see Proposition 2.4.4(2.1)), c0 , c1 and c2 lie on ℓ in this order. Thus we have a2 = ta0 for some t > 0. Let v + (resp. v − ) be the point of Fix(ρ(P1 )) = I(ρ(P0 )) ∩ I(ρ(P1 )) ∩ → I(ρ(P2 )) which lies on the left (resp. right) to the directed line c−− 0 c2 . Then ǫ ǫ ǫ ∠(c0 , c1 , v ) = ∠(v , c1 , c2 ) = π/2 and ∆(c0 , c1 , v ) is similar to ∆(v ǫ , c1 , c2 ). Thus ∆(c0 , v ǫ , c2 ) is also similar to ∆(v ǫ , c1 , c2 ). Let γ be the reﬂection in the line containing π(Axis(ρ(P2 ))), and set wǫ = γ(v ǫ ). Since v ǫ ∈ I(ρ(P2 )), wǫ is equal to the image of v ǫ by ρ(P2 ), and hence wǫ ∈ I(ρ(P2 )) ∩ I(ρ(P3 )) by Lemma 4.1.3(2) (Chain rule). Since ∆(γ(c0 ), wǫ , c2 ) = γ(∆(c0 , v ǫ , c2 )) is similar to ∆(v ǫ , c1 , c2 ), we have c3 = γ(c0 ). Since c3 − c0 = 1 = 0, ∆(c0 , c2 , c3 ) is an isosceles triangle with base c0 c3 . Moreover, c1 lies in the interior of the edge c0 c2 . Hence (ρ, τ ) is an isosceles representation. Thus we have proved that the conditions (1), (2) and (4) are mutually equivalent. By a parallel argument we can see that (1), (3) and (5) are mutually equivalent. Corollary 5.2.5. Let (ρ, τ ) an isosceles representation. Then, under Notation 2.1.14 (Adjacent triangles), we have the following. Ih(ρ(P1 )) ∩ (Eh(ρ(P0 )) ∩ Eh(ρ(P2 ))) = Axis(ρ(P1 )) Ih(ρ(P2′ )) ∩ (Eh(ρ(P1′ )) ∩ Eh(ρ(P3′ ))) = Axis(ρ(P2′ )).

5.2 Isosceles representations and thin labels

109

We also present the following two characterizations of isosceles representations. Lemma 5.2.6. Under Assumption 2.4.6 (σ-NonZero), suppose that the intersection Ih(ρ(Pj−1 )) ∩ Ih(ρ(Pj )) ∩ Ih(ρ(Pj+1 )) is non-empty and is not a singleton. Then one of the following holds. 1. (ρ, τ ) is an isosceles representation, where τ = s[j−1] , s[j+1] . 2. L(ρ, σ) is folded at c(ρ(Pj )) (cf. Deﬁnition 4.8.1). Proof. Note that both Ih(ρ(Pj−1 )) ∩ Ih(ρ(Pj )) and Ih(ρ(Pj )) ∩ Ih(ρ(Pj+1 )) are geodesics by Condition 2.4.6 (σ-NonZero). Suppose that Ih(ρ(Pj−1 )) ∩ Ih(ρ(Pj )) ∩ Ih(ρ(Pj+1 )) is nonempty and is not a singleton. Then it follows that the intersection is equal to the geodesic Ih(ρ(Pj−1 )) ∩ Ih(ρ(Pj )) = Ih(ρ(Pj )) ∩ Ih(ρ(Pj+1 )), and the centers c(ρ(Pj−1 )), c(ρ(Pj )) and c(ρ(Pj+1 )) lie on the perpendicular bisector ℓ of the open interval proj(Ih(ρ(Pj−1 )) ∩ Ih(ρ(Pj ))). Since ρ(Pj ) interchanges Ih(ρ(Pj−1 )) ∩ Ih(ρ(Pj )) and Ih(ρ(Pj )) ∩ Ih(ρ(Pj+1 )), one of the following holds: 1. Axis(ρ(Pj )) = Ih(ρ(Pj−1 )) ∩ Ih(ρ(Pj )). 2. proj(Axis(ρ(Pj ))) is contained in ℓ. If the ﬁrst condition is satisﬁed, then (ρ, τ ), with τ = s[j−1] , s[j+1] , is an isosceles representation by Proposition 5.2.3. If the second condition holds, then we see that L(ρ, σ) is folded at c(ρ(Pj )) by Lemma 4.8.6. This completes the proof. Lemma 5.2.7. Let ρ : π1 (O) → P SL(2, C) be a type-preserving representation which is not a real representation. Then, under Notation 2.1.14 (Adjacent triangles), (ρ, τ ) is an isosceles representation if and only if both φρ (s0 ) and φρ (s2 ) are non-zero real numbers. Proof. Set (x, y, z) = (φρ (s0 ), φρ (s1 ), φρ (s2 )), and let (a0 , a1 , a2 ) be the complex probability of ρ at σ (if it is deﬁned). Recall the identities 2.4 and 2.5. a0 =

x , yz

a1 =

y , zx

a2 =

z , xy

x2 =

1 , a1 a2

y2 =

1 , a2 a0

z2 =

1 . a0 a1

Suppose (ρ, τ ) is an isosceles representation. Then arg a0 ≡ arg a2 ≡ − arg a1 (mod 2π), and hence x and z are non-zero real numbers. Suppose conversely that x and z are non-zero real numbers. Then y = 0 by the Markoﬀ identity, and therefore the complex probability of ρ at σ is deﬁned. Since a1 a2 = 1/x2 is a positive real, we have arg a1 + arg a2 ≡ 0 (mod 2π). Similarly, we have arg a0 + arg a1 ≡ 0 (mod 2π). Thus arg a2 ≡ − arg a1 ≡ arg a0 (mod 2π) and hence a2 = ta0 for some t > 0. On the other hand, since ρ is not real, ℑy = 0 and therefore ℑa1 = 0. By using the facts that arg(a0 + a2 ) = − arg a1 , ℑ(a0 + a1 + a2 ) = 0 and ℑa1 = 0, we see a0 + a2 = a ¯1 . Hence (ρ, τ ) is an isosceles representation.

110

5 Special examples

The following proposition determines when an isosceles representation is quasifuchsian. Proposition 5.2.8. Let (ρ, τ ) be an isosceles representation, and assume that ℑ(c(ρ(P0 ))) ≤ ℑ(c(ρ(P2 ))) under Notation 2.1.14 (Adjacent triangles). Then the following hold. (1) ρ(KP0 ) (resp. ρ(KP2 )) is purely-hyperbolic, parabolic or elliptic according as θ− (ρ, σ; s2 ) (resp. θ+ (ρ, σ; s0 )) is positive, 0 or negative. (2) Suppose θ− (ρ, σ; s2 ) > 0 and θ+ (ρ, σ; s0 ) > 0. Then ρ is a quasifuchsian representation and its Ford domain P h(ρ) is equal to the following polyhedron (see Fig. 5.5): Eh(ρ, τ ) : = {Eh(ρ(P )) | P is an elliptic generator with slope s(P ) ∈ τ (0) } = {Eh(ρ(Pj )) | j ≡ 1 (mod 3)} = {Eh(ρ(Pj′ )) | j ≡ 2 (mod 3)}. Moreover θǫ (ρ, σ) = (+, 0, +) and θǫ (ρ, σ ′ ) = (+, +, 0), and they determine the same point of int τ , for each ǫ ∈ {−, +}.

Remark 5.2.9. If one of ρ(KP0 ) and ρ(KP2 ) is parabolic and the other is purely hyperbolic, then ρ gives a cusp group. If both of them are parabolic, then ρ gives the simplest double cusp group. If one of them becomes elliptic, then ρ is not discrete anymore in general. But it can be regarded as the holonomy representation of a hyperbolic cone manifold whose ‘Ford domain’ is given by the formula in the second assertion. This observation is the starting point to the forthcoming paper [11], whose main results had been announced in [9].

Fρ (P3j , P3j+2 )

Fρ (P3j+2 , P3j+3 ) Fρ (P3j+2 )

Fρ (P3j )

Fρ (P3j+3 )

Fig. 5.5. Ford domain of an isosceles representation

5.2 Isosceles representations and thin labels

111

Proof. (1) Recall that tr(ρ(KPj )) = φρ (sj ) up to sign. Since φρ (s0 ) is real by Proposition 5.2.7, ρ(KP0 ) is purely-hyperbolic, parabolic or elliptic according as |φρ (s0 )| is greater than 2, equal to 2, or less than 2. Since r(ρ(P0 )) = r(ρ(P3 )) = 1/|φρ (s0 )| and c(ρ(P3 )) − c(ρ(P0 )) = 1, the latter conditions are equivalent to the conditions that I(ρ(P0 )) ∩ I(ρ(P3 )) is empty, a singleton, or two points, respectively. To study the intersection of these two isometric circles, consider the quadrangles + − + − + − R ♦L 0 := ∆−1 (ρ, σ)∪∆−1 (ρ, σ), ♦0 := ∆0 (ρ, σ)∪∆0 (ρ, σ)∪∆1 (ρ, σ)∪∆1 (ρ, σ). R L R Let ♦L 3 and ♦3 , respectively, be the images of ♦0 , ♦0 by the translation z → z + 1. Case 1. θ− (ρ, σ; s2 ) > 0. Let xR 0 be the intersection of I(ρ(P0 )) with the horizontal ray emanating from c(ρ(P0 )) which contains c(ρ(P3 )). Similarly, let xL 3 be the intersection of I(ρ(P3 )) with the horizontal ray emanating from c(ρ(P3 )) which contains c(ρ(P0 )). Then I(ρ(P0 )) ∩ I(ρ(P3 )) = ∅ if and only L if the line segments [c(ρ(P0 )), xR 0 ] and [c(ρ(P3 )), x3 ] intersect. To show that these two line segments are disjoint, note that

θ− (ρ, σ; s0 ) = (π/2)−(θ− (ρ, σ; s1 )+θ− (ρ, σ; s2 )) = (π/2)−θ− (ρ, σ; s2 ) < π/2. R L This implies that [c(ρ(P0 )), xR 0 ] is contained in ♦0 . Similarly, [c(ρ(P3 )), x3 ] L is contained in ♦3 . On the other hand, since L(ρ, σ) is convex to the above at c(ρ(P2 )), we have θ+ (ρ, σ; s2 ) > θ− (ρ, σ; s2 ) > 0 by Corollary 4.2.15. This implies ∆ǫ1 (ρ, σ) ∩ ∆ǫ2 (ρ, σ) = {c(ρ(P2 ))} for each ǫ ∈ {−, +} and hence ♦R 0 ∩ R L ♦L 3 = {c(ρ(P2 ))}. Hence [c(ρ(P0 )), x0 ] and [c(ρ(P3 )), x3 ] are disjoint. Thus we have I0R ∩ I3L = ∅. Hence I(ρ(P0 )) ∩ I(ρ(P3 )) = ∅. and therefore ρ(P0 ) is purely-hyperbolic. Case 2. θ− (ρ, σ; s2 ) = 0. Then we have θ− (ρ, σ; s0 ) = π/2 and hence R L [c(ρ(P0 )), xR 0 ] and [c(ρ(P3 )), x3 ] are contained in (the boundary of) ♦0 and + − L ♦3 , respectively. On the other hand, θ (ρ, σ; s2 ) > θ (ρ, σ; s2 ) = 0 as in Case 1. This implies ∆ǫ1 (ρ, σ) ∩ ∆ǫ2 (ρ, σ) = c2 v − , where v − := v − (ρ; P1 , P2 ) = R L − v − (ρ; P2 , P3 ). Hence ♦R 0 ∩ ♦3 = c2 v . Thus we have [c(ρ(P0 )), x0 ] ∩ [c(ρ(P3 )), L − − x3 ] = {v }. This implies I(ρ(P0 )) ∩ I(ρ(P3 )) = {v }. Hence ρ(P0 ) is parabolic. Case 3. θ− (ρ, σ; s2 ) < 0. Then we have θ+ (ρ, σ; s2 ) = π − θ− (ρ, σ; s2 ) − 2α(ρ, σ; s2 ) > π − 2α(ρ, σ; s2 ) > 0 (cf. Notation 4.2.10(2)). Since (ρ, τ ) is thin, we have α(ρ, σ; s1 ) = π/2 and hence α(ρ, σ; s2 ) < π/2. Hence Ih(ρ(P1 )) ∩ Ih(ρ(P2 )) ∩ Ih(ρ(P3 )) is a singleton by Lemma 4.3.6. This singleton lies in Axis(ρ(P2 )) by Lemma 4.1.3(4) (Chain rule). On the other hand, Ih(ρ(P1 )) ∩ Ih(ρ(P2 )) = Axis(ρ(P1 )) by Corollary 5.2.5. Hence Axis(ρ(P1 )) ∩ Axis(ρ(P2 )) = ∅, and therefore ρ(KP0 ) = ρ(P2 )ρ(P1 ) is an elliptic transformation. Thus we have proved the assertion for ρ(P0 ). We can prove the assertion for ρ(P2 ) by a parallel argument. (2) Suppose θ− (ρ, σ; s2 ) > 0 and θ+ (ρ, σ; s0 ) > 0. Since L(ρ, σ) is convex to the above at c(ρ(P2 )) and convex to the below at c(ρ(P0 )), we have

112

5 Special examples

θ− (ρ, σ; s0 ) > θ+ (ρ, σ; s0 ) > 0 and θ+ (ρ, σ; s2 ) > θ− (ρ, σ; s2 ) > 0 by Corollary 4.2.15. As in the latter half of the proof of Proposition 5.1.3, we set: Eh(ρ, τ ) = ∩{Eh(ρ(Pj ) | j ≡ 1 (mod 3)},

Fρ (P3j ) = Ih(ρ(P3j )) ∩ (Eh(ρ(P3j−1 )) ∩ Eh(ρ(P3j+2 ))) , Fρ (P3j+2 ) = Ih(ρ(P3j+2 )) ∩ (Eh(ρ(P3j )) ∩ Eh(ρ(P3j+3 ))) ,

Fρ (P3j , P3j+2 ) = Ih(ρ(P3j )) ∩ Ih(ρ(P3j+2 )), Fρ (P3j−1 , P3j ) = Ih(ρ(P3j−1 )) ∩ Ih(ρ(P3j )). By Corollary 5.2.5, we have Fρ (P3j , P3j+2 ) = Ih(ρ(P3j )) ∩ Ih(ρ(P3j+1 ))

= Ih(ρ(P3j+1 )) ∩ Ih(ρ(P3j+2 )) = Axis(ρ(P3j+1 )).

By using the fact that all components of θ± (ρ, σ) are non-negative, we see the following: 1. The geodesics Fρ (P3j−1 , P3j ) and Fρ (P3j , P3j+2 ) are disjoint, and Fρ (P3j ) is a 2-dimensional convex polyhedron bounded by these two geodesics. 2. The geodesics Fρ (P3j , P3j+2 ) and Fρ (P3j+2 , P3j+3 ) are disjoint, and Fρ (P3j+2 ) is a 2-dimensional convex polyhedron bounded by these two geodesics. Thus ∂Eh(ρ, τ ) consists of the faces Fρ (P3j), Fρ (P3j+2) and the edges Fρ (P3j−1 , P3j ), Fρ (P3j , P3j+2 ) where j runs over Z. The involution ρ(P3j ) preserves the face Fρ (P3j ) and interchanges Fρ (P3j−1 , P3j ) with Fρ (P3j , P3j+2 ), whereas the involution ρ(P3j+2 ) preserves the face Fρ (P3j+2 ) and interchanges Fρ (P3j , P3j+2 ) with Fρ (P3j+2 , P3j+3 ). Thus the set {Fρ (P3j−1 , P3j ), Fρ (P3j , P3j+2 )} forms an “edge cycle modulo ρ(K)” for each integer j. On the other hand, the dihedral angle of E(ρ, τ ) along Fρ (P3j , P3j+2 ) is equal to θ3j + θ3j+1 − π, where θj be the dihedral angle of Eh(ρ(Pj )) ∩ Eh(ρ(Pj+1 )). Hence the sum of the dihedral angles of Eh(ρ, τ ) along the edges in the edge cycle is equal to π by Lemma 4.1.3 (chain rule). Hence we can see that Eh(ρ, τ ) together with the gluing isometries satisﬁes the conditions for the “Poincare’s theorem on fundamental polyhedra modulo ρ(K)”. (See Sect. 6.4 for details.) From this fact, we see that ρ is discrete and faithful, and Eh(ρ, τ ) is equal to the Ford domain P h(ρ). Moreover, by Lemma 4.5.3(2), θǫ (ρ, σ) and θǫ (ρ, σ ′ ), determine the same point of int τ , for each ǫ ∈ {−, +}. This completes the proof of Proposition 5.2.8. Remark 5.2.10. Set ν = (θ− (ρ, σ), θ+ (ρ, σ)) and ρ = (ρ, ν). Then the convex polyhedra Eh(ρ, τ ), Fρ (Pj ), Fρ (P3j−1 , P3j ) and Fρ (P3j , P3j+2 ), introduced in the proof, are equal to Eh(ρ), Fρ (Pj ), Fρ (P3j−1 , P3j ) and Fρ (P3j , P3j+2 ), respectively (cf. Deﬁnitions 3.4.3 and 3.4.6).

5.2 Isosceles representations and thin labels

113

Next, we prove the following proposition, which shows that every thin label is realized as the side parameter of a quasifuchsian representation. Proposition 5.2.11. Let ν = (ν − , ν + ) be a thin label, τ the edge of D containing ν ± , and σ a triangle of D containing τ . Then there is a unique element ρ ∈ QF which realizes ν, that is, the following hold. 1. (ρ, τ ) is an isosceles representation, and the Ford domain P h(ρ) is equal to Eh(ρ, τ ). 2. (θ− (ρ, σ), θ+ (ρ, σ)) is well-deﬁned and represents the point ν. To prove this proposition, we prepare the following lemma. Lemma 5.2.12. Let σ be a triangle of D and let ν ± be points in σ ∩ H2 . Then there is a unique (non-trivial) Markoﬀ map φ which is an algebraic root for ((ν − , σ), (ν + , σ)) (Deﬁnition 4.2.19). Proof. Let φ be a non-trivial Markoﬀ map with (x, y, z) = (φ(s0 ), φ(s1 ), φ(s2 )), where σ = s0 , s1 , s2 . Then φ is an algebraic root for ((ν − , σ), (ν + , σ)) if and only if (x, y, z) is a root of the following system of equations. x2 + y 2 + z 2 = xyz,

x + αǫ y + β ǫ z = 0 (ǫ ∈ {−, +}).

Here αǫ = ǫi exp(ǫiθ2ǫ ) and β ǫ = −ǫi exp(−ǫiθ1ǫ ), where ν ǫ = (θ0ǫ , θ1ǫ , θ2ǫ ) ∈ σ ǫ . By the two linear equations, we have y=

(β + − β − )x , α+ β − − α− β +

z=

(−α+ + α− )x . α+ β − − α− β +

Here α+ β − − α− β + = 0, because ν = (s0 , s0 ). By substituting y and z in the Markoﬀ equation with the above, we obtain: (α+ β − − α− β + )2 x2 + (β + − β − )2 x2 + (−α+ + α− )2 x2 = (β + − β − )(−α+ + α− )x3 .

Hence we have either x = 0 or x=

(α+ β − − α− β + )2 + (β + − β − )2 + (−α+ + α− )2 . (β + − β − )(−α+ + α− )

Here −α+ + α− = 0 and β + − β − = 0, because ν = (s2 , s2 ) and ν = (s1 , s1 ). If x = 0 then y = z = 0 and hence φ is the trivial Markoﬀ map. So x must take the latter value. Hence there is a unique algebraic root for ((ν − , σ), (ν + , σ)). Proof (Proof of Proposition 5.2.11). We may assume τ is the edge s1 , s2 of the triangle σ = s0 , s1 , s2 in the proof of Lemma 5.2.12. Then β ǫ = ǫiαǫ , and hence the Markoﬀ triple (x, y, z) determined by the non-trivial algebraic root for ((ν − , σ), (ν + , σ)) is given by:

114

x=

5 Special examples

4iα+ α− (α+ α− + 1) , (α+ + α− )(−α+ + α− )

y=

2i(α+ α− + 1) , (α+ − α− )

z=

−2(α+ α− + 1) . (α+ + α− )

Thus the complex probability (a0 , a1 , a2 ) at σ is: a0 =

α+ α− , α+ α− + 1

a1 =

(α+ + α− )2 , 4α+ α− (α+ α− + 1)

a2 =

−(α+ − α− )2 . 4α+ α− (α+ α− + 1)

So we have (1 + cos θ) + i sin θ 1 ℜa0 = ℜ = 2(1 + cos θ) 2 ⎞2 ⎛ 2 + − θ + +θ − i sin 2 2 2 a2 −(α+ − α− )2 1 − e−i(θ2 +θ2 ) ⎠ ∈ R+ . = −⎝ =− = − + θ + +θ − a1 (α+ + α− )2 1 + e−i(θ2 +θ2 ) cos 2 2 2

Hence (ρ, τ ) is an isosceles representation. Since ρ is quasifuchsian, Proposition 5.2.8 implies that the Ford domain P h(ρ) is equal to Eh(τ ). Moreover, (θ− (ρ, σ), θ+ (ρ, σ)) is well-deﬁned by Proposition 5.2.3. To show that it is equal to ν, we have only to show that φ is upward at σ, because φ is an algebraic root for ν. To this end, note that +

−

θ + +θ −

i sin 2 2 2 i/x iy α+ − α− i/x 1 − e−i(θ2 +θ2 ) = = + = , = = − + θ + +θ − a2 z/(xy) x α + α− 1 + e−i(θ2 +θ2 ) cos 2 2 2 1 iz α+ − α− i/y i/y 1 1 = = = = − + a0 x/(yz) x 2α+ α− 2 α− α + π 1 i(θ− + π ) e 2 2 − e−i(θ2 + 2 ) = 2 i/z 2 ix −2i(α+ α− ) i/y = = = = − + + − iθ a1 y/(zx) y α +α −e 2 + e−iθ2 i/y i/y Hence we see that arg( i/x a2 ) = π/2 and that both arg( a0 ) and arg( a1 ) belongs to (0, π) by using the fact that θ2± ∈ (0, π/2). So φ is upward at σ and hence (θ− (ρ, σ), θ+ (ρ, σ)) = ν. This completes the proof of Proposition 5.2.11.

Next, we prove the following proposition, which is used in the proof of Proposition 8.3.2. Proposition 5.2.13. Under Notation 2.1.14 (Adjacent triangles), there is a neighborhood, U , of int τ × int τ in (σ ′ ∩ H2 ) × (σ ∩ H2 ) and a continuous map U ∋ ν → φν ∈ Φ which have the following properties. 1. φν is an algebraic root for ((ν − , σ ′ ), (ν + , σ)), where (ν − , ν + ) = ν. 2. The representation induced by φν is quasifuchsian. 3. The representation induced by any other algebraic root for ((ν − , σ ′ ), (ν + , σ)) is not quasifuchsian.

5.2 Isosceles representations and thin labels

115

To prove this proposition, we need the following lemma. Lemma 5.2.14. Under Notation 2.1.14 (Adjacent triangles), let ν = (ν − , ν + ) be a point of (σ ′ ∩ H2 ) × (σ ∩ H2 ). Then the following hold. 1. There are at most two non-trivial Markoﬀ maps, counted with multiplicity, which are algebraic roots for ((ν − , σ ′ ), (ν + , σ)). 2. If ν ± ∈ int τ , then there is a unique non-trivial Markoﬀ map, counted with multiplicity, which is an algebraic root for ((ν − , σ ′ ), (ν + , σ)). Moreover it is equal to the unique non-trivial algebraic root for ((ν − , σ), (ν + , σ)) given in Lemma 5.2.12. Proof. Set ν − = (θ0− , θ1− , θ2− ) ∈ σ ′ and ν + = (θ0+ , θ1+ , θ2+ ) ∈ σ. Then a Markoﬀ map φ is an algebraic root for ((ν − , σ ′ ), (ν + , σ)) if and only if the value (x, y, z, w) = (φ(s0 ), φ(s1 ), φ(s2 ), φ(s′2 )) is a root of the following system of equations: x2 + y 2 + z 2 = xyz,

y + w = xz,

x + α+ y + β + z = 0,

x + α− z + β − w = 0

Here αǫ = ǫi exp(ǫiθ2ǫ ) and β ǫ = −ǫi exp(−ǫiθ1ǫ ). Note that the last two equations are equivalent to the equations y = −α+ x − γ + z,

w = −β − x − γ − z,

where γ ǫ = i exp(iθ0ǫ ). The second equation and these two equations imply xz = −(α+ + β − )x − (γ + + γ − )z,

and hence z =

−(α+ + β − )x . x + (γ + + γ − )

By substituting y in the Markoﬀ equation with the linear combination of x and z, we obtain (1 + (α+ )−2 )x2 + (1 + (γ + )2 )z 2 + xz(α+ x + γ + z + 2α+ γ + ) = 0. By substituting z with

−(α+ +β − )x x+(γ + +γ − ) ,

this equation turns into the equation

x + − α β (x + (γ + + γ − ))

2

f (x) = 0,

where f (x) is a quadratic polynomial in x. By putting x = X − (γ + + γ − ), f (x) turns into the quadratic polynomial g(X) := α+ β − (α+ β − − 1)X 2 + (α+ + β − )(α+ γ + + β − γ − )X + (α+ + β − )2 (1 − γ + γ − ).

The fact that ν = (s′2 , s1 ) implies that the coeﬃcient of X 2 is non-zero, and the fact that ν ∈ / Diagonal(τ (0) ) implies that the constant term is nonzero. Thus the above equation is equivalent to the equation x2 f (x) = 0, and

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5 Special examples

f (x) = 0 has two roots counted with multiplicity. Thus the equation has at most two non-zero roots. Hence we obtain the ﬁrst assertion. To see the second assertion, assume that ν ± ∈ int τ . Then we see f (x) = α+ β − (α+ β − − 1)x2 − 2iα+ β − (α+ + β − )x. Hence it has a unique non-zero root x=

2i(α+ + β − ) . (α+ β − − 1)

By noting that the coordinate of the point ν − in σ is (θ0− , 0, θ1− ), we see that this is equal to the non-zero root in Lemma 5.2.12. Thus we obtain the second assertion. Proof (Proof of Proposition 5.2.14). Let ϕj : (σ ′ ∩H2 )×(σ∩H2 ) → Φ (j = 0, 1) be continuous maps such that ϕ1 (ν) and ϕ2 (ν) correspond to the roots of the quadratic polynomial f (x) for ν in the proof of Lemma 5.2.14. We assume that if ν is thin then ϕ2 (ν) is the trivial Markoﬀ map. Then the type-preserving representation ρ induced by ϕ1 (ν) determines an isosceles representation (ρ, τ ) and ρ is quasifuchsian whenever ν is thin (Proposition 5.2.11). Since QF is open in X , there is a neighborhood U of int τ ×int τ in (σ ′ ∩H2 )×(σ ∩H2 ) such that the representation induced by ϕ1 (ν) is quasifuchsian for every ν ∈ U . Since ϕ2 (ν) is the trivial Markoﬀ map whenever ν is thin, we can choose U so that the representation induced by ϕ2 (ν) is not quasifuchsian for every ν ∈ U . Since ϕ1 (ν) and ϕ2 (ν) are the only (possibly) non-trivial algebraic roots for ν ∈ (σ ′ ∩ H2 ) × (σ ∩ H2 ), we obtain the desired conclusion, by setting φν = ϕ1 (ν). At the end of this section, we point out the following corollary, which gives another characterization of isosceles representations. Corollary 5.2.15. Under the assumption of Proposition 5.2.3, the following conditions are equivalent. 1. (ρ, τ ) is an isosceles representation, where τ = σ ∩ σ ′ . 2. The side parameters θ± (ρ, σ) and θ± (ρ, σ ′ ) are deﬁned, and θ− (ρ, σ ′ ; s′2 ) = θ+ (ρ, σ; s1 ) = 0. 3. The side parameters θ± (ρ, σ) and θ± (ρ, σ ′ ) are deﬁned, and θ− (ρ, σ; s1 ) = θ+ (ρ, σ ′ ; s′2 ) = 0 Proof. By Proposition 5.2.3, the ﬁrst condition implies the remaining conditions. Suppose the second condition is satisﬁed. Then, by Lemma 5.2.14, ρ is induced by the Markoﬀ map which is the non-trivial algebraic root for ((ν − , σ), (ν + , σ)), where ν − and ν + , respectively, are the points of τ determined by θ− (ρ, σ ′ ) and θ+ (ρ, σ). Hence, by Proposition 5.2.11, (ρ, τ ) is an isosceles representation. Similarly, the third condition implies the ﬁrst. This completes the proof.

5.3 Groups generated by two parabolic transformations

117

5.3 Groups generated by two parabolic transformations In this section, we study those type-preserving representations induced by Markoﬀ maps which vanish at some vertex of D. To describe geometric meaning of these representations, let (B 3 , t) be the pair of a 3-ball B 3 and a pair t of arcs properly embedded in B 3 as illustrated in Fig. 5.6, which is called a trivial tangle. A disk D properly embedded in B 3 which separate the two components of t is called a meridian disk of the trivial tangle. We identify D

t Fig. 5.6. Trivial tangle

∂B 3 − ∂t with the 4-times punctured sphere S = (R2 − Z2 )/G introduced in Sect. 2.1, so that the boundary of a meridian disk is identiﬁed with the essential simple loop α ˜ ∞ of slope ∞. Then π1 (B 3 −t) is identiﬁed with the quotient α∞ . Here we identify an (oriented) essential simple loop with group π1 (S)/˜ the (conjugacy class of an) element of the fundamental group represented by 2 the loop. Note that α ˜ ∞ = α∞ ∈ π1 (O), where α∞ is the essential simple loop in O (or in T ) of slope ∞. Let {Pj } be the sequence of elliptic generators associated with the triangle 0, 1, ∞ of D. (Thus (P0 , P1 , P2 ) = (P0 , Q0 , R0 ) in the presentation 2.3 by Convention 2.1.5.) Then, by Deﬁnition 2.1.3 and Proposition 2.1.2, we see α∞ = K −1 P2 = P0 P1 and hence α ˜ ∞ = (P0 P1 )2 . On the other hand, ˜∞ K1 K2 = K P0 K P1 = P0 (P2 P1 P0 )P0 · P1 (P2 P1 P0 )P1 = (P0 P1 )2 = α

−1 −1 ˜∞ P2 . K0 K3 = KK P2 = (P2 P1 P0 ) · P2 (P2 P1 P0 )P2 = P2 (P1 P0 )2 P2 = P2 α

Hence we obtain the following lemma. Lemma 5.3.1. Denote the image in π1 (B 3 − t) of the generator Ki in the presentation 2.2 of π1 (S) by the same symbol Ki (i ∈ {0, 1, 2, 3}). Then π1 (B 3 − t) is the free group freely generated by K0 and K1 . Moreover K2 = K1−1 and K3 = K0−1 in π1 (B 3 − t).

118

5 Special examples

The following lemma describes the meaning of those Markoﬀ maps which vanish at some vertex of D. Lemma 5.3.2. Let ρ : π1 (S) → P SL(2, C) be a type-preserving representation and φ a Markoﬀ map inducing ρ. Then ρ descends to a representation of π1 (B 3 − t) = K0 , K1 , if and only if φ(∞) = 0. Moreover if this condition is satisﬁed, then 10 11 ρ(K0 ) = , , ρ(K1 ) = ω1 01 where ω = −x2 with x = φ(0).

2 Proof. Since π1 (B 3 − t) = π1 (S)/˜ α∞ and since α ˜ ∞ = α∞ , ρ descends to 3 a representation of π1 (B − t) = K0 , K1 , if and only if ρ(α∞ ) is trivial or of order 2. However, ρ(α∞ ) cannot be trivial, because π1 (T ) = α∞ , α0 and ρ is irreducible. Hence the above condition is equivalent to the condition that ρ(α∞ ) is an elliptic transformation of order 2, which in tern is equivalent to the condition φ(∞) = 0. Hence we obtain the ﬁrst assertion. By the Markoﬀ identity, we see (φ(0), φ(1), φ(∞)) = (x, ±ix, 0) for some x ∈ C − {0}. Thus the formula for ρ(K0 ) and ρ(K1 ) are obtained from Lemma 2.3.7(3).

By the above lemma, the image of ρ : π1 (B 3 − t) → P SL(2, C) induced by a type-preserving representation is equal to the group 11 10 Gω = , , 01 ω1 for some ω ∈ C − {0}. Discreteness problem for the non-elementary twoparabolic groups Gω has been studied extensively by various authors, including [20, 48, 1, 45]. In particular, the following facts have been proved. Proposition 5.3.3. (1) (Brenner [20]) If |ω| ≥ 4, then Gω is discrete and free. (2) (Knapp [48]) If ω is real, then Gω is discrete if and only if ω or −ω belongs to [4, ∞) ∪ {4 cos2 (π/n) | n ∈ N}. The following characterization of the discrete two-parabolic groups of co-ﬁnite volume is proved by Adams by using the orbifold uniformization theorem. Proposition 5.3.4 (Adams [1]). Gω is discrete and vol(H3 /Gω ) is ﬁnite, if and only if Gω is isomorphic to the link group, π1 (S 3 − K), of a hyperbolic 2-bridge link K. R. Riley devoted massive eﬀort to identify and to plot those complex numbers corresponding to the 2-bridge link groups and produced a mysterious output [69] (Fig. 0.2a). It also outlines the following domain, which is now called the Riley slice of the Schottky space (Fig. 0.2b).

5.3 Groups generated by two parabolic transformations

119

Deﬁnition 5.3.5 (Riley slice). The Riley slice R of the Schottky space is the subspace of C consisting of those complex numbers ω such that Gω is discrete and free and that the quotient Ω(Gω )/Gω of the domain of discontinuity is homeomorphic to the 4-times punctured sphere S. Keen-Series[45] (see also Komori-Series[49]) introduces the pleating coordinates on the Riley slice R and produced a beautiful picture of R. In the forthcoming paper [11], we add to their picture those complex numbers ω corresponding to the 2-bridge link groups, by extending Jorgensen’s theory to the outside of quasifuchsian space (cf. Fig. 0.2b and the announcement [9]). In the remainder of this section, we describe the Ford domains of the Kleinian groups ρ(π1 (O)) which are ﬁnite extensions of those groups in Proposition 5.3.3. We begin with the following elementary facts. Lemma 5.3.6. Let φ be a nontrivial Markoﬀ map with φ(∞) = 0. Then the following hold. 1. There is a sign ǫ ∈ {−, +} and a complex number x ∈ C − {0}, such that φ(k) = (ǫi)k x for every integer k. 2. Let {Pj } be the sequence of elliptic generators associated with σ = 0, 1, ∞. Then the isometric hemisphere Ih(ρ(Pj )) is deﬁned if and only if j ≡ 2 (mod 3), and ρ(P3j+2 ) is the π-rotation around the vertical geodesic above the point c(ρ(P3j )) + (1/2). Moreover, Ih(ρ(P3j )) = Ih(ρ(P3j+1 )), and ρ(P3j+1 P3j ) is the π-rotation around the vertical geodesic above the point c(ρ(P3j )). 3. Let σ ′ = s′0 , s′1 , s′2 be a triangle of D such that s′0 , s′2 is adjacent to ∞, i.e., s′0 , s′2 = k + 1, k for some k ∈ Z, and s′1 = ∞. Then the complex probability of ρ at σ ′ is equal to (a, 1, −a) for some a ∈ C − {0}. In ′ ′ )), )) = Ih(ρ(P3j+2 particular, L(ρ, σ ′ ) is folded at c(ρ(P1′ )), and Ih(ρ(P3j ′ ′ where {Pj } is the sequence of elliptic generators associated with σ . Proof. (1) As in the proof of Lemma 5.3.2, we see by Markoﬀ identity that (φ(0), φ(1), φ(∞)) = (x, ǫix, 0) for some x ∈ C − {0} and a sign ǫ ∈ {−, +}. Since k, k + 1, ∞ is a triangle of D, we see by Proposition 2.3.4(2) that φ(k − 1) + φ(k + 1) = φ(k)φ(∞) = 0. Hence we obtain the desired formula. (2) This is equivalent to Proposition 2.4.4(2). (3) By (1) we see (φ(s′0 ), φ(s′2 )) = (x, ǫix) for some ǫ ∈ {−, +}. Thus we have φ(s′1 ) = φ(s′0 )φ(s′2 ) − φ(∞) = ǫix2 by Proposition 2.3.4(2). Hence the complex probability (a′0 , a′1 , a′2 ) of ρ at σ ′ is equal to (−1/x2 , 1, 1/x2 ). Thus we obtain the ﬁrst assertion, which implies the remaining assertions. Proposition 5.3.3, together with the above lemma, motivates the following deﬁnition. Deﬁnition 5.3.7. (1) (Brenner representation) A type-preserving represenˆ if φ(s) = 0 and tation ρ is called a Brenner representation of slope s ∈ Q |φ(s′ )| ≥ 2 for some (or every) vertex s′ adjacent to s.

120

5 Special examples

(2) (Knapp representation) A type-preserving representation ρ is called a ˆ and of degree n ≥ 3 if φ(s) = 0 and Knapp representation of slope s ∈ Q ′ φ(s ) = ±2 cos(π/n) or ±2i cos(π/n) for some vertex s′ adjacent to s. Then we have the following propositions. Proposition 5.3.8. Let ρ be a Brenner representation of slope s. Pick a triangle σ = s0 , s1 , s2 of D with s2 = s, and let {Pj } be the sequence of elliptic generators associated with σ. Then ρ is discrete, and the Ford domain of ρ(π1 (O)) is equal to the common exterior ∩j Eh(ρ(P3j )) of mutually disjoint isometric hemispheres {Ih(ρ(P3j ))} (see Fig. 5.7). Moreover, we have the following group presentations: ρ(π1 (O)) ∼ = P0 , P1 , P2 | P02 = P12 = P22 = 1, (P0 P1 )2 = 1, ρ(π1 (S)) ∼ = K0 , K1 . Proof. By Lemma 5.3.6 and the fact that r(ρ(P3j )) = 1/|φ(s0 )| ≤ 1/2, we see that the isometric hemispheres {Ih(ρ(P3j ))} are mutually disjoint. Consider the fundamental domain R0 = {(z, t) ∈ H3 | |ℜ(z − c(ρ(P0 )))| ≤ 1/2,

ℑ(z − c(ρ(P0 ))) ≥ 0}

of the inﬁnite dihedral Kleinian group ρ(K), ρ(P0 P1 ), and set R := R0 ∩ Eh(ρ(P0 )). Then we can check that the polyhedron R, together with the face pairing isometries ρ(K), ρ(P1 P0 ), ρ(P0 ) and ρ(P1 ), satisﬁes the conditions of Poincare’s theorem on fundamental polyhedra (see Fig. 5.8). In the extreme case where φ(s0 ) (resp. φ(s1 )) is real, ρ(P1 ) (resp. ρ(P0 )) does not belong to the set of the face pairing isometries. This implies the discreteness of ρ and the desired group presentation of ρ(π1 (O)). This also implies that ∩j Eh(ρ(P3j )) is a fundamental domain of ρ(π1 (O)) modulo the stabilizer ρ(K), ρ(P0 P1 ) of ∞. Hence we see that ∩j Eh(ρ(P3j )) is the Ford domain of ρ(π1 (O)). The group presentation of ρ(π1 (S)) is obtained through the Reidemeister-Schreier method by using the following isomorphism. ρ(π1 (O))/ρ(π1 (S)) ∼ = P0 | P02 = 1 ⊕ P1 | P12 = 1,

where P2 = P0 P1 .

Proposition 5.3.9. Let ρ be a Knapp representation of slope s and of order n. Pick a triangle σ = s0 , s1 , s2 of D with s2 = s and φ(s0 ) = ±2 cos(π/n), and let {Pj } be the sequence of elliptic generators associated with σ. Then ρ is discrete, and the Ford domain of ρ(π1 (O)) is equal to ∩j Eh(ρ(P3j )) (see Fig. 5.9). Moreover, we have the following group presentations: ρ(π1 (O)) ∼ = P0 , P1 , P2 | P02 = P12 = P22 = 1, (P0 P1 )2 = 1, (P1 P2 )n = 1, ρ(P1 ), ρ(K) ∼ = P0 , K | P02 = (KP0 )n = 1 ∼ = π1 (O(2, n, ∞)), if n:odd, ∼ π1 (O(2, n, ∞)) ρ(π1 (S)) = π1 (O(n/2, ∞, ∞)) if n:even.

5.3 Groups generated by two parabolic transformations

121

K

P2

P0

ℜ

P1

Fig. 5.7. Ford domain of a Brenner representation

P1 P0

P0 P1

ℜ

ℑ K

Fig. 5.8. Fundamental domain of a Brenner representation

122

5 Special examples

Here O(p, q, r) denotes the 2-dimensional hyperbolic orbifold with underlying space S 2 and with cone points of indices p, q, r, where a cone point of index ∞ is regarded as a puncture; in particular, π1 (O(p, q, r)) is the triangle group x, y, z | xp = y q = z r = xyz = 1 of type (p, q, r). Proof. By Lemma 5.3.6 and the fact r(ρ(P3j )) = 1/|φ(s0 )| = 1/(2 cos(π/n)), we see that the dihedral angle of the polyhedron Eh(ρ(P0 )) ∩ Eh(ρ(P3 )) along the edge Ih(ρ(P0 )) ∩ Ih(ρ(P3 )) is equal to 2π/n. Let f := c(ρ(P0 )) + (1/2) be the ﬁxed point in C of the π-rotation ρ(P2 ) (cf. Lemma 5.3.6(2)). Let R0 be the fundamental domain of the inﬁnite dihedral Kleinian group ρ(K), ρ(P0 P1 ) introduced in the proof of Proposition 5.3.8. Then we can check that the intersection R := R0 ∩ Eh(ρ(P0 )), together with the face pairing isometries ρ(K), ρ(P1 P0 ), ρ(P0 ), satisﬁes the conditions of Poincare’s theorem on fundamental polyhedra (see Fig. 5.10). This implies the discreteness of ρ and the desired group presentation of ρ(π1 (O)). This also implies that ∩j Eh(ρ(P3j )) is a fundamental domain of ρ(π1 (O)) modulo the stabilizer ρ(K), ρ(P0 P1 ) of ∞. Hence we see that ∩j Eh(ρ(P3j )) is the Ford domain of ρ(π1 (O)). The remaining group presentations are obtained through the Reidemeister-Schreier method by using the following isomorphisms. ρ(π1 (O))/ρ(P1 ), ρ(K) ∼ = P1 | P12 = 1, where P0 = 1 and P2 = P1 . ρ(π1 (O))/ρ(π1 (S)) P1 | P12 = 1, where P0 = 1 and P2 = P1 if n:odd, ∼ = 2 2 P0 | P0 = 1 ⊕ P1 | P1 = 1, where P2 = P0 P1 if n:even. The following proposition gives a partial converse to Propositions 5.3.8 and 5.3.9. (Though we conjecture that the converse also holds, we have not been able to prove it.) Proposition 5.3.10. Let ρ be a discrete type-preserving representation, and suppose that the Ford domain P h(ρ) is equal to ∩{Ih(ρ(P )) | s(P ) = s} for some slope s. Then the image of ρ is isomorphic to the image of either a Brenner representation or a Knapp representation. In particular, ρ does not belong to QF. Proof. Let {Pj } be a sequence of elliptic generators such that s(P0 ) = s. We ﬁrst show that the stabilizer Γ∞ of the point ∞ in the group Γ := ρ(π1 (O)) is strictly larger than the inﬁnite cyclic group ρ(K). Suppose to the contrary ˜ := that this is not the case. Then a fundamental domain of Γ is given by R ˜ R0 ∩ Eh(ρ(P0 )), where ˜ 0 = {(z, t) ∈ H3 | |ℜ(z − c(ρ(P0 )))| ≤ 1/2} R is a fundamental domain of ρ(K). Then we have the following claim.

˜ conClaim 5.3.11. The set of gluing isometries for the fundamental domain R sists of ρ(K) and ρ(P0 ), and Γ has one of the following presentations.

5.3 Groups generated by two parabolic transformations

123

K KP0

P0 ℜ

P1

Fig. 5.9. Ford domain of a Knapp representation P1 P0

π/n

P0

ℜ ℑ K

Fig. 5.10. Fundamental domain of a Knapp representation

124

5 Special examples

P0 , K | P02 = (KP0 )n = 1 P0 , K

| P02

n

= [K, P0 ] = 1

where n is an integer with n ≥ 3 or n = ∞,

where n is an integer with n ≥ 2 or n = ∞.

˜ is the disjoint union of Proof. Suppose Ih(ρ(P0 )) ∩ Ih(ρ(P3 )) = ∅. Then ∂ R Ih(ρ(P0 )) and two vertical planes. The face Ih(ρ(P0 )) is paired with itself by ρ(P0 ) and the vertical planes are paired with each other by ρ(K). Hence we obtain the ﬁrst assertion. By applying Poincare’s theorem on fundamental polyhedra to this fundamental domain, we have Γ ∼ = = P0 , K | P02 = 1 ∼ ∞ 2 P0 , K | P02 = (KP0 )∞ = 1 ∼ = [K, P ] = 1. P , K | P = 0 0 0 ˜ is the union of the face Suppose Ih(ρ(P0 )) ∩ Ih(ρ(P3 )) = ∅. Then ∂ R ˜ and two vertical faces. If proj(Axis(ρ(P0 ))) is not parallel to Ih(ρ(P0 )) ∩ ∂ R the real axis nor the imaginary axis, then ρ(P0 )(int P h(ρ)) has a non-empty intersection with int P h(ρ), a contradiction. Hence proj(Axis(ρ(P0 ))) is paral˜ lel to the real axis or the imaginary axis, and therefore the face Ih(ρ(P0 ))∩∂ R is paired to itself by ρ(P0 ). Moreover the vertical faces are paired with each other by ρ(K). Hence we obtain the ﬁrst assertion. By applying Poincare’s theorem on fundamental polyhedra to this fundamental domain, we see that Γ is isomorphic to P0 , K | P02 = (KP0 )n = 1

or

P0 , K | P02 = [K, P0 ]n/2 = 1,

according as proj(Axis(ρ(P0 ))) is parallel to the real axis or the imaginary axis. Here n is the integer such that the dihedral angle of Eh(ρ(P0 )) ∩ Eh(ρ(P3 )) along Ih(ρ(P0 )) ∩ Ih(ρ(P3 )) is equal to 2π/n. Since n ≥ 3, we obtain the conclusion. In what follows we show either group presentation leads to a contradiction. Case 1. Γ ∼ = P0 , K | P02 = (KP0 )n = 1 where n is an integer with n ≥ 3 or n = ∞. Set X = KP0 ∈ Γ . Then Γ ∼ = P0 | P02 ∗ X | X n = 1 and H1 (Γ ) ∼ = [P0 ] | 2[P0 ] = 0 ⊕ [X] | n[X] = 0. n

Hence each order 2 element of Γ is conjugate to P0 or K 2 with n even. This implies that the homology class of ρ(K) = ρ(P2 P1 P0 ) is equal to one of the following homology classes. n n n n [P0 ]+[P0 ]+[P0 ] = [P0 ], [P0 ]+[P0 ]+ [X] = ([X]), [P0 ]+ [X]+ [X] = [P0 ]. 2 2 2 2 Since n ≥ 3, none of them is equal to [K] = [P0 ] + [X], a contradiction. Case 2. Γ ∼ = P0 , K | P02 = [K, P0 ]n = 1 where n is an integer with n ≥ 2 or n = ∞. Then H1 (Γ ) ∼ = [P0 ] | 2[P0 ] = 0 ⊕ [K]. Thus the homology classes of ρ(Pj ) is either [P0 ] or 0 for every integer j. In particular the homology class of ρ(K) = ρ(P2 P1 P0 ) is a (possibly trivial) torsion element. This contradicts the fact that [K] is of inﬁnite order.

5.3 Groups generated by two parabolic transformations

125

Thus we have proved that Γ∞ is strictly larger than ρ(K). Next we show that Γ∞ is an inﬁnite dihedral group. To this end, let T be an element of Γ∞ which does not belong to ρ(K). After composing a power of ρ(K) we may assume T preserves Ih(ρ(P0 )). Since T is an Euclidean isometry, this implies that T is a rotation about the vertical geodesic above c(ρ(P0 )). Since T preserves P h(ρ) = ∩j Eh(ρ(P3j )), the angle of rotation of T must be equal to π. Thus Γ∞ is the inﬁnite dihedral group ρ(K), T . Let R0 and R be as in the proofs of Propositions 5.3.8 and 5.3.9. Then R is a fundamental domain of Γ and we can see that the set of gluing isometries consist of T , ρ(K), ρ(P0 ) and ρ(P0 )T . (In the special case when the proj(Axis(ρ(P0 ))) or proj(Axis(ρ(P0 )T )) is parallel to the real line, ρ(P0 ) or T accordingly does not give a face pairing.) By applying Poincare’s theorem on fundamental polyhedra, we see Γ ∼ = P0 , K, T | P02 = T 2 = (P0 T )2 = (T K)2 = (KP0 )n = 1. Set Q = T P0 and R = KP0 Q. Then we see Γ ∼ = P0 , Q, R | P02 = Q2 = R2 = 1, (P0 Q)2 = 1, (QR)n = 1. This completes the proof. At the end of this section, we prove two lemmas for the Markoﬀ maps which vanish at ∞. They are analogies of [44, Proposition 3.1] for the Markoﬀ maps which takes the value 2 at ∞, and it is used in Sect. 9.1. Lemma 5.3.12. Let Φ0 be the space of Markoﬀ maps, including the trivial one, such that (φ(1/0), φ(0/1), φ(1/1)) = (0, x, ix) for some x ∈ C, and identify Φ0 with the complex plane by the correspondence ˆ let Vr : Φ0 → C be the function deﬁned by φ → x = φ(0/1). For each r ∈ Q, Vr (φ) = φ(r). Then Vr is a polynomial in the variable x, which we denote by V [r], and satisﬁes the following conditions. 1. V [r + 1](x) = V [r](ix) and V [−r](x) = V [r](x). Here V [r](x) denotes the polynomial obtained from V [r](x) by converting each coeﬃcient into its complex conjugate. 2. For each rational number q/p ∈ [0, 1/2], V [q/p](x) = iq (xp − cq/p xp−2 + (lower terms)), for some integer cq/p . Here cq/p = 0 if q/p = 0/1 or 1/2. If q/p ∈ (0, 1/2), then cq/p > 0. Proof. (1) is proved by induction on the depth of r, i.e., the edge path distance between ∞ and r in D, by using the identity in Proposition 2.3.4(2). (2) Since V [0/1] = ix and V [1/2] = ix2 , the assertion is valid for 0/1 and 1/0. We show that the assertion is valid for (q1 + q2 )/(p1 + p2 ) ∈ (0, 1/2) by

126

5 Special examples

assuming that it is valid for q1 /p1 , q2 /p2 and (q2 − q1 )/(p2 − p1 ),where p1 , p2 , q q q1 and q2 are integers such that 1 ≤ p1 ≤ p2 , 0 ≤ q1 ≤ q2 and 1 1 = ±1. p1 p2 By Proposition 2.3.4(2), V [(q1 + q2 )/(p1 + p2 )]

= V [q1 /p1 ]V [q2 /p2 ] − V [(q2 − q1 )/(p2 − p1 )] = iq1 (xp1 − cq1 /p1 xp1 −2 + (lower terms))

× iq2 (xp2 − cq2 /p2 xp2 −2 + (lower terms))

− iq2 −q1 (xp2 −p1 + (lower terms))

= iq1 +q2 (xp1 +p2 − (cq1 /p1 + cq2 /p2 )xp1 +p2 −2 − (−1)q1 xp2 −p1 + (lower terms)).

If p1 ≥ 2, then p1 + p2 − 2 > p2 − p1 and hence the assertion holds, and we have c(q1 +q2 )/(p1 +p2 ) = cq1 /p1 + cq2 /p2 . If p1 = 1, then q1 = 0 and p1 + p2 − 2 = p2 − p1 . Thus the coeﬃcient of xp1 +p1 −2 is equal to −iq1 +q2 (cq1 /p1 + cq2 /p2 + 1) = −iq1 +q2 (cq2 /p2 + 1). Hence the assertion also holds, and we have c(q1 +q2 )/(p1 +p2 ) = cq1 /p1 + cq2 /p2 + 1 = cq2 /p2 + 1. Remark 5.3.13. It is easy to see that there is an integral polynomial F [q/p] such that V [q/p](x) is equal to iq xF [q/p](x2 ) or iq x2 F [q/p](x2 ) according as p is odd or even. This polynomial is essentially equal to the polynomial f introduced in [71, Sect. II.5], which gives the hyperbolicity equation for the complement of the 2-bridge link K(q/p). Lemma 5.3.14. For each integer n ≥ 2, we have V [n/(2n + 1)](x) = in (x2n+1 − x2n−1 − (n − 1)x2n−3 + (lower terms)), where V [n/(2n + 1)](x) is as in Lemma 5.3.12. Proof. This is proved by induction on n. As noted in Remark 5.3.13, the corresponding polynomial F [n/(2n + 1)] gives the hyperbolicity equation for the twist knot K(n/(2n + 1)) and detailed study of the polynomial has been made in [37] and [36].

5.4 Imaginary representations Deﬁnition 5.4.1. (Imaginary representation) A type-preserving representation ρ is called an imaginary representation if it is induced by an imaginary Markoﬀ map φ, i.e., a non-real Markoﬀ map whose image is contained in R ∪ iR (see [17, Sect. 5]).

5.5 Representations with accidental parabolic/elliptic transformations

127

By using the identities 2.4 and 2.5, we can easily obtain the following characterization of the imaginary representations. Lemma 5.4.2. Let ρ be a type-preserving representation of π1 (O) and ψ the complex probability map corresponding to ρ. Then the following conditions are equivalent. 1. ρ is an imaginary representation. 2. The values of φ at the vertices of some triangle of D are contained in R ∪ iR, but not in R. 3. The value (a0 , a1 , a2 ) of ψ at some triangle of Dφ consists of real numbers, such that at least one of which is negative. 4. The value (a0 , a1 , a2 ) of ψ at any triangle of Dφ consists of real numbers, such that at least one of which is negative. In particular, if ρ is doubly folded at some triangle, then ρ is an imaginary representation. By using a result of Bowditch [17], we have the following proposition. Proposition 5.4.3. Let ρ be an imaginary representation. Then ρ is discrete if and only if it is either a Brenner representation or a Knapp representation. In particular, no representation in QF is imaginary. Proof. By [17, Proposition 5.1], an imaginary representation ρ can be discrete only when the corresponding Markoﬀ map φ vanish at some vertex, say s. Let s′ be a vertex adjacent to s and put x = φ(s′ ) and ω = −x2 . Then the twoparabolic group Gω is a ﬁnite index subgroup of ρ(π1 (O)) by Lemma 5.3.2, and therefore ρ is discrete if and only if Gω is discrete. On the other hand, ω is real, because ρ is imaginary and x belongs to R ∪ iR. Hence Gω is discrete if and only ω or −ω belongs to [4, ∞) ∪ {4 cos2 (π/n) | n ∈ N}. by Proposition 5.3.3(2). This is equivalent to the condition that ρ is either a Brenner representation or a Knapp representation of slope s. The last assertion follows from the fact that Brenner representations and Knapp representations are not faithful.

5.5 Representations with accidental parabolic/elliptic transformations In this section we study those representations which send a generator of π1 (T ) to a parabolic or elliptic transformation. The results in this section is used in Sects. 6.7 and 8.7.

128

5 Special examples

Lemma 5.5.1. Under Assumption 2.4.6 (NonZero), suppose that L(ρ, σ) is not folded at c(ρ(P1 )) nor c(ρ(P2 )). Then the following conditions are equivalent: (1) ρ(KP0 ) is an elliptic transformation. (2) Axis(ρ(P1 )) ∩ Axis(ρ(P2 )) = ∅. (3) ∩3j=0 Ih(ρ(Pj )) = ∅. Moreover, if the above mutually equivalent conditions are satisﬁed, then ∩3j=0 Ih(ρ(Pj )) = Axis(ρ(P1 )) ∩ Axis(ρ(P2 )) and it is a singleton contained in the axis of the elliptic transformation ρ(KP0 ) (see Fig. 5.11).

Axis(ρ(KP0 )) i/φ(s2 )

i/φ(s1 ) P3

P0

i/φ(s0 )

i/φ(s0 )

a0 a2 P1

a1

P2

Fig. 5.11. Frontier of elliptic type: this ﬁgure is for ǫ = +.

Proof. (1)⇒(2) This follows from the general fact that ρ(KP0 ) = ρ(P2 P1 ) is elliptic, parabolic or loxodromic according as Axis(ρ(P1 )) and Axis(ρ(P2 )) share a common point, share a single common endpoint, or Axis(ρ(P1 )) and Axis(ρ(P2 )) are disjoint. (2)⇒(3) Suppose the axes of ρ(P1 ) and ρ(P2 ) have a common point, say x. Since Axis(ρ(Pj )) ⊂ Ih(ρ(Pj )), we see x ∈ Ih(ρ(P1 )) ∩ Ih(ρ(P2 )). By using Lemma 4.1.3(2) (chain rule) as in the above, we see that x ∈ Ih(ρ(P0 )) ∩ Ih(ρ(P1 )) and x ∈ Ih(ρ(P2 )) ∩ Ih(ρ(P3 )). Hence ∩3j=0 Ih(ρ(Pj )) contains x, and therefore it is non-empty. (3)⇒(1). Suppose that ∩3j=0 Ih(ρ(Pj )) = ∅. Then both ∩2j=0 Ih(ρ(Pj )) and 3 ∩j=1 Ih(ρ(Pj )) are non-empty. Since ρ(P1 ) interchanges Ih(ρ(P0 ))∩Ih(ρ(P1 )) and Ih(ρ(P1 )) ∩ Ih(ρ(P2 )) by Lemma 4.1.3(2) (Chain rule), ∩2j=0 Ih(ρ(Pj )) is preserved by ρ(P1 ). Similarly ∩3j=1 Ih(ρ(Pj )) is preserved by ρ(P2 ). Thus if both of them are singletons, then they are ﬁxed by ρ(P1 ) and ρ(P2 ), respectively. Moreover both of them must be equal to the non-empty set ∩3j=0 Ih(ρ(Pj )). Hence ρ(KP0 ) = ρ(P2 P1 ) ﬁxes the singleton ∩3j=0 Ih(ρ(Pj )) in H3 , and hence ρ(KP0 ) is elliptic. Suppose that either ∩2j=0 Ih(ρ(Pj )) or

5.5 Representations with accidental parabolic/elliptic transformations

129

∩3j=1 Ih(ρ(Pj )) is not a singleton. For simplicity, we assume that ∩2j=0 Ih(ρ(Pj )) is not a singleton. (The other case can be treated similarly.) Then it must be a complete geodesic. Thus by Lemma 5.2.6 and the assumption that L(ρ, σ) is not folded at c(ρ(P1 )), (ρ, τ ) is an isosceles representation, where τ = s0 , s2 . So ∩2j=0 Ih(ρ(Pj )) = Axis(ρ(P1 )) by Proposition 5.2.3. Moreover Ih(ρ(P2 )) ∩ Ih(ρ(P3 )) = Ih(ρ(P1′ )) ∩ Ih(ρ(P3′ )) = Axis(ρ(P2′ )) by Proposition 5.2.3, and therefore it is a geodesic, where Pj′ are as in the proposition. Hence ∩3j=1 Ih(ρ(Pj )) = (Ih(ρ(P1 )) ∩ Ih(ρ(P2 ))) ∩ (Ih(ρ(P2 )) ∩ Ih(ρ(P3 ))) = Axis(ρ(P1 )) ∩ Axis(ρ(P2′ )).

Since (ρ, τ ) is an isosceles representation, this implies that ∩3j=1 Ih(ρ(Pj )) is a singleton. Thus it is ﬁxed by ρ(P2 ) by the previous argument. Since it is also ﬁxed by ρ(P1 ), because it is contained in Axis(ρ(P1 )). Hence ρ(KP0 ) = ρ(P2 P1 ) ﬁxes the singleton, and hence ρ(KP0 ) is elliptic. Thus we have proved that the three conditions are equivalent. The remaining assertion is already proved in the above argument. Remark 5.5.2. In the above lemma, we cannot drop the condition that L(ρ, σ) is not folded at any vertex. In fact we can easily ﬁnd a representation ρ such that L(ρ, σ) is folded at c(ρ(P1 )), ∩3j=0 Ih(ρ(Pj )) = ∅ but that ρ(KP0 ) is not elliptic. There exists also a representation ρ such that L(ρ, σ) is doubly folded, ρ(KP0 ) is elliptic and that ∩3j=0 Ih(ρ(Pj )) is a complete geodesic. By a similar argument, we can also prove the following lemma: Lemma 5.5.3. Under Assumption 2.4.6 (NonZero), suppose that L(ρ, σ) is not folded at c(ρ(P1 )) nor c(ρ(P2 )). Then the following conditions are equivalent: (1) ρ(KP0 ) is a parabolic transformation. (2) Axis(ρ(P1 )) and Axis(ρ(P2 )) share a unique common endpoint. (3) The four isometric circles I(ρ(Pj )) (0 ≤ j ≤ 4) have a unique common point. Moreover, if the above mutually equivalent conditions are satisﬁed, then the parabolic ﬁxed point of ρ(KP0 ) is equal to c(ρ(P0 )) + (1/2). Furthermore, this point is equal to ∩3j=0 I(ρ(Pj )) = Axis(ρ(P1 )) ∩ Axis(ρ(P2 )) (see Fig. 5.12). Proof. All assertions, except the location of the parabolic ﬁxed point, are proved by arguments parallel to the proof of Lemma 5.5.1. To prove the remaining assertion, suppose ρ(KP0 ) is parabolic. Then φ(s0 ) = ±2 and ρ(P0 ) is the π-rotation around the geodesic with endpoints c(ρ(P0 )) ± (i/2) by Proposition 2.4.4(1.1). Hence ρ(KP0 )(c(ρ(P0 )) + (1/2)) = ρ(K)(c(ρ(P0 )) − (1/2)) = c(ρ(P0 )) + (1/2). Hence the parabolic ﬁxed point of ρ(KP0 ) is equal to c(ρ(P0 )) + (1/2).

130

5 Special examples Fix(ρ(KP0 )) P0

i/φ(s1 )

i/φ(s2 )

P3 i/φ(s0 )

i/φ(s0 )

a0 P1

P2

a2

a1

Fig. 5.12. Frontier of parabolic type: this ﬁgure is for ǫ = +.

In the remainder of this section, we present further properties of representations which send KP0 to a parabolic transformation. Lemma 5.5.4. Under Assumption 4.2.4 (σ-Simple), assume that ρ(KP0 ) is parabolic. Then the following hold (see Fig. 5.12). (1) arg a1 = − arg a2 ∈ (−π/2, 0) ∪ (0, π/2), where (a0 , a1 , a2 ) is the complex probability of ρ at σ. (2) For the parabolic ﬁxed point, Fix(ρ(KP0 )), of ρ(KP0 ), the following hold. Fix(ρ(KP0 )) = c(ρ(P0 )) + (1/2) + = Fix+ σ (ρ(P1 )) = Fixσ (ρ(P2 ))

= v + (ρ; P0 , P1 ) = v + (ρ; P1 , P2 ) = v + (ρ; P2 , P3 ), where ǫ = − or + according as arg a1 is contained in (−π/2, 0) or (0, π/2). (3) θǫ (ρ, σ) = (π/2, 0, 0), where ǫ is as in the above. Conversely, if θǫ (ρ, σ) = (π/2, 0, 0) for some ǫ ∈ {−, +}, then ρ(KP0 ) is parabolic. Proof. Under Assumption 4.2.4 (σ-Simple), suppose ρ(KP0 ) is parabolic. Then a1 a2 = 1/φ(s0 )2 = 1/4, we have arg a1 ≡ − arg a2 (mod 2π). We can easily see that if this argument belongs to either [−π, −π/2] or [π/2, π] then L(ρ, σ) is not simple. Hence the argument belongs to (−π/2, π/2). We show that the argument is not equal to 0. Suppose this is not the case. Then a1 , a2 and a0 = 1 − a1 − a2 are non-zero real numbers. Since ρ is not an imaginary representation, a0 , a1 and a2 are positive real numbers by Lemma 5.4.2. Hence √ a1 + a2 ≥ 2 a1 a2 = 2/|φ(s0 )| = 1, and hence a0 = 1 − a1 − a2 ≤ 0. Since a0 = 0, we have a0 < 0, a contradiction. Hence we obtain the ﬁrst assertion, arg a1 = − arg a2 ∈ (−π/2, 0) ∪ (0, π/2). To prove the second and third assertion, set f = c(ρ(P0 )) + (1/2) = c(ρ(P3 )) − (1/2). Then Lemma 5.5.3 says that f is equal to the parabolic ﬁxed point of ρ(KP0 ) and that it is the unique common ﬁxed point of ρ(P1 )

5.5 Representations with accidental parabolic/elliptic transformations

131

and ρ(P2 ). Suppose ﬁrst that arg a1 = − arg a2 ∈ (0, π/2). Then ℑ(c(ρ(P1 ))) and ℑ(c(ρ(P2 ))) are smaller that ℑ(f ), and hence arg

c(ρ(P0 )) − c(ρ(P1 )) ∈ (0, π) f − c(ρ(P1 ))

and

arg

f − c(ρ(P2 )) c(ρ(P3 )) − c(ρ(P2 ))

∈ (0, π).

+ Thus we have f = Fix+ σ (ρ(P1 )) = Fixσ (ρ(P2 )) by deﬁnition. This implies + that the point f is also equal to v (ρ; P1 , P2 ). By Lemma 4.2.7, this in turn implies v + (ρ; P0 , P1 ) = v + (ρ; P1 , P2 ) = v + (ρ; P2 , P3 ). Thus we have the second assertion. This also implies θ+ (ρ, σ; s1 ) = θ+ (ρ, σ; s2 ) = 0, and therefore we obtain the third assertion θ+ (ρ, σ) = (π/2, 0, 0). By a parallel argument we can also show that if arg a1 = − arg a2 ∈ (−π/2, 0) then θ− (ρ, σ) = (π/2, 0, 0). Thus we obtain the assertion (3). Conversely, suppose θǫ (ρ, σ) = (π/2, 0, 0) for some ǫ ∈ {−, +}. Then we have Fixǫσ (ρ(P1 )) = v ǫ (ρ; P1 , P2 ) = Fixǫσ (ρ(P2 )).

Hence ρ(KP0 ) = ρ(P2 P1 ) is parabolic (cf. Lemma 5.5.1). This completes the proof of Lemma 5.5.4. We occasionally need to study the sequence of elliptic generators associated with a chain all of whose triangles share a common vertex, which is explained in the following: Notation 5.5.5 (InﬁniteFan). For the sequence of elliptic generators {Pj } (k) associated with a triangle σ = s0 , s1 , s2 of D, the symbols {Pj }j (k ∈ Z) denote the sequences of elliptic generators obtained from {Pj } by the following recursive formula: (0)

(0)

(0)

(k+1)

) = (P0 , P2 , P2 P1 P2 )

(P0 , P1 , P2 ) = (P0 , P1 , P2 ) (k+1)

(P0

(k+1)

, P1

(k)

, P2

(k)

(k)

(k)

(k)

(k)

(k)

We set σ (k) = s(P0 ), s(P1 ), s(P2 ). It should be noted that (· · · , σ (−1) , σ (0) , σ (1) , · · · ) forms an “inﬁnite fan”, or an “bi-inﬁnite chain” sharing s0 as the common vertex (see Fig. 5.13). Lemma 5.5.6. Let ρ be a type-preserving representation such that ρ(KP0 ) is parabolic, and let φ be a Markoﬀ map inducing ρ. Then the following hold under Notation 5.5.5 (InﬁniteFan). (k) (1) If φ(s(P1 )) = 0 for some k, then ρ is an imaginary representation. (k) (k) (k) (2) Suppose ρ is not an imaginary representation. Let (a0 , a1 , a2 ) be the complex probability of ρ at σ (k) . Then the following hold: (k)

(k)

(i) arg a1 = − arg a2 ∈ (−π, 0) ∪ (0, π) for every k. (k) (k) (ii) The sign of arg a1 = − arg a2 does not depend on k. In other words, (k) the sign of ℑaj does not depend on k for each j = 1, 2 (see Fig. 5.13).

132

5 Special examples (3)

(4)

P2 P0 =

(1)

σ (2)

P2

σ (1) s0

(0)

= P1

(0)

= P2

P2

(2)

P2

(2)

= P1

= P1

(k) P0

(k)

P3 = P3

(0)

(3)

P2 = P2

= P1

(1)

= P1

(0)

P1 = P1

(1)

(1)

P2

(2)

= P1

σ (0) σ (−1) σ

(−2)

P1

(−1)

P1

(−1)

(−3)

P1 P0 =

(−2)

= P2

(−4)

= P2

(k) P0

(k)

P3 = P3

(0)

(−2)

P1

(−3)

= P2 (0) (−1) P1 = P1 = P2 (−1)

P1

P2 = P2 (−2)

= P2

Fig. 5.13. Inﬁnite Fan

(k)

Proof. (1) Suppose φ(s(P1 )) = 0 for some k. Then (k+1)

φ(s(P1

(k)

(k)

)) = φ(s(P2 )) = ±iφ(s(P0 )) = ±iφ(s0 ) ∈ iR.

Hence ρ is an imaginary representation by Lemma 5.4.2. (2) We note that the complex probability of ρ at σ (k) is well-deﬁned by (k) the ﬁrst assertion. We see by the proof of Lemma 5.5.4(1) that arg a1 = (k) (k) (k) − arg a2 . Since ρ is not an imaginary representation, a1 and a2 are not (k) (k) (k) real. Hence arg a1 = − arg a2 ∈ (−π, π) − {0}. In particular, ℑa1 and (k) (k+1) (k) ℑa2 have diﬀerent signs. On the other hand, since a2 = 1 − a1 , we (k+1) (k) (k) (k+1) = −ℑa1 . Thus ℑa1 and ℑa1 have the same sign. Thus have ℑa2 (k) (k) the signs of ℑa1 and ℑa2 do not depend on k. This completes the proof. Remark 5.5.7. The above lemma holds under the weaker assumption that ρ(KP0 ) is parabolic or elliptic with rotation angle = π. Actually the proof is valid under this weaker assumption with a slight modiﬁcation.

6 Reformulation of Main Theorem 1.3.5 and outline of the proof

In this chapter, we give a “2-dimensional” reformulation of the Main Theorem 1.3.5 and present a route map of its proof. In Sect. 6.1, we introduce the deﬁnition of “quasifuchsian” labeled representations and “good” labeled representations (Deﬁnition 6.1.7). A labeled representation is deﬁned to be quasifuchsian if (i) ρ is quasifuchsian, (ii) the Ford domain P h(ρ) is as described by Jorgensen, and (iii) ν is equal to the side parameter of ρ (Deﬁnition 6.1.1), whereas a labeled representation is deﬁned to be good if it satisﬁes the three conditions, Nonzero, Frontier and Duality (Deﬁnition 6.1.7). Theorem 6.1.7 (Good implies quasifuchsian) claims that if a labeled representation is good, then it is quasifuchsian. Though the condition for a labeled representation to be good is rather complicated, it is not diﬃcult to check if a given labeled representation satisfy the condition. By introducing the space J [QF] ⊂ QF × (H2 × H2 ) of all good labeled representations, Jorgensen’s result is replaced with the assertion that the projections µ1 : J [QF] → QF and µ2 : J [QF] → H2 × H2 are homeomorphisms (Modiﬁed Main Theorem 6.1.11). In Sect. 6.2, we give a route map for the proof of Modiﬁed Main Theorem 6.1.11. In Sects. 6.3, 6.4 and 6.5, we prove Theorem 6.1.7 (Good implies quasifuchsian) by using Poincare’s theorem on fundamental polyhedra. Here we actually need a variation of Poincare’s theorem. Since its rigorous proof is rather complicated as is the original Poincare’s theorem [28] and since we also need its generalization when we treat hyperbolic cone-manifolds in the forthcoming paper [11], we present only a sketch of the proof. A detailed proof will be included in [11]. However we believe that readers have no fear concerning the validity of the variation of Poincare’s theorem. (Such a variation is used in [64] without even mentioning that it is diﬀerent from the original one.) In Sect. 6.6, we prove Theorem 6.1.12 which describe the structure of the ideal simplicial complex ∆E (ρ) dual to the Ford domain. As a corollary we prove Proposition 6.6.1. To prove the theorem, we present a general recipe for understanding the structure of ∆E (Γ ) from the Ford domain P h(Γ ) (Lemma

134

6 Reformulation of Main Theorem 1.3.5 and outline of the proof

6.6.2). A variation of the argument for the proof of Lemma 6.6.2 leads to an important Lemma 7.1.3. In Sect. 6.7, we give a characterization of the chain Σ(ν) for a good labeled representation (Propositions 6.7.1 and 6.7.2). They enable us to reconstruct the chain Σ(ν) of a good (or quasifuchsian) labeled representation (ρ, ν) from a single triangle in the chain. This fact provides a key step for checking if a given representation is quasifuchsian, and is used in the Bers’ slice project in [50].

6.1 Reformulation of Main Theorem 1.3.5 Deﬁnition 6.1.1 (Quasifuchsian labeled representation). A labeled representation ρ = (ρ, ν) is said to be quasifuchsian if ρ ∈ QF and the weighted ¯ (ρ(π1 (T ))) is equivalent to Spine(ν) (see Deﬁnition spine Ford(ρ(π1 (T ))) of M 1.3.4). The ﬁrst task for the proof of Main Theorem 1.3.5 is to give a characterization of the quasifuchsian labeled representations in terms of the isometric hemispheres of the images of elliptic generators. For this purpose, we recall Deﬁnition 3.3.4 and introduce two more deﬁnitions. Deﬁnition 6.1.2 (NonZero). A labeled representation ρ = (ρ, ν) is said to (0) satisfy the condition NonZero if φ−1 = ∅. ρ (0) ∩ Σ(ν) Deﬁnition 6.1.3 (Duality). Let ρ = (ρ, ν) be a labeled representation satisfying the condition NonZero. Then we say that ρ satisﬁes the condition Duality, if the following conditions are satisﬁed. 1. For each simplex ξ of L(ν), Fρ (ξ) is contained in Eh(ρ), and it is a convex polyhedron of dimension 2 − dim ξ (see Deﬁnitions 3.4.3 and 3.4.6). 2. If P and P ′ are mutually distinct vertices of L(ν), then Ih(ρ(P )) = Ih(ρ(P ′ )). We shall show that if ρ satisﬁes the conditions NonZero and Duality, then the collection {Fρ (ξ) | ξ ∈ L(ν)(≤2) } gives the cellular structure of ∂Eh(ρ), which is dual to L(ν) (see Proposition 6.3.1). Deﬁnition 6.1.4 (Frontier). Let ρ = (ρ, ν) with ν = (ν − , ν + ) be a labeled representation satisfying the condition NonZero. (1) We say that ρ satisﬁes the condition ǫ-Frontier if the following conditions are satisﬁed. 1. σ ǫ (ν) (see Deﬁnition 3.3.3) is an ǫ-terminal triangle of ρ (see Deﬁnition 4.3.8). 2. ν ǫ = θǫ (ρ, σ ǫ (ν)) (see Deﬁnition 4.2.17). 3. E ǫ (ρ) = E ǫ (ρ, σ ǫ (ν)) (see Deﬁnition 3.4.3).

6.1 Reformulation of Main Theorem 1.3.5

135

(2) We say that ρ = (ρ, ν) satisﬁes the condition Frontier if it satisﬁes ǫ-Frontier for each ǫ = ±. Remark 6.1.5. If ν is thin, then σ ǫ (ν) is not uniquely determined by ν. However, by virtue of Propositions 5.2.3, 5.2.8 and Corollary 5.2.15, if ρ = (ρ, ν) satisﬁes the condition Frontier for some choice of σ ǫ (ν), then it also satisﬁes the condition Frontier for arbitrary choice of σ ǫ (ν). Remark 6.1.6. (1) If a labeled representation satisﬁes the conditions NonZero and Duality, then we can see by an argument in the proof of Proposition 8.3.6 that it also satisﬁes the ﬁrst and the third conditions in Deﬁnition 6.1.4(1) for each ǫ ∈ {−, +}. However, we include these two conditions in the deﬁnition of Frontier, because it is useful in the proof of a key Proposition 6.2.1. (2) Let ρ = (ρ, ν) be a quasifuchsian labeled representation. Then it follows that σ ǫ (ν) is an ǫ-terminal triangle of ρ (see Remark 6.1.9 below). However, the converse does not hold; that is, ρ may have an ǫ-terminal triangle which is diﬀerent from σ ǫ (ν). Let φ be the Markoﬀ map with (φ(s0 ), φ(s1 ), φ(s2 )) = (20, 1.9966 + 0.001i, 19.88056385 − 2.303580328i). Then we see that ρ = ρφ is quasifuchsian and σ is a (+)-terminal triangle of ρ. However, σ + (ν(ρ)) is diﬀerent from σ, where ν(ρ) is the image of ρ by the map ν in Theorem 1.3.2. In fact, for each elliptic generator, P , of slope s0 , Ih(ρ(P )) has the small radius 1/20 and does not support a face of the Ford domain of ρ. One can check this by putting z1 = 0.4972138848 + 0.05736029875i and z2 = 0.5021685589 + 0.05793691669i in OPTi. This phenomenon is related to the existence of an exotic component of the linear slice of quasifuchsian punctured torus space [51]. Deﬁnition 6.1.7 (Good labeled representation). A labeled representation ρ = (ρ, ν) is said to be good if it satisﬁes the conditions NonZero, Frontier and Duality. In Sect. 6.5, we prove the following characterization of quasifuchsian labeled representations. Theorem 6.1.8 (Good implies quasifuchsian). A labeled representation ρ = (ρ, ν) is quasifuchsian if it is good. Remark 6.1.9. The converse to the above theorem also holds. But we do not give a direct proof, because we do not need it for the proof of Main Theorem 1.3.5 and because it is a consequence of the theorem. To reformulate Main Theorem 1.3.5 by using Theorem 6.1.8, we introduce the following notations. Deﬁnition 6.1.10. The symbol J [QF] denotes the subset of X × (H2 × H2 ) consisting of all good labeled representations. By µ1 : J [QF] → X and µ2 : J [QF] → H2 × H2 , respectively, we denote the projection to the ﬁrst and the second factors.

136

6 Reformulation of Main Theorem 1.3.5 and outline of the proof

Then by virtue of Theorem 6.1.8, we see that Main Theorem 1.3.5 is a consequence of the following theorem. Modiﬁed Main Theorem 6.1.11. (1) The projection µ1 : J [QF] → X induces a homeomorphism J [QF] → QF. (2) The projection µ2 : J [QF] → H2 × H2 is a homeomorphism. Proof (Proof of Main Theorem 1.3.5 by assuming Theorem 6.1.8 and Modiﬁed Main Theorem 6.1.11). By Modiﬁed Main Theorem 6.1.11(1), we can deﬁne −1 a continuous map ν : QF → H2 × H2 as the composition µ2 ◦ µ1 |J [QF ] . Let ρ be an element of QF. Then (ρ, ν(ρ)) is a good labeled representation by the deﬁnition of the map ν. Thus (ρ, ν(ρ)) is quasifuchsian by Theorem 6.1.8, and hence the weighted spine Ford(ρ) of M (ρ) is equivalent to Spine(ν(ρ)) (see Deﬁnition 6.1.1). This implies that the map ν : QF → H2 × H2 is equal to the map in Main Theorem 1.3.5 (cf. Corollary 6.2.5). By Modiﬁed Main Theorem 6.1.11(2) and (3), this continuous map is bijective. Finally, we see by Proposition 6.2.1 that the inverse of ν is also continuous. Hence we obtain Main Theorem 1.3.5. A route map of the proof of this theorem is given in Sect. 6.2, and the remaining chapters of this paper are devoted to the proof of this theorem. In Sect. 6.6, we shall also prove the following theorem, which implies Theorem 1.4.2. Theorem 6.1.12. Let ρ be a good labeled representation. Then the ideal polyhedral complex ∆E (ρ) is combinatorially equivalent to Trg(ν).

6.2 Route map of the proof of Modiﬁed Main Theorem 6.1.11 To prove Modiﬁed Main Theorem 6.1.11(1), we need the following Propositions 6.2.1, 6.2.3, 6.2.4 and 6.2.5. Proposition 6.2.1 is proved in Chap. 7, Propositions 6.2.3 and 6.2.4 are proved in Chap. 8, and Proposition 6.2.5 is proved in Sect. 6.4 of this chapter. Proposition 6.2.1 (Openness). For any good labeled representation ρ0 = (ρ0 , ν 0 ), there is an open neighborhood U of ρ0 in X and a continuous map U ∋ ρ → ν(ρ) ∈ H2 × H2 with ν(ρ0 ) = ν 0 , such that (ρ, ν(ρ)) is good for any ρ ∈ U. Deﬁnition 6.2.2 (SameStratum). Let {ρn } be a sequence of labeled representations. We say that the sequence satisﬁes the condition SameStratum if all νnǫ (n ∈ N) is contained in a common (open) cell of D for each ǫ ∈ {−, +}. In this case, we denote the common chain Σ(ν n ) (resp. elliptic generator complex L(ν n )) by Σ0 (resp. L0 ).

6.2 Route map of the proof of Modiﬁed Main Theorem 6.1.11

137

Proposition 6.2.3 (SameStratum). Let {ρn } = {(ρn , ν n )} be a sequence in J [QF] such that {ρn } converges to ρ∞ ∈ QF. Then there is a subsequence of {ρn }, denoted by the same symbol, satisfying the following conditions: (1) {ρn } converges to a labeled representation ρ∞ = (ρ∞ , ν ∞ ) for some ν ∞ ∈ H2 × H2 . (2) {ρn } satisﬁes the condition SameStratum. Proposition 6.2.4 (Closedness). Let {ρn } = {(ρn , ν n )} be a sequence in J [QF] satisfying the following conditions:

(1) {ρn } converges to a labeled representation ρ∞ = (ρ∞ , ν ∞ ) ∈ QF × (H2 × H2 ). (2) {ρn } satisﬁes the condition SameStratum.

Then the limit ρ∞ is a good labeled representation and hence belongs to J [QF]. Proposition 6.2.5 (Uniqueness of good label). For each quasifuchsian representation ρ ∈ QF, there is at most one label ν such that (ρ, ν) is good. Proof (Proof of Modiﬁed Main Theorem 6.1.11(1)). We ﬁrst show that µ1 : J [QF] → X induces a surjection J [QF] → QF. To this end, let QF 0 be the subspace of QF consisting of ρ ∈ QF such that (ρ, ν) ∈ J [QF] for some label ν ∈ H2 × H2 . It is clear that QF 0 is non-empty by Proposition 5.1.5. We show that QF 0 is open and closed in QF. The openness of QF 0 follows from Proposition 6.2.1 (Openness). To prove the closedness, let {ρn } = {(ρn , ν n )} be a sequence in J [QF], such that ρn converges to ρ∞ in QF. By Proposition 6.2.3 (SameStratum), we may assume that {ρn } satisﬁes the condition of Proposition 6.2.4 (Closedness). Hence, (ρ∞ , ν ∞ ) is good by the proposition, where ν ∞ = lim ν n . This proves the closedness of QF 0 . Since QF is connected, we have QF = QF 0 . Hence µ1 : J [QF] → X induces a surjection J [QF] → QF. By Proposition 6.2.5 (Uniqueness of good label), the above map is also injective, and hence bijective. Since it is a restriction of the projection µ1 , it is continuous. Moreover, its inverse map is also continuous by Proposition 6.2.1. Hence µ1 : J [QF] → X induces a homeomorphism J [QF] → QF. Modiﬁed Main Theorem 6.1.11(2) is a consequence of the following proposition, which is proved in Chap. 9. Proposition 6.2.6 (Unique realization). For every label ν ∈ H2 × H2 , there is a unique quasifuchsian representation ρ ∈ QF, such that (ρ, ν) is good. Proof (Proof of Modiﬁed Main Theorem 6.1.11(2)). By Proposition 6.2.6 (Unique realization), µ2 : J [QF] → H2 × H2 is bijective. Since it is a restriction of the projection, it is continuous. Since J [QF] ∼ = R4 by = QF ∼

138

6 Reformulation of Main Theorem 1.3.5 and outline of the proof

Modiﬁed Main Theorem 6.1.11(1), µ2 is regarded as a bijective continuous map from R4 to H2 × H2 ∼ = R4 . This together with the invariance of domain implies that µ2 is a homeomorphism. In fact, the restriction of µ2 to a closed ball B is a homeomorphism and hence µ2 (int B) is a subspace of H2 × H2 homeomorphic to int B. Thus the invariance of domain implies that µ2 (int B) is open in H2 × H2 . Hence µ2 is an open map and therefore it is a homeomorphism. Finally, we note that the following convergence theorem holds a key to the proof of Proposition 6.2.6 (Unique realization), which in tern is proved in Sect. 8.2. Proposition 6.2.7 (Convergence). Let {ρn } = {(ρn , ν n )} be a sequence of good labeled representations satisfying the following conditions: (1) {ν n } converges to ν ∞ ∈ H2 × H2 . (2) {ρn } satisﬁes the condition SameStratum.

Then {ρn } has a convergent subsequence.

6.3 The cellular structure of ∂Eh(ρ) In this section, we describe the cellular structure of the boundary of the virtual Ford domain Eh(ρ) of a good labeled representation ρ. Proposition 6.3.1 (Cellular structure). Let ρ = (ρ, ν) be a labeled representation satisfying the conditions NonZero and Duality. Then the collection {Fρ (ξ) | ξ ∈ L(ν)(≤2) } gives the cellular structure of ∂Eh(ρ), which is dual to L(ν). Namely, we have the following: 1. For each ξ ∈ L(ν)(≤2) , Fρ (ξ) is a convex polyhedron of dimension 2−dim ξ contained in ∂Eh(ρ). 2. For each ξ ∈ L(ν)(≤2) , ∂Fρ (ξ) is the disjoint union of {int Fρ (ξ ′ ) | ξ < ξ ′ ∈ L(ν)(≤2) }. 3. ∂Eh(ρ) is the disjoint union of {int Fρ (ξ) | ξ ∈ L(ν)(≤2) }.

To prove this proposition, we prepare a few lemmas. Throughout this section, ρ denotes a labeled representation satisfying the conditions NonZero and Duality. Lemma 6.3.2. Let ρ = (ρ, ν) be a labeled representation satisfying the conditions NonZero and Duality. Then then following hold. (1) For any ξ ∈ L(ν)(≤2) , we have {Ih(ρ(P )) | P ∈ ξ (0) } ∩ Eh(ρ) Fρ (ξ) = {Ih(ρ(P )) | P ∈ ξ (0) } ∩ ∂Eh(ρ) = (2) For any ξ, ξ ′ ∈ L(ν)(≤2) with ξ < ξ ′ , we have Fρ (ξ) ⊃ Fρ (ξ ′ ).

6.3 The cellular structure of ∂Eh(ρ)

139

Proof. (1) Since ρ satisﬁes Duality, we have Fρ (ξ) ⊂ ∂Eh(ρ) ⊂ Eh(ρ). Hence, we have Fρ (ξ) = Fρ (ξ) ∩ Eh(ρ) {Ih(ρ(P )) | P ∈ ξ (0) } ∩ Eh(ρ; lk(ξ, L(ν))) ∩ Eh(ρ) = = {Ih(ρ(P )) | P ∈ ξ (0) } ∩ Eh(ρ).

Since Ih(ρ(P )) ∩ Eh(ρ) = Ih(ρ(P )) ∩ ∂Eh(ρ) for any P ∈ L(ν)(0) , we also have Fρ (ξ) = {Ih(ρ(P )) | P ∈ ξ (0) } ∩ ∂Eh(ρ). (2) By (1), we see

{Ih(ρ(P )) | P ∈ ξ (0) } ∩ Eh(ρ) {Ih(ρ(P )) | P ∈ (ξ ′ )(0) } ∩ Eh(ρ) ⊃

Fρ (ξ) =

= Fρ (ξ ′ ).

Lemma 6.3.3. Let ρ = (ρ, ν) be a labeled representation satisfying the conditions NonZero and Duality. Then the following hold. (1) For any ξ ∈ L(ν)(≤2) and any X ∈ L(ν)(0) − ξ (0) , we have Fρ (ξ) ∩ Ih(ρ(X)) ⊂ ∂Fρ (ξ). (2) For any ξ, ξ ′ ∈ L(ν)(≤2) with ξ < ξ ′ , we have ∂Fρ (ξ) ⊃ Fρ (ξ ′ ).

Proof. First, we prove (2) by assuming (1). Pick an element X ∈ (ξ ′ )(0) − ξ (0) . Then, by Lemma 6.3.2 (2), Fρ (ξ ′ ) = Fρ (ξ) ∩ Fρ (ξ ′ ) ⊂ Fρ (ξ) ∩ Ih(ρ(X)).

Since the last set is contained in ∂Fρ (ξ) by (1), we have Fρ (ξ ′ ) ⊂ ∂Fρ (ξ). Next, we prove (1). Pick elements ξ ∈ L(ν)(≤2) and X ∈ L(ν)(0) − ξ (0) . Case 1. ξ is a 0-simplex (P ). Suppose contrary that Fρ (P ) ∩ Ih(ρ(X)) contains a point in the interior of the 2-dimensional face Fρ (P ). Then since Fρ (P ) ⊂ ∂Eh(ρ) we must have Ih(ρ(X)) = Ih(ρ(P )). Hence, by the condition Duality, we have X = P , a contradiction. Case 2. ξ is a 1-simplex (P, Q). Suppose contrary that Fρ (ξ) ∩ Ih(ρ(X)) contains a point in the interior of Fρ (ξ). Then Ih(ρ(X)) contains an open circular arc of Ih(ρ(P )) ∩ Ih(ρ(Q)), because Fρ (ξ) is a sub-arc of Ih(ρ(P )) ∩ Ih(ρ(Q)) which is contained in Eh(ρ). Hence, the three isometric hemispheres Ih(ρ(P )), Ih(ρ(Q)) and Ih(ρ(X)) intersect in a common half circle. So, one of the three isometric hemispheres, say Ih(ρ(Y )) (Y ∈ {P, Q, X}), must be contained in the union of the half balls bounded by the remaining two isometric hemispheres. Since the three isometric hemispheres are mutually distinct by the condition Duality, this implies that Fρ (Y ) cannot be a 2-dimensional convex polyhedron contained in ∂Eh(ρ), a contradiction. Case 3. ξ is a 2-simplex (P, Q, R). Then we have the following claim.

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6 Reformulation of Main Theorem 1.3.5 and outline of the proof

Claim 6.3.4. There is an open neighborhood U of the vertex Fρ (ξ) in ∂Eh(ρ) such that U ⊂ ∪{int Fρ (ξ ′ ) | ξ ′ ≤ ξ}. Proof. Note that the following facts are deduced from the proof of (2) and the conclusions for Cases 1 and 2. 1. The 0-dimensional convex polyhedron Fρ (ξ) is a vertex of each of the 1-dimensional convex polyhedra Fρ (P, Q), Fρ (Q, R) and Fρ (P, R). 2. The 1-dimensional convex polyhedron Fρ (P, Q) is an edge of the 2dimensional convex polyhedra Fρ (P ) and Fρ (Q). Similarly Fρ (Q, R) (resp. Fρ (P, R)) is an edge of the 2-dimensional convex polyhedron Fρ (Q) and Fρ (R) (resp. Fρ (P ) and Fρ (R)). Let CP be the convex hull of int Fρ (P, Q), int Fρ (P, R) and Fρ (ξ). Then, by the above facts, we see that CP is a sector of Fρ (P ) contained in the subset int Fρ (ξ) ∪ int Fρ (P, Q) ∪ int Fρ (P, R) ∪ int Fρ (P ). Let CQ and CR ,respectively, be the sectors of Fρ (Q) and Fρ (R), deﬁned in a similar way, and set U := CP ∪ CQ ∪ CR . Then U is contained in ∪{int Fρ (ξ ′ ) | ξ ′ ≤ ξ}. On the other hand, since Fρ (ξ) = Ih(ρ(P )) ∩ Ih(ρ(Q)) ∩ Ih(ρ(R)) is a 0dimensional convex polyhedron, the relative position of Ih(ρ(P )), Ih(ρ(Q)) and Ih(ρ(R)) is as illustrated in Fig. 6.1. Hence we see that U is an open disk in ∂Eh(ρ) containing the vertex Fρ (ξ). This implies that U is an open neighborhood of Fρ (ξ) in ∂Eh(ρ), because ∂Eh(ρ) is a 2-dimensional manifold. Hence we have the claim. Q Fρ (ξ) P

R

Fig. 6.1. Three isometric circles: ξ (0) = {P, Q, R}

Suppose contrary that Fρ (ξ) ∩ Ih(ρ(X)) contains a point in the interior of Fρ (ξ). Then the vertex Fρ (ξ) is contained in Ih(ρ(X)), and hence it is

6.3 The cellular structure of ∂Eh(ρ)

141

contained in the 2-dimensional face Ih(ρ(X)) ∩ Eh(ρ) = Fρ (X) (see Lemma 6.3.2 (1)). For any open neighborhood U of Fρ (ξ) in ∂Eh(ρ), we have U ∩ Fρ (X)−Fρ (ξ) = ∅, because the vertex Fρ (ξ) is contained in the 2-dimensional face Fρ (X). Hence there is a simplex ξ ′ < ξ such that int Fρ (ξ ′ ) ∩ Fρ (X) = ∅ (and hence int Fρ (ξ ′ ) ∩ Ih(ρ(X)) = ∅) by the above sublemma. Since dim ξ ′ ≤ 1, this contradicts the conclusion in Case 1 or Case 2. Lemma 6.3.5. Let ρ = (ρ, ν) be a labeled representation satisfying the conditions NonZero and Duality. Then for any distinct elements ξ and ξ ′ of L(ν)(≤2) , we have int Fρ (ξ) ∩ int Fρ (ξ ′ ) = ∅. Proof. We may assume that there is a vertex X of ξ ′ which is not a vertex of ξ. Then Fρ (ξ) ∩ Fρ (ξ ′ ) ⊂ Fρ (ξ) ∩ Ih(ρ(X)) ⊂ ∂Fρ (ξ) by Lemma 6.3.3 (1). Hence int Fρ (ξ) ∩ int Fρ (ξ ′ ) ⊂ int Fρ (ξ) ∩ (Fρ (ξ) ∩ Fρ (ξ ′ )) ⊂ int Fρ (ξ) ∩ ∂Fρ (ξ) = ∅, and we obtain the conclusion. Lemma 6.3.6. Let ρ = (ρ, ν) be a labeled representation satisfying the conditions NonZero and Duality. Then for any ξ ∈ L(ν)(≤2) , we have ∂Fρ (ξ) =

{Fρ (ξ ′ ) | ξ < ξ ′ ∈ L(ν)(≤2) , dim ξ ′ = dim ξ + 1}.

Proof. We ﬁrst show that ∂Fρ (ξ) =

{Fρ (ξ) ∩ Ih(ρ(X)) | X ∈ lk(ξ, L(ν))(0) }.

By Lemma 6.3.3 (1), the left hand side contains the right hand side. Suppose contrary that the left hand side is not contained in the right hand / Ih(ρ(X)) for side. Then there is a point x ∈ ∂Fρ (ξ) such that x ∈ any X ∈ lk(ξ, L(ν))(0) . Then x lies in the open set ∩{int Eh(ρ(X)) | X ∈ lk(ξ, L(ν))(0) } = int Eh(ρ; lk(ξ, L(ν))). By the deﬁnition of Fρ (ξ), this implies that x ∈ int Fρ (ξ), a contradiction. Thus we have the identity. On the other hand, for each X ∈ lk(ξ, L(ν))(0) , we have Fρ (ξ)∩Ih(ρ(X)) = Fρ (ξ ′ ), where ξ ′ is the simplex of L(ν) spanned by ξ and X. Because Lemma 6.3.2 (1) implies Fρ (ξ) ∩ Ih(ρ(X)) = {Ih(ρ(P )) | P ∈ ξ (0) } ∩ Eh(ρ) ∩ Ih(ρ(X)) {Ih(ρ(P )) | P ∈ (ξ ′ )(0) } ∩ Eh(ρ) = = Fρ (ξ ′ ).

Hence we obtain the desired result. Proof (Proof of Proposition 6.3.1). The ﬁrst assertion directly follows from the condition Duality.

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6 Reformulation of Main Theorem 1.3.5 and outline of the proof

Next, we prove the second assertion. If dim ξ = 2, the assertion is obvious. If dim ξ = 1, the assertion follows from Lemma 6.3.6. The assertion for the case dim ξ = 0 follows from this observation and Lemmas 6.3.5 and 6.3.6. Finally, we prove the third assertion. Since {Ih(ρ(X)) | X ∈ L(ν)(0) } is locally ﬁnite, we have ∂Eh(ρ) = ∪{Ih(ρ(X)) ∩ Eh(ρ) | X ∈ L(ν)(0) }. By Lemma 6.3.2 (1), this implies ∂Eh(ρ) = ∪{Fρ (X) | X ∈ L(ν)(0) }. Hence we obtain the third conclusion by the second conclusion and Lemma 6.3.5. Corollary 6.3.7. Let ρ be a labeled representation satisfying the conditions NonZero and Duality, and let ξ1 and ξ2 be simplices of L(ν). Then the following conditions are equivalent. 1. Fρ (ξ1 ) ∩ Fρ (ξ2 ) = ∅. 2. There is a simplex ξ ′ of L(ν) which contains both ξ1 and ξ2 as faces. Proof. By Proposition 6.3.1, Fρ (ξi ) is the disjoint union of {int Fρ (ξ ′ ) | ξi ≤ ξ ′ ∈ L(ν)(≤2) }. Since ∂Eh(ρ) is the disjoint union of {int Fρ (ξ) | ξ ∈ L(ν)(≤2) }, this implies the desired result.

6.4 Applying Poincare’s theorem on fundamental polyhedra In this section, we prove the following key proposition for the proof of Theorem 6.1.8 Proposition 6.4.1. Let ρ = (ρ, ν) be a good labeled representation. Then ρ is discrete and the Ford domain P h(ρ) is equal to the virtual Ford domain Eh(ρ). First, we note the following consequence of Lemma 4.1.3 (Chain rule). Lemma 6.4.2. Let ρ = (ρ, ν) be a labeled representation satisfying the conditions NonZero and Duality. Then we have the following. (1) For each 2-dimensional face Fρ (P ), the involution ρ(P ) maps Fρ (P ) onto itself. (2) Suppose a face Fρ (P ) contains an edge Fρ (ξ) (ξ = P0 , P1 ), that is, P = P0 or P1 . Then the involution ρ(P ) maps Fρ (ξ) onto Fρ (ξ ′ ) where ξ ′ = P−1 , P0 or P1 , P2 according as P = P0 or P1 . (3) Suppose a face Fρ (P ) contains a vertex Fρ (ξ) (ξ = P0 , P1 , P2 ), that is, P = P0 , P1 or P2 . Then the involution ρ(P ) maps Fρ (ξ) to Fρ (ξ ′ ) where ξ ′ = P−1 , P1P0 , P0 , P0 , P1 , P2 or P2 , P1P2 , P3 according as P = P0 , P1 or P2 . In particular, ρ(P1 ) ﬁxes Fρ (ξ). The above lemma (together with Proposition 6.3.1) implies that the family of the involutions {ρ(P ) | P ∈ L(ν)(0) } determines a “gluing data” for the polyhedron Eh(ρ). The key ingredient of the proof of Proposition 6.4.1 is

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143

to show that Eh(ρ) together with the gluing data satisﬁes the conditions of (a variation of) Poincare’s theorem on fundamental polyhedra. We note that Eh(ρ) is a fundamental domain only modulo the action of the inﬁnite cyclic Kleinian group ρ(K), i.e., an actual fundamental domain is the intersection of Eh(ρ) with a fundamental domain for ρ(K). So, if we were to apply the usual version of Poincare’s theorem on fundamental polyhedra (e.g. [28] and [53]), then we need to work on an artiﬁcial polyhedron and an artiﬁcial cellular structure. To avoid this troublesome phenomena, we work with the quotient of Eh(ρ) by ρ(K). Throughout the remainder of this section, we assume that ρ = (ρ, ν) is a thick labeled representation which satisﬁes the conditions NonZero, Frontier and Duality. To make the above mentioned setting clear, we introduce the following deﬁnition. Deﬁnition 6.4.3. (1) Cusp(K) denotes the hyperbolic manifold H3 /ρ(K). 3 Cusp(K) and ∂Cusp(K) denote H /ρ(K) and C/ρ(K), respectively. 3 (2) Let qK : H → Cusp(K) be the projection, and set EhK (ρ) := qK (Eh(ρ)) ⊂ Cusp(K),

EK (ρ) := qK (E(ρ)) ⊂ ∂Cusp(K),

FK,ρ (ξ) := qK (Fρ (ξ))

(ξ ∈ (L(ν)/K)(≤2) ).

Lemma 6.4.4. Let ρ = (ρ, ν) be a good labeled representation. Then for each ξ ∈ ∪2i=0 L(ν)(i) , the restriction of the projection qK to Fρ (ξ) is injective. Proof. Suppose that the restriction of qK to Fρ (ξ) is not injective. Then Fρ (ξ) ∩ ρ(K n )(Fρ (ξ)) = ∅ for some nonzero integer n. By Corollary 6.3.7, this implies that ξ and K n (ξ) is contained in a simplex of L(ν). But this is impossible, because n = 0. Hence we obtain the desired result. Thus FK,ρ (ξ) can be regarded as a “polyhedron” in Cusp(K), and we obtain a map FK,ρ : (L(ν)/K)(≤2) → {polyhedron in Cusp(K)}. By Proposition 6.3.1, the images {FK,ρ (ξ)} determine a “cellular structure” of the boundary of EhK (ρ). On the other hand, by Lemma 6.4.2, for each 2-dimensional face FK,ρ (P ) of ∂EhK (ρ) ρ(P ) induces an isometric involution on FK,ρ (P ). These involutions determine the “gluing data” for the polyhedron EhK (ρ) in Cusp(K). By Lemma 6.4.2 (2), we can see that each “edge cycle” of the gluing data is of the following type: •

The cycle {FK,ρ (P0 , P1 ), FK,ρ (P1 , P2 ), FK,ρ (P2 , P3 )} of length 3, where {Pj } is the sequence of elliptic generators associate with a triangle of Σ(ν). The sum of the dihedral angles in the cycle is equal to 2π by Lemma 4.1.3 (Chain rule).

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6 Reformulation of Main Theorem 1.3.5 and outline of the proof

Let ∼ be the equivalence relation on EhK (ρ) induced by the gluing data, and let MO (ρ) denote the quotient pseudometric space of the hyperbolic manifold EhK (ρ). Then, by the above facts, we can see that the pseudometric of MO (ρ) is actually a metric, which is locally isometric to H3 or to the quotient of H3 by an isometric involution. Moreover, we can prove that MO (ρ) is complete by using the condition Frontier and the fact that the gluing data has “consistent horoballs”, that is, we can associate each ideal vertex v of an edge of ∂EhK (ρ) a horoball, Hv , with center v so that the following conditions are satisﬁed (cf. [28, Theorem 6.3]). 1. Hv does not intersect the faces of ∂EhK (ρ) whose closure in Cusp(K) does not contain v. 2. For each 2-dimensional face FK,ρ (P ) whose closure contains v, the involution, h, of FK,ρ (P ) induced by ρ(P ) maps Hv ∩FK,ρ (P ) to Hh(v) ∩FK,ρ (P ). (For the details of the above arguments, please see the subsequent paper [11], in which we describe constructions of hyperbolic cone manifolds from polyhedra in Cusp(K) by gluing their faces.) Thus we see that MO (ρ) is a hyperbolic orbifold and that Eh(ρ) is a fundamental domain of the discrete group ρ(π1 (O)) modulo ρ(K). Since Eh(ρ) is bounded by isometric hemispheres of elements of ρ(π1 (O)), the Ford domain P h(ρ(π1 (O))) is contained in Eh(ρ). This implies P h(ρ(π1 (O))) = Eh(ρ), since both of them are fundamental domains of ρ(π1 (O)) modulo ρ(K). Hence we have P h(ρ(π1 (T )) = P h(ρ(π1 (O))) = Eh(ρ) by Lemma 2.2.8. This completes the proof of Proposition 6.4.1. Proof (Proof of Proposition 6.2.5 (Uniqueness of good label)). Assume that two labeled representations (ρ, ν 1 ) and (ρ, ν 2 ) are good for a quasifuchsian representation ρ ∈ QF and two labels ν j = (νj− , νj+ ) (j = 1, 2). Then by Proposition 6.4.1 and Lemma 2.5.4(2-iii), both Σ(ν 1 )(0) and Σ(ν 2 )(0) consist of the slopes of the elliptic generators P such that Ih(ρ(P )) supports a 2dimensional face of the Ford domain of P h(ρ). Hence we have Σ(ν 1 )(0) = Σ(ν 2 )(0) and therefore Σ(ν 1 ) = Σ(ν 2 ). Let σ ǫ be the ǫ-terminal triangle of Σ(ν 1 ) = Σ(ν 2 ). Then we have ν1ǫ = θǫ (ρ, σ ǫ ) = ν2ǫ . Hence we have ν 1 = ν 2 .

6.5 Proof of Theorem 6.1.8 (Good implies quasifuchsian) In this section, we complete the proof of Theorem 6.1.8. Let ρ = (ρ, ν) be a good labeled representation. Then, by Proposition 6.4.1, ρ is discrete and the Ford domain P h(ρ) is equal to the virtual Ford domain Eh(ρ). By using this fact, we show that the Ford complex Ford(ρ) = Ford(ρ(T )) of M (ρ) = M (ρ(π1 (T )) is equivalent to Spine(ν) = Spine(δ − (ν), δ + (ν)). To this end, we give detailed explanation of the relation between the Ford complex and the Ford domain, which are sketched at the end of Sect. 1.2 (see Fig. 1.2). Consider the foliation of each face Fρ (P ) of ∂Eh(ρ) by geodesic

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145

segments orthogonal to Fρ (P ) ∩ Axis(ρ(P )). We say that a leaf of the foliation is inner if it is a (possibly degenerate) geodesic segment with endpoints in ∂Fρ (P ). If a leaf is not inner, then it is a bi-inﬁnite geodesic which joins two points of F ρ (P ) ∩ fr E(ρ). By joining the inner leaves, we obtain a foliation, {Lt }t∈(−1,1) , of an open submanifold of ∂Eh(ρ) by periodic piecewise geodesic lines, because the following observation shows that Lt does not branch at vertices of ∂Eh(ρ): •

For each vertex Fρ (P, Q, R) of ∂Eh(ρ), the following hold (see Fig. 6.2). 1. The vertex Fρ (P, Q, R) is the intersection of the three faces Fρ (P ), Fρ (Q) and Fρ (R). 2. The vertex Fρ (P, Q, R) is contained in Axis(ρ(Q)) but is contained in neither Axis(ρ(P )) nor Axis(ρ(R)). Hence each of Fρ (P ) and Fρ (R) contains a unique inner leaf having the vertex as an endpoint, and the leaf in Fρ (Q) containing the vertex is the singleton {Fρ (P, Q, R)}.

Q

P

R

Fig. 6.2. Fρ (P, Q, R): the dotted broken lines are the leaves of the foliation {Lt }

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6 Reformulation of Main Theorem 1.3.5 and outline of the proof

Let L±1 be the piecewise geodesic line in ∂Eh(ρ) obtained as the limits 3 limt→±1 Lt (in the space of closed subsets of H with Chabauty topology). Consider the piecewise linear (aﬃne) lines π(L±1 ) in C, where π : H3 → C is the projection. Then the interior of the region in C bounded by these two lines is disjoint from E(ρ) and fr E − (ρ) (resp. fr E + (ρ)) lies below π(L−1 ) (resp. above π(L+1 )). By using this observation, we have the following lemma. Lemma 6.5.1. Let ρ = (ρ, ν) be a good labeled representation. Then E(ρ) = E − (ρ) ⊔ E + (ρ) and Ω(ρ)/ρ(π1 (T )) forms a pair of punctured tori. In particular, ρ is quasifuchsian. Proof. The ﬁrst assertion follows from the preceding observation. Since P h(ρ) = Eh(ρ) by Proposition 6.4.1, this implies P (ρ) = E(ρ) = E − (ρ)⊔E + (ρ) (cf. Lemma 3.4.9). By using the fact that σ ǫ (ν) is an ǫ-terminal triangle of ρ (see Deﬁnition 4.3.8), we see that the image of E ǫ (ρ) (⊂ Ω(ρ)) in Ω(ρ)/ρ(π1 (T )) is a once-punctured torus. Hence we obtain the conclusion. Let St := π −1 (π(Lt )) ∩ Eh(ρ) be the piecewise totally geodesic plane in Eh(ρ) lying above Lt , let Tt be the image of St in M (ρ) ∼ = T × (−1, 1). Then Tt is isotopic to a level punctured torus, and the image, Ct , of Lt = ∂St in Tt is a spine of Tt . Moreover, we can observe the following. 1. Suppose Lt is generic, i.e., it does not contain a vertex of ∂Eh(ρ). Then by using Lemma 6.4.2, we can see that there is a triangle σ of Σ(ν) such that Lt is obtained by joining non-degenerate inner leaves of the faces {Fρ (Pj )}, where {Pj } is the sequence of elliptic generators associated with σ. In this case Ct is identiﬁed with the generic spine, spine(σ) (when we identify Tt with a level punctured torus T × {t}). 2. Suppose Lt is not generic, i.e., it contains a vertex, say Fρ (P, Q, R), of ∂Eh(ρ). Then, since (P, Q, R) is a 2-simplex of L(ν), both σ := s(P ), s(Q), s(R) and σ ′ := s(P ), s(R), s(QR ) belong to Σ(ν) (see Deﬁnitions 3.2.3 and 3.3.5). Let τ be the edge σ ∩ σ ′ = s(P ), s(R). Then we see that the spine Ct is identiﬁed with the non-generic spine, spine(τ ). Moreover, after changing the parameter t by −t if necessary, we may assume that Ct−η and Ct+η , respectively, are isotopic to the generic spines, spine(σ) and spine(σ ′ ), for every small enough positive real η. We explain the reason why Ct is identiﬁed with spine(σ) in the ﬁrst observation. Let γ˜j be the geodesic segment Lt ∩ Fρ (Pj ) and β˜j the vertical geodesic γj ) = p(˜ γj+3 ) and ray joining the point γ˜j ∩ Axis(ρ(Pj )) to ∞. Set γj = p(˜ ˜ ˜ βj = p(βj ∪ βj+3 ), where p : St → Tt is the projection. Then Ct consists of the three edges γj (j = 0, 1, 2) and β0 ∪ β1 ∪ β2 gives a topological ideal triangulation of the punctured torus Tt dual to the spine Ct . By the deﬁnition of the slopes of elliptic generators (Deﬁnition 2.1.3), we can see that the slope of the arc βj is equal to the slope s(Pj ) = s(Pj+3 ). Hence β0 ∪ β1 ∪ β2 is identiﬁed with the topological ideal triangulation trg(σ) and hence Ct is identiﬁed with

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147

spine(σ) (see Sect. 1.2). The corresponding assertion in the second observation can be seen similarly. After modifying the continuous family {(Tt , Ct )}t∈(−1,1) near t = ±1, keeping the above two conditions, we can extend it to a continuous family ¯ (ρ) ∼ {(Tt , Ct )}t∈[−1,1] such that {Tt }t∈[−1,1] gives a foliation of M = T × [−1, 1] and that Ford(ρ) = ∪t∈[−1,1] Ct . In particular, Cǫ1 is equal to the image of ¯ (ρ), and hence it is identiﬁed with spine(δ ǫ (ν)). This fr E ǫ (ρ) in Tǫ1 = ∂ ǫ M ¯ (ρ) bounded by T + can be done as follows. Let N + be the submanifold of M t0 + + ¯ and ∂ M (ρ) for a real number t0 < 1 suﬃciently close to 1. Then we can see that (N + , N + ∩ Ford(ρ)) is homeomorphic to (Tt+ , Ct+ ) × [t+ 0 , 1] or its 0 0 quotient by the equivalence relation which collapses (an edge of Ct+ ) × 1 to 0 a point, according as ν + is generic or non-generic (i.e., according as δ + (ν) is a triangle or an edge). Similar assertion holds near t = −1. By using these facts, we obtain the desired family {(Tt , Ct )}t∈[−1,1] . We note that C±1 is identiﬁed with spine(δ ± (ν)) and that {Ct }t∈[−1,1] gives a sequence of elementary transformations relating spine(δ − (ν)) to spine(δ + (ν)). Moreover, the number of Whitehead transformations (= transformations from generic spines to generic spines via a single non-generic spine) that occur when t moves in the open interval (−1, 1) is equal to the number of the vertices of ∂Eh(ρ) modulo the equivalence relation generated by the gluing data, which in tern is equal to the number of triangles in Σ(ν) minus 1. Hence {Ct }t∈[−1,1] realizes the canonical sequence of elementary transformations spine(δ − (ν)) = spine(δ0 ) → spine(δ1 ) → · · · → spine(δm ) = spine(δ + (ν))

introduced in Sect. 1.2. Thus Ford(ρ) is isotopic to Spine(δ − (ν), δ + (ν)). Furthermore, since ρ satisﬁes the condition Frontier, it follows that ν ǫ = θǫ (ρ, σ ǫ (ν)). This completes the proof of Theorem 6.1.8.

6.6 Structure of the complex ∆E and the proof of Theorem 6.1.12 In this section, we prove Theorem 6.1.12 and the following proposition. Proposition 6.6.1. Let ρ = (ρ, ν) be a quasifuchsian labeled representation. Then L(ρ, ν) is simple. To this end, we describe a method for obtaining the complex ∆E (Γ ) from the Ford domain in the general setting where Γ is a non-elementary Kleinian group such that Γ∞ contains parabolic transformations. Let ∼ be the equivalence relation on P h(Γ ) such that M (Γ ) ∼ = P h(Γ )/ ∼, namely two points in P h(Γ ) are equivalent if and only if they project to the same point of M (Γ ). For each p ∈ ∂P h(Γ ), let [p] be the ∼-equivalence class of p, namely [p] = {x ∈ P h(Γ ) | x = A(p) for some A ∈ Γ }.

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6 Reformulation of Main Theorem 1.3.5 and outline of the proof

Lemma 6.6.2. Let e be a cell of ∂P h(Γ ) and p a point in the (relative) interior of e. Then the following hold. 1. [p]/Γ∞ 2. Let {p Ai ∈ Γ dual to

is a ﬁnite set. = p1 , p2 , · · · , pn } ⊂ [p] be a representative for [p]/Γ∞ , and let e be an element such that Ai (p) = pi . Then the ideal polyhedron ∆ e is spanned by the ideal vertices {A(∞) | A ∈

n !

i=1

Γp A−1 i },

where Γp is the stabilizer of p with respect to the action of Γ . Proof. (1) This follows from Lemma A.1.10. e is given by (2) Recall that the ideal vertices Ve of the ideal polyhedron ∆ Ve = {A(∞) | d(p, A(H∞ )) = d(p, Γ∞ H∞ ),

A ∈ Γ }.

On the other hand we have the following equivalence: d(p, A(H∞ )) = d(p, Γ∞ H∞ ), ⇔ d(A−1 (p), H∞ ) = d(A−1 (p), Γ∞ H∞ ),

⇔ A−1 (p) ∈ P h(Γ ),

⇔ A−1 (p) ∈ [p], ⇔ A−1 (p) = BAi (p) for some i ∈ {1, · · · , n} and B ∈ Γ∞ ,

⇔ A−1 = BAi C for some i ∈ {1, · · · , n}, B ∈ Γ∞ and C ∈ Γp , "n ⇔ A ∈ i=1 Γp A−1 i Γ∞ .

Since every element of a double coset Γp A−1 i Γ∞ maps ∞ to Hence we obtain the desired result. We apply the above lemma to the setting of Theorem 6.1.12. We note that, by virtue of Proposition 2.2.8(3), we may work with Γ := ρ(π1 (O)) instead of ρ(π1 (T )). Lemma 6.6.3. Let ρ = (ρ, ν) be a thick good labeled representation. Then for e dual to e is described each cell e of ∂P h(ρ) = ∂Eh(ρ), the ideal polyhedron ∆ as follows: e is the ideal edge spanned by {∞, ρ(P )(∞)}. 1. If e is a face Fρ (P ), then ∆ e is the ideal triangle spanned by 2. If e is an edge Fρ (P, Q), then ∆ {∞, ρ(P )(∞), ρ(Q)(∞)}.

e is the ideal tetrahedron spanned by 3. If e is a vertex Fρ (P, Q, R), then ∆ {∞, ρ(P )(∞), ρ(Q)(∞), ρ(R)(∞)}.

6.6 Structure of the complex ∆E and the proof of Theorem 6.1.12

149

Proof. We prove only (3), because the remaining assertion can be proved similarly. Let p be the vertex Fρ (P, Q, R). Then the (∼)-equivalence class [p] modulo Γ∞ = ρ(K) is equal to {p, ρ(P )(p)} (see Sect. 6.4). Since p lies on the axis of ρ(Q), the image of p in the hyperbolic orbifold H3 /Γ lies in a cone axis of cone angle π. Thus we have Γp = ρ(Q). Hence, by Lemma 6.6.2, the e is spanned by the image of ∞ by the elements in the set ideal polyhedron ∆ $ # ρ {1, Q} ∪ {1, Q}P −1 = ρ{1, P, Q, QP }. Since ρ(QP )(∞) = ρ(RRQP )(∞) = ρ(R)(∞), we obtain the desired result.

Lemma 6.6.4. Let ρ = (ρ, ν) be a thin good labeled representation such that ν ± ∈ int τ under Notation 2.1.14 (Adjacent triangles). Then, modulo the action of ρ(K), ∂P h(ρ) = ∂Eh(ρ) consists of the two 2-dimensional convex polyhedra Fρ (P0 ) and Fρ (P2 ), together with the two 1-dimensional convex e dual polyhedra Fρ (P0 , P2 ) and Fρ (P2 , P3 ). Moreover the ideal polyhedron ∆ to a cell e of ∂P h(ρ) is described as follows: e is the ideal edge spanned 1. If e is the face Fρ (P0 ) (resp. Fρ (P2 )), then ∆ by {∞, ρ(P0 )(∞)} (resp. {∞, ρ(P2 )(∞)}). e is the 2. If e is the edge Fρ (P0 , P2 ) (resp. Fρ (P2 , P3 ) = Fρ (P1′ , P3′ )), then ∆ ideal quadrangle spanned by {∞, ρ(P0 )(∞), ρ(P1 )(∞), ρ(P2 )(∞)} (resp. {∞, ρ(P1′ )(∞), ρ(P2′ )(∞), ρ(P3′ )(∞)}).

Proof. By the deﬁnition of a good labeled representation and the deﬁnition of L(ν) for the thin label ν (Deﬁnition 3.2.7), we obtain the classiﬁcation of the cells of ∂P h(ρ) = ∂Eh(ρ). Since (1) is obtained easily, we prove only (2) for the case when e = Fρ (P0 , P2 ). Let p be a point of Fρ (P0 , P2 ). Then the (∼)equivalence class [p] modulo Γ∞ = ρ(K) is equal to {p, ρ(P2 )(p)} (see Sect. 6.4). On the other hand, we see, by the forthcoming Proposition 6.7.3, that (ρ, τ ) is an isosceles representation. So Fρ (P0 , P2 ) = Ih(ρ(P0 )) ∩ Ih(ρ(P2 )) is identical with Axis(ρ(P1 )) by Proposition 5.2.3. Thus we see Γp = ρ(P1 ) as in the proof of Lemma 6.6.3. Hence, by Lemma 6.6.2, the ideal polyhedron e is spanned by the image of ∞ by the elements in the set ∆ $ # ρ {1, P1 } ∪ {1, P1 }P2−1 = ρ{1, P1 , P2 , P1 P2 }. e is spanned by Since ρ(P1 P2 )(∞) = ρ(P0 P0 P1 P2 )(∞) = ρ(P0 )(∞), ∆ {∞, ρ(P0 )(∞), ρ(P1 )(∞), ρ(P2 )(∞)}.

Since (ρ, τ ) is an isosceles representation, the convex hull of these four points is an ideal quadrangle. As a corollary to Lemmas 6.6.3 and 6.6.4, we obtain the following. Corollary 6.6.5. For every good labeled representation ρ = (ρ, ν), L(ρ) is equal to the projection to C of the cross section of the ideal polyhedral complex E (ρ) with the horizontal horosphere ∂H∞ . ∆

150

6 Reformulation of Main Theorem 1.3.5 and outline of the proof

E (ρ) which has ∞ as an Proof. We ﬁrst note that any ideal polyhedron of ∆ ideal vertex is the geometric dual to a cell of ∂P h(ρ) ⊂ Ford(ρ). Thus the E (ρ) with the horizontal horosphere ∂H∞ consists of the cross section of ∆ cross sections of the ideal polyhedra which appear in Lemmas 6.6.3 or 6.6.4, according as ν is thick or thin. Suppose ν is thick. Then (the image by the projection to C of) the e with ∂H∞ is equal to the Euclidean simplex ρ(P )(∞), cross section of ∆ ρ(P )(∞), ρ(Q)(∞) or ρ(P )(∞), ρ(Q)(∞), ρ(R)(∞) according as e is Fρ (P ), Fρ (P, Q) or Fρ (P, Q, R). Hence these (images of the) cross sections constitute L(ρ). Suppose ν is thin. Then (the image by the projection to C of) the cross e with ∂H∞ is equal to the Euclidean simplex ρ(P0 )(∞), section of ∆ ρ(P2 )(∞), ρ(P0 )(∞), ρ(P2 )(∞) or ρ(P2 )(∞), ρ(P3 )(∞) according as e is Fρ (P0 ), Fρ (P2 ), Fρ (P0 , P2 ) or Fρ (P2 , P3 ). Hence, again, these (images of the) cross sections constitute L(ρ). This completes the proof of the corollary. Proof (Proof of Proposition 6.6.1). By Corollary 6.6.5, L(ρ) is identiﬁed with E (ρ) with the horizontal the cross section of the ideal polyhedral complex ∆ horosphere ∂H∞ . Hence L(ρ, ν) satisﬁes the ﬁrst two conditions in Deﬁnition 3.2.2 (Simple). So we have only to show that the last condition is satisﬁed. If the length m of the chain Σ(ν) is equal to 1, then we have nothing to prove. So we assume m ≥ 2. By the condition Frontier, E − (ρ) = E − (ρ, σ1 ), and hence all components of θ− (ρ, σ1 ) are non-negative. Let s1 be the vertex of σ1 = s0 , s1 , s2 which is not a vertex of σ2 , and let {Pj } be the sequence of elliptic generators associated with σ1 . Then (P0 , P1 , P2 ) is a 2simplex of L(ν) and hence Fρ (P0 , P1 , P2 ) = ∩2j=0 Ih(ρ(Pj )) is a vertex of ∂Eh(ρ). Hence ∩2j=0 Ih(ρ(Pj )) is a singleton. Thus by Lemma 4.3.6, we have either θ+ (ρ, σ1 ; s1 ) < 0 < θ− (ρ, σ1 ; s1 ) or θ− (ρ, σ1 ; s1 ) < 0 < θ+ (ρ, σ1 ; s1 ). Since θ− (ρ, σ1 ; s1 ) ≥ 0 as observed in the above, we must have θ+ (ρ, σ1 ; s1 ) < 0 < θ− (ρ, σ1 ; s1 ). Hence L(ρ, σ1 ) is convex to the below at c(ρ(P1 )) by Corollary 4.2.15. Hence the edge c(ρ(P0 ))c(ρ(P2 )) of L(ρ, σ2 ) lies above L(ρ, σ1 ) by Lemma 3.1.7. Hence L(ρ, σ2 ) lies above L(ρ, σ1 ). Since the open 2-simplices of L(ρ, ν) are disjoint, L(ρ, σ3 ) lies above L(ρ, σ2 ). By repeating this argument, we see that the last condition is satisﬁed. Hence L(ρ, ν) is simple. Proof (Proof of Theorem 6.1.12). Let ρ = (ρ, ν) be a good labeled representation, and set Γ = ρ(π1 (T )). Then Γ is quasifuchsian and Ford(Γ ) is equivalent to Spine(ν) by Theorem 6.1.8. By the construction of Trg(ν) in Sect. 1.4, we can see that Trg(ν) is a “topological dual” to Ford(Γ ). (See Fig. 1.1. By collapsing each vertical line contained in the side of the cube into a point in the ﬁgure, we obtain a topological ideal tetrahedron dual to the vertex of Spine(σ1 , σ2 ).) On the other hand, ∆E (Γ ) is the geometric dual to Ford(Γ ) (cf. Sect. 1.1), which in tern implies that ∆E (Γ ) is a topological dual to Trg(ν). Thus ∆E (Γ ) should be combinatorially equivalent to Spine(ν). This is the idea of the proof. To make this idea explicit, we give more precise

6.7 Characterization of Σ(ν) for good labeled representations

151

descriptions of Trg(ν) and ∆E (Γ ). For simplicity we assume ν is thick. (The thin case can be easily treated.) Set Σ(ν) = (σ1 , σ2 , · · · , σn ). Then we have the following one-to-one correspondences for Trg(ν). • • •

The 3-cell of Trg(ν) corresponding to (σi , σi+1 ) is the image of any of the i , σi+1 ). topological ideal tetrahedra in Trg(σ The 2-cell of Trg(ν) corresponding to (s0 , s1 ) (resp (s1 , s2 )) is the image of i ), the face (0, 0), (p0 , q0 ), (p1 , q1 ) (resp. (0, 0), (p1 , q1 ), (p2 , q2 )) of trg(σ where σi = s0 , s1 , s2 = q0 /p0 , q1 /p1 , q2 /p2 . (0) The 1-cell of Trg(ν) corresponding to a slope s ∈ σi ⊂ Σ(ν)(0) is the i ) of slope s. image of any of the edges of trg(σ

For ∆E (Γ ), we have the following one-to-one correspondences. •

The 3-cell of ∆E (Γ ) corresponding to (σi , σi+1 ) is the image of the ideal tetrahedron ∞, ρ(P )(∞), ρ(Q)(∞), ρ(R)(∞), where (P, Q, R) is an elliptic generator triple such that σi = s(P ), s(Q), s(R) and σi+1 = s(P ), s(R), s(QR ). • The 2-cell of ∆E (Γ ) corresponding to (s0 , s1 ) (resp (s1 , s2 )) is the image of the ideal triangle ∞, ρ(P )(∞), ρ(Q)(∞) (resp. ∞, ρ(Q)(∞), ρ(R)(∞)), where (P, Q, R) is an elliptic generator triple such that σi = s(P ), s(Q), s(R). (0) • The 1-cell of Trg(ν) corresponding to a slope s ∈ σi ⊂ Σ(ν)(0) is the image of the ideal edge ∞, ρ(P )(∞), where P is an elliptic generator of slope s.

Thus we have a natural correspondence between cells of Trg(ν) and ∆E (Γ ). Moreover we can easily check that it respects the incidence relation. Hence Trg(ν) and ∆E (Γ ) are combinatorially equivalent.

6.7 Characterization of Σ(ν) for good labeled representations In this section, we prove Propositions 6.7.1 and 6.7.2 below, by reﬁning the proof of Proposition 6.6.1. These are useful in the actual computation of the Ford domains, because they enable us to reconstruct the chain Σ(ν) of a good (or quasifuchsian) labeled representation (ρ, ν) from a single triangle in the chain. To be precise, if we know that a triangle σ belongs to the chain Σ(ν) associated with a quasifuchsian labeled representation (ρ, ν), then we can reconstruct the chain Σ(ν) only from the representation ρ and the triangle σ. This is because these propositions enable us to ﬁnd out the slope of σ which is not a slope of the successor (resp. predecessor) of σ in Σ(ν) and hence the successor (resp. predecessor) itself. This holds the key to the successful computer program OPTi [78] (cf. [79]) developed by the third author and is the starting point of the numerous computer experiments made by the last two authors (see [50] [79] [82]).

152

6 Reformulation of Main Theorem 1.3.5 and outline of the proof

Proposition 6.7.1. Let ρ = (ρ, ν) be a thick good labeled representation, and let (σ1 , σ2 , · · · , σm ) be the chain Σ(ν). Then for each triangle σi in the chain, L(ρ, σi ) is simple and ρ satisﬁes the triangle inequality at σi . In particular θǫ (ρ, σi ) is deﬁned. Suppose 1 ≤ i < m and let s1 be the vertex of σ := σi = s0 , s1 , s2 which is not a vertex of the successor σi+1 of σi . Then the following hold. 1. θ+ (ρ, σ; s1 ) < 0 < θ− (ρ, σ; s1 ). 2. ∩2j=0 Ih(ρ(Pj )) is a singleton which is not contained in Dh(ρ(P−1 )) nor Dh(ρ(P3 )), where {Pj } is the sequence of elliptic generators associated with σ. Moreover, s1 is characterized by these properties among the vertices of σ, Namely, the following hold for the remaining vertices s0 and s2 of σ. 1. Either (i) θ+ (ρ, σ; s0 ) ≥ 0 or (ii) θ+ (ρ, σ; s0 ) < 0 and ∩1j=−1 Ih(ρ(Pj )) is a singleton which is contained in int Dh(ρ(P2 )). 2. Either (i) θ+ (ρ, σ; s2 ) ≥ 0 or (ii) θ+ (ρ, σ; s2 ) < 0 and ∩3j=1 Ih(ρ(Pj )) is a singleton which is contained in int Dh(ρ(P0 )) Proof. By Proposition 6.6.1, L(ρ, σ) is simple. The condition Duality implies that Fρ (Pj , Pj+1 ) is a 1-dimensional convex polyhedron and hence Ih(ρ(Pj ))∩ Ih(ρ(Pj+1 )) is a geodesic for every integer j. Thus ρ satisﬁes the triangle inequality at σ by Lemma 4.2.1. Now suppose that 1 ≤ i < m and s1 is the vertex of σ = σi which is not a vertex of σi+1 . Then (P0 , P1 , P2 ) is a 2-simplex of L(ν) and hence Fρ (P0 , P1 , P2 ) = ∩2j=0 Ih(ρ(Pj )) is a vertex of ∂Eh(ρ). Hence ∩2j=0 Ih(ρ(Pj )) is a singleton. Thus by Lemma 4.3.6, we have either θ+ (ρ, σ; s1 ) < 0 < θ− (ρ, σ; s1 ) or θ− (ρ, σ; s1 ) < 0 < θ+ (ρ, σ; s1 ). On the other hand, since L(ρ) is simple by Proposition 6.6.1 (and Theorem 6.1.8), L(ρ, σi+1 ) lies above L(ρ, σ) = L(ρ, σi ) (cf. Deﬁnition 3.2.2(3)). Since c(ρ(P0 ))c(ρ(P2 )) is an edge of L(ρ, σi+1 ), this implies that L(ρ, σ) is convex to the below at c(ρ(P1 )) by Lemma 3.1.7. Hence we see θ+ (ρ, σ; s1 ) < θ− (ρ, σ; s1 ) by Corollary 4.2.15, and therefore θ+ (ρ, σ; s1 ) < 0 < θ− (ρ, σ; s1 ) by the preceding observation. Next, we show that the singleton v(0, 1, 2) := ∩2j=0 Ih(ρ(Pj )) is not contained in Dh(ρ(P−1 )) nor Dh(ρ(P3 )). Suppose to the contrary that v(0, 1, 2) is contained in, say Dh(ρ(P3 )). Since v(0, 1, 2) is contained in ∂Eh(ρ) by the condition Duality, it must be contained Ih(ρ(P3 )) and hence ∩3j=0 Ih(ρ(Pj )) = ∅. Thus ρ(KP0 ) is elliptic by Lemma 5.5.1. This is a contradiction (cf. Lemma 2.5.4(1)), because ρ is quasifuchsian by Theorem 6.1.8. We come to a similar contradiction if we assume that v(0, 1, 2) is contained in Dh(ρ(P−1 )). Hence ∩2j=0 Ih(ρ(Pj )) is not contained in Dh(ρ(P−1 )) nor Dh(ρ(P3 )). Next we prove the assertions for the remaining slopes s0 and s2 . Since the arguments are parallel, we study only the slope s2 . Suppose that θ+ (ρ, σ; s2 ) < 0. Then we have θ− (ρ, σ; s2 ) > 0 by Lemma 4.3.5, because the condition Duality implies that Fρ (P2 ) is a 2-dimensional convex polyhedron and hence Ih(ρ(P2 )) ∩ Eh(ρ(P1 )) ∩ Eh(ρ(P2 )) has dimension 2. Thus v(1, 2, 3) :=

6.7 Characterization of Σ(ν) for good labeled representations

153

∩3j=1 Ih(ρ(Pj )) is a singleton by Lemma 4.3.6, and we can see as in the previous argument appealing to Lemma 2.5.4(1) that it is diﬀerent from v(0, 1, 2). On the other hand, since θ+ (ρ, σ; s2 ) < 0, we see by Lemma 4.3.1 that v + (1, 2) := v + (ρ; P1 , P2 ) lies in int D(ρ(P3 )). Hence the interior of the half geodesic [v(1, 2, 3), v + (1, 2)] is contained in int Dh(ρ(P3 )). Since v(0, 1, 2) = Fρ (P0 , P1 , P2 ) lies in ∂Eh(ρ), it cannot be contained in the interior of the half geodesic. Thus the points v − (1, 2) := v − (ρ; P1 , P2 ), v(0, 1, 2), v(1, 2, 3), and v + (1, 2) lie in the complete geodesic Ih(ρ(P1 ))∩Ih(ρ(P2 )) in this order. In particular, v(1, 2, 3) is contained in the interior of the half geodesic [v(0, 1, 2), v + (1, 2)], which in tern is contained in int Dh(ρ(P0 )), because the condition θ+ (ρ, σ; s1 ) < 0 implies that v + (1, 2) is contained in int D(ρ(P0 )) by Lemma 4.3.1. Hence v(1, 2, 3) is contained in int Dh(ρ(P0 )). This completes the proof of Proposition 6.7.1. By parallel arguments, we obtain the following twin to the above proposition. Proposition 6.7.2. Let ρ = (ρ, ν) be a thick good labeled representation, and let (σ1 , σ2 , · · · , σm ) be the chain Σ(ν). Suppose 1 < i ≤ m and let s1 be the vertex of σ := σi = s0 , s1 , s2 which is not a vertex of the predecessor σj−1 of σi . Then the following hold. 1. θ− (ρ, σ; s1 ) < 0 < θ+ (ρ, σ; s1 ). 2. Ih(ρ(P0 )) ∩ Ih(ρ(P1 )) ∩ Ih(ρ(P2 )) is a singleton which is not contained in Dh(ρ(P−1 )) nor Dh(ρ(P3 )), where {Pj } is the sequence of elliptic generators associated with σ. Moreover, s1 is characterized by these properties among the vertices of σ. Namely, the following hold for the remaining vertices s0 and s2 of σ. 1. Either (i) θ− (ρ, σ; s0 ) ≥ 0 or (ii) θ− (ρ, σ; s0 ) < 0 and Ih(ρ(P−1 )) ∩ Ih(ρ(P0 )) ∩ Ih(ρ(P1 )) is a singleton which is contained in int Dh(ρ(P2 )). 2. Either (i) θ− (ρ, σ; s2 ) ≥ 0 or (ii) θ− (ρ, σ; s2 ) < 0 and Ih(ρ(P1 )) ∩ Ih(ρ(P2 )) ∩ Ih(ρ(P3 )) is a singleton which is contained in int Dh(ρ(P0 )) For thin good labeled representations, we have the following proposition. Proposition 6.7.3. For a type-preserving representation ρ, the following conditions are equivalent under Notation 2.1.14 (Adjacent triangles). 1. There is a thin label ν = (ν − , ν + ) with ν ± ∈ int τ such that (ρ, ν) is a good labeled representation. 2. The side parameter θǫ (ρ, σ) is deﬁned and θǫ (ρ, σ) = (+, 0, +) for each ǫ ∈ {−, +}. 3. The side parameter θǫ (ρ, σ ′ ) is deﬁned and θǫ (ρ, σ ′ ) = (+, +, 0) for each ǫ ∈ {−, +}.

154

6 Reformulation of Main Theorem 1.3.5 and outline of the proof

Moreover if the above mutually equivalent conditions are satisﬁed, then θǫ (ρ, σ) and θǫ (ρ, σ ′ ) determine the same point, ν ǫ , of int τ for each ǫ ∈ {−, +}, and ν = (ν − , ν + ) is the unique label such that (ρ, ν) is a good labeled representation. Furthermore (ρ, τ ) is an isosceles representation. Proof. (1) → (2). Suppose (ρ, ν) is a good labeled representation for some thin label ν = (ν − , ν + ) with ν ± ∈ int τ . Then θǫ (ρ, σ) gives the barycentric coordinate of ν ǫ . Since ν ǫ ∈ int τ , this implies that θǫ (ρ, σ) = (+, 0, +). (2) → (1). Suppose the condition (2) is satisﬁed. Then (ρ, τ ) is an isosceles representation by Proposition 5.2.3. Since θǫ (ρ, σ) = (+, 0, +) for each ǫ, Proposition 5.2.8 implies that ρ is quasifuchsian and its Ford domain is equal to Eh(τ ). Hence we see that (ρ, ν) is a good labeled representation, where ν = (ν − , ν + ) is the thin label with ν ǫ = θǫ (ρ, σ) ∈ int τ . Similarly we see the equivalence among the conditions (1) and (3). The additional assertions follow from the above proof and Proposition 6.2.5 (Uniqueness).

7 Openness

This chapter is devoted to the proof of Proposition 6.2.1 (Openness). The main ingredient of the proof is the study of the behavior of hidden isometric hemispheres, namely those isometric hemispheres which intersect the Ford domain but do not support 2-dimensional faces of the Ford domain (Deﬁnition 7.1.1). In Sect. 7.1, we give a recipe to list all hidden isometric hemispheres in the general setting (Lemma 7.1.3). By using the result, we show that the Ford domain of a good labeled representation (ρ, ν) has a hidden isometric hemisphere if and only if either one of the components of ν is non-generic (Deﬁnition 7.1.4) or ν is thin (Lemma 7.1.6). In Sects. 7.2 and 7.3, respectively, we prove Proposition 6.2.1 (Openness) around thick and thin good labeled representation.

7.1 Hidden isometric hemispheres Throughout this section, except the last part, Γ denotes a Kleinian group such that Γ∞ contains parabolic transformations. Deﬁnition 7.1.1. We call an isometric hemisphere Ih(A) (A ∈ Γ − Γ∞ ) a hidden isometric hemisphere for the Ford domain P h(Γ ), if P h(Γ )∩Ih(A) = ∅ while Ih(A) does not support a 2-dimensional face of the convex polyhedron P h(Γ ). We begin by explaining the following key observation, which enables us to determine all hidden isometric hemispheres for the Ford domain. This is a reformulation of [72, Lemma 5.45], and is essentially equal to Lemma 6.6.2. Lemma 7.1.2. Let x be a point in P h(Γ ), and let A be an element of Γ − Γ∞ such that x ∈ Ih(A). Then A(x) is also contained in P h(Γ ).

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7 Openness

Proof. We begin by presenting a characterization of P h(Γ ) which is suitable 3 for our purpose. For each A ∈ P SL(2, C) and x ∈ H , the derivative 3 3 dAx : Tx H → TA(x) H is equal to a positive constant times an orthogo3

nal transformation with respect to the Euclidean metric of H . We denote the positive constant by |A′ (x)|. Then we have the following characterization of P h(Γ ). 3 P h(Γ ) = {x ∈ H | |A′ (x)| ≤ 1 for every A ∈ Γ }. Now let x be a point in P h(Γ ) and let A be an element of Γ such that x ∈ Ih(A). Then for every B ∈ Γ , |B ′ (A(x))| =

|(BA)′ (x)| |A′ (x)|

= |(BA)′ (x)|

because x ∈ Ih(A)

≤1

because x ∈ P h(Γ ).

Hence A(x) ∈ P h(Γ ) by the preceding characterization of P h(Γ ). This completes the proof. By using this lemma, we can determine all isometric hemispheres containing a point of P h(Γ ). To describe the result, we use a slight generalization of the notation in Sect. 6.6. For each point p in P h(Γ ), set [p] := {x ∈ P h(Γ ) | x = A(p) for some A ∈ Γ }. Then we have the following companion to Lemma 6.6.2. Lemma 7.1.3. Suppose that Γ is geometrically ﬁnite. Then for each point p ∈ P h(Γ ), the following hold. 1. [p]/Γ∞ is a ﬁnite set. 2. Let {p = p1 , p2 , · · · , pn } ⊂ [p] be a representative for [p]/Γ∞ , and let Ai ∈ Γ be an element such that Ai (p) = pi and A1 = 1. Then the set of the isometric hemispheres of elements of Γ − Γ∞ containing the point p is equal to n % ! Ih(A) | A ∈ Ai Γp − {1} . i=1

where Γp is the stabilizer of p with respect to the action of Γ . In particular, the number of the isometric hemispheres containing p is inﬁnite if and only if p is a parabolic ﬁxed point. If p is not a parabolic ﬁxed point, then the number is equal to n|Γp | − 1.

Proof. (1) follows from Lemma A.1.10. (2) For each p ∈ P h(Γ ) and A ∈ Γ −Γ∞ , we have the following equivalence.

7.1 Hidden isometric hemispheres

157

p ∈ Ih(A),

⇔ A(p) ∈ P h(Γ ),

⇔ A(p) ∈ [p], ⇔ A(p) = BAi (p) for some i ∈ {1, · · · , n} and B ∈ Γ∞

⇔ A = BAi C for some i ∈ {1, · · · , n}, B ∈ Γ∞ and C ∈ Γp , ⇔ Ih(A) = Ih(Ai C) for some some i ∈ {1, · · · , n} and C ∈ Γp . Here the ﬁrst equivalence follows from Lemma 7.1.2, and the last equivalence follows from Lemma 4.1.1(3). Hence we obtain the ﬁrst assertion. Moreover, Lemma 4.1.1(3) also implies that the isometric hemispheres Ih(Ai C) (i ∈ {1, · · · , n} and C ∈ Γp ) are mutually distinct. Hence there are inﬁnitely many isometric hemispheres which contain p if and only if Γp is inﬁnite. Since Γ is geometrically ﬁnite and p ∈ P (Γ ), this is equivalent to the condition that p is a parabolic ﬁxed point by Lemma A.1.4. This completes the proof. By using the above lemma, we can completely determine the hidden isometric hemispheres for good labeled representations. Roughly speaking, those hidden isometric hemispheres described in Lemma 4.5.4(4) and Corollary 5.2.5 are the only hidden isometric hemispheres for good labeled representations. To present the explicit statement, we introduce the following deﬁnition and notation. Deﬁnition 7.1.4. Let ν = (ν − , ν + ) ∈ H2 × H2 be a thick label. (1) The component ν ǫ of the label ν is said to be non-generic or generic according as it is contained in the 1-skeleton of the Farey triangulation or not, namely, dim δ ǫ (ν) = 1 or 2. Here δ ǫ (ν) is the edge or the triangle of D whose (relative) interior contains ν ǫ (Deﬁnition 1.3.4(2)). (2) Let ǫ ∈ {−, +}. If ν ǫ is non-generic, then σ ǫ,∗ (ν) denotes the triangle of D such that σ ǫ (ν) ∩ σ ǫ,∗ (ν) = δ ǫ (ν). If ν ǫ is generic, σ ǫ,∗ (ν) is undeﬁned. Notation 7.1.5 (Non-generic label). Let ν = (ν − , ν + ) ∈ H2 × H2 be a thick label such that ν ǫ is non-generic. Then the symbols sj and s′j denote ˆ such that σ ǫ (ν) = s0 , s1 , s2 and σ ǫ,∗ (ν) = s′ , s′ , s′ such the elements of Q 0 1 2 ′ that s0 = s0 and s′1 = s2 . As in Notation 2.1.14 (Adjacent triangles), {Pj } and {Pj′ }, respectively, denote the sequences of elliptic generators associated with σ ǫ (ν) and σ ǫ,∗ (ν) such that P0′ = P0 , P1′ = P2 and P2′ = P1P2 . (See the upper right ﬁgure in Fig. 7.3). Then we have the following classiﬁcation of the hidden isometric hemispheres for good labeled representations (cf. [72, Corollary 6.11]). Lemma 7.1.6 (Hidden isometric hemisphere). Let ρ = (ρ, ν) be a good labeled representation. Then the following is the complete list of the hidden isometric hemispheres for P h(ρ) modulo ρ(K).

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7 Openness

1. If ν is thick and ν ǫ is non-generic, then Ih(ρ(P2′ )) is a hidden isometric hemisphere for P h(ρ) under Notation 7.1.5 (Non-generic label). In fact Ih(ρ(P2′ )) ∩ P h(ρ) = v ǫ (ρ; P2 , P3 ) (see Fig. 7.1(a)). 2. If ν is thin and ν ± ∈ τ under Notation 2.1.14 (Adjacent triangles). Then Ih(ρ(P1 )) and Ih(ρ(P2′ )) are hidden isometric hemispheres for P h(ρ). In fact Ih(ρ(P1 )) ∩ P h(ρ) = Axis(ρ(P1 )) and Ih(ρ(P2′ )) ∩ P h(ρ) = Axis(ρ(P2′ )) (see Fig. 7.1(b)).

P3

P0 P2′ P0

P2 P3

P2 P1

P1 (a)

(b)

P2′

Fig. 7.1. Hidden isometric hemispheres

Proof. Let p be a point in P h(ρ), and let Hidden(p) (resp. V isible(p)) be the number of the hidden isometric hemispheres for P h(ρ) containing p (resp. the number of the isometric hemispheres containing p which support a 2dimensional face of P h(ρ)). Then by Lemma 7.1.3, we have the following formula. Hidden(p) = |[p]/Γ∞ | · |Γp | − V isible(p) − 1. Then we obtain the following table (cf. Proof of Lemmas 6.6.3 and 6.6.4), which implies the desired result. In the row starting from Fρ (P ) in the table, for example, the ﬁrst line is for the case when p is generic, i.e., p ∈ / Axis(ρ(P )), and the second line is for the case when p ∈ Axis(ρ(P )). Similar conventions are employed for the rows with two lines. The case when ν is thick The face whose interior contains p |Γp | |[p]/Γ∞ | visible(p) hidden(p) Fρ (P, Q, R) 2 2 3 0 Fρ (P, Q) 1 3 2 0 Fρ (P ) 1 2 1 0 2 1 1 0 non-degenerate outer edge fρǫ (P ) 1 2 1 0 2 1 1 0 outer vertex fρǫ (Pj , Pj+1 ) (ν ǫ : generic) 1 3 2 0 degenerate outer edge fρǫ (P1 ) 2 2 3 0 outer vertex fρǫ (P2 , P3 ) (ν ǫ : non-generic) 2 2 2 1

7.2 Proof of Proposition 6.2.1 (Openness) - Thick Case -

159

The case when ν is thin The face whose interior contains p |Γp | |[p]/Γ∞ | visible(p) hidden(p) Fρ (P0 , P2 ) 2 2 2 1 Fρ (P2 , P3 ) 2 2 2 1 Fρ (Pj ) j ≡ 1 (mod 3) 1 2 1 0 2 1 1 0 fρǫ (Pj ) j ≡ 1 (mod 3) 1 2 1 0 2 1 1 0 fρǫ (P0 , P2 ) 2 2 2 1 fρǫ (P2 , P3 ) 2 2 2 1

7.2 Proof of Proposition 6.2.1 (Openness) - Thick Case In this section, we prove Proposition 6.2.1 (Openness) in the case when ρ0 = (ρ0 , ν 0 ) is thick. The essence of the proof is the study the behavior of the hidden isometric hemispheres under small deformation. To describe these isometric hemispheres, we introduce the “augmentation” L∗ (ν) of the simplicial complex L(ν) = L(Σ(ν)) deﬁned in Deﬁnitions 3.2.3 and 3.3.2 (Deﬁnition 7.2.1). Its vertex set L∗ (ν)(0) consists of those elliptic generators which are ‘involved’ in the Ford domain P h(ρ0 ), and the proof of Proposition 6.2.1 (Openness) consists of detailed analysis of the behavior of the isometric hemispheres of ρ(P ) with P ∈ L∗ (ν)(0) for nearby representations ρ. Here is a rough sketch of the idea of the proof. By Lemmas 4.6.1 and 4.6.2, we can construct a continuous map ν from a neighborhood U0 of ρ0 to H2 ×H2 , such that σ ǫ (ν(ρ)) is an ǫ-terminal triangle of ρ for every ρ ∈ U0 . By using Lemma 7.2.3 (Disjointness), which is obtained as a corollary to Lemma 7.1.6 (Hidden isometric hemisphere), we can ﬁnd a smaller neighborhood U2 such that (ρ, ν(ρ)) satisﬁes the conditions NonZero and Frontier for every ρ ∈ U2 (Lemma 7.2.5). We note that L(ν(ρ)) is a subcomplex of the augmentation L∗ (ν) every ρ ∈ U2 . Finally, by using Lemma 7.2.3 (Disjointness) again, we ﬁnd a smaller neighborhood U3 such that (ρ, ν(ρ)) also satisﬁes the condition Duality (Lemma 7.2.7). The idea of this last step is as follows. For each ρ ∈ U2 , L(ν(ρ)) is a subcomplex of L∗ (ν 0 ). •

•

If ξ is a simplex of L(ν 0 ) then Fρ0 (ξ) is a transversal intersection of hyperplanes and and half spaces. Thus Fρ (ξ) with ρ = (ρ, ν(ρ)) continues to be a convex polyhedron of the same dimension. On the other hand, Lemma 7.2.3 (Disjointness) guarantees that those isometric hemispheres which are not involved in the deﬁnition of Fρ0 (ξ), except the hidden isometric hemispheres, are disjoint from Fρ (ξ). Moreover the behavior of the hidden isometric hemispheres are controlled by the side parameter. These imply that Fρ (ξ) is contained in Eh(ρ), after choosing a smaller neighborhood U3 . If ξ is not contained in L(ν 0 ), then hidden isometric hemispheres are involved in the deﬁnition of Fρ (ξ). However, the behavior of the hidden

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7 Openness

isometric hemispheres are controlled by the side parameter. This fact enables us to show that Fρ (ξ) is a convex polyhedron contained in Eh(ρ) of the correct dimension, after choosing a smaller neighborhood U3 . Before starting the formal proof of Proposition 6.2.1 (Openness), we prepare a few notations. Deﬁnition 7.2.1. Let ν = (ν − , ν + ) ∈ H2 × H2 be a thick label. (1) ∂ ǫ L(ν) denotes the 1-dimensional subcomplex of L(ν) spanned by the vertices (P ) with s(P ) ∈ σ ǫ (ν). (2) Σ ∗ (ν) denotes the chain of triangles obtained from Σ(ν) by adding ǫ,∗ σ (ν) (ǫ ∈ {−, +}) whenever it is deﬁned (Deﬁnition 7.1.4). (3) L∗ (ν) denotes the abstract simplicial complex L(Σ ∗ (ν)) (see Deﬁnition 3.1.1). ǫ (4) ∂aug L∗ (ν) denotes the subcomplex L(σ ǫ (ν)) or L((σ ǫ (ν), σ ǫ,∗ (ν))) of ∗ L (ν) according as ν ǫ is generic or non-generic. ǫ L∗ (ν) spanned by the vertices (5) L(δ ǫ (ν)) denotes the subcomplex of ∂aug ǫ P such that s(P ) ∈ δ (ν). We note that these deﬁnitions are also motivated by the results in Secǫ tions 4.5 and 4.6. In particular, ∂aug L∗ (ν) corresponds to the complex ′ ǫ L(ρ, {σ, σ }) ⊂ C in these sections. If ν is generic, then δ ǫ (ν) = σ ǫ (ν) and ǫ L∗ (ν) = L(σ ǫ (ν)) = L(δ ǫ (ν)). If ν ǫ is non-generic, then these three com∂aug plexes are related as illustrated in Fig. 7.2. In the proof of Proposition 6.2.1 (Openness) we need to study the behavior of (the closure of) the isometric hemispheres near the ideal boundary of Eh(ρ) systematically. To this end we introduce the following deﬁnition. Deﬁnition 7.2.2. Let ρ = (ρ, ν) be a labeled representation, such that ∗ (0) φ−1 = ∅. Then, for ξ ∈ L∗ (ν)(≤2) and ǫ ∈ {−, +}, fρǫ (ξ) denotes ρ (0) ∩ L (ν) the (possibly empty) compact subset of C deﬁned by fρǫ (ξ) = {I(ρ(P )) | P ∈ ξ (0) } ∩ E ǫ (ρ).

The above deﬁnition is related to Deﬁnition 4.2.5 and Notation 4.3.7 as follows. If {Pj } is the sequence of elliptic generators associated with σ ǫ (ν), then: fρǫ (Pj ) = eǫ (ρ, σ ǫ (ν); Pj ), fρǫ (Pj , Pj+1 ) = v ǫ (ρ; Pj , Pj+1 ). The following corollary to Lemma 7.1.6 (Hidden isometric hemisphere) plays an important role in the proof of Proposition 6.2.1 (Openness). Lemma 7.2.3 (Disjointness). Let ρ = (ρ, ν) be a thick good labeled representation. Then the following hold.

7.2 Proof of Proposition 6.2.1 (Openness) - Thick Case -

161

L∗ (ν)

σ +,∗

ν+ σ+

+ ∂aug L∗ (ν) ∂ + L(ν)

− ∂aug L∗ (ν) = ∂ − L(ν)

σ−

ν−

L(δ + (ν))

L(δ − (ν)) ± Fig. 7.2. L∗ (ν), ∂aug L∗ (ν) and L(δ ± (ν)) in the case when ν + is non-generic and − ν is generic.

P1′ P0′ s′2

s0 = s′0

σ ǫ,∗ (ν)

P2

P0

P1

s2 = s′1 P2′

σ ǫ (ν) νǫ

s1

P2′

δ ǫ (ν) P0

P3 P2 P1

ǫ Fig. 7.3. ∂aug L∗ (ν) and Eh(ρ, ν)

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7 Openness

(1) If ξ ∈ L(ν)(≤2) and X ∈ L∗ (ν)(0) − st0 (ξ, L∗ (ν))(0) , then Dh(ρ(X)) is disjoint from F ρ (ξ). ǫ ǫ (2) If ξ ∈ ∂aug L∗ (ν)(≤2) and X ∈ L∗ (ν)(0) − st0 (ξ, ∂aug L∗ (ν))(0) , then D(ρ(X)) is disjoint from fρǫ (ξ). ǫ L∗ (ν))(0) , then D(ρ(X)) is disjoint from (3) If X ∈ L∗ (ν)(0) − (∂aug ǫ fr E (ρ). We now start the formal proof of Proposition 6.2.1 (Openness). Lemma 7.2.4. Let ρ0 = (ρ0 , ν 0 ) be a thick good labeled representation. Then ∗ (0) there is a neighborhood U1 of ρ0 in X such that φ−1 = ∅ for ρ (0) ∩ Σ (ν 0 ) every ρ ∈ U1 . Proof. Recall that ρ0 is quasifuchsian by Theorem 6.1.8. So we may set U1 = QF (cf. Lemma 2.5.4). Lemma 7.2.5. Let ρ0 and U1 be as in Lemma 7.2.4. Then there is a neighborhood U2 of ρ0 in U1 and a continuous map U2 ∋ ρ → ν(ρ) ∈ H2 × H2 , such that ν(ρ0 ) = ν 0 and that (ρ, ν(ρ)) satisﬁes the conditions NonZero and Frontier for every ρ ∈ U2 . Proof. We ﬁrst choose a neighborhood U1ǫ of ρ0 in X and construct a continuous map U1ǫ ∋ ρ → ν ǫ (ρ) ∈ H2 , for each ǫ, as follows: Case 1. ν0ǫ is generic, i.e., ν0ǫ ∈ int σ ǫ (ν 0 ). Then by Lemma 4.6.1, there is a neighborhood U1ǫ of ρ0 in U1 , such that σ ǫ (ν 0 ) is an ǫ-terminal triangle of ρ for every ρ ∈ U1ǫ . We deﬁne the continuous map U1ǫ ∋ ρ → ν ǫ (ρ) ∈ H2 by ν ǫ (ρ) := θǫ (ρ, σ ǫ (ν 0 )) ∈ σ ǫ (ν 0 ) ∩ H2 . Case 2. ν0ǫ is non-generic, i.e., δ ǫ (ν 0 ) is the edge σ ǫ (ν 0 ) ∩ σ ǫ,∗ (ν 0 ). Then by Lemma 4.6.2, there is a neighborhood U1ǫ of ρ0 in U1 , such that, for each ρ ∈ U1ǫ , σ ǫ (ν 0 ) or σ ǫ,∗ (ν 0 ) is an ǫ-terminal triangle of ρ according as θǫ (ρ, σ ǫ (ν 0 ); s1 ) ≥ 0 or θǫ (ρ, σ ǫ (ν 0 ); s1 ) < 0, under Notation 7.1.5 (Nongeneric label). We deﬁne the map U2ǫ ∋ ρ → ν ǫ (ρ) ∈ H2 by θǫ (ρ, σ ǫ (ν 0 )) ∈ σ ǫ (ν 0 ) ∩ H2 if θǫ (ρ, σ ǫ (ν 0 ); s1 ) ≥ 0, ν ǫ (ρ) := θǫ (ρ, σ ǫ,∗ (ν 0 )) ∈ σ ǫ,∗ (ν 0 ) ∩ H2 if θǫ (ρ, σ ǫ (ν 0 ); s1 ) ≤ 0. This map is well-deﬁned and continuous by Lemma 4.5.3(2), because it implies that if θǫ (ρ, σ ǫ (ν 0 ); s1 ) = 0 then θǫ (ρ, σ ǫ (ν 0 )) and θǫ (ρ, σ ǫ,∗ (ν 0 )) determine the same point of the interior of the edge δ ǫ (ν 0 ) = σ ǫ (ν 0 ) ∩ σ ǫ,∗ (ν 0 ). Set U1′ = U1− ∩ U1+ , and deﬁne the continuous map ν : U1′ → H2 × H2 by ν(ρ) = (ν − (ρ), ν + (ρ)). Then σ ǫ (ν(ρ)) is an ǫ-terminal triangle of ρ for every ρ ∈ U1′ and ǫ ∈ {−, +}. Thus the pair (ρ, ν(ρ)) satisﬁes the ﬁrst condition of the condition ǫ-Frontier (Deﬁnition 6.1.4). It is obvious that (ρ, ν(ρ)) also satisﬁes the second condition. To show that (ρ, ν(ρ)) satisﬁes the last condition, we need the following lemma.

7.2 Proof of Proposition 6.2.1 (Openness) - Thick Case -

163

Claim 7.2.6. There is a neighborhood U2 of ρ0 in U1′ such that D(ρ(X)) ∩ E ǫ (ρ, σ ǫ (ν(ρ))) = ∅ for every ρ ∈ U2 and X ∈ L(ν(ρ))(0) − L(σ ǫ (ν(ρ)))(0) . Proof. Pick an element X ∈ L∗ (ν 0 )(0) . We show that there is a neighborhood U2 (X) of ρ0 in U1′ , such that if ρ ∈ U2 (X) and X ∈ L(ν(ρ))(0) −L(σ ǫ (ν(ρ)))(0) then D(ρ(X)) ∩ E ǫ (ρ, σ ǫ (ν(ρ))) = ∅. ǫ L∗ (ν 0 )(0) for some ǫ. Then we may set U2 (X) = U1′ . To Suppose X ∈ ∂aug see this, pick an element ρ ∈ U1′ such that X ∈ L(ν(ρ))(0) − L(σ ǫ (ν(ρ)))(0) . Then ν0ǫ is non-generic, s(X) = s1 and ν ǫ (ρ) ∈ int σ ǫ,∗ (ν 0 ) under Notation 7.1.5 (Non-generic label). So θǫ (ρ, σ ǫ (ν 0 ); s1 ) < 0 by the deﬁnition of ν(ρ). Thus D(ρ(X)) is disjoint from E ǫ (ρ, σ ǫ (ν(ρ))) by Lemma 4.6.2(2). ǫ Suppose X ∈ / ∂aug L∗ (ν 0 )(0) for each ǫ ∈ {−, +}. Then D(ρ0 (X)) is disjoint ǫ ǫ from E (ρ0 , σ (ν(ρ))) by Lemma 7.2.3(3) (Disjointness). Hence we can ﬁnd a desired neighborhood U2 (X). We may assume U2 (X) depends only on the slope s(X) and hence the intersection, U2 , of all U2 (X) (X ∈ L∗ (ν 0 )(0) ) is a neighborhood of ρ0 . Since L(ν(ρ)) is a subcomplex of L∗ (ν 0 ), we obtain the claim. The above claim implies that E ǫ (ρ) = E ǫ (ρ, σ ǫ (ν(ρ))) for every ρ ∈ U2 , where ρ = (ρ, ν(ρ)). Hence (ρ, ν(ρ)) satisﬁes the condition Frontier. This completes the proof of Lemma 7.2.5. The following lemma enables us to treat the condition Duality (Deﬁnition 6.1.3). Lemma 7.2.7. Let ρ0 = (ρ0 , ν 0 ), U2 and ν : U2 → H2 × H2 be as in Lemma 7.2.5. Then there is a neighborhood U3 of ρ0 in U2 , such that the following condition is satisﬁed for every ρ ∈ U3 : Fρ (ξ) is a convex polyhedron of dimension 2 − dim ξ contained in Eh(ρ) for every ξ ∈ L(ν(ρ))(≤2) , where ρ denotes the labeled representation (ρ, ν(ρ)). Proof. We show that for every ξ ∈ L∗ (ν 0 )(≤2) there is a neighborhood U3 (ξ) of ρ0 in U2 , such that if ρ ∈ U3 (ξ) and ξ ∈ L(ν(ρ))(≤2) then Fρ (ξ) is a convex polyhedron of dimension 2 − dim ξ contained in Eh(ρ). Case 1. ξ = (P, Q, R) ∈ L(ν 0 )(2) . Then Fρ0 (ξ) is the transversal intersection of the three isometric hemispheres Ih(ρ0 (P )), Ih(ρ0 (Q)), and Ih(ρ0 (R)). Hence we can ﬁnd a neighborhood U3 (ξ) of ρ0 in U2 , such that Fρ (ξ) = Ih(ρ(P )) ∩ Ih(ρ(Q)) ∩ Ih(ρ(R)) is a 0-dimensional convex polyhedron for every ρ ∈ U3 (ξ). On the other hand, by Lemma 7.2.3(1) (Disjointness), Fρ0 (ξ) is disjoint from Dh(ρ0 (X)) for every X ∈ L∗ (ν 0 )(0) − ξ (0) = L∗ (ν 0 )(0) − st0 (ξ, L∗ (ν))(0) . Hence we can choose U3 (ξ), so that if ρ ∈ U3 (ξ) and ξ ∈ L(ν(ρ))(2) , then Fρ (ξ) is disjoint from Dh(ρ(X)) for every X ∈ L∗ (ν 0 )(0) −st0 (ξ, L∗ (ν 0 ))(0) . Since L(ν(ρ))(0) −st0 (ξ, L(ν(ρ)))(0) ⊂

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7 Openness

L∗ (ν 0 )(0) − st0 (ξ, L∗ (ν 0 ))(0) , we can conclude that Fρ (ξ) ⊂ Eh(ρ) whenever ρ ∈ U3 (ξ) and ξ ∈ L(ν(ρ))(2) . Case 2. ξ ∈ L∗ (ν 0 )(2) − L(ν 0 )(2) . Then we may assume that ν0ǫ is non-generic for some ǫ and ξ = (P0 , P1 , P2 ) or (P1′ , P2′ , P3′ ) under Notation 7.1.5 (Non-generic label). For simplicity we assume ξ = (P0 , P1 , P2 ). Pick an element ρ ∈ U2 such that ξ ∈ L(ν(ρ))(2) . Then ν ǫ (ρ) ∈ int σ ǫ,∗ (ν 0 ). Hence, by the deﬁnition of ν ǫ (ρ), we have θǫ (ρ, σ ǫ (ν 0 ); s1 ) < 0 and therefore Fρ (ξ) = ∩2j=0 Ih(ρ(Pj )) is a 0-dimensional convex polyhedron by Lemma 4.6.2(2). By Lemma 7.2.3(2) (Disjointness), Dh(ρ0 (X)) is disjoint from the singleton fρǫ 0 (ξ) ⊂ ∩2j=0 I(ρ0 (Pj )) for every X ∈ L∗ (ν 0 )(0) − {P0 , P1 , P2 }. Hence we can ﬁnd a neighborhood U3 (ξ) of ρ0 in U2 , such that Fρ (ξ) ⊂ Eh(ρ) whenever ρ ∈ U3 (ξ) and ξ ∈ L(ν(ρ))(2) . Case 3. ξ = (P, Q) ∈ L(ν 0 )(1) . Since Fρ0 (ξ) is a 1-dimensional convex polyhedron obtained as the transversal intersection Fρ (ξ) = (Ih(ρ0 (P )) ∩ Ih(ρ0 (Q))) ∩ (∩{Eh(ρ0 (X)) | X ∈ lk(ξ, L(ν(ρ)))}) , there is a neighborhood U3 (ξ) of ρ0 in U2 such that Fρ (ξ) continues to be a 1-dimensional convex polyhedron for every ρ ∈ U3 (ξ). By Lemma 7.2.3(1) (Disjointness), Dh(ρ0 (X)) is disjoint from F ρ0 (ξ) for every X ∈ L∗ (ν 0 )(0) − st0 (ξ, L∗ (ν 0 ))(0) . Thus we can choose U3 (ξ), so that every ρ ∈ U3 (ξ) has the same property. Since L(ν(ρ))(0) − st0 (ξ, L(ν(ρ)))(0) ⊂ L∗ (ν 0 )(0) − st0 (ξ, L∗ (ν 0 ))(0) , we can conclude that Fρ (ξ) ⊂ Eh(ρ) for every ρ ∈ U3 (ξ). Case 4. ξ = (P, Q) ∈ L∗ (ν 0 )(1) − L(ν 0 )(1) . As in Case 2, we may assume ′ ǫ ) for some j ∈ {0, 1, 2} under ν0 is non-generic for some ǫ and ξ = (Pj′ , Pj+1 Notation 7.1.5 (Non-generic label). First, we study the case when ξ = (P0′ , P1′ ) = (P0 , P2 ). Note that ξˆ := (P0 , P1 , P2 ) belongs to L∗ (ν 0 )(2) . By the argument in Case 2, there is a ˆ of ρ0 in U2 , such that if ρ ∈ U3 (ξ) ˆ and ξ ∈ L(ν(ρ))(1) (and neighborhood U3 (ξ) (2) ˆ ˆ hence ξ ∈ L(ν(ρ)) ) then Fρ (ξ) is a 0-dimensional convex polyhedron. We ˆ To this end, pick an element ρ ∈ U3 (ξ) ˆ show that we may set U3 (ξ) = U3 (ξ). (1) (2) ǫ ′ ′ ǫ such that ξ ∈ L(ν(ρ)) . Then ξˆ ∈ L(ν(ρ)) and hence v (ρ; P0 , P1 ) ∈ E (ρ) ˆ in H3 is conby Lemma 7.2.5. Hence the convex hull of v ǫ (ρ; P0′ , P1′ ) and Fρ (ξ) tained in Eh(ρ) and has dimension 1. Since lk(ξ, L(ν(ρ))) = {P1 }, the convex hull is equal to F ρ (ξ) = Ih(ρ(P0′ )) ∩ Ih(ρ(P1′ )) ∩ Eh(ρ(P1 )). ˆ has the desired property for ξ = (P ′ , P ′ ). Hence U3 (ξ) = U3 (ξ) 1 0 Next, we show that the same neighborhood also satisﬁes the desired con′ ′ dition for (P1′ , P2′ ) and (P2′ , P3′ ). Since ρ(Pj′ )(Fρ (Pj′ , Pj+1 )) = Fρ (Pj−1 , Pj′ ) ′ ′ ′ ′ by Lemma 4.1.3 (Chain rule), we see dim Fρ (P2 , P3 ) = dim Fρ (P1 , P2 ) =

7.3 Proof of Proposition 6.2.1 (Openness) - Thin case -

165

dim Fρ (P0′ , P1′ ) = 1 for every ρ in the neighborhood such that ν(ρ) ∈ int σ ǫ,∗ (ν 0 ). Moreover, F ρ (P1′ , P2′ ) (resp. F ρ (P2′ , P3′ )) is the convex hull of 3 v ǫ (ρ; P1′ , P2′ ) (resp. v ǫ (ρ; P2′ , P3′ )) and Fρ (P1′ , P2′ , P3′ ) in H . Hence we can con′ ′ ′ ′ clude that both F ρ (P1 , P2 ) and F ρ (P2 , P3 ) are 1-dimensional convex polyhedra in Eh(ρ) as in the above. Case 5. ξ = (P ) ∈ L(ν 0 )(0) . As in Case 3, we can ﬁnd a neighborhood U3 (ξ) of ρ0 in U2 such that dim Fρ (ξ) = 2 for every ρ ∈ U3 (ξ). By using Lemma 7.2.3 (Disjointness)(1), we can choose U3 (ξ), so that Fρ (ξ) ⊂ Eh(ρ) for every ρ ∈ U3 (ξ). Case 6. ξ = (P ) ∈ L∗ (ν 0 )(0) − L(ν 0 )(0) . As in Case 2, we may assume that ν0ǫ is non-generic for some ǫ and that ξ = (P2′ ) under Notation 7.1.5 (Nonˆ generic label). Let ξˆ = (P1′ , P2′ , P3′ ). We show that we may set U3 (ξ) = U3 (ξ). (0) ˆ ˆ To this end, pick an element ρ ∈ U3 (ξ) such that ξ ∈ L(ν(ρ)) . Then ξ ∈ ˆ is a 0-dimensional convex polyhedron in Eh(ρ) by L(ν(ρ))(2) and hence Fρ (ξ) ˆ Moreover eǫ (ρ, σ ǫ,∗ (ν 0 ); P ′ ) ⊂ E ǫ (ρ) by Lemma 7.2.5. the deﬁnition of U3 (ξ). 2 ˆ in H3 is contained in Hence the convex hull of eǫ (ρ, σ ǫ,∗ (ν 0 ); P2′ ) and Fρ (ξ) Eh(ρ) and has dimension 2. Since lk(ξ, L(ν(ρ)))(0) = {P1′ , P3′ }, the convex hull is equal to F ρ (ξ) = Ih(ρ(P2′ )) ∩ Eh(ρ(P1′ )) ∩ Eh(ρ(P3′ )) . ˆ has the desired property. Hence U3 (ξ) = U3 (ξ) We have thus constructed the desired neighborhood U3 (ξ) for every ξ ∈ L∗ (ν 0 )(≤2) . We may assume U3 (ξ) depends only on the image of ξ in L(ν 0 )/K. Hence the intersection U3 = ∩{U3 (ξ) | ξ ∈ L∗ (ν 0 )(2) } is a neighborhood of ρ0 in X . Since L(ν(ρ)) is a subcomplex of L∗ (ν 0 ) for every ρ ∈ U3 , U3 has the desired property. This completes the proof of Lemma 7.2.7.

Proof (Proof of Proposition 6.2.1 (Openness)). Let U be the neighborhood U3 of ρ0 in Lemma 7.2.7 and ν : U → H2 ×H2 be the restriction of the continuous map in Lemma 7.2.5. Then by Lemmas 7.2.4, 7.2.5 and 7.2.7, (ρ, ν(ρ)) satisﬁes the conditions NonZero, Frontier and Duality for every ρ ∈ U . Hence (ρ, ν(ρ)) is good.

7.3 Proof of Proposition 6.2.1 (Openness) - Thin case In this section, we prove Proposition 6.2.1 (Openness) in the case when ρ0 = (ρ0 , ν 0 ) is thin. The diﬀerence between the thick and thin cases is that there is interaction between the two components of the side parameter. Because of this interaction, various patterns appear after a small perturbation of ρ0 , as illustrated in Fig. 7.4. Though various patterns appear, they are controlled by the side parameter. However, the idea of the proof is essentially the same as that for the thick case. By using Lemma 4.6.7, we construct a continuous map ν from

166

7 Openness s′2

P2 ν

s0 = s′0

+

P0

s2 = s′1 ν− s1 P1

(0) ν+

ν+

ν−

ν−

(2)

(1) ν+

ν+

ν−

ν−

(4)

(3) ν+

ν+

ν−

ν−

(5)

(6)

ν+

ν−

ν+

ν− (7)

(8)

Fig. 7.4. Ford domains of representation in a neighborhood of an isosceles representation

7.3 Proof of Proposition 6.2.1 (Openness) - Thin case -

167

a neighborhood U0 of ρ0 to H2 × H2 , such that σ ǫ (ν(ρ)) is an ǫ-terminal triangle of ρ for every ρ ∈ U0 . By using Lemma 7.3.1 (Disjointness), an analogy of Lemma 7.2.3 (Disjointness), we can ﬁnd a smaller neighborhood U2 such that (ρ, ν(ρ)) satisﬁes the conditions NonZero and Frontier for every ρ ∈ U2 (Lemma 7.3.3). Finally, by using Lemma 7.3.1 (Disjointness) again, we ﬁnd a smaller neighborhood U3 such that (ρ, ν(ρ)) also satisﬁes the condition Duality (Lemma 7.3.5). We now give a formal proof for the thin case. Throughout the proof, we employ Notation 2.1.14 (Adjacent triangles), and assume that ν0± ∈ τ = σ∩σ ′ , where ν 0 = (ν0− , ν0+ ). We adapt Deﬁnition 7.1.4 for the thin case, and deﬁne Σ ∗ (ν 0 ) to be the chain (σ, σ ′ ) and L∗ (ν 0 ) := L(Σ ∗ (ν 0 )). Then the following corollary to Lemma 7.1.6 (Hidden isometric hemisphere) plays the role of Lemma 7.2.3 (Disjointness). Lemma 7.3.1 (Disjointness). Let ρ0 = (ρ0 , ν 0 ) be a thin good labeled representation as above. Then (ρ0 , τ ) is an isosceles representation and the following holds. 1. If X ∈ L∗ (ν 0 )(0) −{P0 , P1 , P2 }, then Dh(ρ(X)) is disjoint from F ρ (P0 , P2 ). 2. If X ∈ L∗ (ν 0 )(0) −{P1′ , P2′ , P3′ }, then Dh(ρ(X)) is disjoint from F ρ (P1′ , P3′ ) = F ρ (P2 , P3 ). The proof of Lemma 7.2.4 works in the thin case, and we have the following lemma. Lemma 7.3.2. Let ρ0 = (ρ0 , ν 0 ) be a thin good labeled representation as above. Then there is a neighborhood U1 of ρ0 in X such that φ−1 ρ (0) ∩ Σ ∗ (ν 0 )(0) = ∅ for every ρ ∈ U1 . We also have the following analogy of Lemma 7.2.5. Lemma 7.3.3. Let ρ0 and U1 be as in Lemma 7.3.2. Then there is a neighborhood U2 of ρ0 in U1 and a continuous map U2 ∋ ρ → ν(ρ) ∈ H2 × H2 , such that ν(ρ0 ) = ν 0 and ρ = (ρ, ν(ρ)) satisﬁes the conditions NonZero and Frontier for every ρ ∈ U2 . Moreover, if ν(ρ) is thin, then ρ is good. Proof. As in the proof of Lemma 7.2.5, by using Lemma 4.6.7 instead of Lemma 4.6.2, we can ﬁnd a neighborhood U2 and a continuous map ν : U2 → H2 × H2 such that σ ǫ (ν(ρ)) is an ǫ-terminal triangle of ρ = (ρ, ν(ρ)) for every ρ ∈ U1′ . If ν(ρ) is thin. Then (ρ, τ ) is an isosceles representation by Proposition 5.2.3. Thus, since we have chosen U2 so that it is contained in QF, it follows that ρ = (ρ, ν(ρ)) is good by Proposition 5.2.8(2). Thus we may assume ν(ρ) is thick. Claim 7.3.4. If ρ ∈ U2 and and ν(ρ) is thick, then D(ρ(X)) ∩ E ǫ (ρ, σ ǫ (ν(ρ))) = ∅ for every X ∈ L(ν(ρ))(0) − L(σ ǫ (ν(ρ)))(0) . (See Fig. 7.4(7) and (8).)

168

7 Openness

Proof (Proof of Claim 7.3.4). Suppose ν(ρ) is thick and X ∈ L(ν(ρ))(0) − L(σ ǫ (ν(ρ)))(0) . Then the components, ν ǫ (ρ) and ν −ǫ (ρ), of ν(ρ) are contained in the interior of distinct triangles. (Otherwise, L(ν(ρ)) = L(σ ǫ (ν(ρ))) and hence such X does not exist.) So we may assume ν ǫ (ρ) ∈ int σ, ν −ǫ (ρ) ∈ int σ ′ and X = P2′ . Then θǫ (ρ, σ) = (+, +, +) and hence θǫ (ρ, σ ′ ; s′2 ) < 0 by Lemma 4.5.3(1). Hence D(ρ(P2′ )) is disjoint from E ǫ (ρ, σ) = E ǫ (ρ, σ ǫ (ν(ρ))) by Lemma 4.6.7(1). By using the above claim, we see as in the proof of Lemma 7.2.5 that (ρ, ν(ρ)) satisﬁes the condition Frontier (Deﬁnition 6.1.4) for every ρ ∈ U2 whenever ν(ρ) is thick, where ρ = (ρ, ν(ρ)). Thus we have proved that ρ = (ρ, ν(ρ)) satisﬁes the condition Frontier for every ρ ∈ U2 . This completes the proof of Lemma 7.3.3. Now the proof of Proposition 6.2.1 for the thin case is completed by the following analogy of Lemma 7.2.7. Lemma 7.3.5. Let ρ0 = (ρ0 , ν 0 ), U2 and ν : U2 → H2 × H2 be as in Lemma 7.3.3. Then there is a neighborhood U3 of ρ0 in U2 , such that the following condition is satisﬁed for every ρ ∈ U3 : Fρ (ξ) is a convex polyhedron of dimension 2 − dim ξ contained in Eh(ρ) for every ξ ∈ L(ν(ρ))(≤2) , where ρ denotes the labeled representation (ρ, ν(ρ)). Proof. As in the proof of Lemma 7.2.7, we show that for every ξ ∈ L∗ (ν 0 )(≤2) there is a neighborhood U3 (ξ) of ρ0 in U2 , such that if ρ ∈ U3 (ξ) and ξ ∈ L(ν(ρ))(≤2) , then Fρ (ξ) is a convex polyhedron of dimension 2 − dim ξ contained in Eh(ρ). Case 1. ξ ∈ L∗ (ν 0 )(2) . Then we may assume ξ = (P0 , P1 , P2 ) or ′ (P1 , P2′ , P3′ ). For simplicity, we assume ξ = (P0 , P1 , P2 ). Pick an element ρ ∈ U2 such that ξ ∈ L(ν(ρ))(2) . Then by the deﬁnition of L(ν(ρ)), we have Σ(ν(ρ)) = {σ, σ ′ }, θǫ (ρ, σ) = (+, +, +) and θ−ǫ (ρ, σ ′ ) = (+, +, +) for some ǫ ∈ {−, +}. Since θ−ǫ (ρ, σ ′ ; s′2 ) > 0, we have θ−ǫ (ρ, σ; s1 ) < 0 by Lemma 4.5.3(1). Since θǫ (ρ, σ; s1 ) > 0, this implies that Fρ (ξ) = ∩2j=0 Ih(ρ(Pj )) is a 0-dimensional convex polyhedron by Lemma 4.3.6. By Lemma 7.3.1 (Disjointness) (1), Ih(ρ0 (X)) is disjoint from the 1-dimensional convex polyhedron Fρ0 (P0 , P2 ) = Ih(ρ0 (P0 )) ∩ Ih(ρ0 (P2 )) for every X ∈ L(ν(ρ))(0) − {P0 , P1 , P2 } = L∗ (ν 0 )(0) − {P0 , P1 , P2 }. Hence we can ﬁnd a neighborhood U3 (ξ) of ρ0 in U2 , such that Fρ (ξ) ⊂ Eh(ρ) whenever ρ ∈ U3 (ξ) and ξ ∈ L(ν(ρ))(≤2) . Case 2. ξ ∈ L(ν 0 )(1) . Then we may assume ξ = (P0 , P2 ) or (P2 , P3 ). For simplicity we assume ξ = (P0 , P2 ) = (P0′ , P1′ ). Pick an element ρ ∈ U2 such that ξ ∈ L(ν(ρ))(1) . By Lemma 7.3.3, we may assume ν(ρ) is thick. Then Σ(ν(ρ)) = {σ, σ ′ } or {σ ′ }. Suppose Σ(ν(ρ)) = {σ, σ ′ }. Then by the argument in Case 1, ∩2j=0 Ih(ρ(Pj )) is a 0-dimensional convex polyhedron. Since lk(ξ, L(ν(ρ))) = {P1 }, Fρ (ξ) = Ih(ρ(P0 )) ∩ Ih(ρ(P2 )) ∩ Eh(ρ(P1 )) is a 1-dimensional convex polyhedron. By

7.3 Proof of Proposition 6.2.1 (Openness) - Thin case -

169

using Lemma 7.3.1 (Disjointness)(1) as in Case 1, we can ﬁnd a neighborhood U3′ (ξ) of ρ0 in U2 , such that Fρ (ξ) ⊂ Eh(ρ) whenever ρ ∈ U3′ (ξ) and Σ(ν(ρ)) = {σ, σ ′ }. Suppose Σ(ν(ρ)) = {σ ′ }. Since Ih(ρ0 (P0 )) ∩ Ih(ρ0 (P2 )) is a 1-dimensional subspace of H3 , there is a neighborhood U2 (ξ) of ρ0 in U2 such that Fρ (ξ) = Ih(ρ(P0 )) ∩ Ih(ρ(P2 )) continues to be a 1-dimensional subspace of H3 for every ρ ∈ U2 (ξ) such that Σ(ν(ρ)) = {σ ′ }. (Here we use the fact that lk(ξ, L(ν(ρ))) = ∅.) By using Lemma 7.3.1 (Disjointness)(1) as in Case 1, we can ﬁnd a neighborhood U3′′ (ξ) of ρ0 in U2 , such that Fρ (ξ) ⊂ Eh(ρ) whenever ρ ∈ U3′′ (ξ) and Σ(ν(ρ)) = {σ ′ }. Thus U3 (ξ) := U3′ (ξ) ∩ U3′′ (ξ) satisﬁes the desired property. Case 3. ξ ∈ L∗ (ν 0 )(1) − L(ν 0 )(1) . Then we may assume ξ = (P0 , P1 ), (P1 , P2 ), (P1′ , P2′ ) or (P2′ , P3′ ). For simplicity we assume ξ = (P0 , P1 ). If ξ ∈ L(ν(ρ))(1) , then we have Σ(ν(ρ)) = {σ, σ ′ } or {σ}; thus Fρ (ξ) = Ih(ρ(P0 )) ∩ Ih(ρ(P1 )) ∩ Eh(ρ(P2 )) or Ih(ρ(P0 )) ∩ Ih(ρ(P1 )) according as the former or latter holds. By using the argument in Case 1 and Lemma 7.3.1 (Disjointness)(1) as in Case 2, we can ﬁnd the desired neighborhood U3 (ξ) of ρ0 . Case 4. ξ ∈ L(ν 0 )(0) , i.e., ξ = (Pj ) for some j such that j ≡ 1 (mod 3). For simplicity we assume P = P0 . Pick an element ρ ∈ U2 . By Lemma 7.3.3, ′ , P1′ } we may assume ν(ρ) is thick. Then lk(ξ, L(ν(ρ)))(0) = {P−1 , P1 }, {P−1 ′ ′ or {P−1 , P−1 , P1 , P1 }. By using the argument in Case 1 and Lemma 7.3.1 (Disjointness) as in Case 2, we can ﬁnd the desired neighborhood U3′ (ξ) of ρ0 . Case 5. ξ ∈ L∗ (ν 0 )(0) − L(ν 0 )(0) . Then ξ = (Pj ) (j ≡ 1 (mod 3)) or ξ = (Pj′ ) (j ≡ 2 (mod 3)). For simplicity we assume ξ = (P1 ). Pick an element ρ ∈ U2 such that ξ ∈ L(ν(ρ))(2) . Then lk(ξ, L(ν(ρ)))(0) = {P0 , P2 }. By using the argument in Case 3, we can ﬁnd the desired neighborhood U3′ (ξ) of ρ0 . We have thus constructed the desired neighborhood U3 (ξ) for every ξ ∈ L∗ (ν 0 )(≤2) . As in the proof of Lemma 7.2.7, the intersection U3 = ∩{U3 (ξ) | ξ ∈ L∗ (ν 0 )(2) } is a neighborhood of ρ0 satisfying the desired property. This completes the proof of Lemma 7.3.5.

8 Closedness

In this chapter, we study what happens at the limit of a sequence {ρn } = {(ρn , ν n )} of good labeled representations. In Sect. 8.1, we prove Proposition 6.2.3 (SameStratum) which guarantees that if lim ρn ∈ QF exists then some subsequence {ρn } satisﬁes the condition SameStratum (Deﬁnition 6.2.2) and therefore we can talk about the “behavior of a face (or an edge or a vertex) of P h(ρn ) as n → ∞”. The proof is based on the fact that the convergence ρn → ρ∞ is strong and Lemma 8.1.1 due to Jorgensen, which is a prototype of Minsky’s pivot theorem [58]. Proposition 6.2.3 (SameStratum) enables us to use Proposition 6.2.4 (Closedness), in the proof of the closedness of the image of the projection µ1 : J [QF] → QF in QF. In Sect. 8.2, we prove Proposition 6.2.7 (Convergence), which guarantees that, for a sequence of good labeled representations {(ρn , ν n )}, if ν ∞ := lim ν n ∈ H2 × H2 exists (and if it satisﬁes the condition SameStratum), then it has a subsequence such that the corresponding subsequence of {ρn } converges in QF. To this end, we study the behavior of the complex probability (a0,n , a1,n , a2,n ) of ρn at a triangle σ0 in the common chain Σ0 . The ShimizuLeutbecher lemma (Lemma 2.5.4(2-ii)) implies that the sequence is bounded and hence some subsequence converges to a triple (a0,∞ , a1,∞ , a2,∞ ) in C3 . Our task is to show that no component of the triple is 0. This is done by contradiction, by showing that if some component is equal to 0, then the limit label ν ∞ is equal to (s, s) for some vertex s of σ0 , and hence ν ∞ cannot be contained in H2 × H2 . This is proved by studying the shape of the Ford domains under the wrong assumption. Proposition 6.2.7 (Convergence) is used in Chap. 9, together with Propositions 6.2.2 (SameStratum) and 6.2.4 (Closedness) to prove the bijectivity of µ2 : J [QF] → H2 × H2 . The rest of this chapter (Sects. 8.3–8.12) is devoted to the proof of Proposition 6.2.4 (Closedness), which guarantees that if {ρn } converges to a labeled representation ρ∞ = (ρ∞ , ν ∞ ) ∈ QF × (H2 × H2 ) and if {ρn } satisﬁes the condition SameStratum, then the limit ρ∞ is a good labeled representation and hence belongs to J [QF]. The main task in the proof is to show that no

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8 Closedness

unexpected degeneration of a face of P h(ρn ) happens as n → ∞. This is the most involved part of this paper, and a route map is given in Sect. 8.3. A reason why it is so involved is that we have to list all possible degenerations, before showing degenerations do not happen. However, as is found in Jorgensen’s original argument [40], the idea to prohibit degenerations consists of the following three observations. 1. If a ‘generic’ edge of the Ford domain shrinks to a point at the limit, then ρ∞ has an ‘accidental’ elliptic or parabolic transformation. Since ρ∞ ∈ QF, an accidental elliptic transformation does not exist. The existence of an accidental parabolic transformation is also prohibited by Proposition 8.7.1, which shows that those elements of ρ∞ (π1 (T )) which support faces of the Ford domain cannot be parabolic. (If we know that ρ∞ ∈ QF in advance, then it is obvious that an accidental parabolic transformation does not exist. Thus we do not need Proposition 8.7.1 in the proof of surjectivity of µ1 : J [QF] → QF in QF. But in the proof of the bijectivity of µ2 : J [QF] → H2 × H2 , we do need Proposition 8.7.1.) 2. If a vertex of the Ford domain P h(ρn ) drops onto the complex plane which forms an isolated point of the Ford region P (ρ∞ ) ⊂ C of ρ∞ , then we have a contradiction to the chain rule of isometric circles (Fig. 8.9). 3. By virtue of the chain rule and by a topological argument, no unexpected degeneration occur on the boundary of the Ford region P (ρ∞ ) in the complex plane (Sect. 8.12). (The proof of the corresponding assertion in [72, Lemma 6.14] seems not to be correct.) Another reason for the complication in this step (and the previous step) lies in the treatment of the ‘thin’ case, i.e., the case when both components of ν ∞ = lim ν n belong to the interior of a single edge τ of the Farey triangulation. In this case, it turns out that the pair (ρ∞ , τ ) forms an ‘isosceles representation’, which is extensively studied in Sect. 5.2. In a neighborhood of isosceles representations, various phenomena occur. However, we can see by a direct calculation that these are essentially controlled by side parameters (Proposition 5.2.13) and we treat the special case by using this proposition.

8.1 Proof of Proposition 6.2.3 (SameStratum) The following lemma is due to Jorgensen [40, Lemma 4.3]. Bowditch’s Theorem 1 in [17] and Minsky’s pivot theorem in [58] may be regarded as a reﬁned variation of this lemma. Lemma 8.1.1. Let (ρ, ν) be an element of J [QF], s a pivot √ of Σ(ν), and P an elliptic generator with s(P ) = s. Then r(ρ(P )) > 1/(4 + 2 5). Proof. For completeness, we give a proof following Jorgensen (cf. [40, Lemma 4.3]). From the assumption that s is a pivot of Σ(ν), there exists an elliptic

8.1 Proof of Proposition 6.2.3 (SameStratum)

173

generator triple (P0 , P1 , P2 ) such that s(P0 ) = s and that both (P0 , P1 , P2 ) and (P1 , P2 , P3 ) are contained in L(ν)(2) (cf. Lemma 3.2.4(2)). For simplicity, we denote φ(s(Pj )) by xj (j = 0, 1, 2). In what follows we prove that |x0 | < √ 4 + 2 5. This completes the proof because r(ρ(P )) = r(ρ(P0 )) = 1/|x0 |. Since (P0 , P1 ) (resp. (P1 , P3 )) is contained in L(ν)(1) , Ih(ρ(P0 ))∩Ih(ρ(P1 )) (resp. Ih(ρ(P1 )) ∩ Ih(ρ(P3 ))) is nonempty (cf. Deﬁnition 6.1.3). Thus we have 2(r(ρ(P0 )) + r(ρ(P1 ))) = (r(ρ(P0 )) + r(ρ(P1 ))) + (r(ρ(P1 )) + r(ρ(P3 ))) > |c(ρ(P1 )) − c(ρ(P0 ))| + |c(ρ(P3 )) − c(ρ(P1 ))| ≥ |c(ρ(P3 )) − c(ρ(P0 ))| = 1.

Hence |x0 |−1 + |x1 |−1 > 1/2. Since both (P0 , P2 ) and (P2 , P3 ) are contained in L(ν), we also have |x0 |−1 + |x2 |−1 > 1/2. Thus, for each j = 1, 2, it follows that (|x0 | − 2)|xj | < 2|x0 |. √ In what follows we may suppose that |x0 | > 2, otherwise |x0 | ≤ 2 < 4+2 5 which is the desired inequality. Then |xj | < 2|x0 |/(|x0 | − 2) for each j = 1, 2. From the Markoﬀ equation, we have −x20 = x21 + x22 − x0 x1 x2 . Thus |x0 |2 = |x21 + x22 − x0 x1 x2 | ≤ |x1 |2 + |x2 |2 + |x0 ||x1 ||x2 | 2 2 2 2|x0 | 2|x0 | 2|x0 | < + + |x0 | |x0 | − 2 |x0 | − 2 |x0 | − 2 |x0 | + 2 = 4|x0 |2 . (|x0 | − 2)2 √ Therefore |x0 |2 − 8|x0 | − 4 < 0, and hence |x0 | < 4 + 2 5. Notation 8.1.2. Let {ρn } = {(ρn , ν n )} be a sequence in J [QF]. Then we will use the following notation under Notation 5.5.5. (1) φn denotes the upward Markoﬀ map inducing ρn . (k) (k) (k) (2) (a0,n , a1,n , a2,n ) denotes the complex probability of ρn at σ (k) . (3) When the sequence {ρn } converges to ρ∞ ∈ QF, φ∞ denotes the limit of (k) (k) (k) the sequence {φn }, and (a0,∞ , a1,∞ , a2,∞ ) denotes the complex probability of ρ∞ at σ (k) . We begin the proof of Proposition 6.2.3 by showing the following lemma. Lemma 8.1.3. Let {(ρn , ν n )} be a sequence in J [QF], such that {ρn } converges to ρ∞ ∈ QF. Let X be an element of π1 (O) such that Ih(ρ∞ (X)) supports a face of the Ford domain P h(ρ∞ ). Then Ih(ρn (X)) supports a face of the Ford domain P h(ρn ) for all suﬃciently large n. Moreover, there is an elliptic generator, P , such that Ih(ρ∞ (X)) = Ih(ρ∞ (P )) and Ih(ρn (X)) = Ih(ρn (P )) for every n.

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8 Closedness

Proof. Suppose the ﬁrst assertion does not hold, that is, there is a subsequence of {ρn }, which we denote by the same symbol, such that each Ih(ρn (X)) does not support a face of P h(ρn ). Let x∞ be a point in the interior of the face of P h(ρ∞ ) supported by Ih(ρ∞ (X)). Then we can ﬁnd a sequence {xn } in H3 converging to x∞ , such that each xn is contained in Ih(ρn (X)). Since (ρn , ν n ) ∈ J [QF], there is an elliptic generator P (n) , such that xn ∈ Dh(ρn (P (n) )) and that Ih(ρn (P (n) )) supports a face of P h(ρn ). Then Ih(ρn (P (n) )) = Ih(ρn (X)), because Ih(ρn (X)) does not support a face of P h(ρn ). Since {xn } converges to x∞ ∈ H3 and since the radii r(ρn (P (n) )) are bounded above by 1 (see Lemma 2.5.2(1)), we may assume, by taking a subsequence, that {c(ρn (P (n) ))} converges to some point in the complex plane and that {r(ρn (P (n) ))} converges to some positive number. Hence, by taking a subsequence again, we may assume that the axis of the elliptic transformation ρn (P (n) ) converges to some geodesic as n → ∞. This means that the sequence {ρn (P (n) )} in P SL(2, C) converges to an elliptic transformation, say Y ′ , of order 2. Since {ρn } converges strongly to ρ∞ (see for example [55, Proposition 7.39]), we may assume, after taking a subsequence, that P (n) is equal to a ﬁxed element, Y , and Y ′ = ρ∞ (Y ). Note that x∞ = lim xn ∈ lim Dh(ρn (P (n) )) = Dh(ρ∞ (Y )). n→∞

n→∞

This implies that Ih(ρ∞ (X)) = Ih(ρ∞ (Y )), because x∞ is contained in the interior of the face of P h(ρ∞ ) supported by Ih(ρ∞ (X)). In particular, we have limn→∞ |c(ρn (P (n) )) − c(ρn (X))| = 0. On the other hand, since limn→∞ r(ρn (P (n) )) = r(ρ∞ (X)), r(ρn (P (n) )) is bounded below by a positive number. Hence, by Corollary 2.5.3, we have Ih(ρn (P (n) )) = Ih(ρn (X)) for suﬃciently large n. This is a contradiction. Hence, the ﬁrst assertion of the lemma holds, that is, Ih(ρn (X)) supports a face of the Ford domain P h(ρn ) for all suﬃciently large n. Since (ρn , ν n ) ∈ J [QF], there is an elliptic generator P (n) such that Ih(ρn (X)) = Ih(ρn (P (n) )). Since the image of two diﬀerent elliptic generators by a quasifuchsian representation cannot share the same isometric hemisphere (see Lemma 2.5.4 (2-3)), the above elliptic generator P (n) is uniquely determined by X and does not depend on n. Hence we also obtain the second assertion. Corollary 8.1.4. Under the assumption of Lemma 8.1.3, any face of the Ford domain P h(ρ∞ ) is supported by the isometric hemisphere Ih(ρ∞ (P )) for some elliptic generator P . Lemma 8.1.5. Under the assumption of Lemma 8.1.3, there is an edge τ of D such that Σ(ν n ) contains τ for all suﬃciently large n. Proof. Let V be the set of slopes of elliptic generators P such that Ih(ρ∞ (P )) supports a face of the Ford domain P h(ρ∞ ). Then V is non-empty by Corollary 8.1.4. Since ρ∞ ∈ QF, we see by Lemma 5.3.10 and Corollary 8.1.4 that

8.1 Proof of Proposition 6.2.3 (SameStratum)

175

V contains at least two elements, say s1 and s2 . By Lemma 8.1.3, Σ(ν n ) contains s1 and s2 for suﬃciently large n. If s1 and s2 span an edge of D, then let τ be the edge. Otherwise, let τ be an edge of a triangle of σ whose interior intersects the geodesic in H2 connecting s1 and s2 . Since (ρn , ν n ) ∈ J [QF], τ is an edge of Σ(ν n ). Proof (Proof of Proposition 6.2.3). We ﬁrst show that there is a subsequence of {(ρn , ν n )}, which we denote by the same symbol, such that Σ(ν n ) does not depend on n. Suppose this does not hold. Then inﬁnitely many mutually diﬀerent chains appear in any subsequence of {Σ(ν n )}. By using this fact, we construct an ascending sequence {Σk } of ﬁnite chains and a descending series (k) of subsequences {Σn } of {Σ(ν n )} (n is the suﬃx for the members of each subsequence and k is the suﬃx for the subsequences), inductively as follows. Step 1. Let τ be the edge of D satisfying the conclusion of Lemma 8.1.5. By the assumption, Σ(ν n ) must contain τ for all suﬃciently large n. So, at least one of the two triangles containing τ is contained in inﬁnitely many members (1) (1) of the sequence {Σ(ν n )}. Let σ1 be such a triangle. Put Σ1 = {σ1 }, and let (1) {Σn } be the subsequence of {Σ(ν n )} consisting of those Σ(ν n ) containing (1) the triangle σ1 . (k) (k) (k) Step k+1. Suppose we have constructed the chain Σk = (σ1 , σ2 , · · · , σk ) (k) (k) and the subsequence {Σn } of {Σ(ν n )}, such that Σk ⊂ Σn for any n. Since (k) inﬁnitely many mutually diﬀerent chains appear in the subsequence {Σn }, at (k) least one of the four triangles adjacent to the union ∪ki=1 σi and not contained (k) in Σk is contained in inﬁnitely many members of the subsequence {Σn }. If a (k) triangle σ adjacent to σk satisﬁes this condition, then deﬁne (k+1)

(σ1

(k+1)

, · · · , σk

(k+1)

(k)

(k)

, σk+1 ) := (σ1 , · · · , σk , σ). (k)

If not, then let σ be a triangle adjacent to σ1 satisfying the condition, and deﬁne (k+1) (k+1) (k+1) (k) (k) (σ1 , σ2 · · · , σk+1 ) := (σ, σ1 , · · · , σk ). (k+1)

Set Σk+1 = (σ1 of

(k) {Σn }

(k+1)

, σ2

(k+1)

(k+1)

· · · , σk+1 ), and let {Σn (k) Σn

} be the subsequence

consisting of those containing the chain Σk+1 . Now, let Σ∞ be the inﬁnite chain obtained as the union of the ascend(∞) ing chains {Σk }, and let {Σn } be the subsequence of {Σ(ν n )}, deﬁned by (∞) (n) Σn := Σn . Consider the subsequence of {(ρn , ν n )} corresponding to the (∞) (∞) subsequence {Σn }, and denote it by {(ρn , ν n )} so that Σn = Σ(ν n ). Then, from the construction of the chains and the fact that (ρn , ν n ) ∈ J , for each k and n with k ≤ n, Σk is a subchain of Σ(ν n ). In particular, if an elliptic generator P satisﬁes s(P ) ∈ Σk , then Ih(ρn (P )) supports a face of P h(ρn ) for any n with n ≥ k. These properties together with the geometrical ﬁniteness of ρ∞ implies the following claim.

176

8 Closedness

Claim 8.1.6. (1) The number of pivots of Σ∞ is ﬁnite. (2) For each pivot of Σ∞ , the number of triangles in Σ∞ containing the pivot is ﬁnite. Proof. (1) Suppose to the contrary that Σ∞ contains inﬁnitely many pivots sk (k ∈ N). Then, for each k ∈ N, we can ﬁnd an elliptic generator P (k) with s(P (k) ) = sk , such that c(ρ∞ (P (k) )) is contained in the strip {z ∈ C | 0 ≤ ℜz ≤ 1}. By Lemma 2.5.4(2.3), c(ρ∞ (P (k) )) (k ∈ N) are distinct from one another. Since sk is a pivot of Σ(ν n ) (Deﬁnition 3.2.1(4)) for all suﬃciently √ for any large n, we see r(ρ∞ (P (k) )) ≥ 1/(4 + 2 5) by Lemma 8.1.1. √ Thus, (k′ ) 2 k = k ′ , we have |c(ρ∞ (P (k) )) − c(ρ (P ))| ≥ 1/(4 + 2 5) by Lemma ∞ √ 2.5.2. Put R = 1/{3(4 + 2 5)2 }, and let D(k) be the disk in C with center c(ρ∞ (P (k) )) and radius R. Then, by the above observation, D(k) (k ∈ N) are mutually disjoint. On the other hand, since ρ∞ is geometrically ﬁnite, there are two lines l± = {z ∈ C | ℑz = t± } (t− < t+ ) such that the limit set Λ(Im ρ∞ ) of Im ρ∞ is contained in the region bounded by l− and l+ . Note that each c(ρ∞ (P (k) )) is contained in Λ(Im ρ∞ ), for it is the ﬁxed point of the parabolic element ρ∞ (P (k) KP (k) ). Thus each D(k) is contained in the rectangular region {z ∈ C | − R ≤ ℜz ≤ 1 + R, t− − R ≤ ℑz ≤ t+ + R}. This is a contradiction. (2) Suppose to the contrary that there exists an elliptic generator P0 such that s(P0 ) is contained in inﬁnitely many triangles in Σ∞ . Choose elliptic generators P1 and P2 so that the triplet (P0 , P1 , P2 ) is an elliptic generator triple and that the triangle s(P0 ), s(P1 ), s(P2 ) is contained in Σ∞ . In what follows, we follow Notation 8.1.2. Then we may suppose that the triangle σ (k) is contained in Σ∞ for all positive integer k. (Actually we must also consider the case when σ (k) is contained in Σ∞ for all negative integer k. However a parallel argument works for the case.) From the construction of Σ∞ , every σ (k) (k ∈ N) is contained in Σ(ν n ) for all suﬃciently large n. Thus, for any k ∈ N, ρn satisﬁes the triangle inequality at σ (k) for all suﬃciently (k) large n by Proposition 6.7.1. Hence I(ρ∞ (P0 )) ∩ I(ρ∞ (P1 )) = ∅ for any (k) k ∈ N (cf. Lemma 4.2.1). Therefore |c(ρ∞ (P0 )) − c(ρ∞ (P1 ))| ≤ 1 + 1 = 2 by (k) Lemma 2.5.2(1). We can see that limk→∞ r(ρ∞ (P1 )) = 0 as follows. Suppose (k ) to the contrary that r(ρ∞ (P1 l )) ≥ r0 for some subsequence {kl } ⊂ {k} and for some r0 > 0. Then, by Lemma 2.5.4(2-3), the closed disks Dl with (k ) center c(ρ∞ (P1 l )) and with radius 1/(3r02 ) are disjoint from one another. On the other hand, they are contained in a compact region in C because (k) |c(ρ∞ (P0 )) − c(ρ∞ (P1 ))| ≤ 2. This is a contradiction. (k) (k+1) (k) Since P2 = P1 by deﬁnition, we also have limk→∞ r(ρ∞ (P2 )) = 0. (k) (k) Since σ (k) is contained in Σ∞ , the edge (P1 , P2 ) is contained in (k) (k) Σ(ν n ) for all suﬃciently large n. Thus I(ρ∞ (P1 )) ∩ I(ρ∞ (P2 )) = ∅. (k) (k) Since both r(ρ∞ (P1 )) and r(ρ∞ (P2 )) tend to 0 as k → ∞, this im(k) (k) (k) plies that a0,∞ = c(ρ∞ (P2 ))) − c(ρ∞ (P1 ))) tends to 0 as k → ∞. Since

8.1 Proof of Proposition 6.2.3 (SameStratum)

177

& & & |a(k) | − |a(k) | ≤ |a(k) | by Lemma 4.2.1, we see lim |a(k) | − |a(k) | = 1,∞ 2,∞ 0,∞ 1,∞ 2,∞ & (k) (k) 0. On the other hand, we have |a1,∞ a2,∞ | = r(ρ∞ (P0 )) ≤ 1 by Lemma (k)

(k)

2.5.4. Thus both |a1,∞ | and |a2,∞ | are bounded, and therefore by taking a (k)

(k)

(∞)

subsequence, a1,∞ and a2,∞ converge to some complex numbers, a1,∞ and (∞)

(∞)

(∞)

(k)

(k)

a2,∞ , respectively. Since |a1,∞ | − |a2,∞ | = lim(|a1,∞ | − |a2,∞ |) = 0 and since (∞)

(∞)

(k)

(∞)

(∞)

a1,∞ + a2,∞ = 1 − limk→∞ a0,∞ = 1, we have a1,∞ = 12 + it and a2,∞ = for some t ∈ R. Thus the Markoﬀ map φ∞ satisﬁes & & (k) (k) (∞) (∞) φ∞ (s(P0 )) = ±1/

a1,∞ a2,∞ = ±1/

a1,∞ a2,∞ = ±1/

1 2

− it

(1/4)2 + t2 ∈ [−2, 2].

Hence ρ∞ (KP0 ) is either parabolic or elliptic. This contradicts the assumption that ρ∞ ∈ QF.

The above claim implies Σ∞ is ﬁnite, a contradiction. Hence there is a subsequence of {(ρn , ν n )}, which we denote by the same symbol, such that Σ(ν n ) does not depend on n. Hence, we may assume, by taking a subsequence again, that {(ρn , ν n )} satisﬁes the condition SameStratum. Thus we have proved the second assertion of Proposition 6.2.3. To show the ﬁrst assertion, we note that we may assume, by taking a 2 ˆ ˆ ∪Q) = further subsequence, that {ν n } converges to a label ν ∞ in (H2 ∪Q)×(H 2 2 D×D. We show that the limit ν ∞ belongs to H ×H . Suppose to the contrary ǫ / H2 × H2 . Then ν∞ = s for some ǫ = ± and for some vertex s of that ν ∞ ∈ ǫ σ . This contradicts the following Lemma 8.1.7. Thus we have completed the proof of Proposition 6.2.3. Lemma 8.1.7. Let {ρn } = {(ρn , ν n )} be a sequence in J [QF] such that {ρn } 2 2 converges to ρ∞ = (ρ∞ , ν ∞ ) ∈ QF × (H × H ). Suppose that the sequence ǫ ˆ for each ǫ ∈ {−, +}. ∈ Q {ρn } satisﬁes the condition SameStratum. Then ν∞ ǫ ˆ Then there is a sequence of Proof. Suppose to the contrary that ν∞ ∈ Q. elliptic generators {Pj } associated with the ǫ-terminal triangle, σ ǫ , of the ǫ . By the assumption that the common chain Σ(ν n ) such that s(P0 ) = ν∞ ǫ ǫ sequence {νn } converges to ν∞ = s(P0 ), the sequence of side parameters {θǫ (ρn , σ)} converges to the triplet (π/2, 0, 0) (cf. Deﬁnition 4.2.9). Since the radius r(ρn (Pj )) is bounded above by 1 (Lemma 2.5.2(1)), the Euclidean length of eǫ (ρn , σ ǫ ; Pj ) is bounded above by 2θǫ (ρn , σ ǫ ; s(Pj )). Thus the union eǫ (ρn , σ ǫ ; P1 ) ∪ eǫ (ρn , σ ǫ ; P2 ) converges to a point in C as n → ∞ with respect to the Hausdorﬀ topology. The limit point is a common ﬁxed point of the involutions ρn (P1 ) and ρn (P2 ), because eǫ (ρn , σ ǫ ; Pj ) contains a ﬁxed point of ρn (Pj ). Hence ρ∞ (KP0 ) = ρ∞ (P2 P1 ) is either parabolic or the identity. This contradicts the assumption that ρ∞ ∈ QF.

178

8 Closedness

8.2 Proof of Proposition 6.2.7 (Convergence) Proof (Proof of Proposition 6.2.7). Let {ρn } = {(ρn , ν n )} be a sequence of good labeled representations satisfying the following conditions: 1. {ν n } converges to ν ∞ ∈ H2 × H2 . 2. {ρn } satisﬁes the condition SameStratum. First, suppose that ν ∞ is thin. Then, by Proposition 5.2.11, there is a unique element ρ∞ ∈ QF which realizes ν ∞ . By Proposition 5.2.13, we can see that the sequence {ρn } converges to ρ∞ . This completes the proof of Proposition 6.2.7 in the case when ν ∞ is thin. Next, suppose that ν ∞ , and hence all ν n , is thick. Pick an arbitrary triangle σ0 in the chain Σ(ν n ). Let {Pj } be the sequence of elliptic generators associated with σ0 , and let (a0,n , a1,n , a2,n ) be the complex probability of ρn at σ0 . Since σ0 is contained in Σ(ν n ), the 1-dimensional convex polyhedron Fρn (Pj , Pj+1 ) is contained in Ih(ρn (Pj )) ∩ Ih(ρn (Pj+1 )) for every j ∈ Z, and hence ρn satisﬁes the triangle inequality at σ0 by Lemma 4.2.1. This implies that any two successive isometric hemispheres Ih(ρn (Pj+1 )) and Ih(ρn (Pj+2 )) have a non-trivial intersection (cf. Lemma 4.2.1). Hence |aj,n | = |c(ρn (Pj+2 )) − c(ρn (Pj+1 ))| < 2 by Lemma 2.5.4(2-2). So, we may assume, by taking a subsequence if necessary, that the sequence {(a0,n , a1,n , a2,n )} of complex probabilities converges to some triplet (a0,∞ , a1,∞ , a2,∞ ) in C3 which also satisﬁes a0,∞ + a1,∞ + a2,∞ = 1. In what follows, we will see that the triplet (a0,∞ , a1,∞ , a2,∞ ) is in fact contained in (C − {0})3 . Suppose one of the components, say a0,∞ , is equal to 0. Then the following claim holds. Claim 8.2.1. Every triangle of Σ(ν n ) contains the slope s(P0 ) as a vertex. Moreover, we have limn→∞ θǫ (ρn , σ ǫ ) = (π/2, 0, 0) for each ǫ ∈ {−, +}. Proof. Suppose to the contrary that Σ(ν n ) contains a neighbor σ ′ of σ0 with s(P0 ) ∈ (σ ′ )(0) . Then σ ′ = s(P1 ), s(P1 P0 P1 ), s(P2 ). Let (a′0,n , a′1,n , a′2,n ) be the complex probability of ρn at σ ′ . Then we have: a0,n a1,n 0 a0,n a2,n 0 → , a′1,n = a1,n + a2,n → 1, a′2,n = → . a1,n + a2,n 1 a1,n + a2,n 1 & & & Hence the triplet ( |a′0,n |, |a′1,n |, |a′2,n |) does not satisfy the triangle inequality for suﬃciently large n ∈ N. On the other hand, ρn must satisfy the condition σ ′ -Simple (cf. Assumption 4.2.4) because σ ′ ∈ Σ(ν n ) and ρn = (ρn , ν n ) ∈ J [QF]. This is a contradiction by Lemma 4.2.1. Thus any neighbor of σ in Σ(ν n ) contains s(P0 ) as a vertex, and hence we can see inductively that every triangle of Σ(ν n ) contains s(P0 ) as a vertex. In what follows, we will follow Notation 8.1.2. By the observation in the preceding paragraph, we have Σ(ν n ) = {σ (k) | k − ≤ k ≤ k + } for some k ± ∈ Z. We also have, for each k, a′0,n =

8.2 Proof of Proposition 6.2.7 (Convergence) (k) (k)

(k) (k)

(k+1) a0,n

(k)

a0,n =

=

a0,n a1,n (k) a0,n

+

(k) a2,n

179

,

(k+1) (k+1) a0,n a2,n , (k+1) (k+1) a0,n + a1,n

a1,n a2,n

(k+1) a1,n

=

(k)

(k+1)

(k) a0,n

+

(k) a2,n

,

(k+1)

a2,n

(k)

(k)

= a0,n + a2,n , (k+1) (k+1)

a1,n = a0,n

(0)

(k+1)

+ a1,n , (0)

a1,n a2,n

(k)

a2,n =

(k+1)

a0,n

(k+1)

+ a1,n

.

(0)

From the assumption, we have (a0,n , a1,n , a2,n ) = (a0,n , a1,n , a2,n ). Thus (0) a0,n

= a0,n tends to 0 as n → ∞ by assumption. Hence we can see inductively (k)

(k)

(k)

that every complex probability (a0,n , a1,n , a2,n ) converges to the common (a0,∞ , a1,∞ , a2,∞ ) = (0, a1,∞ , a& 2,∞ ) as n → ∞. In particular, we can see (k)

for any k that r(ρn (P1 )) =

(k) (k)

|a0,n a2,n | → 0 as n → ∞. Since each pair (k)

(k)

(k+1)

)) of isometric hemispheres Ih(ρn (P1 )) and Ih(ρn (P2 )) = Ih(ρn (P1 (k − ≤ k ≤ k + ) have a non-trivial intersection, the union of isometric cir(k) k+ +1 )) converges to a point in C in the Hausdorﬀ topology cles ∪k=k − I(ρn (P1 +

(k)

k +1 as n → ∞. On the other hand, the union ∪k=k )) has a non-trivial − I(ρn (P1 ± intersection with both E (ρn ) for every n ∈ N, because ρn satisﬁes the condition Frontier (cf. Deﬁnition 6.1.4). Hence the distance dE (E − (ρn ), E + (ρn )) converges to 0 as n → ∞.

Fig. 8.1. Complex probability taken from OPTi

In what follows, we prove a1,∞ = a2,∞ = 1/2 by using the above obser vation. By Lemma 4.2.1, we have |a1,n | − |a2,n | < |a0,n | for any

n ∈ N and hence |a1,∞ | = |a2,∞ |. Since a1,∞ + a2,∞ = 1 − a0,∞ = 1, it follows that ℜ(a that ℑ(a2,∞ ) = −ℑ(a1,∞ ). Thus 1,∞ ) = ℜ(a2,∞ ) = 1/2 and r(ρn (P0 )) = |a1,n a2,n | converges to 1/4 + y 2 where y = |ℑ(a1,∞ )| = |ℑ(a2,∞ )|. So the distance between the two components of the common exterior ∩k∈Z E(ρn (P3k )) is at least y for all suﬃciently large n ∈ N, because it converges to ∩k∈Z E(ρ∞ (P3k )) in the Hausdorﬀ topology whose components are separated by the y-neighborhood of the real axis. Since

180

8 Closedness

E ± (ρn ) are contained in the diﬀerent components of ∩k∈Z E(ρn (P3k )), we see dE (E − (ρn ), E + (ρn )) ≥ y for suﬃciently large n ∈ N. This implies y = 0, because dE (E − (ρn ), E + (ρn )) converges to 0 as observed in the preceding paragraph. Hence a0,∞ = a2,∞ = 1/2. Since a0,∞ = a2,∞ = 1/2, it follows that r(ρn (P0 )) = |a0,n a2,n | → 1/2 (k) as n → ∞. Thus, by using also the fact that that r(ρn (P1 )) converges to ǫ ǫ 0 for every k, we can see that limn→∞ θ (ρn , σ ; s(P0 )) = π/2 for each ǫ ∈ {−, +}. Hence we have limn→∞ θǫ (ρn , σ ǫ ) = (π/2, 0, 0) for each ǫ ∈ {−, +}, by the inequality θǫ (ρn , σ ǫ ; s(Pj )) ≥ 0 (cf. Deﬁnition 4.3.8(1)) and by the '2 identity j=0 θǫ (ρn , σ ǫ ; s(Pj )) = π/2 (cf. Proposition 4.2.16). This completes the proof of Claim 8.2.1. By Claim 8.2.1, the sequence {ν n } converges to the pair (s(P0 ), s(P0 )) ∈ ∂H2 × ∂H2 . This contradicts the assumption that the limit is contained in H2 ×H2 . Hence the triplet (a0,∞ , a1,∞ , a2,∞ ) is contained in (C−{0})3 . So the triplet (a0,∞ , a1,∞ , a2,∞ ) deﬁnes a representation, ρ∞ , in X . By Proposition 2.4.2, the sequence {ρn } converges to ρ∞ . Since the sequence {ν n } converges in H2 × H2 by assumption, it follows that the sequence {ρn } converges in QF × (H2 × H2 ). This completes the proof of Proposition 6.2.7.

8.3 Route map of the proof of Proposition 6.2.4 (Closedness) In this section, we give a route map of the proof of Proposition 6.2.4. We ﬁrst recall the assumption of the proposition. Assumption 8.3.1. We assume that {ρn } = {(ρn , ν n )} is a sequence in J [QF] satisfying the following conditions:

(1) {ρn } converges to a labeled representation ρ∞ = (ρ∞ , ν ∞ ) ∈ QF × (H2 × H2 ). (2) {ρn } satisﬁes the condition SameStratum, i.e., every chain Σ(ν n ) is equal to a ﬁxed chain Σ0 , and hence every L(ν n ) is equal to the ﬁxed elliptic generator complex L0 = L(Σ0 ).

For each sign ǫ ∈ {−, +}, σ ǫ = sǫ0 , sǫ1 , sǫ2 denotes the ǫ-terminal triangle of Σ0 and {Pjǫ } denotes the sequence of elliptic generators associated with σ ǫ .

We note that Proposition 6.2.4 in the case where ν ∞ is thin can be easily proved. Proposition 8.3.2 (ThinGoodConvergence implies GoodLimit). Under Assumption 8.3.1, assume that ν ∞ ∈ H2 × H2 is thin. Then the limit ρ∞ is a good labeled representation and hence belongs to J [QF].

8.3 Route map of the proof of Proposition 6.2.4 (Closedness)

181

Proof. Since ν ∞ is thin, we see by Proposition 5.2.11 that there is a unique representation ρ′ ∈ QF such that (ρ′ , ν ∞ ) is a good labeled representation. ± Let τ be the edge of D such that ν∞ ∈ int τ . Since lim ν n = ν ∞ , we may assume ν n lies in the neighborhood U of int τ ×int τ constructed in Proposition 5.2.13. Since ρn (resp. ρ′ ) is induced by an algebraic root for ν n (resp. ν ∞ ), Proposition 5.2.13 guarantees that ρn converges to ρ′ . Thus we obtain ρ′ = ρ∞ , and hence ρ∞ = (ρ∞ , ν ∞ ) = (ρ′ , ν ∞ ) is a good labeled representation. Thus Proposition 6.2.4 is reduced to the following proposition. Proposition 8.3.3 (ThickGoodConvergence implies GoodLimit). Under Assumption 8.3.1, assume that ν ∞ ∈ H2 × H2 is thick. Then the limit ρ∞ is a good labeled representation and hence belongs to J [QF]. In the remainder of this section, we give an outline for the proof of Proposition 8.3.3. Reduction of Proposition 8.3.3. We introduce a condition which we call HausdorﬀConvergence in Deﬁnition 8.4.1, and reduce Proposition 8.3.3 to the following three Propositions 8.3.4, 8.3.5 and 8.3.6. Proposition 8.3.4 (ThickGoodConvergence implies HausdorﬀConvergence). Under Assumption 8.3.1, {ρn } contains a subsequence which satisﬁes the condition HausdorﬀConvergence. Proposition 8.3.5 (ThickGoodConvergence implies Duality). Under Assumption 8.3.1, assume that ν ∞ is thick and that {ρn } satisﬁes the condition HausdorﬀConvergence. Then ρ∞ satisﬁes the condition Duality. Proposition 8.3.6 (ThickGoodConvergence implies Frontier). Under Assumption 8.3.1, assume that ν ∞ is thick and that {ρn } satisﬁes the condition HausdorﬀConvergence. Then ρ∞ satisﬁes the condition Frontier. Assuming the above three propositions, the proof of Proposition 8.3.3 is completed as follows: Proof (Proof of Proposition 8.3.3). Under Assumption 8.3.1, assume that ν ∞ is thick. Since ρ∞ ∈ QF, ρ∞ satisﬁes the condition NonZero (cf. Lemma 2.5.4). By Proposition 8.3.4, we may assume after taking a subsequence that {ρn } also satisﬁes the condition HausdorﬀConvergence. Thus, by Propositions 8.3.5 and 8.3.6, ρ∞ satisﬁes the conditions Duality and Frontier. Hence ρ∞ is good (cf. Deﬁnitions 6.1.7). The proof of Proposition 8.3.4 is given in Sect. 8.4. An outline of the proof of Proposition 8.3.5 is given at the end of Sect. 8.4, where it is reduced to Propositions 8.4.4 and 8.4.5: Proposition 8.4.4 is proved in Sect. 8.6, and Proposition 8.4.5 is proved in Sects. 8.8–8.11. Finally, the proof of Proposition 8.3.6 is given in Sect. 8.12.

182

8 Closedness

8.4 Reduction of Proposition 8.3.5 - The condition HausdorﬀConvergence We refer the deﬁnitions of Hausdorﬀ metric and Chabauty topology, to [15, Sect. E.1] and [16, Sect. 9.11]. Deﬁnition 8.4.1 (HausdorﬀConvergence). Suppose {ρn } = {(ρn , ν n )} satisﬁes the condition SameStratum. Then we say that {ρn } satisﬁes the condition HausdorﬀConvergence if the following conditions are satisﬁed with 3 respect to the Hausdorﬀ metric on the space of compact sets of H , with respect 3 to the Euclidean metric of H . 1. For each simplex ξ of L0 (Deﬁnition 6.2.2), the sequence {F ρn (ξ)} converges. We denote the limit by F ∞ (ξ). 2. For each simplex ξ of ∂ ǫ L0 (cf. Deﬁnition 7.2.1), the sequence {fρǫ n (ξ)} ǫ converges. We denote the limit by f∞ (ξ). Remark 8.4.2. As a matter of fact, this is a condition for a sequence in Homtp (π1 (O), P SL(2, C)) and not for a sequence in X . Thus the precise meaning of the above deﬁnition is that {ρn } contains a subsequence which lifts to a sequence in Homtp (π1 (O), P SL(2, C)) satisfying the required conditions. To prove Proposition 8.3.4, we make the following simple observation. Lemma 8.4.3. Under Assumption 8.3.1, the following hold for any elliptic generator P . 1. Ih(ρ∞ (P )) is well-deﬁned, and the sequence {Ih(ρn (P ))} converges to Ih(ρ∞ (P )) with respect to the Hausdorﬀ metric. In particular, there is a 3 compact subset, say Z, of H which contains Ih(ρ∞ (P )) and Ih(ρn (P )) for all n ∈ N. 2. Eh(ρ∞ (P )) is well-deﬁned, and the sequence {Eh(ρn (P ))} converges to Eh(ρ∞ (P )) with respect to the Chabauty topology on the closed sets of 3 H . Moreover, for any compact set Z, {Eh(ρn (P )) ∩ Z} converges to Eh(ρ∞ (P )) ∩ Z with respect to the Hausdorﬀ metric. Proof. Recall that we employ Convention 2.2.7(3) and hence ρn converges to ρ∞ as elements of Homtp (π1 (O), P SL(2, C)) (not only as elements of X ). By Lemma 2.5.4, φn (s(P )) (n ∈ N) and φ∞ (s(P )) are non-zero. Hence I(ρn (P )) and I(ρ∞ (P )) are well-deﬁned by Lemma 2.4.4(1.1). Thus {c(ρn (P ))} and {r(ρn (P ))} converge to c(ρ∞ (P )) and r(ρ∞ (P )), respectively. This implies the desired results. Proof (Proof of Proposition 8.3.4). Let ξ be a simplex of L0 . Pick a vertex, P , of ξ. By Lemma 8.4.3, Ih(ρ∞ (P )) is well-deﬁned, {Ih(ρn (P ))} converges 3 to Ih(ρ∞ (P )), and there is a compact subset, say Z, of H which contains Ih(ρ∞ (P )) and Ih(ρn (P )) for all n ∈ N. Since F ρn (ξ) ⊂ Ih(ρn (P )), F ρn (ξ)

8.4 Reduction of Proposition 8.3.5 - The condition HausdorﬀConvergence -

183

is contained in Z for all suﬃciently large n ∈ N. Hence {F ρn (ξ)} contains a convergent subsequence. Similarly, we see that, for each simplex ξ of ∂ ǫ L0 , {fρǫ n (ξ)} contains a convergent subsequence. Since L0 and ∂ ± L0 contain only ﬁnitely many simplices modulo the action of K, we obtain the desired result. Proposition 8.3.5 is reduced to the following Propositions 8.4.4 and 8.4.5. Proposition 8.4.4 is proved in Sect. 8.6. Propositions 8.4.5 is divided into two cases according as the length of the chain Σ0 is equal to 1 or ≥ 2; the length 1 case is treated in Sect. 8.8 and the length ≥ 2 case is treated in Sects. 8.9–8.11. Proposition 8.4.4 (F ∞ (ξ) ⊂ ∂Eh(ρ∞ , L0 )). Under Assumption 8.3.1, assume that {ρn } satisﬁes the condition HausdorﬀConvergence. Then the following hold. (1) For any simplex ξ of L0 , F ∞ (ξ) is contained in ∂Eh(ρ∞ , L0 ). ǫ (ξ) is contained in fr E(ρ∞ , L0 ). (2) For any simplex ξ of ∂ ǫ L0 , f∞ Proposition 8.4.5. Under Assumption 8.3.1, assume that ν ∞ is thick and that {ρn } satisﬁes the condition HausdorﬀConvergence. Then the following hold: (1) For any ξ ∈ L(ν ∞ )(i) , F ρ∞ (ξ) is equal to F ∞ (ξ), and it is a convex polyhedron of dimension 2 − i. (0) (2) For any vertex (P ) ∈ L0 − L(ν ∞ )(0) , Ih(ρ∞ (P )) is invisible in Eh(ρ∞ ). Namely, Eh(ρ∞ (P )) ⊃ Eh(ρ∞ ). Assuming Propositions 8.4.4 and 8.4.5, we can prove Proposition 8.3.5 as follows. Proof (Proof of Proposition 8.3.5 assuming Propositions 8.4.4 and 8.4.5). Since ρ∞ ∈ QF by the assumption, ρ∞ satisﬁes the condition NonZero (cf. Lemma 2.5.4). Thus we have: (0)

Eh(ρ∞ , L0 ) = ∩{Eh(ρ∞ (P )) | P ∈ L0 } = ∩{Eh(ρ∞ (P )) | P ∈ L(ν ∞ )(0) } (0) ∩ ∩{Eh(ρ∞ (P )) | P ∈ L0 − L(ν ∞ )(0) } (0) = Eh(ρ∞ ) ∩ ∩{Eh(ρ∞ (P )) | P ∈ L0 − L(ν ∞ )(0) } = Eh(ρ∞ ).

Here, the last equality follows from Proposition 8.4.5(2). Let ξ be any element of L(ν ∞ )(i) . Then, by Proposition 8.4.5, F ρ∞ (ξ) 3

is equal to F ∞ (ξ) and it is a (2 − i)-dimensional convex polyhedron in H . By Proposition 8.4.4, F ∞ (ξ) is contained in ∂Eh(ρ∞ , L0 ), which is equal to

184

8 Closedness

∂Eh(ρ∞ ) by the preceding observation. Thus ρ∞ satisﬁes the ﬁrst condition in Deﬁnition 6.1.3 (Duality). Next we prove that ρ∞ satisﬁes the second condition in Deﬁnition 6.1.3 (Duality). For mutually distinct vertices P and P ′ of L(ν ∞ ), suppose to the contrary that Fρ∞ (P ) = Fρ∞ (P ′ ). Then the two isometric hemispheres Ih(ρ∞ (P )) and Ih(ρ∞ (P ′ )) coincide, because dim Fρ∞ (P ) = dim Fρ∞ (P ′ ) = 2. This contradicts Lemma 2.5.4(3). Thus ρ∞ also satisﬁes the second condition in Deﬁnition 6.1.3 (Duality). This completes the proof of Proposition 8.3.5.

8.5 Classiﬁcation of simplices of L(ν) In this section, we divide the edges of the abstract simplicial complex L(ν) into three classes, generic edges and ǫ-extreme edges (ǫ ∈ {−, +}), and give certain characterizations of the three classes. This result is used in the following sections. Deﬁnition 8.5.1 (Classiﬁcation of the edges of L(ν)). Let ν = (ν − , ν + ) be a thick label, and let ξ = (Pj , Pj+1 ) be a 1-cell of L(ν), where {Pj } is the sequence of elliptic generators associated with a triangle σ in the chain Σ(ν). (1) (ǫ-extreme edge) We say that ξ is an ǫ-extreme edge, if it is an edge of ∂ ǫ L(ν). namely, σ is the ǫ-terminal triangle of Σ(ν). (2) (Generic edge) We say that ξ is a generic edge, if ξ is not an ǫ-extreme edge for any ǫ = ±, namely σ ∈ Σ(ν) − {σ − (ν), σ + (ν)}. P2P1

P2P1 P1

P1

P0

P3

P0 P2 P1P2

P2 P1P2

Fig. 8.2. Generic edges: the thick broken line consists of generic edges

Lemma 8.5.2. Let ξ be a generic edge of L(ν). Then there is a triangle σ of Σ(ν) − {σ − (ν), σ + (ν)} satisfying the following conditions (see Fig. 8.2). 1. ξ = (Pj , Pj+1 ) for some j, where {Pj } is the sequence of elliptic generators associated with σ. 2. The elliptic generator triples (P0 , P1 , P2 ), (P1 , P2 , P3 ), (P0 , P2P1 , P1 ), and (P2 , P1P2 , P3 ) determine 2-simplices of L(ν).

8.6 Proof of Proposition 8.4.4 (F ∞ (ξ) ⊂ ∂Eh(ρ∞ , L0 ))

185

3. Set σ ′ = s(P0 ), s(P2P1 ), s(P1 ) and σ ′′ = s(P2 ), s(P1P2 ), s(P3 ). If σ ′ lies on the ǫ-side of σ, then σ ′′ lies on the (−ǫ)-side of σ and the following hold. a) (P1 , P2 , P3 ) and (P0 , P2P1 , P1 ) lie on the ǫ-side of L(σ). b) (P0 , P1 , P2 ) and (P2 , P1P2 , P3 ) lie on the (−ǫ)-side of L(σ). Proof. Let ξ be a generic edge of L(ν). Then, by deﬁnition, there is a triangle σ of Σ(ν) − {σ − , σ + } satisfying the ﬁrst condition, namely, ξ = (Pj , Pj+1 ) for some j where {Pj } is the sequence of elliptic generators associated with σ. Since σ is diﬀerent from σ ± (ν), we may assume, after a shift of indices, that s(P0 ), s(P1 ) (resp. s(P0 ), s(P2 )) is an edge of a triangle, say σ ′ (resp. σ ′′ ), in Σ(ν) diﬀerent from σ. We see that the sequence of elliptic generators associated with σ ′ (resp. σ ′′ ) is as follows (cf. Proposition 2.1.6(1)): P0 · · · , P−2 , P0 , P−1 = P2P1 , P1 , P3 , P2P3 , · · ·

(resp. · · · , P0 , P2 , P1P2 , P3 , P5 , P4P5 , · · · ). Hence we see that (P0 , P1 , P2 ), (P1 , P2 , P3 ), (P0 , P2P1 , P1 ), and (P2 , P1P2 , P3 ) are 2-simplices of L(ν). Thus the second condition is satisﬁed. The last assertion follows from the deﬁnition (Deﬁnition 3.2.5). We can easily observe the following lemma. Lemma 8.5.3. Let ρ = (ρ, ν) be a good labeled representation, and let ξ be an edge of L(ν). Then the following hold. 1. If ξ is generic, then F ρ (ξ) is a geodesic segment in H3 . 2. If ξ is ǫ-extreme for an unique sign ǫ, then F ρ (ξ) is a half geodesic. 3. If ξ is ǫ-extreme for for each sign ǫ, then F ρ (ξ) is a complete geodesic.

8.6 Proof of Proposition 8.4.4 (F ∞(ξ) ⊂ ∂Eh(ρ∞, L0 )) Lemma 8.6.1. Under Assumption 8.3.1, assume that {ρn } satisﬁes the condition HausdorﬀConvergence. Then the following hold. ǫ (Pjǫ ) is equal to the (possibly degenerate) subarc of I(ρ∞ (Pjǫ )) ∩ (1) f∞ ǫ ǫ )) ∩ E(ρ∞ (Pj+1 ))) which contains lim Fixǫσǫ (ρn (Pjǫ )) and which (E(ρ∞ (Pj−1 ǫ ǫ ǫ ǫ ǫ is bounded by f∞ (Pj−1 , Pjǫ ) ∈ I(ρ∞ (Pj−1 )) ∩ I(ρ∞ (Pjǫ )) and f∞ (Pjǫ , Pj+1 )∈ ǫ ǫ (Pjǫ ) is equal to the double I(ρ∞ (Pjǫ ))∩I(ρ∞ (Pj+1 )). Moreover, the angle of f∞ ǫ of the [j]-th barycentric coordinate of the point ν∞ = θǫ (ρ, σ ǫ ) in the ǫ-terminal triangle σ ǫ of Σ0 . (i) (2) Let ξ be an element of L0 , where i = 0 or 1. Then F ∞ (ξ) is equal to 3 the convex polyhedron in H spanned by the limits of the vertices of F ρn (ξ) and the limits of the ideal boundaries of F ρn (ξ). To be precise, F ∞ (ξ) is the 3

convex polyhedron in H spanned by the union, V , of the following subsets:

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8 Closedness

1. F ∞ (η) where η is a 2-simplex of L0 containing ξ. ǫ ǫ ǫ (Pj−1 , Pjǫ ) where (Pj−1 , Pjǫ ) is an ǫ-extreme edge of L0 containing ξ. 2. f∞ ǫ 3. f∞ (Pjǫ ) where (Pjǫ ) is an ǫ-extreme vertex of L0 containing ξ (i.e., ξ = (Pjǫ )). Proof. (1) This follows from the fact that fρǫ n (Pjǫ ) is the component of ǫ ǫ I(ρn (Pjǫ )) ∩ (E(ρn (Pj−1 )) ∩ E(ρn (Pj+1 ))) containing Fixǫσǫ (ρn (Pjǫ )), and has the angle π times the [j]-th component of νnǫ . (2) We prove only the assertion for vertices of L0 , because the assertion for edges can be proved similarly. Let ξ = (P ) be a vertex of L0 . Let Vn be the union of the vertices and ideal boundaries of F ρn (ξ), that is, Vn is the union of the following sets: 1. F ρn (η) where η is a 2-simplex of L0 containing ξ. ǫ ǫ 2. fρǫ n (Pj−1 , Pjǫ ) where (Pj−1 , Pjǫ ) is an ǫ-extreme edge of L0 containing ξ. ǫ ǫ ǫ 3. fρn (Pj ) where (Pj ) is an ǫ-extreme vertex of L0 containing ξ. 3

Then F ρn (ξ) is the closed convex hull of Vn in H . Since Vn ⊂ F ρn (ξ) and V = lim Vn , V is contained in F ∞ (ξ) = lim F ρn (ξ). On the other hand, since F ρn (ξ) is convex, its limit, F ∞ (ξ), is also convex. Hence the closed convex hull of V is contained in F ∞ (ξ). To see the converse, we show that the complement 3 (in H ) of the closed convex hull of V is contained in the complement of F ∞ (ξ). Let x be a point in the complement of the closed convex hull of V . 3 Then there is a closed half space H of H such that V ⊂ H and x ∈ / H. We can ﬁnd a closed half space H ′ such that H ⊂ int H ′ and x ∈ / H ′ . Since lim Vn = V ⊂ H ⊂ int H ′ , Vn is contained in int H ′ for all suﬃciently large n. Since F ρn (ξ) is the convex hull of Vn , this implies F ρn (ξ) ⊂ int H ′ for all suﬃciently large n. Since x ∈ / H ′ , this implies that x is not contained in F ∞ (ξ) = lim F ρn (ξ), i.e., x is contained in the complement of F ∞ (ξ). Thus we have proved that F ∞ (ξ) is equal to the closed convex hull of V . Since V is a subset of Ih(ρ∞ (P )) which consists of ﬁnitely many points in Ih(ρ∞ (P )) and one or two (possibly degenerate) circular arcs in I(ρ∞ (P )), we see that F ∞ (ξ) is a convex polyhedron (cf. Deﬁnition 3.4.1). Proof (Proof of Proposition 8.4.4). Let ξ be a k-simplex of L0 and let {Pj | 0 ≤ j ≤ k} be the vertices of ξ. Then F ∞ (ξ) ⊂ ∩kj=0 Ih(ρ∞ (Pj )). Since Ih(ρ∞ (Pj )) ∩ int Eh(ρ∞ , L0 ) = ∅, we have F ∞ (ξ) ∩ int Eh(ρ∞ , L0 ) = ∅. So, we have only to show that F ∞ (ξ) ⊂ Eh(ρ∞ , L0 ). Suppose to the contrary that there is a point, x∞ , of F ∞ (ξ) which is not contained in Eh(ρ∞ , L0 ). Then x∞ ∈ int Dh(ρ∞ (X)) for some vertex (X) of L0 . Let ǫ > 0 be the Euclidean distance dE (x∞ , Ih(ρ∞ (X))). Since x∞ ∈ 3 F ∞ (ξ) = lim F ρn (ξ), there is a sequence {xn } in H converging to x∞ such that xn ∈ F ρn (ξ). Then we have dE (xn , x∞ ) < ǫ/2 for all suﬃciently large n. On the other hand, we see dE (Ih(ρn (X)), Ih(ρ∞ (X))) < ǫ/2 for all suﬃciently large n (cf. Lemma 8.4.3). Hence xn ∈ int Dh(ρn (X)) for all suﬃciently large

8.7 Accidental parabolic transformation

187

n. This contradicts the fact that xn ∈ F ρn (ξ) ⊂ ∂Eh(ρn ) = ∂Eh(ρn , L0 ) ⊂ Eh(ρn (X)). Hence we have F ∞ (ξ) ⊂ Eh(ρ∞ , L0 ). This completes the proof of the ﬁrst assertion of Proposition 8.4.4. The second assertion is proved by a similar argument. Lemma 8.6.2. Under Assumption 8.3.1, assume that {ρn } satisﬁes the condition HausdorﬀConvergence. Then, for any simplex ξ of L(ν ∞ ), the following holds: F ∞ (ξ) ⊂ F (ρ∞ ,L0 ) (ξ) ⊂ F ρ∞ (ξ). Proof. Since L0 ⊃ L(ν ∞ ), we have lk(ξ, L0 ) ⊃ lk(ξ, L(ν ∞ )) and therefore Eh(ρ∞ , lk(ξ, L0 )) ⊂ Eh(ρ∞ , lk(ξ, L(ν ∞ ))). Hence we obtain the second inclusion as follows: F (ρ∞ ,L0 ) (ξ) = {Ih(ρ∞ (P )) | P ∈ ξ (0) } ∩ Eh(ρ∞ , lk(ξ, L0 )) ⊂ {Ih(ρ∞ (P )) | P ∈ ξ (0) } ∩ Eh(ρ∞ , lk(ξ, L(ν ∞ ))) = F ρ∞ (ξ).

Next we prove the ﬁrst inclusion. Since st0 (ξ, L0 )(0) is ﬁnite, Lemma 8.4.3 3 shows the existence of a compact subset Z of H which contains Ih(ρ∞ (P )) (0) and Ih(ρn (P )) for every P ∈ st0 (ξ, L0 ) and every n ∈ N. Hence we see by lemma 8.4.3 F ∞ (ξ) = F ∞ (ξ) ∩ Z

= lim(F ρn (ξ) ∩ Z) {Ih(ρn (P )) | P ∈ ξ (0) } = lim ∩ {Eh(ρn (X)) ∩ Z | X ∈ lk(ξ, L0 )(0) } {lim Ih(ρn (P )) | P ∈ ξ (0) } ⊂ ∩ {lim(Eh(ρn (X)) ∩ Z) | X ∈ lk(ξ, L0 )(0) } {Ih(ρ∞ (P )) | P ∈ ξ (0) } = ∩ {Eh(ρ∞ (X)) ∩ Z | X ∈ lk(ξ, L0 )(0) } = F (ρ∞ ,L0 ) (ξ) ∩ Z

= F (ρ∞ ,L0 ) (ξ).

8.7 Accidental parabolic transformation Proposition 8.7.1. Under Assumption 8.3.1, ρ∞ (KP ) is not parabolic for (0) any elliptic generator satisfying s(P ) ∈ Σ0 .

188

8 Closedness

Proof. By Proposition 8.3.2, we may suppose that ν ∞ is thick. Suppose to the contrary that ρ∞ (KP0 ) is parabolic for an elliptic generator triple (P0 , P1 , P2 ) such that s(P0 ), s(P1 ), s(P2 ) is a triangle in Σ0 . In what follows, we will use Notations 5.5.5 (InﬁniteFan) and 8.1.2. Then, by Lemma (k) (k) 5.5.6, either (i) arg a1,∞ = − arg a2,∞ ∈ (0, π) for every k ∈ Z, or (ii) (k)

(k)

arg a1,∞ = − arg a2,∞ ∈ (−π, 0) for every k ∈ Z. For simplicity, we sup(k)

(k)

pose that arg a1,∞ = − arg a2,∞ ∈ (0, π) for every k ∈ Z. (A similar argument works for the remaining case.) Claim 8.7.2. The following hold for every integer k. (k)

(k)

1. If σ (k) ∈ Σ0 , then arg a1,∞ = − arg a2,∞ ∈ (0, π/2]. 2. If σ (k) ∈ Σ0 and σ (k) is not the (+)-terminal triangle of Σ0 , then its successor is equal to σ (k−1) or σ (k+1) . (k)

(k)

Proof. Suppose σ (k) ∈ Σ0 . Suppose to the contrary that arg a1,∞ = − arg a2,∞ ∈ (π/2, π), Then we can see that L(ρn , σ (k) ) is not simple for suﬃciently large n. This contradicts the assumption that σ (k) ∈ Σ0 = Σ(ν n ) and (ρn , ν n ) is good. Hence we obtain the ﬁrst assertion. To see the second assertion, note that the ﬁrst assertion implies that (k) L(ρn , σ (k) ) is convex to the above at c(ρn (P0 )) for all suﬃciently large n. (k) (k) Thus θ+ (ρn , σ (k) ; s(P0 )) > θ− (ρn , σ (k) ; s(P0 )) by Corollary 4.2.15. There(k) fore, by Proposition 6.7.1, if σ is not the (+)-terminal triangle of Σ0 , its (k) successor in Σ0 = Σ(ν n ) must contain s(P0 ) (cf. Deﬁnition 3.2.1(1)). Hence it is equal to either σ (k−1) or σ (k+1) . + Claim 8.7.3. ν∞ is equal to s(P0 ).

Proof. By Claim 8.7.2, the (+)-terminal triangle of Σ0 is equal to σ (k) for (k) (k) some k ∈ Z. Reset σ := σ (k) , sj = sj , aj,∞ = aj,∞ (j = 0, 1, 2) and Pj = (k)

Pj (k ∈ Z). Since ρ∞ (P2 P1 ) = ρ∞ (KP0 ) is parabolic by the assumption, ρ∞ (P1 ) and ρ∞ (P2 ) share the unique ﬁxed point f := c(ρ∞ (P0 )) + 1/2, and it is the parabolic ﬁxed point of ρ∞ (KP0 ) (cf. Lemma 5.5.3). We show that + f = lim Fix+ σ (ρn (P1 )) = lim Fixσ (ρn (P2 )). For simplicity we prove only f = + lim Fixσ (ρn (P1 )). Note that there is a sequence {ǫn } in {−, +} such that f = lim Fixǫσn (ρn (P1 )). Our task is to prove ǫn = + for all suﬃciently large n. Recall that arg a1,∞ = − arg a2,∞ ∈ (0, π/2] by Claim 8.7.2. This implies ℑ(f ) = ℑ(c(ρ∞ (P0 ))) > ℑ(c(ρ∞ (P1 ))). Hence arg

Thus

c(ρ∞ (P0 )) − c(ρ∞ (P1 )) f − c(ρ∞ (P1 ))

∈ (0, π).

8.8 Proof of Proposition 8.4.5 - length 1 case -

arg

c(ρn (P0 )) − c(ρn (P1 )) Fixǫσn (ρn (P1 )) − c(ρn (P1 ))

189

∈ (0, π)

for all suﬃciently large n. This implies ǫn = + for all suﬃciently large n, and therefore f = lim Fix+ σ (ρn (P1 )). Since f = I(ρ∞ (P1 )) ∩ I(ρ∞ (P2 )), this in + turn implies f = lim v + (ρn ; P0 , P1 ). Thus lim Fix+ σ (ρn (P1 )) = v (ρn ; P0 , P1 ), and hence lim θǫ (ρn , σ; s(P1 )) = 0. Similarly, by using f = lim Fix+ σ (ρn (P2 )), + we see lim θǫ (ρn , σ; s(P2 )) = 0. Hence, ν∞ is equal to the point s(P0 ) in σ represented by (π/2, 0, 0) = lim θ+ (ρn , σ). Claim 8.7.3 contradicts Assumption 8.3.1(1). This completes the proof of Proposition 8.7.1. Proposition 8.7.4. Under Assumption 8.3.1, assume that ν ∞ is thick and that {ρn } satisﬁes the condition HausdorﬀConvergence. Then (ρ∞ , τ ) is not an isosceles representation for any edge τ of Σ0 . Proof. Suppose to the contrary that there is an edge τ of Σ0 such that (ρ∞ , τ ) is an isosceles representation. Set σ = s0 , s1 , s2 = s(P0 ), s(P1 ), s(P2 ) and τ = s0 , s2 . Since ρ∞ ∈ QF, none of ρ∞ (KP0 ) and ρ∞ (KP2 ) is elliptic (cf. Lemma 2.5.4). Thus, by Proposition 5.2.8(1), they are either purely-hyperbolic or parabolic. Hence, by Proposition 8.7.1, both ρ∞ (KP0 ) and ρ∞ (KP2 ) must be purely-hyperbolic. Then ρ∞ ∈ QF and (ρ∞ , µ) ∈ J [QF] for some thin label µ ∈ int τ × int τ by Proposition 5.2.8(2). By Proposition 6.2.1, there are a neighborhood U of ρ∞ in QF and a continuous map U ∋ ρ → ν(ρ) ∈ H2 × H2 with ν(ρ∞ ) = µ such that (ρ, ν(ρ)) is good for any ρ ∈ U . We may assume, by taking a subsequence, that ρn ∈ U for all n ∈ N. Since both (ρn , ν n ) and (ρn , ν(ρn )) are good, we have ν n = ν(ρn ) by Corollary 6.2.5. Hence ν ∞ = lim ν n = lim ν(ρn ) = ν(ρ∞ ) = µ. This contradicts the assumption that ν ∞ is thick.

8.8 Proof of Proposition 8.4.5 - length 1 case In this section, we prove Proposition 8.4.5 in the case where the length of Σ0 is 1. We begin by proving the following general fact. Proof (Proof of Proposition 8.4.5 - the length 1 case -). Under the assumption of Proposition 8.4.5, assume further that the length of the common chain Σ0 is equal to 1. Then Σ0 consists of precisely one triangle, say σ = s0 , s1 , s2 , L0 contains no 2-simplices, and L(ν ∞ ) is equal to L0 . Thus the assertion (2) obviously holds. To prove the assertion (1), let {Pj } be the sequence of elliptic generators (0) (1) associated with σ. Then L0 = {Pj | j ∈ Z} and L0 = {(Pj , Pj+1 ) | j ∈ Z}. First, we prove the assertion (1) for an edge ξ = (Pj , Pj+1 ) of L0 = L(ν ∞ ). We may assume j = 0 without loss of generality. We show that

190

8 Closedness 3

F ∞ (ξ) is a 1-dimensional convex polyhedron in H and that F ρ∞ (ξ) is equal to F ∞ (ξ). To this end, we have only to show that Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) is a complete geodesic. In fact, if this is proved, then I(ρ∞ (P0 )) ∩ + − (P0 , P1 )}. This implies that F ∞ (ξ) is equal (P0 , P1 ), f∞ I(ρ∞ (P1 )) = {f∞ to the 1-dimensional convex polyhedron Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )), because − + F ∞ (ξ) = [f∞ (P0 , P1 ), f∞ (P0 , P1 )] by Lemma 8.6.1(2). Since F ρ∞ (ξ) is also equal to Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) by deﬁnition, we have F ρ∞ (ξ) = F ∞ (ξ). In what follows we prove by contradiction that Ih(ρ∞ (P0 ))∩Ih(ρ∞ (P1 )) is a complete geodesic. So, suppose this is not the case. Then either I(ρ∞ (P0 )) = I(ρ∞ (P1 )) or I(ρ∞ (P0 )) and I(ρ∞ (P1 )) are tangent at a point, because I(ρn (P0 ))∩I(ρn (P1 )) = ∅ for every n ∈ N and hence I(ρ∞ (P0 ))∩I(ρ∞ (P1 )) = ∅. By Lemma 2.5.4(3), we have I(ρ∞ (P0 )) = I(ρ∞ (P1 )). Therefore I(ρ∞ (P0 )) and I(ρ∞ (P1 )) are tangent at a point, and hence we have (i) I(ρ∞ (P0 )) ⊂ D(ρ∞ (P1 )), (ii) I(ρ∞ (P1 )) ⊂ D(ρ∞ (P0 )), or (iii) int(D(ρ∞ (P0 ))) ∩ int(D(ρ∞ (P1 ))) = ∅. If the condition (i) holds, then I(ρ∞ (P0 ))∩E(ρ∞ (P1 )) consists of precisely one

lies in int(D(ρ∞ (P1 ))). point. Thus one of the two ﬁxed points of ρ∞ (P0 ) in C ǫ Hence Fixσ (ρn (P0 )) ∈ int(D(ρn (P1 ))) for some ǫ ∈ {−, +} and for suﬃciently large n ∈ N. On the other hand, since Σ(ν n ) = Σ0 = {σ} and since ρn = (ρn , ν n ) is good, σ is the ǫ-terminal triangle of ρn and hence Fixǫσ (ρn (P0 )) ∈ E(ρn (P1 )). This is a contradiction. We have a similar contradiction if the condition (ii) holds. If the condition (iii) holds, then we see that lim (θ− (ρn , σ; sj ) + θ+ (ρn , σ; sj )) = π for j = 0, 1 (see Fig. 8.3). On the v + (ρn ; P0 , P1 ) Fix+ σ (ρn (P0 ))

Fix+ σ (ρn (P1 ))

lim v + (ρn ; P0 , P1 ) = lim v − (ρn ; P0 , P1 ) lim Fix+ σ (ρn (P1 )) lim Fix+ σ (ρn (P0 )) I(ρ∞ (P1 ))

I(ρn (P1 )) I(ρn (P0 )) I(ρ∞ (P0 )) Fix− σ (ρn (P0 ))

Fix− σ (ρn (P1 )) − v (ρn ; P0 , P1 )

lim Fix− σ (ρn (P0 ))

lim Fix− σ (ρn (P1 ))

Fig. 8.3. Case (iii): int(D(ρ∞ (P0 ))) ∩ int(D(ρ∞ (P1 ))) = ∅.

other hand, since σ is the ǫ-terminal triangle of ρn for each ǫ ∈ {−, +}, we have 0 ≤ θǫ (ρn , σ; sj ) ≤ π/2 for j = 0, 1, 2 by Proposition 4.2.16. Hence '2 ǫ ǫ lim θ (ρn , σ; sj ) = π/2 for j = 0, 1. Therefore lim j=0 θ (ρn , σ; sj ) ≥ π >

8.9 Proof of Proposition 8.4.5 - length ≥ 2 case - (Step 1)

191

π/2. This contradicts Proposition 4.2.16. Hence Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) is a complete geodesic. Thus we have proved that the assertion (1) holds for any edge of L0 . Next, we prove the assertion (1) for a vertex ξ = (Pj ) of L0 = L(ν ∞ ). We may suppose j = 1 without loss of generality. By Lemma 8.6.1, F ∞ (P1 ) is the convex − + ǫ ǫ ǫ (P1 ) ∪ f∞ (P1 ). Since f∞ (P1 ) contains f∞ (P0 , P1 ) and f∞ (P1 , P2 ) for hull of f∞ each ǫ, F ∞ (P1 ) contains the 1-dimensional convex polyhedra F ∞ (P0 , P1 ) and F ∞ (P1 , P2 ). Hence dim F ∞ (P1 ) = 2 if and only if F ∞ (P0 , P1 ) = Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) is equal to F ∞ (P1 , P2 ) = Ih(ρ∞ (P1 )) ∩ Ih(ρ∞ (P2 )). Suppose to the contrary that this happens. Then Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) ∩ Ih(ρ∞ (P2 )) ǫ ǫ is a complete geodesic, and f∞ (P1 ) is equal to the singleton f∞ (P0 , P1 ) = ǫ ǫ f∞ (P1 , P2 ), which in turn is equal to the singleton lim Fixσ (ρn (P1 )). Thus Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) ∩ Ih(ρ∞ (P2 )) is equal to the geodesic joining the two ﬁxed points of ρ∞ (P1 ). Hence we have Axis(ρ∞ (P1 )) = Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P2 )). Thus (ρ∞ , τ ) with τ = s(P0 ), s(P2 ) is an isosceles representation by Proposition 5.2.3. This contradicts Proposition 8.7.4. Hence dim F ∞ (P1 ) = 2. In order to prove F ρ∞ (P1 ) = F ∞ (P1 ), we note that F ∞ (P1 ) is the subspace of Ih(ρ∞ (P1 )) bounded by the two geodesics Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) and Ih(ρ∞ (P1 )) ∩ Ih(ρ∞ (P2 )). By Lemma 8.6.2, F ∞ (P1 ) ⊂ F ρ∞ (P1 ), and by deﬁnition F ρ∞ (P1 ) = Ih(ρ∞ (P1 ))∩Eh(ρ∞ (P0 ))∩Eh(ρ∞ (P2 )). Thus F ρ∞ (P1 ) is also the subspace of Ih(ρ∞ (P1 )) bounded by the two geodesics Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) and Ih(ρ∞ (P1 ))∩Ih(ρ∞ (P2 )). Hence we have F ρ∞ (P1 ) = F ∞ (P1 ). Thus we have proved that the assertion (1) holds for any vertex of L0 . This completes the proof of Proposition 8.4.5 in the case when the length of Σ0 is 1.

8.9 Proof of Proposition 8.4.5 - length ≥ 2 case (Step 1) Sections 8.9–8.11 are devoted to the proof of Proposition 8.4.5 in the case where the length of Σ0 is ≥ 2. Thus we presume the following assumption throughout these sections. Assumption 8.9.1. We assume that {ρn } = {(ρn , ν n )} is a sequence in J [QF] satisfying the following conditions:

(1) {ρn } converges to a labeled representation ρ∞ = (ρ∞ , ν ∞ ) ∈ QF × (H2 × H2 ). (2) {ρn } satisﬁes the condition SameStratum, i.e., every chain Σ(ν n ) is equal to a ﬁxed chain Σ0 , and hence every L(ν n ) is equal to the ﬁxed elliptic generator complex L0 = L(Σ0 ). (3) {ρn } = {(ρn , ν n )} satisﬁes the condition HausdorﬀConvergence. (4) ν ∞ is thick. (5) The length of Σ0 is ≥ 2, or equivalently, L0 contains a 2-simplex.

192

8 Closedness

The purpose of this section is to study the behavior of the limit of three isometric hemispheres whose intersection constitute a vertex of Eh(ρn ). Thus we also presume the following assumption throughout this section (except in Lemma 8.9.9) Assumption 8.9.2. Under Assumption 8.9.1, ξ = (P0 , P1 , P2 ) denotes a 2simplex of L0 , where {Pj } is the sequence of elliptic generators associated with a triangle σ = s0 , s1 , s2 of D. As in Notation 2.1.14 (Adjacent triangles), σ ′ = s′0 , s′1 , s′2 = s0 , s2 , s′2 denotes the triangle of D sharing the edge s0 , s2 = s′0 , s′1 with σ. We note that both σ and σ ′ belong to the chain Σ0 by Lemma 3.2.4(2). The symbol ǫ0 denotes the sign such that σ lies in the ǫ0 -side of σ ′ . We ﬁrst prove the following consequence of Lemma 4.8.7, which plays a key role in the proof of Lemma 8.9.8. Lemma 8.9.3 (NoBadFold). Under Assumptions 8.9.1 and 8.9.2, L(ρ∞ , σ) is not folded at s(P1 ). Namely, the centers c(ρ∞ (Pj )) (j = 0, 1, 2) cannot sit on the line containing π(Axis(ρ∞ (P1 ))). Proof. From the assumption, either (σ, σ ′ ) or (σ ′ , σ) forms a subchain, Σ, of (0) = ∅, the complex probability (a0,∞ , a1,∞ , a2,∞ ) ∈ Σ0 . Since φ−1 ∞ (0) ∩ σ 3 (C − {0}) of ρ∞ at σ is deﬁned. Suppose to the contrary that c(ρ∞ (Pj )) (j = 0, 1, 2) sit on the line containing π(Axis(ρ∞ (P1 ))). Then we have a2,∞ = λa0,∞ for some negative real number λ. By Proposition 5.4.3, ρ∞ is not pure-imaginary, and therefore L(ρ∞ , σ) is not doubly folded by Lemma 5.4.2. Hence, if λ = −1, then L(ρn , Σ) is not simple for all suﬃciently large n by Lemma 4.8.7. So we have λ = −1 and hence I(ρ∞ (P0 )) = I(ρ∞ (P2 )). This contradicts Lemma 2.5.4(2-iii). To state the main result of this section, we need the following lemma. Lemma 8.9.4. Under Assumptions 8.9.1 and 8.9.2, precisely one of the following three situations occurs, (1) ξ ∈ L(ν ∞ )(2) . ǫ0 ∈ ints0 , s2 , s1 ∈ / Σ(ν ∞ )(0) and ξ ∈ L(ν ∞ )(2) . In this case σ and (2) ν∞ ′ σ , respectively, are the ǫ0 -terminal triangles of Σ0 and Σ(ν ∞ ) (see Fig. 8.4(a)). −ǫ0 ∈ ints0 , s2 , s1 ∈ Σ(ν ∞ )(0) and ξ ∈ L(ν ∞ )(2) . In this case σ ′ and σ, (3) ν∞ respectively, are the (−ǫ0 )-terminal triangles of Σ0 and Σ(ν ∞ ) (see Fig. 8.4(b)). Proof. Since ξ = (P0 , P1 , P2 ) is a 2-simplex of L0 , both σ and σ ′ belong to the chain Σ0 by Lemma 3.2.4(2). Suppose that the ﬁrst condition does not hold, i.e., ξ ∈ / L(ν ∞ )(2) . Then σ or σ ′ does not belong to Σ(ν ∞ ). If σ does − + not belong to Σ(ν ∞ ), then the geodesic segment [ν∞ , ν∞ ] is disjoint from

8.9 Proof of Proposition 8.4.5 - length ≥ 2 case - (Step 1)

193

int σ, whereas each [νn− , νn+ ] intersects both int σ and int σ ′ . Thus νnǫ0 ∈ σ ǫ0 ∈ σ ∩ σ ′ = s0 , s2 , because σ lies in the ǫ0 -side of σ ′ . In particular, and ν∞ s1 ∈ / Σ(ν ∞ )(0) . Since ν ∞ is thick by assumption, the triangle σ ′ must be ǫ0 ∈ H2 , we see that the contained in Σ(ν ∞ ). Hence, by using the fact that ν∞ second condition in the lemma holds. If σ ′ does not belong to Σ(ν ∞ ), then by a similar argument we see that the third condition in the lemma holds. Under Assumptions 8.9.1 and 8.9.2, the three isometric hemispheres Ih(ρn (Pj )) (j = 0, 1, 2) intersect transversely as illustrated in Fig. 8.5, because ρn = (ρn , ν n ) is good. Namely, the following hold. 1. Ih(ρn (P0 )) ∩ Ih(ρn (P1 )) ∩ Ih(ρn (P2 )) is equal to the singleton F ρn (ξ) in H3 . 2. For each k ∈ {0, 1, 2} set Ln (k) := Ih(ρn (Pi )) ∩ Ih(ρn (Pj )) ∩ Eh(ρn (Pk )) where {i, j, k} = {0, 1, 2} and i < j. Then Ln (k) is equal to the half geodesic [F ρn (ξ), v ǫ(k) (ρn ; Pi , Pj )] contained in the complete geodesic Ih(ρn (Pi )) ∩ Ih(ρn (Pj )), where ǫ(k) = ǫ0 or −ǫ0 according as k ∈ {0, 2} or k = 1. Proposition 8.9.5 describes what happens to the conﬁguration of the three isometric hemispheres at the limit. Roughly speaking, we show that the limit conﬁguration is combinatorially equivalent to the original one or that illustrated in Figs. 8.6 or 8.7 according to the three situations in Lemma 8.9.4. Throughout Sects. 8.9 and 8.10 we denote lim v ǫ (ρn ; P, Q) (resp. lim Fixǫσ (ρn (P ))) by ǫ (P, Q) (resp. Fixǫσ (ρ∞ (P ))). v∞ s1

νn+

s1 σ

s0 = s′0

s2 = s′1 σ

′

[νn− , νn+ ] s′2

[νn− , νn+ ]

σ s0 = s′0

s2 = s′1 σ′

νn− s′2

(a)

(b)

ǫ0 −ǫ0 Fig. 8.4. (a) ν∞ ∈ ints0 , s2 , (b) ν∞ ∈ ints0 , s2 ; the ﬁgures are for ǫ0 = +.

Proposition 8.9.5. Under Assumptions 8.9.1 and 8.9.2, set L∞ (k) := Ih(ρ∞ (Pi ))∩Ih(ρ∞ (Pj ))∩Eh(ρ∞ (Pk )) for each k ∈ {0, 1, 2}, where {i, j, k} = {0, 1, 2} and i < j. Let ǫ(k) be the sign deﬁned in the above, i.e., ǫ(k) = ǫ0 or −ǫ0 according as k ∈ {0, 2} or k = 1 (1) If ξ ∈ L(ν ∞ )(2) , then the combinatorial structure of the conﬁguration of the three isometric hemispheres does not change at the limit. Namely, the following hold (cf. Fig. 8.5).

194

8 Closedness v + (ρn ; P0 , P1 ) Ih(ρn (P1 ))

v + (ρn ; P1 , P2 ) Ln (0)

Ln (2) Ih(ρn (P2 )) Ih(ρn (P0 )) F ρ n (ξ)

Ln (1)

v − (ρn ; P0 , P2 )

Fig. 8.5. The combinatorial structure in the case when ξ = (P0 , P1 , P2 ) ∈ L0 ; the ﬁgure is for ǫ0 = +. + + F ∞ (ξ) = v∞ (P0 , P1 ) = v∞ (P1 , P2 ) = L∞ (0) = L∞ (2)

Ih(ρ∞ (P1 ))

Ih(ρ∞ (P0 ))

L∞ (1)

Ih(ρ∞ (P2 ))

+ ν∞

s0 = s′0

s′2

s1 σ σ′

s2 = s′1

− + [ν∞ , ν∞ ]

− v∞ (P0 , P2 ) ǫ0 Fig. 8.6. The degeneration of the combinatorial structure in the case when ν∞ ∈ ints0 , s2 ; the ﬁgure is for ǫ0 = +.

(i) Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) ∩ Ih(ρ∞ (P2 )) is equal to the singleton F ∞ (ξ) in H3 . (ii) For each k ∈ {0, 1, 2}, L∞ (k) is equal to the half geodesic [F ∞ (ξ), ǫ(k) v∞ (Pi , Pj )] contained in the complete geodesic Ih(ρ∞ (Pi ))∩Ih(ρ∞ (Pj )). ǫ0 (2) If ν∞ ∈ ints0 , s2 , i.e., if the second condition in Lemma 8.9.4 holds, then as n → ∞, F ρn (ξ) drops onto the point Fixǫσ0 (ρ∞ (P1 )) in C, Ih(ρn (P1 )) becomes invisible, and Ln (0) and Ln (2) shrink to the point in C, whereas Ln (1) turns into a complete geodesic. To be precise, the following hold (see Fig. 8.6).

(i) Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) ∩ Ih(ρ∞ (P2 )) is equal to the singleton F ∞ (ξ) in C, and

8.9 Proof of Proposition 8.4.5 - length ≥ 2 case - (Step 1) L∞ (2)

195

Ih(ρ∞ (P1 )) L∞ (0)

+ v∞ (P0 , P1 )

+ v∞ (P1 , P2 )

s1 σ s0 = s′0 ′ σ Ih(ρ∞ (P2 )) s′2

Ih(ρ∞ (P0 ))

− + [ν∞ , ν∞ ] s2 = s′1 − ν∞

− F ∞ (ξ) = v∞ (P0 , P2 ) = L∞ (1) −ǫ0 Fig. 8.7. The degeneration of the combinatorial structure in the case when ν∞ ∈ ints0 , s2 ; the ﬁgure is for ǫ0 = +.

ǫ0 ǫ0 ǫ0 F ∞ (ξ) = Fixǫσ0 (ρ∞ (P1 )) = v∞ (P0 , P2 ). (P1 , P2 ) = v∞ (P0 , P1 ) = v∞

(ii) Ih(ρ∞ (P1 )) is covered by the remaining two isometric hemispheres. To be precise: (a) Dh(ρ∞ (P1 )) ∩ Eh(ρ∞ (P0 )) ∩ Eh(ρ∞ (P2 )) is equal to the singleton F ∞ (ξ) in C. (b) Eh(ρ∞ (P1 )) ⊃ Eh(ρ∞ (P0 )) ∩ Eh(ρ∞ (P2 )). ǫ(k) (iii) L∞ (k) = [F ∞ (ξ), v∞ (Pi , Pj )]. Thus L∞ (0) and L∞ (2) are equal to the singleton F ∞ (ξ) in C, whereas L∞ (1) is equal to the complete geodesic Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P2 )).

−ǫ0 (3) If ν∞ ∈ ints0 , s2 , i.e., if the third condition in Lemma 8.9.4 holds, 0 then, as n → ∞, F ρn (ξ) drops onto the point Fix−ǫ σ (ρ∞ (P1 )) in C, and Ln (1) shrinks to the point in C, whereas Ln (0) and Ln (2) turn into complete geodesics. To be precise, we have the following (see Fig. 8.7).

(i) Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) ∩ Ih(ρ∞ (P2 )) is equal to the singleton F ∞ (ξ) in C, and −ǫ0 −ǫ0 −ǫ0 0 F ∞ (ξ) = Fix−ǫ σ (ρ∞ (P1 )) = v∞ (P0 , P1 ) = v∞ (P1 , P2 ) = v∞ (P0 , P2 ). ǫ(k)

(ii) L∞ (k) = [F ∞ (ξ), v∞ (Pi , Pj )]. Thus L∞ (1) is equal to the singleton F ∞ (ξ) in C, whereas L∞ (2) is equal to the complete geodesic Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) and L∞ (0) is equal to the complete geodesic Ih(ρ∞ (P1 )) ∩ Ih(ρ∞ (P2 )). As an immediate corollary, we obtain the following. Corollary 8.9.6. Under Assumptions 8.9.1 and 8.9.2, we have ǫ(k) (Pi , Pj )]. L∞ (k) = lim Ln (k) = [F ∞ (ξ), v∞

for each k ∈ {0, 1, 2}.

196

8 Closedness

We begin the proof of this proposition by proving a few lemmas. Lemma 8.9.7. Under Assumptions 8.9.1 and 8.9.2, the following hold. (1) F ∞ (ξ) ⊂ Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) ∩ Ih(ρ∞ (P2 )). ǫ(k) (2) [F ∞ (ξ), v∞ (Pi , Pj )] ⊂ L∞ (k). Proof. (1) This is proved as follows. F ∞ (ξ) = lim F ρn (ξ) = lim Ih(ρn (P0 )) ∩ Ih(ρn (P1 )) ∩ Ih(ρn (P2 )) ⊂ Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) ∩ Ih(ρ∞ (P2 )). ǫ(k)

(2) Since F ∞ (ξ) = lim F ρn (ξ) and v∞ (Pi , Pj ) = lim v ǫ(k) (ρn ; Pi , Pj ), we have: ǫ(k) [F ∞ (ξ), v∞ (Pi , Pj )] = lim[F ρn (ξ), v ǫ(k) (ρn ; Pi , Pj )] = lim Ih(ρn (Pi )) ∩ Ih(ρn (Pj )) ∩ Eh(ρn (Pk ))

⊂ Ih(ρ∞ (Pi )) ∩ Ih(ρ∞ (Pj )) ∩ Eh(ρ∞ (Pk )) = L∞ (k).

Lemma 8.9.8. Under Assumptions 8.9.1 and 8.9.2, the following hold. (1) Ih(ρ∞ (P0 ))∩Ih(ρ∞ (P1 )) and Ih(ρ∞ (P1 ))∩Ih(ρ∞ (P2 )) are complete geodesics. (2) Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) = Ih(ρ∞ (P1 )) ∩ Ih(ρ∞ (P2 )). (3) Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) ∩ Ih(ρ∞ (P2 )) is equal to the singleton F ∞ (ξ). Proof. (1) By Lemma 8.9.7(1), we have F ∞ (ξ) ⊂ Ih(ρ∞ (P0 ))∩Ih(ρ∞ (P1 ))∩Ih(ρ∞ (P2 )) ⊂ Ih(ρ∞ (P0 ))∩Ih(ρ∞ (P1 )). Since Ih(ρ∞ (P0 )) = Ih(ρ∞ (P1 )) (cf. Lemma 2.5.4(2-iii)), the above inclusion implies that Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) is either the singleton F ∞ (ξ) or a complete geodesic. Suppose to the contrary that Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) is the singleton F ∞ (ξ). Then, by Lemma 4.1.3(2) (Chain rule), Ih(ρ∞ (P1 )) ∩ Ih(ρ∞ (P2 )) = ρ∞ (P1 )(Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 ))) is also a singleton. Since F ∞ (ξ) is contained in Axis(ρ∞ (P1 )), this implies that the centers c(ρ∞ (Pj )) (j = 0, 1, 2) lie in the line in C containing proj(Axis(ρ∞ (P1 ))). This contradicts Lemma 8.9.3 (NoBadFold). Hence Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) is a geodesic. By Lemma 4.1.3(2), Ih(ρ∞ (P1 )) ∩ Ih(ρ∞ (P2 )) is also a geodesic. (2) Suppose to the contrary that Ih(ρ∞ (P0 ))∩Ih(ρ∞ (P1 )) = Ih(ρ∞ (P1 ))∩ Ih(ρ∞ (P2 )). Then, by Lemma 5.2.6, either (ρ∞ , τ ) is an isosceles representation where τ = s0 , s2 or L(ρ∞ , σ) is folded at c(ρ∞ (P1 )). By Lemma 8.9.3 (NoBadFold), the latter cannot happen. So, (ρ∞ , τ ) is an isosceles representation. This contradicts Proposition 8.7.4.

8.9 Proof of Proposition 8.4.5 - length ≥ 2 case - (Step 1)

197

(3) By (1) and (2), Ih(ρ∞ (P0 ))∩Ih(ρ∞ (P1 )) and Ih(ρ∞ (P1 ))∩Ih(ρ∞ (P2 )) are diﬀerent geodesics, and hence Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) ∩ Ih(ρ∞ (P2 )) is either empty or a singleton. Hence we obtain the desired result by Lemma 8.9.7(1). Lemma 8.9.9. Under Assumption 8.9.1, let η be a generic edge of L0 (Deﬁnition 8.5.1). Then η is also an edge of L(ν ∞ ), and F ∞ (η) is a 1-dimensional 3 convex polyhedron in H . Proof. Since η is a generic edge of L0 , we see by Lemma 8.5.2 that there is a triangle σ of Σ0 − {σ − , σ + } such that (i) η = (Pj , Pj+1 ) for some j and (2) (ii) both (P0 , P1 , P2 ) and (P1 , P2 , P3 ) belong to L0 . Here {Pj } is the sequence of elliptic generators associated with σ = s0 , s1 , s2 . Since σ belongs to Σ0 −{σ − , σ + }, it also belongs to Σ(ν ∞ ). Hence η is also an edge of L(ν ∞ ). Suppose that F ∞ (η) is not a 1-dimensional convex polyhedron. Then it must be a singleton, because it is the convex hull of two (possibly identical) points by Lemma 8.6.1(2). By Lemma 6.4.2(2), this implies that F ∞ (Pj , Pj+1 ) is equal to a singleton for every j. Hence, we have F ∞ (P0 , P1 , P2 ) = F ∞ (P1 , P2 , P3 ), because both points are contained in the singleton F ∞ (P1 , P2 ). On the other hand, F ∞ (P0 , P1 , P2 ) and F ∞ (P1 , P2 , P3 ), respectively, are contained in Axis(ρ∞ (P1 )) and Axis(ρ∞ (P2 )) by Lemma 6.4.2(3). Thus Axis(ρ∞ (P1 )) and Axis(ρ∞ (P2 )) share the point F ∞ (P0 , P1 , P2 ) = F ∞ (P1 , P2 , P3 ). Hence ρ∞ (KP0 ) = ρ∞ (P2 P1 ) is elliptic, parabolic or the identity. This contradicts Lemma 2.5.4(2) and Proposition 8.7.1. Lemma 8.9.10. Under Assumptions 8.9.1 and 8.9.2, F ∞ (ξ) belongs to C if and only if ξ ∈ L(ν ∞ )(2) . Moreover, in this case, precisely one of the following holds. ǫ0 ǫ0 (1) F ∞ (ξ) = Fixǫσ0 (ρ∞ (P1 )) and it is equal to v∞ (P1 , P2 ) = (P0 , P1 ) = v∞ (0) ǫ0 ǫ0 / Σ(ν ∞ ) , and the condition v∞ (P0 , P2 ). Moreover ν∞ ∈ ints0 , s2 , s1 ∈ in Lemma 8.9.4(2) holds (see Fig. 8.6). −ǫ0 −ǫ0 0 (2) F ∞ (ξ) = Fix−ǫ σ (ρ∞ (P1 )) and it is equal to v∞ (P0 , P1 ) = v∞ (P1 , P2 ) = (0) −ǫ0 −ǫ0 v∞ (P0 , P2 ). Moreover ν∞ ∈ ints0 , s2 , s1 ∈ Σ(ν ∞ ) , and the condition in Lemma 8.9.4(3) holds (see Fig. 8.7).

Proof. Note that L(ρn , σ) lies above or below L(ρn , σ ′ ) according as ǫ0 = + or −, because σ lies on the ǫ0 -side of σ ′ and because ρn satisﬁes the condition Simple (cf. Deﬁnition 3.2.2). Since the three isometric hemispheres Ih(ρn (Pj )) (j = 0, 1, 2) intersect at the singleton Fρn (ξ), the following hold by Lemma 4.3.6 and Corollary 4.2.15. 1. θǫ0 (ρn , σ; s1 ) > 0 and θ−ǫ0 (ρn , σ; s1 ) < 0. 0 / D(ρn (P0 )) ∪ D(ρn (P2 )) and Fix−ǫ 2. Fixǫσ0 (ρn (P1 )) ∈ σ (ρn (P1 )) int(D( ρn (P0 )) ∩ D(ρn (P2 ))).

∈

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8 Closedness

Note that the three points v ǫ0 (ρn ; P0 , P1 ), v ǫ0 (ρn ; P1 , P2 ), and v −ǫ0 (ρn ; P0 , P2 ) lie on the boundary of ∪2j=0 D(ρn (Pj )) (cf. Fig. 8.5). First, suppose that F ∞ (ξ) belongs to C. Then F ∞ (ξ) is equal to Fixǫσ0 (ρ∞ (P1 )) or Fixσ−ǫ0 (ρ∞ (P1 )), because F ∞ (ξ) ∈ Axis(ρ∞ (P1 )) ∩ C = {Fix± σ (ρ∞ (P1 ))}. Case 1. Suppose that F ∞ (ξ) = Fixǫσ0 (ρ∞ (P1 )). Claim 8.9.11. The 2-dimensional convex polyhedron Ih(ρn (P1 ))∩Eh(ρn (P0 ))∩ Eh(ρn (P2 )) shrinks to the singleton F ∞ (ξ) = Fixǫσ0 (ρ∞ (P1 )) as n → ∞ (cf. Fig. 8.6). Namely, lim Ih(ρn (P1 )) ∩ Eh(ρn (P0 )) ∩ Eh(ρn (P2 )) = F ∞ (ξ). In particular, the following hold.

(1) The geodesic segments [Fρn (ξ), v ǫ0 (ρn ; P0 , P1 )] and [Fρn (ξ), v ǫ0 (ρn ; P1 , P2 )] shrink to the point F ∞ (ξ) as n → ∞. In particular F ∞ (P0 , P1 ) and F ∞ (P1 , P2 ) are singletons. (2) The circular arc eǫ0 (ρn , σ; P1 ) shrinks to the point F ∞ (ξ) as n → ∞. Proof. Let ℓn and ℓ′n , respectively, be the perpendicular to proj(Axis(ρn (P1 ))) at Fixǫσ0 (ρn (P1 )) and at proj(F ρn (ξ)) (see Fig. 8.8). Then proj(Ih(ρn (P1 )) ∩ ℓn ℓ′n

Fig. 8.8. Figure of three isometric circles and two lines

Eh(ρn (P0 ))∩Eh(ρn (P2 ))) is contained in the region in D(ρn (P1 )) bounded by ℓn and ℓ′n . Since lim F ρn (ξ) = F ∞ (ξ) = Fixǫσ0 (ρ∞ (P1 )) = lim Fixǫσ0 (ρn (P1 )), both ℓn and ℓ′n converge to the perpendicular, ℓ∞ , to proj(Axis(ρ∞ (P1 ))) at F ∞ (ξ). Hence proj(Ih(ρn (P1 )) ∩ Eh(ρn (P0 )) ∩ Eh(ρn (P2 ))) converges to ℓ∞ ∩ D(ρ∞ (P1 )) = F ∞ (ξ). This implies that Ih(ρn (P1 )) ∩ Eh(ρn (P0 )) ∩ Eh(ρn (P2 )) converges to F ∞ (ξ). The remaining assertions follow from the fact that [Fρn (ξ), v ǫ0 (ρn ; P0 , P1 )], [Fρn (ξ), v ǫ0 (ρn ; P1 , P2 )] and eǫ0 (ρn , σ; P1 ) are contained in the 2-dimensional convex polyhedron Ih(ρn (P1 )) ∩ Eh(ρn (P0 )) ∩ Eh(ρn (P2 )).

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199

By Claim 8.9.11(1) and Lemma 8.9.9, (P0 , P1 ) is not a generic edge of L0 . Hence the interior of the half geodesic [Fρn (ξ), v ǫ0 (ρn ; P0 , P1 )] does not contain a vertex of Eh(ρn ) by Lemma 8.5.3. Thus v ǫ0 (ρn ; P0 , P1 )∈fr E ǫ0 (ρn ). Similarly, we also see that (P1 , P2 ) is not a generic edge of L0 and v ǫ0 (ρn ; P1 , P2 ) ∈ fr E ǫ0 (ρn ). Hence the ǫ0 -terminal triangle of Σ0 is equal to σ, and therefore νnǫ0 ∈ σ. On the other hand, we have lim θǫ0 (ρn , σ; s1 ) = 0 by Claim 8.9.11(2). ǫ0 ∈ ints0 , s2 and the condition in Lemma 8.9.4(2) holds. We also Hence ν∞ ǫ0 ǫ0 see that F ∞ (ξ) = Fixǫσ0 (ρ∞ (P1 )) = v∞ (P1 , P2 ). Thus we have (P0 , P1 ) = v∞ ǫ0 shown that if F ∞ (ξ) = Fixσ (ρ∞ (P1 )) then the condition in Lemma 8.9.10(1) holds. 0 Case 2. Suppose that F ∞ (ξ) = Fix−ǫ σ (ρ∞ (P1 )). First, we prove that −ǫ0 (P0 , P2 ). Suppose this is not the case. Then we have F ∞ (ξ) = F ∞ (ξ) = v∞ ǫ0 −ǫ0 ǫ0 (P0 , P2 ) (see Fig. 8.9), because (P0 , P2 ) = v∞ (P0 , P2 ) and that v∞ v∞ F ∞ (ξ) ∈ Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P2 )) ∩ C

± = I(ρ∞ (P0 )) ∩ I(ρ∞ (P2 )) = {v∞ (P0 , P2 )}.

Let θ(i, j) be the dihedral angle of E(ρ∞ (Pi )) ∩ E(ρ∞ (Pj )). Then θ(0, 2) > 0, ǫ0 −ǫ0 (P0 , P2 ). Moreover we can see that θ(0, 1) + (P0 , P2 ) = v∞ because v∞ θ(1, 2) + θ(0, 2) = π. Hence we have θ(0, 1) + θ(1, 2) < π. This contraI(ρ∞ (P1 )) θ(1, 2) θ(0, 1) I(ρ∞ (P0 ))

I(ρ∞ (P2 ))

θ(0, 2) F ∞ (ξ) =

ǫ0 v∞ (P0 , P2 )

Fig. 8.9. θ(0, 1)+θ(1, 2)+θ(0, 2) = π and hence θ(0, 1)+θ(1, 2) < π. This contradicts Lemma 4.1.3(3) (Chain rule). −ǫ0 dicts Lemma 4.1.3(3) (Chain rule). Hence F ∞ (ξ) = v∞ (P0 , P2 ). This im−ǫ0 plies that the geodesic segment [Fρn (ξ), v (ρn ; P0 , P2 )] shrinks to the point F ∞ (ξ) ∈ C as n → ∞ (see Fig. 8.7). We can see by using Lemmas 8.5.3 and 8.9.9 that the interior of the geodesic segment [Fρn (ξ), v −ǫ0 (ρn ; P0 , P2 )]

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8 Closedness

does not contain a vertex of Eh(ρn ). So, v −ǫ0 (ρn ; P0 , P2 ) ∈ fr E −ǫ0 (ρn ), and hence s0 and s2 are vertices of the (−ǫ0 )-terminal triangle of Σ0 . On the other hand, since θ−ǫ0 (ρn , σ; s1 ) < 0, σ is not the (−ǫ0 )-terminal triangle of Σ0 . Hence the (−ǫ0 )-terminal triangle of Σ0 is equal to σ ′ and νn−ǫ0 is contained in σ ′ . Notice that ρn (P1′ ) = ρn (P2 ) acts on Ih(ρn (P1′ )) = Ih(ρn (P2 )) as a Euclidean isometry. Since ρn (P1′ )([Fρn (ξ), v −ǫ0 (ρn ; P0′ , P1′ )]) = ρn (P1′ )(F ρn (P0′ , P1′ )) = F ρn (P1′ , P2′ ) = [Fρn (P1′ , P2′ , P3′ ), v −ǫ0 (ρn ; P1′ , P2′ )] by Lemma 6.4.2(2), and since [Fρn (ξ), v −ǫ0 (ρn ; P0 , P2 )] shrinks to a point, we see that [Fρn (P1′ , P2′ , P3′ ), v −ǫ0 (ρn ; P1′ , P2′ )] also shrinks to a point. Similarly, we also see that [Fρn (P1′ , P2′ , P3′ ), v −ǫ0 (ρn ; P2′ , P3′ )] shrinks to a point. Thus −ǫ0 −ǫ0 we have F ∞ (P1′ , P2′ , P3′ ) = v∞ (P2′ , P3′ ). On the other hand, (P1′ , P2′ ) = v∞ ′ ′ ′ ′ since F ρn (P1 , P2 , P3 ) ∈ Axis(ρn (P2 )) for every n, the point is equal to + ′ ′ ′ Fix− σ ′ (ρ∞ (P2 )) or Fixσ ′ (ρ∞ (P2 )). Let ℓn be the perpendicular at c(ρn (P2 )) ′ −ǫ0 ′ ′ −ǫ0 ′ to proj(Axis(ρn (P2 ))). Since 0 ≤ θ (ρn , σ ; P2 ) ≤ π/2, e (ρn , σ ; P2′ ) is contained in the closure of the component of I(ρn (P2′ )) − ℓn contain′ ′ ′ ′ −ǫ0 ′ −ǫ0 0 ing Fix−ǫ σ ′ (ρn (P2 )). Thus v∞ (P1 , P2 ) = v∞ (P2 , P3 ) must be equal to −ǫ0 −ǫ0 ′ ′ −ǫ0 ′ ∈ Fixσ′ (ρ∞ (P2 )). Therefore lim θ (ρn , σ ; P2 ) = 0, and hence ν∞ ints0 , s2 . We also see that the condition in Lemma 8.9.4(3) holds. Hence 0 we have shown that if F ∞ (ξ) = Fix−ǫ σ (ρ∞ (P1 )) then the condition in Lemma 8.9.10(2) holds. We have shown that if F ∞ (ξ) belongs to C then precisely one of the conditions (1) and (2) holds and hence ξ ∈ L(ν ∞ )(2) . In what follows we prove that the condition ξ ∈ L(ν ∞ )(2) implies the condition F ∞ (ξ) ∈ C. So, suppose that ξ ∈ L(ν ∞ )(2) . Then precisely one of the following holds by Lemma 8.9.4. ǫ0 ∈ ints0 , s2 . 1. νnǫ0 ∈ σ ∩ H2 and ν∞ ′ 2 −ǫ0 −ǫ0 ∈ ints0 , s2 . 2. νn ∈ σ ∩ H and ν∞

In the ﬁrst case, we have lim θǫ0 (ρn , σ; s(P1 )) = 0. Thus eǫ0 (ρn , σ; P1 ) shrinks to the point Fixǫσ0 (ρ∞ (P1 )). This implies that Fixǫσ0 (ρ∞ (P1 )) ∈ I(ρ∞ (P0 )) ∩ I(ρ∞ (P1 )) ∩ I(ρ∞ (P2 )) ⊂ Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) ∩ Ih(ρ∞ (P2 )). Thus Fixǫσ0 (ρ∞ (P1 )) = F ∞ (ξ) by Lemma 8.9.8(3). Hence F ∞ (ξ) ∈ C. In the second case, we can see by a similar argument that F ∞ (P1′ , P2′ , P3′ ) is con(2 tained in C. Since i=0 Ih(ρ∞ (Pj )) is the singleton F ∞ (ξ) by Lemma 8.9.8(3), (3 ′ ′ i=1 Ih(ρ∞ (Pj )) is equal to the singleton ρ∞ (P1 )(F ∞ (ξ)) by Lemma 4.1.3(4) (3 (Chain rule). Therefore i=1 Ih(ρ∞ (Pj′ )) must be equal to F ∞ (P1′ , P2′ , P3′ ) in C, and hence F ∞ (ξ) = ρ∞ (P1′ )(F ∞ (P1′ , P2′ , P3′ )) ∈ C. This completes the proof Lemma 8.9.10. Proof (Proof of Proposition 8.9.5(1)). Suppose that ξ is contained in L(ν ∞ ). (1-i) By Lemma 8.9.8(3), Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) ∩ Ih(ρ∞ (P2 )) is equal to the singleton F ∞ (ξ). By Lemma 8.9.10, this lies in H3 . Hence we obtain the conclusion. (1-ii) By Lemma 8.9.8(1), Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) is a complete geodesic. Moreover, by the above (1-i), it intersects the hyperplane Ih(ρ∞ (P2 )) =

8.9 Proof of Proposition 8.4.5 - length ≥ 2 case - (Step 1)

201

∂Eh(ρ∞ (P2 )) at the single point F ∞ (ξ) in H3 . Thus the intersection L∞ (2) = Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) ∩ Eh(ρ∞ (P2 )) is a half geodesic in Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) which has F ∞ (ξ) as the endpoint in H3 . On the other hand, ǫ(2) L∞ (2) contains the half geodesic [F ∞ (ξ), v∞ (P0 , P1 )] by Lemma 8.9.7(2). ǫ(2) Hence we have L∞ (2) = [F ∞ (ξ), v∞ (P0 , P1 )]. By a parallel argument, we can prove the assertion for L∞ (0). We show the assertion for L∞ (1). Since F ∞ (ξ) is a subset of Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P2 )) and since F ∞ (ξ) is contained in H3 , we see that Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P2 )) contains a point in H3 . Since these two isometric hemispheres are not identical by Lemma 2.5.4(2), this implies that Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P2 )) is a complete geodesic. So, we can obtain the assertion for L∞ (1) by the above argument for L∞ (2). ǫ0 Proof (Proof of Proposition 8.9.5(2)). Suppose ν∞ ∈ ints0 , s2 . Then ξ ∈ / (2) L(ν ∞ ) and hence we see that the conclusion of Lemma 8.9.10(1) holds. Hence we obtain the assertion (2-i). ǫ0 ∈ ints0 , s2 , we have We show the assertions (2-ii) and (2-iii). Since ν∞ ǫ0 lim θ (ρn , σ; s1 ) = 0. On the other hand, we see lim θ−ǫ0 (ρn , σ; s1 ) ≤ 0 by Lemma 4.3.6 (cf. the ﬁrst paragraph in the proof of Lemma 8.9.10). Hence, by Lemma 4.3.5, we have Ih(ρ∞ (P1 )) ⊂ Dh(ρ∞ (P0 )) ∪ Dh(ρ∞ (P2 )). Thus Ih(ρ∞ (P1 ))∩Eh(ρ∞ (P0 ))∩Eh(ρ∞ (P2 )) is equal to Ih(ρ∞ (P0 ))∩Ih(ρ∞ (P1 ))∩ Ih(ρ∞ (P2 )), which in turn is equal to F ∞ (ξ) by Lemma 8.9.8(3). Hence we see that Dh(ρ∞ (P1 )) ∩ Eh(ρ∞ (P0 )) ∩ Eh(ρ∞ (P2 )) is equal to the singleton F ∞ (ξ) in C. This implies that Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P2 )) lies in Eh(ρ∞ (P1 )), and hence Eh(ρ∞ (P1 )) ⊃ Eh(ρ∞ (P0 )) ∩ Eh(ρ∞ (P2 )).

Thus we have proved the assertion (2-ii). Moreover, the above argument shows L∞ (1) = Ih(ρ∞ (P0 ))∩Ih(ρ∞ (P2 ))∩Eh(ρ∞ (P1 )) = Ih(ρ∞ (P0 ))∩Ih(ρ∞ (P2 )). −ǫ(1)

ǫ0 Since F ∞ (ξ) = v∞ (P0 , P2 ) = v∞

(P0 , P2 ) by (2-i), this implies

ǫ(1) L∞ (1) = Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P2 )) = [F ∞ (ξ), v∞ (P0 , P2 )].

Thus we have obtained the assertion (2-iii) for L∞ (1). We note that the above argument also implies L∞ (2) = Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) ∩ Eh(ρ∞ (P2 ))

⊂ Eh(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) ∩ Eh(ρ∞ (P2 )) = F ∞ (ξ). ǫ(2)

ǫ0 (P0 , P1 ) = v∞ (P0 , P1 ) by (2-i), Hence L∞ (2) = F ∞ (ξ). Since F ∞ (ξ) = v∞ ǫ(2) we also have L∞ (2) = [F ∞ (ξ), v∞ (P0 , P1 )]. This proves the assertion (2-iii) for L∞ (2). By a parallel argument, we can also prove the assertion (2-iii) for L∞ (0).

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8 Closedness

−ǫ0 ∈ ints0 , s2 . Then ξ ∈ / Proof (Proof of Proposition 8.9.5(3)). Suppose ν∞ L(ν ∞ )(2) and hence we see that the conclusion of Lemma 8.9.10(2) holds. Hence we obtain the assertion (3-i). We show the assertion (3-ii) for L∞ (1). If Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P2 )) is a singleton, then the assertion obviously holds. So, we assume that Ih(ρ∞ (P0 ))∩ Ih(ρ∞ (P2 )) is a complete geodesic. Since Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) ∩ 0 Ih(ρ∞ (P2 )) is equal to the singleton F ∞ (ξ) = Fix−ǫ σ (ρ∞ (P1 )) in C, we see that L∞ (1) = Ih(ρ∞ (P0 ))∩Ih(ρ∞ (P2 ))∩Eh(ρ∞ (P1 )) is either the entire geodesic or the singleton F ∞ (ξ). Since v ǫ0 (ρn ; P0 , P2 ) is contained in D(ρn (P1 )) ǫ0 (P0 , P2 ) of Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P2 )) is for every n ∈ N, the endpoint v∞ −ǫ0 contained in D(ρ∞ (P1 )). Since v∞ (P0 , P2 ) = F ∞ (ξ) is the unique point in ǫ0 (P0 , P2 ) Ih(ρ∞ (P0 ))∩Ih(ρ∞ (P1 ))∩Ih(ρ∞ (P2 )) and since it is distinct from v∞ ǫ0 by the assumption, v∞ (P0 , P2 ) is not contained in Eh(ρ∞ (P1 )). Therefore −ǫ0 (P0 , P2 ) = L∞ (1) is equal to the singleton F ∞ (ξ). Since F ∞ (ξ) = v∞ ǫ(1) ǫ(1) v∞ (P0 , P2 ) by (3-i), we also have L∞ (1) = [F ∞ (ξ), v∞ (P0 , P2 )]. Thus we have obtained the assertion (3-ii) for L∞ (1). We show the assertion (3-ii) for L∞ (2). By Lemma 8.9.8(1), Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) is a complete geodesic. This complete geodesic is equal to [F ∞ (ξ), −ǫ(2) ǫ(2) −ǫ0 v∞ (P0 , P1 )], because F ∞ (ξ) = v∞ (P0 , P1 ) = v∞ (P0 , P1 ) by (3-i). Since ǫ(2) L∞ (2) contains [F ∞ (ξ), v∞ (P0 , P1 )] by Lemma 8.9.7(2), we see ǫ(2) L∞ (2) = [F ∞ (ξ), v∞ (P0 , P1 )] = Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )).

This proves the assertion (3-ii) for L∞ (2). We obtain the assertion for L∞ (0) by a parallel argument. This completes the proof of Proposition 8.9.5(3). Proposition 8.9.5 is used in the following forms in Sects. 8.10 and 8.11. Corollary 8.9.12. Under Assumptions 8.9.1 and 8.9.2, the following hold. (1) Ih(ρ∞ (P0 ))∩Ih(ρ∞ (P1 ))∩Ih(ρ∞ (P2 )) is equal to the singleton F ∞ (ξ) 3 in H . ǫ (2) Suppose ν∞ ∈ ints0 , s2 for some ǫ. Then ξ ∈ / L(ν ∞ )(2) and F ∞ (ξ) = ǫ (0) Fixσ (ρ∞ (P1 )). Moreover, if s1 ∈ / Σ(ν ∞ ) , or equivalently (P1 ) ∈ / L(ν ∞ )(0) , then Ih(ρ∞ (P1 )) is covered by the remaining two isometric hemispheres. To be precise, the following hold (see Fig. 8.5). (i) Dh(ρ∞ (P1 ))∩Eh(ρ∞ (P0 ))∩Eh(ρ∞ (P2 )) is equal to the singleton F ∞ (ξ) in C. (ii) Eh(ρ∞ (P1 )) ⊃ Eh(ρ∞ (P0 )) ∩ Eh(ρ∞ (P2 )).

ǫ ∈ ints0 , s2 for some ǫ. Then Proof. We have only to prove (2). Suppose ν∞ (2) (0) / Σ(ν ∞ ) by Lemma 8.9.4(2), and we obtain ξ∈ / L(ν ∞ ) . If ǫ = ǫ0 , then s1 ∈ the desired result by Proposition 8.9.5(2). If ǫ = −ǫ0 , then s1 ∈ Σ(ν ∞ )(0) by Lemma 8.9.4(3), and we see by Proposition 8.9.5(3-i) that ǫ 0 F ∞ (ξ) = Fix−ǫ σ (ρ∞ (P1 )) = Fixσ (ρ∞ (P1 )).

This completes the proof.

8.10 Proof of Proposition 8.4.5 - length ≥ 2 case - (Step 2)

203

Corollary 8.9.13. Under Assumption 8.9.1, let ξ be a triangle of L0 , and let {P, Q, R} be the vertex set of ξ, i.e., ξ = ((P, Q, R)) (cf. Deﬁnition 3.2.6). Then we have the following. (1) Ih(ρ∞ (P )) ∩ Ih(ρ∞ (Q)) ∩ Ih(ρ∞ (R)) is equal to the singleton F ∞ (ξ). (2) Set L∞ (R) := Ih(ρ∞ (P )) ∩ Ih(ρ∞ (Q)) ∩ Eh(ρ∞ (R)). Then L∞ (R) is ǫ(R) equal to the (possibly degenerate) geodesic segment [F ∞ (ξ), v∞ (P, Q)], where ǫ(R) is the sign such that ((P, Q)) lies on the ǫ(R)-side of ξ (cf. Deﬁnition 3.2.5). Moreover, we have the following: (i) Suppose ξ ∈ L(ν ∞ )(2) . Then F ∞ (ξ) ∈ H3 and L∞ (R) is a half geodesic. (ii) Suppose ξ ∈ / L(ν ∞ )(2) and ((P, Q)) ∈ / L(ν ∞ )(1) . Then F ∞ (ξ) = ǫ(R) ǫ(R) v∞ (P, Q) and L∞ (R) is the singleton F ∞ (ξ) = v∞ (P, Q) in C (see Figs. 8.5 and 8.6). (iii) Suppose ξ ∈ / L(ν ∞ )(2) and ((P, Q)) ∈ L(ν ∞ )(1) . Then F ∞ (ξ) = −ǫ(R) v∞ (P, Q) and L∞ (R) is the complete geodesic Ih(ρ∞ (P ))∩Ih(ρ∞ (Q)) (see Figs. 8.5 and 8.6). (3) If ((P, Q)) ∈ L(ν ∞ )(1) , then Ih(ρ∞ (P )) ∩ Ih(ρ∞ (Q)) is a complete geodesic. Proof. We may assume ξ is equal to the 2-simplex ξ in Assumption 8.9.2, namely {P, Q, R} = {P0 , P1 , P2 }. Then (P0 , P1 ) and (P1 , P2 ) lie on the ǫ0 -side of ξ, whereas (P0 , P2 ) lies on the (−ǫ0 )-side of (P0 , P2 ). This means that the sign ǫ(Pk ) introduced in the corollary is equal to the sign ǫ(k) in Proposition ǫ(k) 8.9.5. Since L∞ (k) = [F ∞ (ξ), v∞ (Pi , Pj )] for every k ∈ {0, 1, 2} by Corolǫ(R) lary 8.9.6, we have L∞ (R) = [F ∞ (ξ), v∞ (P, Q)]. The remaining assertions follows from Proposition 8.9.5 and the fact that ǫ(Pk ) = ǫ(k).

8.10 Proof of Proposition 8.4.5 - length ≥ 2 case (Step 2) In this section, we prove the following Propositions 8.10.1 and 8.10.2, which show that Proposition 8.4.5(1), in the case where the length of Σ0 is ≥ 2, is valid for the edges of L0 . Proposition 8.10.1. Under Assumption 8.9.1, let ξ be a generic edge of L0 . Then ξ is an edge of L(ν ∞ ), F ρ∞ (ξ) is equal to F ∞ (ξ), and it is a 3

1-dimensional convex polyhedron in H .

Proposition 8.10.2. Under Assumption 8.9.1, let ξ be an ǫ-extreme edge of L0 for some sign ǫ ∈ {−, +}. Then the following hold. (1) If ξ is an edge of L(ν ∞ ), then F ρ∞ (ξ) is equal to F ∞ (ξ), and it is a 3

1-dimensional convex polyhedron in H . (2) If ξ is not contained in L(ν ∞ ), then F ∞ (ξ) is a singleton in C.

204

8 Closedness

Proof (Proof of Proposition 8.10.1). Under Assumption 8.9.1, let ξ be a generic edge of L0 . Then ξ = (Pj , Pj+1 ) where {Pj } is the sequence of elliptic generators associated with a triangle σ = s0 , s1 , s2 in Σ0 − {σ − , σ + }, ′ ′ , resand σ+ where σ ǫ (ǫ ∈ {−, +}) is the ǫ-terminal triangle of Σ0 . Let σ− pectively, be the predecessor and the successor of σ in Σ0 . We assume that ′ ′ ∩σ = s0 , s2 and σ+ ∩σ = s0 , s1 . (To be precise, we also need to consider σ− ′ ′ ∩ σ = s0 , s1 , because of Convention ∩ σ = s0 , s2 and σ+ the case where σ− 2.1.12. But this case is equivalent to the above case after changing the roles of K and K −1 .) Then we have the following by Lemma 8.5.2. 1. (P1 , P2 , P3 ) and (P0 , P2P1 , P1 ) are 2-simplices of L0 which lie on the (+)P1 + ′ side of L(σ), and we have σ+ = s0 , s+ 2 , s1 where s2 = s(P2 ). 2. (P0 , P1 , P2 ) and (P2 , P1P2 , P3 ) are 2-simplices of L0 which lie on the (−)P2 − ′ side of L(σ), and we have σ− = s0 , s2 , s− 1 where s1 = s(P1 ). By Lemma 8.9.9, ξ is an edge of L(ν ∞ ), and F ∞ (ξ) is a 1-dimensional 3 convex polyhedron in H . In what follows we prove that F ρ∞ (ξ) is equal to F ∞ (ξ). To this end we show that F ρ∞ (ξ) = F (ρ∞ ,L0 ) (ξ) and F (ρ∞ ,L0 ) (ξ) = F ∞ (ξ). We ﬁrst show F ρ∞ (ξ) = F (ρ∞ ,L0 ) (ξ). In order to use Corollary 8.9.13, we set P = Pj and Q = Pj+1 . Let ξˆǫ (ǫ ∈ {−, +}) be the 2-simplex of L0 containing ξ and lying on the ǫ-side of ξ. Since (P, Q) ∈ L(ν n )(1) and ρn is good, we see that Ln := Ih(ρn (P ))∩Ih(ρn (Q)) is a complete geodesic and that the four points v − (ρn ; P, Q), F ρn (ξˆ− ), F ρn (ξˆ+ ) and v + (ρn ; P, Q) lie in the geodesic Ln in this order. Since ξ = (P, Q) ∈ L(ν ∞ )(1) , we see by Corollary 8.9.13(3) that L∞ := Ih(ρ∞ (P )) ∩ Ih(ρ∞ (Q)) is a complete geodesic. By − continuity we can also see that the four points v∞ (P, Q), F ∞ (ξˆ− ), F ∞ (ξˆ+ ) + and v∞ (P, Q) sit in the geodesic L∞ in this order, though some of them may coincide. Note that F ∞ (ξˆ− ) and F ∞ (ξˆ+ ) are diﬀerent because they are the endpoints of a 1-dimensional polyhedron F ∞ (ξ) (cf. Lemma 8.9.9). Let Rǫ be the elliptic generator such that (ξˆǫ )(0) = {P, Q, Rǫ }. By deﬁnition F (ρ∞ ,L0 ) (ξ) is equal to L∞ ∩ Eh(ρ∞ (R− )) ∩ Eh(ρ∞ (R+ )), and hence it is equal to the intersection of the two subsets L∞ ∩ Eh(ρ∞ (R− )) and L∞ ∩ Eh(ρ∞ (R+ )) of L0 . Since (P, Q) lies on the (−ǫ)-side of ξˆǫ , we see by Corollary 8.9.13(2) −ǫ (P, Q)] ⊂ L∞ L∞ ∩ Eh(ρ∞ (Rǫ )) = [F ∞ (ξˆǫ ), v∞ − for each ǫ ∈ {−, +}. By considering the order of the four points v∞ (P, Q), − + + ˆ ˆ F ∞ (ξ ), F ∞ (ξ ) and v∞ (P, Q) in L∞ , we can conclude that F (ρ∞ ,L0 ) (ξ) is equal to F ∞ (ξ) = [F ∞ (ξˆ− ), F ∞ (ξˆ+ )]. Next we show F ρ∞ (ξ) = F (ρ∞ ,L0 ) (ξ). We divide the proof into four cases. ′ ′ , σ− ∈ Σ(ν ∞ ). Then lk(ξ, L(ν ∞ )) = lk(ξ, L0 ) = {R− , R+ }. Case 1. σ+ Thus F (ρ∞ ,L0 ) (ξ) is by deﬁnition equal to F ρ∞ (ξ). ′ ′ Case 2. σ+ ∈ Σ(ν ∞ ) and σ− ∈ / Σ(ν ∞ ). Then ξˆ− ∈ L(ν ∞ )(2) and lk(ξ, L(ν ∞ )) = {R+ }. Since (P, Q) ∈ L(ν ∞ )(1) , we see by Corollary 8.9.13

8.10 Proof of Proposition 8.4.5 - length ≥ 2 case - (Step 2)

205

(2-iii) that Ih(ρ∞ (P )) ∩ Ih(ρ∞ (Q)) ∩ Eh(ρ∞ (R− )) = Ih(ρ∞ (P )) ∩ Ih(ρ∞ (Q)). Hence F ρ∞ (ξ) = Ih(ρ∞ (P )) ∩ Ih(ρ∞ (Q)) ∩ Eh(ρ∞ (R+ ))

= Ih(ρ∞ (P )) ∩ Ih(ρ∞ (Q)) ∩ Eh(ρ∞ (R− )) ∩ Eh(ρ∞ (R+ )) = F (ρ∞ ,L0 ) (ξ).

′ ′ Case 3. σ− ∈ Σ(ν ∞ ) and σ+ ∈ / Σ(ν ∞ ). The proof for this case is parallel to that for Case 2. ′ ′ ∈ / Σ(ν ∞ ) and σ− ∈ / Σ(ν ∞ ). Then ξˆ± ∈ L(ν ∞ )(2) and Case 4. σ+ lk(ξ, L(ν ∞ )) = ∅. Since (P, Q) ∈ L(ν ∞ )(1) , we see by Corollary 8.9.13(2-iii) that

Ih(ρ∞ (P )) ∩ Ih(ρ∞ (Q)) ∩ Eh(ρ∞ (R± )) = Ih(ρ∞ (P )) ∩ Ih(ρ∞ (Q)). Hence F ρ∞ (ξ) = Ih(ρ∞ (P )) ∩ Ih(ρ∞ (Q))

= Ih(ρ∞ (P )) ∩ Ih(ρ∞ (Q)) ∩ Eh(ρ∞ (R− )) ∩ Eh(ρ∞ (R+ ))

= F (ρ∞ ,L0 ) (ξ).

Proof (Proof of Proposition 8.10.2). Let ξ = ((P, Q)) be an ǫ-extreme edge of L0 . Then there is a unique 2-simplex ξˆ = ((P, Q, R)) containing ξ. Moreover ˆ ξ lies on the ǫ-side of ξ. Case 1. ξ ∈ L(ν ∞ )(1) and ξˆ ∈ L(ν ∞ )(2) . Then lk(ξ, L(ν ∞ )) = {R}. Thus by using Corollary 8.9.13(2-i) and Lemma 8.6.1(2), we see F ρ∞ (ξ) = Ih(ρ∞ (P )) ∩ Ih(ρ∞ (Q)) ∩ Eh(ρ∞ (R)) ˆ v ǫ (P, Q)] = [F ∞ (ξ), ∞

= F ∞ (ξ). Case 2. ξ ∈ L(ν ∞ )(1) and ξˆ ∈ / L(ν ∞ )(2) . Then lk(ξ, L(ν ∞ )) = ∅. Thus by using Corollary 8.9.13(2-iii) and Lemma 8.6.1(2), we see F ρ∞ (ξ) = Ih(ρ∞ (P )) ∩ Ih(ρ∞ (Q)) ˆ v ǫ (P, Q)] = [F ∞ (ξ), ∞

= F ∞ (ξ). Case 3. ξ ∈ / L(ν ∞ )(1) . Then we have F ρ∞ (ξ) is a singleton in C by Corollary 8.9.13(2-ii).

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8 Closedness

8.11 Proof of Proposition 8.4.5 - length ≥ 2 case (Step 3) In this section, we prove the following Proposition 8.11.1, which shows that Proposition 8.4.5, in the case where the length of Σ0 is ≥ 2, is valid for the vertices of L0 . Proposition 8.11.1. Under Assumption 8.9.1, the following hold for each (0) P ∈ L0 .

(1) If P ∈ L(ν ∞ )(0) , then F ρ∞ (P ) is equal to F ∞ (P ) and it is a convex 3

polyhedron in H of dimension 2. (2) If P ∈ / L(ν ∞ )(0) , then Ih(ρ∞ (P )) is invisible in Eh(ρ∞ ). Namely, Eh(ρ∞ (P )) ⊃ Eh(ρ∞ ).

Proof. (1) Let P be an element of L(ν ∞ )(0) . First, we prove that F ∞ (P ) is 3 a 2-dimensional convex polyhedron in H . Since F ∞ (P ) is a convex polyhe3 dron in H of dimension at most 2 by Lemma 8.6.1, it suﬃces to prove that dim F ∞ (P ) ≥ 2. Case 1. Suppose that there is a 2-simplex ξ = (P0 , P1 , P2 ) of L(ν ∞ ) such that P = P1 . Then (P0 , P1 ) and (P1 , P2 ) are edges of L(ν ∞ ), and therefore F ∞ (P0 , P1 ) and F ∞ (P1 , P2 ) are 1-dimensional convex polyhedra by Propositions 8.10.1 and 8.10.2. Since F ∞ (P ) = F ∞ (P1 ) contains F ∞ (P0 , P1 ) and F ∞ (P1 , P2 ), we see that dim F ∞ (P ) < 2 only when F ∞ (P0 , P1 ) and 3 F ∞ (P1 , P2 ) are contained in a 1-dimensional subspace in H . If this happens, then Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 )) and Ih(ρ∞ (P1 )) ∩ Ih(ρ∞ (P2 )) are identical geodesics. This contradicts Corollary 8.9.12(1) (cf. Lemma 8.9.8(2)). Hence we have dim F ∞ (P ) = 2. Case 2. Suppose that no 2-simplex (P0 , P1 , P2 ) of L(ν ∞ ) satisﬁes P = P1 . Claim 8.11.2. Every triangle of Σ(ν ∞ ) contains s(P ) as a vertex. Proof. Suppose that Σ(ν ∞ ) contains a triangle, say σ ′′ , which does not contain s(P ) as a vertex. Then, by considering a path in Σ(ν ∞ ) joining s(P ) to σ ′′ , we can ﬁnd a pair of triangles σ = s0 , s1 , s2 and σ ′ = s′0 , s′1 , s′2 sharing the edge s0 , s2 = s′0 , s′1 such that s1 = s(P ) = s′2 (see Fig. 8.10). Then the elliptic generator triple (P0 , P1 , P2 ), with P = P1 , associated with σ determines a 2-simplex of L(ν ∞ ) by Lemma 3.2.4(2), a contradiction. It should also be noted that some triangle of Σ0 may not contain s(P ) as a vertex. Since ν ∞ is thick, Σ(ν ∞ ) contains at least one triangle, σ = s0 , s1 , s2 with s1 = s(P ). Claim 8.11.3. F ∞ (P ) contains Fixǫσ (ρ∞ (P )) for each ǫ ∈ {−, +}, and hence F ∞ (P ) ⊃ Axis(ρ∞ (P )).

8.11 Proof of Proposition 8.4.5 - length ≥ 2 case - (Step 3) s0 =

207

s′0

σ′ σ

s2 = s′1

s1 = s(P )

Fig. 8.10. Figure of Σ(ν ∞ ) for Claim 8.11.2

Proof. To show the claim, we have only to show that F ∞ (P ) contains both Fix± σ (ρ∞ (P )), because F ∞ (P ) is convex. Let ǫ be an arbitrary sign. First, suppose s(P ) is a vertex of the ǫ-terminal triangle σ ǫ of Σ0 . Then Fixǫσ (ρn (P )) = Fixǫσǫ (ρn (P )) ∈ F ρn (P ) for every n ∈ N, and therefore Fixǫσ (ρ∞ (P )) = lim Fixǫσ (ρn (P )) ∈ lim F ρn (P ) = F ∞ (P ). Next, suppose s(P ) is not a vertex of σ ǫ . Let σ ′ = s′0 , s′1 , s′2 be the ǫ-terminal triangle of Σ(ν ∞ ), where s′1 = s(P ) (see Claim 8.11.2). Then σ ǫ ∩σ ′ is equal to the edge s′0 , s′2 (see Fig. 8.11). Let {Pj′ } be the sequence of elliptic generators associated with the triangle σ ′ such that P = P1′ . Then ξ := (P0′ , P1′ , P2′ ) is a 2-simplex of L0 . By the assumption, ξ is not a 2-simplex of L(ν ∞ ), and ǫ therefore ν∞ ∈ ints′0 , s′2 . Thus we see by Corollary 8.9.12(2) that F ∞ (ξ) = Fixǫσ′ (ρ∞ (P1′ )) = Fixǫσ (ρ∞ (P )). Hence, Fixǫσ (ρ∞ (P )) = F ∞ (ξ) ⊂ F ∞ (P ). Thus we have shown that F ∞ (P ) contains both Fix± σ (ρ∞ (P )). This completes the proof of the claim. Suppose to the contrary that dim F ∞ (P ) < 2. Then the above claim implies F ∞ (P ) = Axis(ρ∞ (P )) and hence Ih(ρ∞ (P0 )) ∩ Ih(ρ∞ (P1 ))) ∩ Ih(ρ∞ (P2 ))) = Axis(ρ∞ (P1 )). This contradicts Corollary 8.9.12(1). Hence we have dim F ∞ (P ) = 2. In what follows we prove the assertion that F ∞ (P ) is equal to F ρ∞ (P ). By Lemma 8.6.2, F ∞ (P ) is contained in F ρ∞ (P ). Hence it suﬃces to prove that F ρ∞ (P ) is contained in F ∞ (P ). Suppose this does not hold. Then there is a point, x∞ , contained in the interior of the 2-dimensional convex polyhedron F ρ∞ (P ) which is not contained in F ∞ (P ) = lim F ρn (P ). Take a sequence {xn } consisting of points xn ∈ Ih(ρn (P )) and converging to x∞ .

208

8 Closedness s′0 σǫ σ′

s′2

s′1 = s(P )

Fig. 8.11. Figure of Σ0 for Claim 8.11.3

/ F ρn (P ) for every Since x∞ ∈ / F ∞ (P ) = lim F ρn (P ), we may assume xn ∈ n ∈ N, after taking a subsequence. Thus for each n ∈ N, there is some Xn ∈ lk(P, L0 )(0) such that int(Dh(ρn (Xn ))) contains xn . Since lk(P, L0 )(0) is ﬁnite, we may suppose, by further taking a subsequence, that every xn is contained in int(Dh(ρn (X))) for some X ∈ lk(P, L0 )(0) . Note that x∞ = lim xn is contained in Dh(ρ∞ (X)) = lim Dh(ρn (X)). First, suppose that X ∈ L(ν ∞ )(0) . Then X ∈ lk(P, L(ν ∞ ))(0) . Since x∞ is contained in F ρ∞ (P ), it is contained in Eh(ρ∞ (X)). Hence x∞ is contained in Ih(ρ∞ (X)) = Dh(ρ∞ (X)) ∩ Eh(ρ∞ (X)). Thus x∞ is contained in Ih(ρ∞ (P )) ∩ Ih(ρ∞ (X)), which is a geodesic by Corollary 8.9.13(2-iii), because (P, X) is an edge of L(ν ∞ ). Thus x∞ cannot be contained in the interior of the 2-dimensional convex polyhedron F ρ∞ (P ), a contradiction. Next, suppose that X ∈ / L(ν ∞ )(0) . Then s(X) is a vertex of the ǫ-terminal ǫ triangle σ of Σ0 for some ǫ ∈ {−, +}, and any other triangle of Σ0 does not contain s(X). Set σ ǫ = s0 , s1 , s2 where s1 = s(X), and let {Pj } be the sequence of elliptic generators associated with σ ǫ such that P1 = X. Then, since the length of Σ0 is ≥ 2 by the assumption, (P0 , P1 , P2 ) is a 2-simplex of L0 . Moreover, (P0 , P2 ) is an edge of L(ν ∞ ). Since X is contained in lk(P, L0 ), P is equal to either P0 or P2 . We may assume without loss of generality that P = P0 . Then, since P2 ∈ lk(P0 , L(ν ∞ ))(0) , we have x∞ ∈ F ρ∞ (P ) ∩ Dh(ρ∞ (X)) ⊂ Ih(ρ∞ (P0 )) ∩ Eh(ρ∞ (P2 )) ∩ Dh(ρ∞ (P1 )) ⊂ Dh(ρ∞ (P1 )) ∩ Eh(ρ∞ (P0 )) ∩ Eh(ρ∞ (P2 )).

/ Σ(ν ∞ )(0) , Since ξ := (P0 , P1 , P2 ) is a 2-simplex of L0 , and since s1 = s(X) ∈ Corollary 8.9.12(2-i) implies that the last term in the above formula is equal

8.12 Proof of Proposition 8.3.6

209

to the singleton F ∞ (ξ) in C. This contradicts the assumption that x∞ is contained in the interior of F ρ∞ (P ). This completes the proof Proposition 8.11.1(1). (0)

(2) Let P be an element of L0 − L(ν ∞ )(0) . Then s(P ) is a vertex of the ǫ-terminal triangle σ ǫ of Σ0 for some ǫ ∈ {−, +}, and s(P ) does not belong to Σ(ν ∞ )(0) . Set σ ǫ = s0 , s1 , s2 where s1 = s(P ), and let {Pj } be the sequence of elliptic generators associated with σ ǫ such that P1 = P . Since the length of Σ0 is ≥ 2 by the assumption, we see that ξ := (P0 , P1 , P2 ) is a 2-simplex of ǫ / Σ(ν ∞ )(0) , we have ν∞ ∈ ints0 , s2 . Hence we see by L0 . Since s1 = s(P ) ∈ Corollary 8.9.12(2-iii) that Eh(ρ∞ (P )) = Eh(ρ∞ (P1 )) ⊃ Eh(ρ∞ (P0 )) ∩ Eh(ρ∞ (P2 ))

⊃ ∩{Eh(ρ∞ (X)) | X ∈ L(ν ∞ )(0) } = Eh(ρ∞ ).

This completes the proof of Proposition 8.11.1(2).

8.12 Proof of Proposition 8.3.6 In this section, we give a proof of Proposition 8.3.6. Throughout this section, we presume the following assumption. Assumption 8.12.1. Under Assumption 8.3.1, assume that ν ∞ is thick and ǫ that {ρn } satisﬁes the condition HausdorﬀConvergence. Moreover, if ν∞ ∈ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ∂σ (ν ∞ ), then ν∞ ∈ s0 , s1 . In this case σ ∩ σ (ν ∞ ) = s0 , s1 . Lemma 8.12.2. Under Assumption 8.12.1, L(ρ∞ , σ ǫ (ν ∞ )) is simple.

Proof. Let {Pj } be the sequence of elliptic generators associated with σ ǫ (ν ∞ ). Then (Pj , Pj+1 ) is an edge of L(ν ∞ ) for every j ∈ Z. By Proposition 8.3.5, ρ∞ satisﬁes the condition Duality. Hence Fρ∞ (Pj , Pj+1 ) is a well-deﬁned 1dimensional convex polyhedron in ∂Eh(ρ∞ ). Now suppose to the contrary that L(ρ∞ , σ ǫ (ν ∞ )) is not simple. Then, by Lemma 4.8.2, it is folded at some vertex c(ρ∞ (Pj )). Hence Ih(ρ∞ (Pj−1 )) ∩ Ih(ρ∞ (Pj )) = Ih(ρ∞ (Pj )) ∩ Ih(ρ∞ (Pj+1 )) by Lemma 4.8.5, and therefore we can see that Fρ∞ (Pj−1 , Pj ) = Fρ∞ (Pj , Pj+1 ). This contradicts Proposition 6.3.1. ǫ Lemma 8.12.3. Under Assumption 8.12.1, f∞ (Pjǫ ) ⊂ fr E(ρ∞ ) for every ǫ j ∈ Z. Here {Pj } is the sequence of elliptic generators associated with the ǫ-terminal triangle, σ ǫ , of the common chain Σ0 . ǫ Proof. By Proposition 8.4.4, we have f∞ (Pjǫ ) ⊂ fr E(ρ∞ , L0 ). On the other hand, we see E(ρ∞ , L0 ) = E(ρ∞ ) by the proof of of Proposition 8.3.5 in Sect. 8.4. Hence we obtain the desired result.

210

8 Closedness

Lemma 8.12.4. Under Assumption 8.12.1, there exists a local homeomorphism h : R → C, with the symmetry h(t + 1) = h(t) + 1 (t ∈ R), which satisﬁes the following condition. ǫ ǫ 1. If ν∞ (Pjǫ ) for every j ∈ Z. ∈ int σ ǫ (ν ∞ ), then h([j/3, (j + 1)/3]) = f∞ ǫ ǫ 2. If ν∞ ∈ ∂σ (ν ∞ ), then for any k ∈ Z, ǫ ǫ h([k, k + 1/2]) = f∞ (P3k ), ǫ ǫ h([k + 1/2, k + 1]) = f∞ (P3k+1 ). ǫ ǫ Proof. Suppose ﬁrst that ν∞ ∈ int σ ǫ (ν ∞ ). Then, for every j ∈ Z, f∞ (Pjǫ ) is a non-degenerate circular arc by Lemma 8.6.1(1). We deﬁne the restriction of the map h to the interval [j/3, (j + 1)/3] to be the unit speed path, with ǫ (Pjǫ ) connecting h(j/3) := respect to the Euclidean metric on C, onto f∞ ǫ ǫ ǫ ǫ ǫ ǫ v (ρ∞ ; Pj−1 , Pj ) and h((j+1)/3) := v (ρ∞ ; Pj , Pj+1 ). Then h is a well-deﬁned continuous map from R to C, whose restriction to each interval [j/3, (j + 1)/3] ǫ ǫ is a homeomorphism onto f∞ (Pjǫ ). Since I(ρ∞ (Pj−1 )) and I(ρ∞ (Pjǫ )) are diﬀerent circles (cf. Lemma 2.5.4(2)) and h(j/3) is an isolated intersection point of these two circles, we see that h is a local homeomorphism near the point j/3. Hence h is a local homeomorphism. It is clear from the construction that h has the symmetry h(t + 1) = h(t) + 1 (t ∈ R). ǫ ∈ ∂σ ǫ (ν ∞ ), by noticing A similar argument works for the case where ν∞ ǫ the fact that f∞ (Pjǫ ) is degenerated if and only if j ≡ 2 (mod 3).

Lemma 8.12.5. Under Assumption 8.12.1, h is a homeomorphism onto its image. Proof. We have only to show that h is injective. In fact, if h is injective then h ¯ from R/Z to C/Z. Since R/Z is compact, induces a continuous injective map h ¯ h is a homeomorphism onto its image. This implies that h is a homeomorphism onto its image. ǫ We give a proof of the injectivity of h only for the case when ν∞ ∈ ǫ ǫ ǫ int σ (ν ∞ ). A similar argument works for the case when ν∞ ∈ ∂σ (ν ∞ ). Suppose to the contrary that h is not injective. Then δ := min{|s − t| | s = t, h(s) = h(t)} is a positive real number, because h is a local homeomorphism. Take t0 ∈ R such that h(t0 ) = h(t0 + δ). Then h([t0 , t0 + δ]) is a circle in C and hence it bounds a disk, D0 , in C. Claim 8.12.6. The disk D0 is contained in E(ρ∞ ). Proof. Note that C − E(ρ∞ ) = ∪{int D(ρ∞ (P )) | P ∈ L(ν ∞ )(0) } = proj(∂Eh(ρ∞ )). Since ρ∞ satisﬁes the condition Duality by Proposition 8.3.5, we see that ∂Eh(ρ∞ ) is arcwise connected by Proposition 6.3.1. Hence proj(∂Eh(ρ∞ )) is arcwise connected. Now suppose that the claim does not hold. Then there is a

8.12 Proof of Proposition 8.3.6

211

point, x0 , in D0 such that x0 ∈ proj(∂Eh(ρ∞ )). Since D0 is compact, there is an integer k such that x0 + k ∈ D0 . Since proj(∂Eh(ρ∞ )) is invariant by the translation z → z + 1, the point x0 + k is also contained in proj(∂Eh(ρ∞ )). Since this set is arcwise connected, there is a path, γ, in proj(∂Eh(ρ∞ )) connecting x0 and x0 + k. Since x0 ∈ D0 and x0 + k ∈ D0 , γ intersects ∂D0 ⊂ ǫ ǫ (Pjǫ ) | j ∈ Z}. Thus some f∞ (Pjǫ ) contains a point in proj(∂Eh(ρ∞ )) = ∪{f∞ ǫ (Pjǫ ) ⊂ E(ρ∞ ), a C − E(ρ∞ ). On the other hand, Lemma 8.12.3 implies f∞ contradiction. We may assume without loss of generality that t0 ∈ [0, 1/3). Let k be the integer such that t0 + δ ∈ (k/3, (k + 1)/3]. Then k = 0, because the restriction of h to the subinterval [0, 1/3] is injective by the construction of h. ǫ ∈ int σ ǫ (ν ∞ ), We show that k = 1. By the temporary assumption that ν∞ ǫ ǫ ǫ we see that (Pj , Pj+1 ) is an edge of L(ν ∞ ), and hence Fρ∞ (Pjǫ , Pj+1 ) is a well-deﬁned 1-dimensional convex polyhedron for every j ∈ Z, because ρ∞ satisﬁes the condition Duality by Proposition 8.3.5. In particular, I(ρ∞ (P0ǫ )) ǫ ǫ (P1ǫ ), respectively, (P0ǫ ) and f∞ and I(ρ∞ (P1ǫ )) intersect in two points, and f∞ ǫ ǫ are subarcs of the arcs I(ρ∞ (P0 )) ∩ E(ρ∞ (P1 )) and E(ρ∞ (P0ǫ )) ∩ I(ρ∞ (P1ǫ )). ǫ ǫ (P1ǫ ) interNow suppose to the contrary that that k = 1. Then f∞ (P0ǫ ) and f∞ ǫ ǫ ǫ sect in two points, and therefore we have f∞ (P0 ) = I(ρ∞ (P0 )) ∩ E(ρ∞ (P1ǫ )) ǫ (P1ǫ ) = E(ρ∞ (P0ǫ )) ∩ I(ρ∞ (P1ǫ )). Thus the disk D0 is equal to the and f∞ union D(ρ∞ (P0ǫ )) ∪ D(ρ∞ (P1ǫ )). This contradicts Claim 8.12.6. We show that k = 2. Suppose to the contrary that k = 2. Then the boundary of the disk D0 consists of three circular arcs. Since D0 ⊂ E(ρ∞ ) by Claim 8.12.6, the sum of the angles of D0 at the three vertices is less than ǫ )). Then two π. Now let θj be the dihedral angle of E(ρ∞ (Pjǫ )) ∩ E(ρ∞ (Pj+1 of the three angles of the circular triangle D0 are equal to the angles θ0 and θ1 , respectively. Hence we have θ0 + θ1 < π. This contradicts Lemma 4.1.3(3) (Chain rule). We show that k = 3. Suppose to the contrary that k = 3. Then as in the preceding argument, the boundary of the disk D0 consists of four circular arcs, and the sum of the angles of D0 at the four vertices is less than 2π. Three of the four angles of the circular triangle D0 are equal to the angles θ0 , θ1 and θ2 , respectively. Hence we have θ0 + θ1 + θ2 < 2π. This contradicts Lemma 4.1.3(3). Finally suppose that k ≥ 4. Then δ > 1, and hence x0 := h(t0 + 1) ∈ ∂D0 − {h(t0 )}. By the symmetry, we have h(t0 + δ + 1) = h(t0 + δ) + 1 = h(t0 ) + 1 = h(t0 + 1) = x0 . Since ∂D0 is a ﬁnite union of circular arcs, we can ﬁnd a small disk neighborhood U of x0 in C, such that U − ∂D0 consists of two components, U ǫ and U −ǫ . We may assume U ǫ ⊂ int D0 . Then U ǫ ⊂ int E(ρ∞ ) by Claim 8.12.6. Moreover, we may also assume U −ǫ ⊂ ∪j int D(ρ∞ (Pjǫ )) ⊂ C − E(ρ∞ ). This is obvious when x0 lies in the interior of a circular edge of ∂D0 . The assertion for the case when x0 is a vertex of ∂D0 follows from the fact that two successive isometric circles I(ρ∞ (Pjǫ )) and

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8 Closedness

ǫ I(ρ∞ (Pj+1 )) intersect transversely as observed in the proof of k = 1. On the other hand, t0 + δ + 1 is an isolated point of the inverse image by h of the intersection ∂D0 ∩ h([t0 + δ, ∞)), because ∂D0 is contained in a ﬁnite union ǫ (Pjǫ )} and because {I(ρ∞ (Pjǫ ))} are mutually distinct (cf. of circular arcs {f∞ Lemma 2.5.4). Hence there is an open interval J having t0 + δ + 1 as an endpoint, such that

h(J) ⊂ U − ∂D0 = U ǫ ⊔ U −ǫ ⊂ (int E(ρ∞ )) ∪ (C − E(ρ∞ )) = C − fr E(ρ∞ ). This contradicts Lemma 8.12.3. Thus k ≥ 4 cannot happen. Hence we have proved that h is injective. This completes the proof of Lemma 8.12.5. ǫ (Pjǫ ). Then ℓ∞ projects to an essential simple Set ℓ∞ = h(R) = ∪j f∞ loop in C/Z by Lemma 8.12.5, and hence it separates C into two connected components. For each sign η ∈ {−, +}, let E η (ℓ∞ ) be the closure of the component which contains the set {z | ηℑz > L} for suﬃciently large L.

Lemma 8.12.7. D(ρ∞ (P )) ⊂ E −ǫ (ℓ∞ ) for every vertex (P ) of L(ν ∞ ). Proof. Suppose this is not the case. Then D(ρ∞ (P )) ∩ int E ǫ (ℓ∞ ) = ∅ and hence there is a point x ∈ int D(ρ∞ (P )) which is contained in int E ǫ (ℓ∞ ). Since x ∈ int E ǫ (ℓ∞ ), there is a path, γ, in C − ℓ∞ joining x to a point in a region {z | ǫℑz > L} ⊂ int E ǫ (ℓ∞ ). By the periodicity of ℓ∞ , the Euclidean distance between the (ﬁnite) path γ and ℓ∞ is positive. Since fr E ǫ (ρn ) converges to ℓ∞ , this implies that γ is disjoint from fr E ǫ (ρn ) for all suﬃciently large n. Moreover if we choose L suﬃciently large in advance, then the endpoint of γ lies in E ǫ (ρn ) for all n. Hence it follows that x ∈ int E ǫ (ρn ) for all suﬃciently large n. On the other hand, since x ∈ int D(ρ∞ (P )) and D(ρ∞ (P )) = lim D(ρn (P )), we have x ∈ int D(ρn (P )) for all suﬃciently large n. This is a contradiction. Hence we obtain the desired result. Lemma 8.12.8. E ǫ (ℓ∞ ) = E ǫ (ρ∞ ) = E ǫ (ρ∞ , σ ǫ (ν ∞ )) and hence ℓ∞ = fr E ǫ (ρ∞ , σ ǫ (ν ∞ )). Proof. By Lemma 8.12.7, ∪{D(ρ∞ (P )) | (P ) ∈ L(ν ∞ )(0) } is contained in E −ǫ (ℓ∞ ). Hence int E ǫ (ℓ∞ ) is disjoint from int(∪{D(ρ∞ (P )) | (P ) ∈ L(ν ∞ )(0) }) = ∪{int D(ρ∞ (P )) | (P ) ∈ L(ν ∞ )(0) } = C − E(ρ∞ ). Thus int E ǫ (ℓ∞ ) and hence E ǫ (ℓ∞ ) are contained in a connected component of E(ρ∞ ). Since both E ǫ (ℓ∞ ) and E ǫ (ρ∞ ) contain a region {z | ǫℑz > L} for suﬃciently large L, E ǫ (ℓ∞ ) is contained in E ǫ (ρ∞ ). On the other hand, every point x ∈ ℓ∞ has a neighborhood U in C such that U − E ǫ (ℓ∞ ) ⊂ ∪{int D(ρ∞ (P )) | (P ) ∈ L(ν ∞ )(0) }. (This is obvious if x lies in the interior ǫ of a nondegenerate circular arc f∞ (Pjǫ ). The assertion for the case when x =

8.12 Proof of Proposition 8.3.6

213

ǫ ǫ ǫ f∞ (Pjǫ , Pj+1 ) follows from the fact that int D(ρ∞ (Pjǫ )) ∩ int D(ρ∞ (Pj+1 )) = ∅.) Thus E ǫ (ℓ∞ ) is open in E ǫ (ρ∞ ). Since E ǫ (ℓ∞ ) is also closed in E ǫ (ρ∞ ), it follows that E ǫ (ℓ∞ ) is equal to a component of E ǫ (ρ∞ ). Since E ǫ (ℓ∞ ) is contained in E ǫ (ρ∞ ), this implies that E ǫ (ℓ∞ ) = E ǫ (ρ∞ ). We can also prove E ǫ (ℓ∞ ) = E ǫ (ρ∞ , σ ǫ (ν ∞ )) by an argument parallel to the above, where we replace ∪{D(ρ∞ (P )) | (P ) ∈ L(ν ∞ )(0) } with ∪{D(ρ∞ (P )) | (P ) ∈ L(σ ǫ (ν ∞ ))(0) }. This completes the proof.

Lemma 8.12.9. σ ǫ (ν ∞ ) is an ǫ-terminal triangle of ρ∞ . Proof. We ﬁrst check that (ρ∞ , σ ǫ (ν ∞ )) satisﬁes Assumption 4.2.4 (σ-Simple). Since ρ∞ ∈ QF, the corresponding Markoﬀ map φ∞ satisﬁes the condition σ ǫ (ν ∞ )-NonZero. Since ρ∞ = (ρ∞ , ν ∞ ) satisﬁes the condition Duality by Proposition 8.3.5, φ∞ satisﬁes the triangle inequality at σ ǫ (ν ∞ ). By Lemma 8.12.2, L(ρ∞ , σ ǫ (ν ∞ )) is simple. Hence (ρ∞ , σ ǫ (ν ∞ )) satisﬁes Assumption 4.2.4 (σ-Simple). We show that σ ǫ (ν ∞ ) satisﬁes the ﬁrst condition in Deﬁnition 4.3.8. ǫ ∈ int σ ǫ (ν ∞ ). In this case σ ǫ (ν ∞ ) is equal to the ǫ-terminal Case 1. ν∞ ǫ triangle σ of the common chain Σ0 . Moreover, since ρn = (ρn , ν n ) with ν n = (νn− , νn+ ) is good, θǫ (ρn , σ ǫ ) is identiﬁed with the point νnǫ (cf. Deﬁnition 4.2.17). Hence we have ǫ θǫ (ρ∞ , σ ǫ (ν ∞ )) = θǫ (ρ∞ , σ ǫ ) = lim θǫ (ρn , σ ǫ ) = lim νnǫ = ν∞ ∈ int σ ǫ (ν ∞ ).

Thus all components of θǫ (ρ∞ , σ ǫ (ν ∞ )) are positive and therefore the ﬁrst condition in Deﬁnition 4.3.8 is satisﬁed. ǫ ∈ intsǫ0 , sǫ1 ⊂ ∂σ ǫ (ν ∞ ). In this case σ ǫ (ν ∞ ) ∩ σ ǫ = sǫ0 , sǫ1 . Case 2. ν∞ Since ǫ ∈ intsǫ0 , sǫ1 , θǫ (ρ∞ , σ ǫ ) = lim θǫ (ρn , σ ǫ ) = lim νnǫ = ν∞

we have θǫ (ρ∞ , σ ǫ ; sǫ2 ) = 0. Hence θǫ (ρ∞ , σ ǫ (ν ∞ )) and θǫ (ρ∞ , σ ǫ ) determine the same point in intsǫ0 , sǫ1 by Lemma 4.5.3. Thus the ﬁrst condition in Deﬁnition 4.3.8 is satisﬁed. We do not need to check the second condition in Deﬁnition 4.3.8, because it is a consequence of the fourth condition (Remark 4.3.9(1)). We show that σ ǫ (ν ∞ ) satisﬁes the third condition in Deﬁnition 4.3.8. By ǫ Lemma 8.12.8, fr E ǫ (ρ∞ , σ ǫ (ν ∞ )) = ℓ∞ = h(R) = ∪j f∞ (Pjǫ ). Moreover this is homeomorphic to R by Lemma 8.12.5. Thus the third condition follows from the following fact, which in tern is a consequence of Lemma 8.6.1. ǫ ǫ (Pjǫ ). ∈ int σ ǫ (ν ∞ ). Then eǫ (ρ∞ , σ ǫ ; Pjǫ ) = f∞ 1. Suppose ν∞ ǫ ǫ ǫ ǫ 2. Suppose ν∞ ∈ ints0 , s1 ⊂ ∂σ (ν ∞ ), and let {Pj } be the sequence of elliptic generators associated with σ ǫ (ν ∞ ). Then eǫ (ρ∞ , σ ǫ ; Pj ) is equal ǫ ǫ ǫ (P1ǫ ) according as j = 0, 1 or 2. (P0ǫ , P1ǫ ) or f∞ to f∞ (P0ǫ ), f∞

Finally we show that the fourth condition in Deﬁnition 4.3.8 is satisﬁed. Let {Pj } be the sequence of elliptic generators associated with σ ǫ (ν ∞ ). Since ρ∞ satisﬁes the condition Duality by Proposition 8.3.5, each Fρ (Pj , Pj+1 ) is

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8 Closedness

a 1-dimensional convex polyhedron contained in Eh(ρ∞ ). In fact it is proved that F ρ∞ (Pj , Pj+1 ) is contained in Eh(ρ∞ ). Since F ρ∞ (Pj , Pj+1 ) is either a complete or a half geodesic which has v ǫ (ρ∞ ; Pj , Pj+1 ) as an endpoint, we see that the fourth condition is satisﬁed. Proof (Proof of Proposition 8.3.6). We check that the conditions in Deﬁnition 6.1.4(1) are satisﬁed for each ǫ ∈ {−, +}. By Lemma 8.12.9, ρ∞ satisﬁes the ﬁrst condition. In the proof of Lemma 8.12.9, we have already observed that the second condition is also satisﬁed. The last condition is a consequence of Lemma 8.12.8. This completes the proof of Proposition 8.3.6.

9 Algebraic roots and geometric roots

The purpose of this chapter is to prove Proposition 6.2.6 (Unique realization), which implies the bijectivity of the map µ2 : J [QF] → H2 × H2 . To this end, we ﬁrst make a careful study of the algebraic curves in the algebraic surface Φ ∼ = {(x, y, z) ∈ C3 | x2 + y 2 + z 2 = xyz} determined by the equations in Deﬁnition 4.2.19, and ﬁnd the irreducible components which contain the geometric roots (Deﬁnition 9.1.2) for a given label ν = (ν − , ν + ) ∈ H2 × H2 (Lemmas 9.1.8 and 9.1.12). We also observe that the number of the algebraic roots for ν is ﬁnite (Proposition 9.1.13). Thus the problem is how to single out the geometric roots among the algebraic roots. Our answer is to appeal to the idea of the geometric continuity. By using the idea, we show that all geometric roots for a given label ν are obtained by continuous deformation of the unique geometric root for a special label corresponding to a fuchsian group. This implies the desired result that each label has the unique geometric root. To realize this idea, we introduce the concept of the “geometric degree” dG (ν) of a label ν, and then show that dG (ν) = 1 for every ν by using the argument of geometric continuity (Proposition 9.2.3).

9.1 Algebraic roots In this chapter, we make a slight change of notation and denote by Φ the space of all Markoﬀ maps, including the trivial one. Deﬁnition 4.2.19 is rephrased as follows. Deﬁnition 9.1.1. (1) Let σ = s0 , s1 , s2 be a triangle of D and ν = (θ0 , θ1 , θ2 ) be a point in σ ∩ H2 . ǫ 1. ζν,σ : Φ → C denotes the map deﬁned by:

ǫ (φ) = φ(s0 ) + αǫ φ(s1 ) + β ǫ φ(s2 ), ζν,σ

where αǫ = ǫi exp (ǫiθ2 ) and β ǫ = −ǫi exp (−ǫiθ1 ).

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2. Φǫν,σ denotes the subset of Φ deﬁned by ǫ (φ) = 0}. Φǫν,σ = {φ ∈ Φ | ζν,σ

(2) Let σ ǫ be a triangle of D and ν ǫ be a point in σ ǫ ∩ H2 for each ǫ ∈ {−, +}. Then a Markoﬀ map φ is called an algebraic root for ((ν − , σ − ), + (ν + , σ + )) if φ ∈ Φ− ν − ,σ − ∩ Φν + ,σ + . (3) For a given ν = (ν − , ν + ) ∈ H2 × H2 , a Markoﬀ map φ is called an algebraic root for ν if it is an algebraic root for ((ν − , σ − (ν)), (ν + , σ + (ν))), where σ ǫ (ν) is the ǫ-terminal triangle of Σ(ν) (Deﬁnition 3.3.3). Deﬁnition 9.1.2. Let ν = (ν − , ν + ) be an element of H2 ×H2 . Then a Markoﬀ map φ is called a geometric root for ν if (ρ, ν) is a good labeled representation and φ is upward at every triangle of Σ(ν), where ρ is the type-preserving representation induced by φ. If ν is thin, then we require that φ is upward at each of the two triangles of D which contain the edge that contains ν ± . Lemma 9.1.3. (1) Let σ ǫ be a triangle of D and ν ǫ be a point in σ ǫ ∩ H2 for each ǫ ∈ {−, +}. Then every geometric root for ν = (ν − , ν + ) is an algebraic root for ((ν − , σ − ), (ν + , σ + )). (2) Let (ρ, ν) with ν = (ν − , ν + ) be a good labeled representation. Then there is a unique Markoﬀ map φ inducing ρ which is a geometric root for ν. Proof. (1) Let φ be a geometric root for ν = (ν − , ν + ) and ρ the typepreserving representation induced by φ. Then (ρ, ν) is a good labeled representation and hence it satisﬁes the condition Frontier (Deﬁnition 6.1.4). Thus ν ǫ is equal to the point θǫ (ρ, σ ǫ ) ∈ σ ǫ . (Here we use Lemmas 4.5.3, 4.5.4 and 4.5.6 when ν ǫ is non-generic and σ ǫ does not belong to Σ(ν).) Moreover φ is upward at σ ǫ . Hence φ is an algebraic root for ((ν − , σ − ), (ν + , σ + )) by Lemma 4.2.18. (2) Let (ρ, ν) be a good labeled representation. Then (ρ, ν) is quasifuchsian by Theorem 6.1.8 and L(ρ, ν) is simple by Proposition 6.6.1. Suppose ﬁrst that ν is thick. Let φ be the unique Markoﬀ map inducing ρ which is upward at σ − (ν) (Lemma 3.1.4). Then φ is upward at every triangle of Σ(ν) by Lemma 4.5.2. Thus φ is a geometric root for ν, and hence it is the unique geometric root for ν inducing the given representation ρ. Next, suppose that ν is thin. Let τ be the edge containing ν ± and σ and σ ′ be the triangles containing τ . Then we see that L(ρ, σ) and L(ρ, σ ′ ) are simple, and there is a unique Markoﬀ map φ inducing ρ which is upward at σ and σ ′ by Lemma 4.5.6. Then φ is a geometric root for ν, and hence it is the unique geometric root for ν inducing the given representation ρ. If σ = s0 , s1 , s2 is a triangle of D, then the correspondence Φ ∋ φ → (φ(s0 ), φ(s1 ), φ(s2 )) ∈ C3 gives an identiﬁcation of Φ with the aﬃne algebraic variety deﬁned by x2 + y 2 + z 2 = xyz. Repeated application of Proposition 2.3.4(2) shows that this

9.1 Algebraic roots

217

identiﬁcation does not depend on the choice of σ modulo post-composition ǫ : Φ → C is a by a biregular automorphism of C3 . It also follows that ζν,σ polynomial function. Thus we may identify the set Φǫν,σ in Deﬁnition 9.1.1 with the following subvariety of Φ. Φǫν,σ = {(x, y, z) ∈ C3 | x2 + y 2 + z 2 = xyz, x + αǫ y + β ǫ z = 0} ǫ (y, z) = 0, x = −(αǫ y + β ǫ z)} = {(x, y, z) ∈ C3 | Fν,σ ∼ = {(y, z) ∈ C2 | F ǫ (y, z) = 0}, ν,σ

where αǫ = ǫi exp (ǫiθ2 ) and β ǫ = −ǫi exp (−ǫiθ1 ) and ǫ (y, z) = y 2 + z 2 + (αǫ y + β ǫ z)2 + yz(αǫ y + β ǫ z). Fν,σ ǫ of Φǫν,σ is given by The prime factorization of the deﬁning polynomial Fν,σ the following lemma. ǫ (y, z) is irreducible if and only if ν ∈ Lemma 9.1.4. The polynomial Fν,σ ǫ int σ. If ν ∈ ∂σ, then the prime factorization of Fν,σ (x, y) is given by the following formula.

1. If ν ∈ ints0 , s1 , then ǫ (y, z) = z{(β ǫ y + (β ǫ )2 + 1)z + ǫiy(y + 2β ǫ )}. Fν,σ

2. If ν ∈ ints1 , s2 , then ǫ Fν,σ (y, z) = {z − ǫiy}{(β ǫ y + (β ǫ )2 + 1)z + ǫi(1 − (β ǫ )2 )y}.

3. If ν ∈ ints2 , s0 , then ǫ (y, z) = y{(αǫ z + (αǫ )2 + 1)y − ǫiz(z + 2αǫ )}. Fν,σ

Before proving this lemma, we make the following simple observation. Sublemma 9.1.5. (1) If ν ∈ int σ, then none of αǫ , αǫ β¯ǫ and β ǫ is equal to ±i. (2) If ν ∈ ∂σ, then we have the following equivalences. 1. ν ∈ ints0 , s1 ⇐⇒ αǫ = ǫi ⇐⇒ αǫ = ±i. 2. ν ∈ ints1 , s2 ⇐⇒ αǫ = −ǫiβ ǫ ⇐⇒ αǫ = ±iβ ǫ . 3. ν ∈ ints2 , s0 ⇐⇒ β ǫ = −ǫi ⇐⇒ β ǫ = ±i.

Proof. Since (1) is a consequence of (2), we prove (2). The ﬁrst and the last assertions are direct consequence of the deﬁning identities αǫ = ǫi exp (ǫiθ2 ) and β ǫ = −ǫi exp (−ǫiθ1 ) and the facts that θj ∈ [0, π/2] for each j. To see the second assertion, note that ν ∈ ints1 , s2 if and only if θ1 + θ2 = π/2. Since αǫ β¯ǫ = − exp(ǫi(θ1 + θ2 )), this is equivalent to the condition αǫ β¯ǫ = −ǫi. Thus we obtain the second assertion.

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ǫ Proof (Proof of Lemma 9.1.4). Throughout the proof, we abbreviate Fν,σ , αǫ and β ǫ , respectively, as F , α and β. Note that

F (y, z) = y 2 + z 2 + (αy + βz)2 + yz(αy + βz) = (1 + β 2 + βy)z 2 + (2αβy + αy 2 )z + (1 + α2 )y 2 = (1 + α2 + αz)y 2 + (2αβz + βz 2 )y + (1 + β 2 )z 2 . Suppose that F has a nontrivial factorization F = F1 F2 . Since degz F = 2, the pair (degz F1 , degz F2 ) is equal to (1, 1) or (2, 0) up to permutation of indices. Case 1. (degz F1 , degz F2 ) = (1, 1). Note that the discriminant of F (y, z) as a polynomial in the variable z is equal to # $ α2 y 2 (2β +y)2 −4y 2 (1+α2 )(1+β 2 +βy) = y 2 α2 y 2 − 4βy − 4(α2 + β 2 + 1) . By the assumption, it must be equal to the square of a polynomial in C[y]. Hence, the discriminant of its second factor vanishes, i.e., (2β)2 + 4α2 (α2 + β 2 + 1) = 4(α2 + β 2 )(α2 + 1) = 0. Hence α = ±iβ or α = ±i. By Sublemma 9.1.5, α = ±iβ if and only if ν ∈ ints1 , s2 . Moreover if this holds, then α = −ǫiβ and F (y, z) = {z − ǫiy}{(βy + β 2 + 1)z + ǫi(1 − β 2 )y}. By Sublemma 9.1.5, the condition α = ±i is equivalent to the condition ν ∈ ints0 , s1 . Moreover, if this condition is satisﬁed, then α = ǫi and F (y, z) = z{(βy + β 2 + 1)z + ǫiy(y + 2β)}. Case 2. (degz F1 , degz F2 ) = (2, 0). Then (degy F1 , degy F2 ) is equal to (1, 1) or (0, 2), because F2 is not a constant. Suppose that (degy F1 , degy F2 ) = (1, 1). Then we see, by an argument similar to the above, that ν lies in either s1 , s2 or s2 , s0 . Conversely, if this condition is satisﬁed, then F (x, y) has the non-trivial factorization explained in the lemma. Suppose (degy F1 , degy F2 ) = (0, 2). Then F1 and F2 , respectively, are elements of C[z] and C[y] of degree 2. Thus F2 must divide the three polynomials, 1 + β 2 + βy,

αy(2β + y),

(1 + α2 )y 2 .

This is a contradiction, because the degree in y of the ﬁrst one is 1. Hence, we see that (degz F1 , degz F2 ) = (2, 0) cannot occur. Thus we have proved that F is irreducible if and only if ν ∈ int σ. To complete the proof, we need to show that the second factor in each of the factorizations of F for the case when ν ∈ ∂σ is actually irreducible. We shall prove this only for the ﬁrst case. Namely, we show that

9.1 Algebraic roots

219

G(y, z) := (βy + β 2 + 1)z + ǫiy(y + 2β) is irreducible if ν ∈ ints0 , s1 . Suppose this has a nontrivial factorization G = G1 G2 . Since degz G = 1, we may assume that degz G1 = 0. Then G1 is an element of C[z] and hence it must be a common factor of the polynomials βy + β 2 + 1 and y(y + 2β). Since ν ∈ ints0 , s1 , we have θ1 = 0 and hence β 2 + 1 = 0 (cf. Sublemma 9.1.5). So, we see that the two polynomials βy + β 2 + 1 and y + 2β coincide up to multiplication of a non-zero complex number. This implies that β 2 = 1 and hence θ1 = π/2, or equivalently, ν = s1 , a contradiction. The remaining cases can be treated similarly. As a corollary to Lemmas 9.1.4, we obtain the following. Corollary 9.1.6. Let σ = s0 , s1 , s2 be a triangle of D and ν = (θ0 , θ1 , θ2 ) be a point in σ ∩ H2 . (1) If ν ∈ int σ, then Φǫν,σ is irreducible. (2) If ν ∈ ∂σ, then Φǫν,σ has precisely two irreducible components, Φˇǫν,σ ǫ , which are deﬁned as follows. and Ψν,σ 1. If ν ∈ ints0 , s1 , then Φˇǫν,σ := {(x, y, z) | (β ǫ y + (β ǫ )2 + 1)z + ǫiy(y + 2β ǫ ) = 0, ǫ Ψν,σ

x = −(ǫiy + β ǫ z)},

:= {(x, y, z) | z = 0, x = −(ǫiy + β ǫ z)} = {(x, ǫix, 0) | x ∈ C}.

2. If ν ∈ ints1 , s2 , then Φˇǫν,σ := {(x, y, z) | (β ǫ y + (β ǫ )2 + 1)z + ǫi(1 − (β ǫ )2 )y = 0, x = −(αǫ y + β ǫ z)},

ǫ Ψν,σ := {(x, y, z) | z − ǫiy = 0, x = −(αǫ y + β ǫ z)} = {(0, y, ǫiy) | y ∈ C}.

3. If ν ∈ ints2 , s0 , then Φˇǫν,σ := {(x, y, z) | (αǫ z + (αǫ )2 + 1)y − ǫiz(z + 2αǫ ) = 0, x = −(αǫ y − ǫiz)},

ǫ Ψν,σ := {(x, y, z) | y = 0, x = −(αǫ y + β ǫ z)} = {(ǫiz, 0, z) | z ∈ C}.

Here αǫ = ǫi exp (ǫiθ2 ) and β ǫ = −ǫi exp (−ǫiθ1 ), and (x, y, z) represents the point φ ∈ Φ such that (x, y, z) = (φ(s0 ), φ(s1 ), φ(s2 )). ǫ Remark 9.1.7. In the case when ν ∈ ∂σ, the component Ψν,σ of Φǫν,σ is disǫ ǫ ˇ tinguished from the component Φν,σ by the property that Ψν,σ is contained in the subvariety {φ ∈ Φ | φ(s) = 0}, where s is the vertex of σ which is not ǫ induces a a vertex of the edge containing ν. In particular, no element of Ψν,σ quasifuchsian representation.

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9 Algebraic roots and geometric roots

Lemma 4.5.3 suggests the following lemma, which shows that the subvariety Φˇǫν,σ of Φ in Corollary 9.1.6(2) does not depend on the choice of σ, though ǫ Ψν,σ does. Lemma 9.1.8. Let σ and σ ′ be triangles of D sharing an edge τ , and let ν be ˇǫ ′ ⊂ Φ. a point in int τ . Then Φˇǫν,σ = Φ ν,σ Proof. We may assume σ = s0 , s1 , s2 , σ ′ = s′0 , s′1 , s′2 , τ = s1 , s2 = s′2 , s′1 , where s′1 = s2 and s′2 = s1 . Then the coordinates of ν in σ and σ ′ , respectively, are of the form (0, θ1 , θ2 ) and (0, θ1′ , θ2′ ) where θ1′ = θ2 and θ2′ = θ1 . We identify Φ with the aﬃne algebraic variety deﬁned by x2 + y 2 + z 2 = xyz through the correspondence Φ ∋ φ → (φ(s0 ), φ(s1 ), φ(s2 )) ∈ C3 . Then Φˇǫν,σ is the subvariety of Φ ⊂ C3 determined by the equations G(y, z) = 0, x = −(αy + βz), where G(y, z) = (βy +β 2 +1)z +ǫi(1−β 2 )y, α = ǫi exp (ǫiθ2 ) = −ǫiβ and β = −ǫi exp (−ǫiθ1 ). On the other hand, Φǫν,σ′ is the subvariety of Φ determined by the equation φ(s′0 ) + α′ φ(s′1 ) + β ′ (s′2 ) = 0, where α′ = ǫi exp (ǫiθ2′ ) = β¯ and ¯ Since (φ(s′ ), φ(s′ ), φ(s′ )) = (yz − x, z, y), Φǫ ′ β ′ = −ǫi exp (−ǫiθ1′ ) = ǫiβ. 2 1 0 ν,σ is determined by the equations ¯ + ǫiβy ¯ = 0. (yz − x)2 + y 2 + z 2 − (yz − x)yz = 0, (yz − x) + βz

(9.1)

Let H(y, z) be the polynomial obtained from the ﬁrst polynomial by putting ¯ + ǫiβy. ¯ Then it has the prime factorization x = yz + βz H(x, y) = β¯2 (z + ǫiy)G(y, z). Thus Φǫν,σ′ has two irreducible components corresponding to the factors z +ǫiy and G(y, z) respectively. However the component corresponding to the factor z + ǫiy is contained in the subvariety φ(s′0 ) = 0 and hence it is equal to the ǫ ˇǫ component Ψν,σ ′ by Remark 9.1.7. Hence Φν,σ ′ is equal to the component corresponding to the factor G(y, z). The equation G(y, z) = 0 implies yz = ¯ 2 + 1)z + ǫi(1 − β 2 )y}, which in tern implies that the second equation in −β{(β ˇǫ ′ = Φ ˇǫν,σ . (9.1) is equivalent to the equation x = −(αy+βz). Hence we have Φ ν,σ ǫ Remark 9.1.9. By the above proof, we see that Ψν,σ ′ is the subvariety of Φ ⊂ C3 determined by the equations z + ǫiy = 0 and x = ǫiz 2 . On the other hand, ǫ ǫ ǫ is determined by the equations z +ǫiy = 0 and x = 0. Thus Ψν,σ and Ψν,σ Ψν,σ ′ share the same equation z + ǫiy = 0, but they are not identical. In fact every ǫ has the properties φ(s0 ) = 0 and φ(s′0 ) = 0, nontrivial Markoﬀ map φ ∈ Ψν,σ ′ ǫ whereas every nontrivial Markoﬀ map φ′ ∈ Ψν,σ ′ has the properties φ (s0 ) = 0 and φ(s′0 ) = 0,

9.1 Algebraic roots

221

By Lemma 9.1.8, we may introduce the following deﬁnition. ˇǫ be the subvariety of Deﬁnition 9.1.10. For each ν ∈ H2 and ǫ = ±, let Φ ν Φ deﬁned as follows. 1. If ν lies in the interior of a triangle σ of D, then Φˇǫν := Φǫν,σ . ˇǫν,σ = Φˇǫ ′ , where 2. If ν lies in the interior of an edge τ of D, then Φˇǫν := Φ ν,σ ′ σ and σ are the triangles of D sharing the edge τ . Then we have the following lemma. Lemma 9.1.11. The singular set of the irreducible variety Φˇǫν is as follows: ˇǫν is singular only at the 1. If ν lies in the interior of a triangle of D, then Φ trivial Markoﬀ map. 2. If ν lies in the interior of an edge of D, say ν = (θ0 , θ1 , 0) ∈ σ = s0 , s1 , s2 , then the trivial Markoﬀ map and the Markoﬀ map φ such that (φ(s0 ), φ(s1 ), φ(s2 )) = (−2ǫiβ ǫ , −2β ǫ , 0) are the only singular points of Φˇǫν . Here β ǫ = −ǫi exp (−ǫiθ1 ). Proof. (1) Suppose that ν lies in the interior of a triangle σ of D. Then Φˇǫν = Φǫν,σ is deﬁned by the polynomial ǫ F (y, z) = Fν,σ (y, z) = y 2 + z 2 + (αy + βz)2 + yz(αy + βz),

where α = αǫ and β = β ǫ are as in Deﬁnition 9.1.1. We can see F−

y ∂F z ∂F yz − = − (αy + βz). 2 ∂y 2 ∂z 2

Let (y0 , z0 ) be a singular point of Φˇǫν , that is, F (y0 , z0 ) =

∂F ∂F (y0 , z0 ) = (y0 , z0 ) = 0. ∂y ∂z

Then we have

y 0 z0 (αy0 + βz0 ) = 0. 2 So, one of αy0 +βz0 , y0 and z0 is equal to 0. On the other hand, since ν ∈ int σ, Sublemma 9.1.5 implies that none of α, β, and α/β are equal to ±i. Through elementary calculation by using this fact. we obtain (y0 , z0 ) = (0, 0). Hence, the origin is the unique singular point of Φˇǫν . ˇǫν is deﬁned by (2) Suppose that ν = (θ0 , θ1 , 0) ∈ σ = s0 , s1 , s2 . Then Φ −

G(y, z) := (βy + β 2 + 1)z + ǫiy(y + 2β), where β = β ǫ = −ǫi exp (−ǫiθ1 ) (see Corollary 9.1.6). Note that G−z

∂G = ǫiy(y + 2β). ∂z

222

9 Algebraic roots and geometric roots

Let (y0 , z0 ) be a singular point of Φˇǫν . Then we have either y0 = 0 or y0 = −2β. On the other hand, since ν lies in the interior of an edge of D, we have 0 < θ1 < π/2 and hence β = −ǫi exp (−ǫiθ1 ) = ±1. By using this fact and the identity G(y0 , z0 ) = 0, we see (y0 , z0 ) is equal to (0, 0) or (−2β, 0). Lemma 9.1.12. Let ρ = (ρ, ν) with ν = (ν − , ν + ) be a good labeled representation, and let φ be the geometric root for ν inducing ρ. Then the following hold. 1. If ν ǫ is non-generic and σ ǫ is a triangle containing ν ǫ , then φ is not ǫ contained in the algebraic variety Ψν,σ ǫ. 2. φ is a smooth point of the algebraic variety Φˇǫν ǫ for each ǫ = ±. Proof. Let σ ǫ be a triangle containing σ ǫ . Then φ ∈ Φǫν ǫ ,σǫ by Lemma 9.1.3. Suppose that ν ǫ is non-generic, i.e., ν ǫ ∈ ∂σ ǫ . Since ρ is good and hence ρ is quasifuchsian (Theorem 6.1.8), we see that φ cannot be contained in the ǫ ˇǫ component Ψν,σ ǫ by Remark 9.1.7. Hence φ is contained in Φν ǫ . Moreover ǫ ˇ Lemma 9.1.11 implies that φ is a smooth point of Φν ǫ . Suppose next that ν ǫ ∈ int σ ǫ . Then Φˇǫν ǫ = Φǫν ǫ ,σǫ and hence φ ∈ Φˇǫν ǫ . Since φ is nontrivial (because ρ is good), it is a smooth point of Φˇǫν ǫ by Lemma 9.1.11. Next, we prove the following proposition. Proposition 9.1.13. Let σ ǫ = sǫ0 , sǫ1 , sǫ2 be a triangle of D, and let ν ǫ = + (θ0ǫ , θ1ǫ , θ2ǫ ) be a point in σ ǫ ∩ H2 for each ǫ ∈ {−, +}. Then Φ− ν − ,σ − and Φν + ,σ + do not share a common component. We begin the proof of the proposition, by proving the following lemma. Lemma 9.1.14. Under the assumption of Proposition 9.1.13, let Φ0 be the subvariety of Φ deﬁned by + + ∼ Φ0 = {φ ∈ Φ | (φ(s+ 0 ), φ(s1 ), φ(s2 )) = (0, x, ix) for some x ∈ C} = C.

Then the restriction of ζν−− ,σ− to Φ0 is a polynomial in the variable x = φ(s+ 1) of positive degree. Moreover, it is not a monomial except when 1. σ − = σ + or 2. σ − ∩ σ + is an edge of D and its interior contains ν − . + + Proof. After a coordinate change, we may assume that s+ 0 , s1 , s2 = 1/0, − 0/1, 1/1 and that all vertices of σ are contained in [0, 1] ∪ {1/0}. Suppose ﬁrst that σ − = σ + . Then for each φ ∈ Φ0 ,

ζν−− ,σ− (φ) = 0 + α− x + β − (ix) = (α− + iβ − )x, where α− = −i exp (−iθ2− ) and β − = i exp (iθ1− ). Since α− + iβ − = 0 by Sublemma 9.1.5, the above polynomial is a monomial of degree 1.

9.1 Algebraic roots

223

Suppose that σ − ∩ σ + is an edge of D, namely σ − = 1/2, 1/1, 0/1. Then for each φ ∈ Φ0 , (φ(1/2), φ(1/1), φ(1/0)) = (ix2 , ix, x) and hence ζν−− ,σ− (φ) = ix2 + α− (ix) + β − x = ix(x + (α− − iβ − )). By Sublemma 9.1.5, α− − iβ − = 0 if and only if ν − ∈ 0/1, 1/1, namely the second condition in Lemma 9.1.14 holds. Hence the desired result holds. Suppose σ − = σ + and σ − ∩ σ + is not an edge. Then, by virtue of Lemma 5.3.12(1), we may further assume that all vertices of σ − are contained in the interval [0, 1/2]. Then we may assume σ − = (q1 + q2 )/(p1 + p2 ), q1 /p1 , q2 /p2 where p1 , p2 , q1 and q2 are integers such that 0 ≤ qj ≤ pj (j = 1, 2), 1 ≤ p1 ≤ q q p2 and 1 2 = ±1. Then by Lemma 5.3.12, for every φ ∈ Φ0 , we have p1 p2 ζν−− ,σ− (φ) = V [(q1 + q2 )/(p1 + p2 )](x) + α− V [q1 /p1 ](x) + β − V [q2 /p2 ](x) = iq1 +q2 (xp1 +p2 − cxp1 +p2 −2 + (lower terms))

+ iq1 α− (xp1 + (lower terms)) + iq2 β − (xp2 + (lower terms)),

where c = c(q1 +q2 )/(p1 +p2 ) is a positive integer. Thus ζν−− ,σ− (φ) is a polynomial in x of positive degree p1 + p2 . In the following we show that it is not a monomial. If p1 ≥ 3, then both p1 and p2 are smaller than p1 + p2 − 2 and hence the coeﬃcient of xp1 +p2 −2 is equal to −iq1 +q2 c = 0. So the polynomial is not a monomial. If p1 = 1, then q1 = 0 and p2 = p1 + p2 − 1, and therefore the coeﬃcient of xp1 +p2 −1 is iq2 β − = 0. Thus the polynomial is not a monomial. If p1 = 2, then q1 = 1 and q2 /p2 = n/(2n + 1) for some integer n ≥ 1. Thus p1 < p1 + p2 − 2 and p2 = p1 + p2 − 2, and therefore the coeﬃcient of xp1 +p2 −2 = xp2 is equal to −iq1 +q2 c + iq2 β − = −iq1 +q2 (c − i−q1 β − ) = −i1+q2 (c + iβ − ). Hence the polynomial is not a monomial if β − = i. Since c = c(n+1)/(2n+3) = 1 by Lemma 5.3.14, the coeﬃcient of xp1 +p2 −2 vanishes if β − = i. So we study the coeﬃcient of xp1 +p2 −4 = x2n−1 in case β − = i. By Lemma 5.3.14, we see ζν−− ,σ− (φ) = V [(n + 1)/(2n + 3)](x) + α− V [1/2](x) + β − V [n/2n + 1](x) = in+1 (x2n+3 − x2n+1 − nx2n−1 + (lower terms))

+ iα− x2 + in β − (x2n+1 − x2n−1 + (lower terms)).

If β − = i, then the coeﬃcient of xp1 +p2 −4 = x2n−1 is equal to −in+1 n − in β − = −in+1 (n + 1) = 0. Hence the polynomial is not a monomial in this case, too. This completes the proof of Lemma 9.1.14.

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9 Algebraic roots and geometric roots

By the fundamental theorem of algebra, we obtain the following corollary, which is used in the proof of Lemma 9.1.16. Corollary 9.1.15. Under the assumption of Proposition 9.1.13, assume that σ − = σ + and that ν − ∈ σ − ∩ σ + . Then there is a non-trivial Markoﬀ map + + + φ ∈ Φ− ν − ,σ − such that (φ(s0 ), φ(s1 ), φ(s2 )) = (0, x, ix) for some x ∈ C − {0}. Lemma 9.1.16. Under the assumption of Proposition 9.1.13, the restriction of the function ζν++ ,σ+ to Φ− ν − ,σ − is not a constant function. + Proof. If σ − = σ + or σ − ∩σ + is an edge, then Φ− ν − ,σ − ∩Φν + ,σ + contains at most two non-trivial Markoﬀ maps counted with multiplicity by Lemmas 5.2.12 and 5.2.14. So we obtain the desired result. Hence we may assume that σ − = σ + and that σ − ∩ σ + is not an edge. Since the image of the trivial Markoﬀ map by ζν++ ,σ+ is 0, we have only to show that ζν++ ,σ+ (φ) = 0 for some φ ∈ Φ− ν − ,σ − . To this end we use Corollary 9.1.15, which guarantees the existence of a non+ + + trivial Markoﬀ map φ ∈ Φ− ν − ,σ − such that (φ(s0 ), φ(s1 ), φ(s2 )) = (0, x, ix) for some x ∈ C − {0}. Then + + + + + + ζν++ ,σ+ (φ) = φ(s+ 0 ) + α φ(s1 ) + β φ(s2 ) = (α + iβ )x, + / ints+ where α+ = i exp (iθ2+ ) and β + = −i exp (−iθ1+ ). If ν + ∈ 1 , s2 , we + have α+ + iβ + = 0 by Sublemma 9.1.5. Thus ζν + ,σ+ (φ) = 0. If ν + ∈ + ′ ints+ 1 , s2 , then we make use of another non-trivial Markoﬀ map φ such that ′ + ′ + ′ + (φ (s0 ), φ (s1 ), φ (s2 )) = (x, ix, 0) for some x ∈ C − {0}, whose existence is guaranteed by Corollary 9.1.15. Then + ′ + + ′ + + ζν++ ,σ+ (φ′ ) = φ′ (s+ 0 ) + α φ (s1 ) + β φ (s2 ) = (1 + iα )x. + + + ∈ ints+ Since ν + ∈ ints+ 0 , s1 . Thus 1 , s2 by the assumption, we have ν + 1 + iα+ = 0 by Sublemma 9.1.5. So ζν + ,σ+ (φ′ ) = 0. This completes the proof of Lemma 9.1.16.

Before proceeding to the next lemma, we prepare the following sublemma. ˆ → k(x) be a map to the rational function ﬁeld Sublemma 9.1.17. Let V : Q k(x) over a ﬁeld k with the property that V (s0 ) + V (s′2 ) = V (s0 )V (s2 ) for any pair of adjacent triangles s0 , s1 , s2 and s′0 , s′1 , s′2 of D with s′0 = s0 and s′1 = s2 . Denote the degree of an element f ∈ k(x) by deg f , i.e., if f = f1 /f2 where f1 , f2 belong to the polynomial ring k[x], then deg f = deg f1 − deg f2 . Suppose that deg V (1/0) ≤ 1 and that deg V (0/1) = deg V (1/1) = 1. Then deg V (q/p) = p for any pair of coprime integers (p, q) with 0 ≤ q ≤ p. Proof. This is proved by an inductive argument using the following facts as in the proof of Lemma 5.3.14 (cf. [44, Proposition 3.1]). •

deg(f g) = deg f + deg g for any f, g ∈ k(x).

9.1 Algebraic roots

•

225

If deg f > deg g, then deg(f + g) = deg f .

Lemma 9.1.18. Under the assumption of Proposition 9.1.13, suppose that ν − ∈ ∂σ − . Then the restriction of the function ζν++ ,σ+ to Φˇ− ν − is not a constant does not share a common component with function. In particular, Φˇ− ν − ,σ − + Φν + ,σ+ . Proof. As in the proof of Lemma 9.1.16, we may assume that σ − = σ + and that σ − ∩σ + is not an edge by Lemmas 5.2.12 and 5.2.14. We may also assume σ − = 1/0, 0/1, 1/1, and identify Φ with the aﬃne algebraic variety deﬁned by x2 + y 2 + z 2 = xyz through the correspondence Φ ∋ φ → (x, y, z) := (φ(1/0), φ(0/1), φ(1/1)) ∈ C3 . Case 1. σ − is equal to the (−)-terminal triangle σ − (ν) of ν. Then may assume that ν − ∈ 1/0, 0/1 and that all vertices of σ + belong to the interval [0, 1]. (To be precise, all vertices of σ + belong to either [0, 1/2] or [1/2, 1].) Then Φˇ− ν − is determined by the equations (β − y + (β − )2 + 1)z − iy(y + 2β − ) = 0,

x = iy − β − z,

where β − = i exp (iθ1− ) (see Corollary 9.1.6). Thus the parameter y gives a coordinate of the open set ˇ−− | β − y + (β − )2 + 1 = 0} U := {(x, y, z) ∈ Φ ν ˇ−− . In fact, the equality (β − y + (β − )2 + 1)z − iy(y + 2β − ) = 0 implies of Φ ν z=

iy(y + 2β − ) . + (β − )2 + 1

β−y

Then it follows that x = iy − β − z =

−iy(y + 2β − − 1) . y + (β − + (β − )−1 )

ˆ let Vq/p : Φ → C be the function deﬁned by Vq/p (φ) = Now, for each q/p ∈ Q, φ(q/p). Then the above equalities show that the restrictions of the functions V0/1 (φ) = x, V1/1 (φ) = y and V1/0 (φ) = z to U are rational functions in y over C of degree 1. Thus, by Proposition 2.3.4(2) and Sublemma 9.1.17, it follows that the restriction of Vq/p to U is a rational function in y of degree p for any pair of coprime integers (p, q) with 0 ≤ q < p. Since all vertices of σ + belong to the interval [0, 1] by the assumption, we + + p1 , p2 , q 1 may assume s+ 0 , s1 , s2 = (q1 + q2 )/(p1 + p2 ), q1 /p1 , q2 /p2 where q1 q 2 = 1. For each and q2 are integers such that 0 ≤ qj ≤ pj (j = 1, 2) and p1 p2 φ ∈ U , we have

226

9 Algebraic roots and geometric roots + + + + ζν++ ,σ+ (φ) = φ(s+ 0 ) + α φ(s1 ) + β φ(s2 )

= V(q1 +q2 )/(p1 +p2 ) (φ) + α+ Vq1 /p1 (φ) + β + Vq2 /p2 (φ). On the other hand, since both p1 and p2 are positive, they are smaller than ˇ−− → p1 +p2 . Hence, by the observation in the preceding paragraph, ζν++ ,σ+ : Φ ν C is a rational function in y of degree p1 +p2 > 0, and hence it is not a constant function. Case 2. σ − = σ − (ν). Then may assume that σ − = 1/0, 0/1, 1/1, ν − ∈ 0/1, 1/1, and all vertices of σ + belong to either [0, 1/2] or [1/2, 1]. Then Φˇ− ν− is determined by the equations (β − y + (β − )2 + 1)z − i(1 − (β − )2 )y = 0,

x = −(α− y + β − z),

where α− = −i exp (−iθ2− ) = iβ − and β − = i exp (iθ1− ) (see Corollary 9.1.6). Thus, as in the previous case, the parameter y gives a coordinate of some open ˇ−− , and the parameters x and z are given by the following rational set of Φ ν functions in y. z=

i(1 − (β − )2 )y , β − y + (β − )2 + 1

x = −(α− y + β − z) =

Thus w := yz − x =

−iβ − y(β − y + 2) . β − y + (β − )2 + 1

−iy(y + 2β − ) . β − y + (β − )2 + 1

Note that the parameters x, y, z and w are rational functions in y of degree 1, 1, 0 and 1, respectively. Suppose that all vertices of σ + belong to [0, 1/2]. Then apply a coordinate change so that the triangles σ − and 0/1, 1/2, 1/1, respectively, become −1/1, 0/1, 1/0 and 0/1, 1/1, 1/0 in the new coordinate. Then all vertices of σ − belong to [0, 1], and the parameters y, z and w, respectively, represent ˇ−− ∋ φ → φ(q/p) ∈ C where q/p = 0/1, 1/0 and 1/1. Since the functions Φ ν they are rational functions in y of degree 1, 0 and 1, respectively, we can prove ˇ−− → C is a rational function in y of positive degree, by using that ζν++ ,σ+ : Φ ν Sublemma 9.1.17 as in the previous case. The same argument works in the case when all vertices of σ + belong to [1/2, 1]. This completes the proof of the main assertion of Lemma 9.1.18. The ˇ−− − remaining assertion follows from the main assertion and the facts that Φ ν ,σ

+ −1 (0). is irreducible and Φ+ ν + ,σ + = (ζν + ,σ + )

Proof (Proof of Proposition 9.1.13). Suppose ν − ∈ int σ − . Then Φ− ν − ,σ − is + + −1 irreducible by Corollary 9.1.6. Since Φν + ,σ+ = (ζν + ,σ+ ) (0) and since the restriction of ζν++ ,σ+ to Φ− ν − ,σ − is not a constant function, this implies that + Φ− and Φ do not share a common component. ν − ,σ − ν + ,σ + − − Suppose ν ∈ ∂σ . Then Φ− ν − ,σ − consists of two irreducible components, − Φˇ − − and Ψ −− − , by Corollary 9.1.6. By Lemma 9.1.18, Φˇ−− − does not ν ,σ

ν ,σ

ν ,σ

9.2 Unique existence of the geometric root

227

share a common component with Φ+ ν + ,σ + . Finally we show the other compo− nent Ψν − ,σ− has the same property. Since the argument is symmetric, we may + + prove that Ψν++ ,σ+ does not share a component with Φ− ν − ,σ − when ν ∈ ∂σ . + + + We may assume ν ∈ s1 , s2 after a cyclic permutation of vertices. Then Ψν++ ,σ+ coincides with the variety Φ0 in Lemma 9.1.14 (see Corollary 9.1.6). Thus the restriction of ζν−− ,σ− to Φ0 = Ψν++ ,σ+ is a non-constant function. Hence we obtain the desired result.

9.2 Unique existence of the geometric root Throughout this section, σ ǫ denotes a triangle of D, and ν ǫ denotes a point in + σ ǫ ∩ H2 for each ǫ ∈ {−, +}. By Proposition 9.1.13, Φ− ν − ,σ − ∩ Φν + ,σ + consists ˇ+ of ﬁnitely many points, and so is Φˇ− ν − ∩ Φν + . In particular, there are only − ﬁnitely many geometric roots for ν = (ν , ν + ), and they are smooth points ˇǫν ǫ for each ǫ = ± (Lemma 9.1.12). of Φ Deﬁnition 9.2.1. The geometric multiplicity, dG (ν), of ν is deﬁned to be the number of geometric roots for ν counted with multiplicity. Namely, if {φ1 , · · · , φk } is the set of the geometric roots for ν, then dG (ν) :=

k j=1

ˇ+ Int(φj , Φˇ− ν − ∩ Φν + ),

ˇ+ ˇ− where Int(φj , Φˇ− ν − ∩ Φν + ) denotes the intersection multiplicity at φj of Φν − ˇ++ in Φ. (See for example [31] and [73] for the deﬁnition and basic facts and Φ ν concerning the intersection multiplicity.) By Lemma 9.1.12, the following hold. 'k + Lemma 9.2.2. dG (ν) = j=1 Int(φj , Φ− ν − ,σ − ∩ Φν + ,σ + ).

Proposition 6.2.6 (Unique realization) is a direct consequence of the following proposition.

Proposition 9.2.3. The geometric multiplicity dG (ν) is equal to 1 for every ν ∈ H2 × H2 . The remainder of this section is devoted to the proof of this proposition. We assume after a coordinate change that σ − = 1/0, 0/1, 1/1 and σ + = q0 /p0 , q1 /p1 , q2 /p2 , where pj and qj are non-negative integers such that qj /pj ∈ [0, 1] ∪ {1/0} (j = 0, 1, 2) and (p0 , q0 ) = (p1 + p2 , q1 + q2 ). We ﬁx an aﬃne embedding Φ ∋ φ → (x, y, z) := (φ(1/0), φ(0/1), φ(1/1)) ∈ C3

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9 Algebraic roots and geometric roots

⊂ CP 3 . Let Φ ǫν ǫ ,σǫ be the subvariety and consider its projective completion Φ ǫ of Φ obtained by the projective completion of Φν ǫ ,σǫ . We note that these projective completions depend on the choice of the triangle σ − deﬁning the aﬃne embedding, because the transformation (x, y, z) → (x, y, xy − z) does not extend to a map on CP 3 . −− − Lemma 9.2.4. Under the above situation, the intersection number of Φ ν ,σ ++ + in Φ is equal to 3p0 . and Φ ν ,σ

Proof. Set (x, y, z) = (φ(1/0), φ(0/1), φ(1/1)). Then we can see inductively as in the proof of Lemma 5.3.12 that φ(qj /pj ) is an integral polynomial, Wj (x, y, z), in x, y, z of degree pj and it has a unique term of the maximal degree pj . Let P − be the aﬃne variety deﬁned by x + α− y + β − z = 0, and let P + be the aﬃne variety deﬁned by W0 (x, y, z) + α+ W1 (x, y, z) + β + W2 (x, y, z) = 0. − and P + , respectively, be the subvarieties of CP 3 obtained as the Let P − with the projective completions of P − and P + . Then we may identify P completion of the (y, z)-plane via the extension of the linear isomorphism C2 ∋ (y, z) → (−α− y − β − z, y, z) ∈ P − .

− ∩Φ − deﬁned by the is an algebraic curve in the projective plane P Then P following polynomial of degree 3 in the variables y and z. (α− y + β − z)2 + y 2 + z 2 + (α− y + β − z)yz = 0.

− ∩P + is the algebraic curve in the projective plane P − On the other hand, P deﬁned by the polynomial in y, z obtained from the polynomial W0 (x, y, z) + α+ W1 (x, y, z)+β + W2 (x, y, z) by substituting x with −(α− y +β − z). Since the above three-variable polynomial has the unique term of the maximal degree − ∩ Φ p0 , the resulting two-variable polynomial has degree p0 . Moreover, P − + and P ∩ P do not share a component by Proposition 9.1.13, because

− − + + = P − ∩ Φ ∩Φ ∩P ∩Φ ∩ P ∩ P ++ + . =Φ −− − ∩ Φ P ν ,σ ν ,σ

− ∩Φ − ∩P + and P Hence, by Bezout theorem, the intersection number of P − is equal to 3p0 . Since the above intersection number is equal to that of in P + ∩ Φ − ∩ Φ −− − = P in Φ (see [73, p. 235, Exercise 3]), and Φ ++ + = P Φ ν ,σ ν ,σ we obtain the desired result.

9.3 Continuity of roots and continuity of intersections

229

++ + −− − and Φ The above proof shows that the intersection number of Φ ν ,σ ν ,σ − − + − in Φ is equal to that of the projective curves P ∩ Φ and P ∩ P in P ∼ = CP 2 . Hence, we obtain the following lemma by virtue of Lemma 9.3.3, which is proved in the next section. Lemma 9.2.5. Let ν t = (ν − , νt+ ) (t ∈ [0, 1]) be a continuous path in H2 × H2 such that νt+ ∈ σ + for every t ∈ [0, 1] and ν − ∈ σ − . Then there are continuous ˜ (1 ≤ j ≤ 3p0 ) such that ϕ1 (t), · · · , ϕ3p (t) form the maps ϕj : [0, 1] → Φ 0 − ++ counted with multiplicity for every t ∈ [0, 1]. intersection of Φν − ,σ− and Φ ν ,σ + t

Lemma 9.2.6. Under the setting of Lemma 9.2.5, for each j ∈ {1, 2, · · · , 3p0 }, if ϕj (t) is a geometric root for ν t for some t ∈ [0, 1], then ϕj (t) is a geometric root for ν t for every t ∈ [0, 1].

Proof. Fix an integer j ∈ {1, 2, · · · , 3p0 }, and let J be the subset of [0, 1] consisting of the point t such that ϕj (t) is a geometric root for ν t . Then J is open by Proposition 6.2.1 (Openness). To show the closedness of J, let t∞ ∈ [0, 1] be a limit point of J, i.e., there is a sequence {tn } in J such that t∞ = lim tn . Then, by Proposition 6.2.7 (Convergence), we may assume ϕj (t∞ ) = ˜ Hence, by Proposition 6.2.4 lim ϕj (tn ) is contained in the aﬃne part Φ of Φ. (Closedness), we see that (ρ∞ , ν t∞ ) is a good labeled representation, where ρ∞ is the type-preserving representation induced by ϕj (t∞ ). Thus ϕj (t∞ ) is a geometric root for ν t∞ , and therefore J is also closed. Hence we obtain the desired result. As a corollary to Lemmas 9.2.2, 9.2.5 and 9.2.6, we obtain the following result. Corollary 9.2.7. Let ν j = (νj− , νj+ ) (j = 0, 1) be elements of H2 × H2 such that ν0− = ν1− and that ν0+ and ν1+ are contained in a common triangle for each ǫ ∈ {0, 1}. Then dG (ν 0 ) = dG (ν 1 ). We can now complete the proof of Proposition 9.2.3 as follows. Let Σ(ν) = {σ1 , · · · , σm }, and pick points νj+ ∈ int(σj ∩ σj+1 ) for each j ∈ {1, · · · , m − 1}. By repeatedly using Corollary 9.2.7, we have + + ) = dG (ν − , νm−2 ) = · · · = dG (ν − , ν1+ ) dG (ν − , ν + ) = dG (ν − , νm−1

= dG (ν − , ν − ).

By Proposition 5.1.5 and Lemma 5.2.12, we see dG (ν − , ν − ) = 1. Hence we have dG (ν) = 1. This completes the proof of Proposition 9.2.3.

9.3 Continuity of roots and continuity of intersections In this section, we prove Lemma 9.3.3, which is used in Sect. 9.2. Though it is certainly well-known to the experts, we could not ﬁnd a proof in the literature.

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Lemma 9.3.1. Let ft (z) = z n + a1 (t)z n−1 + · · · + an (t) (t ∈ [0, 1]) be a continuous family of polynomials with complex coeﬃcients of a ﬁxed degree n. Then there are continuous maps ψj : [0, 1] → C (1 ≤ j ≤ n) such that {ψ1 (t), · · · , ψn (t)} is equal to the set of roots of ft (z) counted with multiplicity for every t ∈ [0, 1]. Proof. Let Pn be the space of monic polynomials of degree n and identify Pn with Cn by the correspondence f (z) = z n + a1 z n−1 + · · · + an → (a1 , a2 , · · · , an ). Recall that if ω1 , · · · , ωn are roots of f (z), then the coeﬃcient ad is equal to (−1)d sd (ω1 , · · · , ωn ), where sd is the n-variable elementary symmetric function of degree d, namely zj1 zj2 · · · zjd . sd (z1 , · · · , zn ) = 1≤j1 <j2 <···<jd ≤n

Consider the map Cn → Cn = Pn deﬁned by (z1 , · · · , zn ) → (−s1 (z1 , · · · , zn ), s2 (z1 , · · · , zn ), · · · , (−1)n sn (z1 , · · · , zn )). Then by the fundamental theorem of algebra (and the fact that the polynomial ring C[z] is a unique factorization domain), it induces a homeomorphism from Cn /Sn , the quotient of Cn by the canonical action of the symmetric group Sn of degree n, to Pn . By the path lifting theorem [19, Theorem 6.2 in Chap. II], every path in Pn ∼ = Cn /Sn lifts to a path in Cn . The lifted path gives rise to the desired family of continuous roots of the polynomials. Remark 9.3.2. The authors got the idea of the proof from [62]. However, Theorem 1 in the paper, which claims the existence of a global continuous section to the projection Cn → Cn /Sn , is not correct. We thank Michael Heusener for informing us of the paper and pointing out the error. We also thank Norbert A’Campo for proving the path lifting property for this special case, before we found the reference [19]. Lemma 9.3.3. Let Ft (x1 , x2 , x3 ) (t ∈ [0, 1]) be a continuous family of homogeneous polynomials with complex coeﬃcients of a ﬁxed degree, and let Vt be the projective curve deﬁned by Ft (x1 , x2 , x3 ). Let W be a projective curve which does not share a component with Vt for every t ∈ [0, 1], and assume that the intersection number of Vt and W is a constant number, n, independent of t. Then there are continuous maps ϕj : [0, 1] → CP 2 (1 ≤ j ≤ n) such that {ϕ1 (t), · · · , ϕn (t)} is equal to the intersection Vt ∩W counted with multiplicity for every t ∈ [0, 1]. Proof. We ﬁrst show that the lemma is valid locally, namely, for every t0 ∈ [0, 1], there are a connected neighborhood U of t0 in [0, 1] and continuous

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maps ϕj : U → CP 2 (1 ≤ j ≤ n) such that {ϕ1 (t), · · · , ϕn (t)} is equal to the intersection Vt ∩ W counted with multiplicity for every t ∈ U . For notational simplicity we prove this local assertion for t0 = 0. Let p be a point in V0 ∩ W , B a branch of W at p, and d be the intersection multiplicity of V0 ∩ B at p. Let w be a local parameter of W at p, namely, w is an injective holomorphic map from D(ǫ1 ) := {z ∈ C | |z| < ǫ} for some ǫ1 > 0 to B ⊂ W ⊂ CP 2 such that w(0) = p. We may assume that w is given by w(z) = [x1 (z) : x2 (z) : 1] ∈ CP 2 . Then Ft (x1 (z), x2 (z), 1) is a continuous function [0, 1] × D(ǫ1 ) → C which is holomorphic in the variable z. By (the proof of) Weierstrass preparation theorem, there are positive numbers δ1 and ǫ2 (< ǫ1 ), a continuous map u : [0, δ1 ] × D(ǫ2 ) → C and continuous maps a1 , · · · , ad : [0, δ1 ] → C, such that Ft (x1 (z), x2 (z), 1) = u(t, z)ft (z) where ft (z) = z d + a1 (t)z d−1 + · · · + ad (t), u(0, 0) = 0, aj (0) = 0 (1 ≤ j ≤ d), and u(t, z) is holomorphic in z. By Lemma 9.3.1, there are continuous maps ψj : [0, δ1 ] → C (1 ≤ j ≤ d) such that {ψ1 (t), · · · , ψd (t)} is equal to the set of roots of ft (z) counted with multiplicity for every t ∈ [0, 1]. Pick small positive numbers δ2 (< δ1 ) and ǫ3 (< ǫ2 ) such that (i) u(t, z) = 0 for every t ∈ [0, δ2 ) and z ∈ D(ǫ3 ) and (ii) {ψ1 (t), · · · , ψd (t)} ⊂ D(ǫ3 ) for every t ∈ [0, δ2 ). Though δ2 depends on the point p ∈ V0 ∩ W and the branch B of W at p a priori, we may choose it so that it is common to all such p and B. Then the union of the sets {w(ψ1 (t)), · · · , w(ψd (t))} for all p and B form the set of the intersection Vt ∩W counted with multiplicity for every t ∈ [0, δ2 ). This completes the proof of the local assertion. To prove the global assertion, let J be the maximal connected subset of [0, 1] containing 0 for which the conclusion of the lemma holds, i.e., there are continuous maps {ϕj : J → CP 2 } (1 ≤ j ≤ n) such that Vt ∩ W = {ϕ1 (t), · · · , ϕn (t)} counted with multiplicity for every t ∈ J. By the local assertion proved in the above, J is open. To show that J is closed, set t∞ = sup J. Since the intersection number of Vt and W is the constant n, there are n points pj,∞ (1 ≤ j ≤ n) of CP 2 , such that Vt ∩ W = {p1,∞ , · · · , pn,∞ } counted with multiplicity. By the local assertion, there are continuous maps {ϕj,∞ : (t∞ − δ, t∞ ] → CP 2 } for some δ > 0 such that ϕj,∞ (t∞ ) = pj,∞ for each j and Vt ∩ W = {ϕ1,∞ (t), · · · , ϕn,∞ (t)} counted with multiplicity for each t ∈ (t∞ − δ, t∞ ]. Put t′∞ = t∞ − (δ/2) ∈ (t∞ − δ, t∞ ]. Then after changing the indices, we may assume ϕj (t′∞ ) = ϕj,∞ (t′∞ ) for every j. Redeﬁne ϕj : [0, t∞ ] → CP 2 so that its restriction to [0, t′∞ ] is equal to the original one and its restriction to [t′∞ , t∞ ] is equal to the restriction of ϕj,∞ . Then each ϕj is continuous and Vt ∩ W = {ϕ1 (t), · · · , ϕn (t)} counted with multiplicity for every t ∈ [0, t∞ ]. Thus J is closed. Hence J is equal to the interval [0, 1] and we obtain the desired result.

A Appendix

A.1 Basic facts concerning the Ford domain In this appendix, we give a proof to some of the basic facts concerning the Ford domain, the proof of which could not be found in the literature. Throughout this appendix, Γ denotes a non-elementary Kleinian group, such that the stabilizer Γ∞ of ∞ contains parabolic transformations, and H∞ denotes a ﬁxed horoball centered at ∞ which is precisely (Γ, Γ∞ )-invariant. The following well-known observation plays an important role in this section. Lemma A.1.1. (1) The Euclidean radii of the horoballs in Γ H∞ − {H∞ } is bounded from the above. (2) For any compact subset K in H3 , only ﬁnitely many horoballs in Γ H∞ intersect K. Proof. (1) follows from the fact that all horoballs in Γ H∞ − {H∞ } are disjoint from H∞ and hence their Euclidean radii are less than the half of the “Euclidean height”, t, of ∂H∞ , where t is the positive real number such that ∂H∞ = C × {t} ⊂ H3 . (2) Since K is a compact subset of H3 , its Euclidean height is bounded below, i.e., there is a positive constant c such that K ⊂ C × [c, ∞) ⊂ H3 . On the other hand, by virtue of (1), there is a compact subset, L, of C which contains the centers of all horoballs in Γ H∞ − {H∞ } intersecting K. (If r is the upper bound obtained in (1), then we may set L to be the closed r-neighborhood of proj(K) in C with respect to the Euclidean metric.) Thus if a horoball in Γ H∞ − {H∞ } intersects K, then its Euclidean radius is ≥ c/2 and its center is contained in the compact set L. Since Γ H∞ consists of disjoint horoballs, only ﬁnitely many of its members satisfy these conditions. Hence we obtain the conclusion. We rephrase Proposition 1.1.3. part of which is proved in [72, Lemma 5.20] under the additional assumption that ∞ is a bounded parabolic ﬁxed point.

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Proposition A.1.2. The Ford domain P h(Γ ) is a “fundamental polyhedron of Γ modulo Γ∞ ”, in the following sense. 1. H3 = ∪{A(P h(Γ )) | A ∈ Γ }. 2. int P h(Γ ) is precisely (Γ, Γ∞ )-invariant. 3. For any compact set K of H3 , only ﬁnitely many images A(P h(Γ )) (A ∈ Γ ) can intersect K, namely the set {AΓ∞ ∈ Γ/Γ∞ | A(P h(Γ )) ∩ K = ∅} is ﬁnite. 4. P h(Γ ) is a closed convex polyhedron (Deﬁnition 3.4.1(2)). Proof. (1) By applying Lemma A.1.1(1), to a (hyperbolic) closed ball B(x, r) with center x ∈ H3 and radius r > 0, we see that the minimal distance d(x, Γ H∞ ) := min{d(x, AH∞ ) | A ∈ Γ } is well-deﬁned, i.e., there is an element Ax ∈ Γ such that d(x, Ax H∞ ) ≤ d(x, AH∞ ) for every A ∈ Γ . This implies that A−1 x (x) ∈ P h(Γ ) = {x ∈ H3 | d(x, H∞ ) = d(x, Γ H∞ )}. Hence we have H3 = ∪{A(P h(Γ )) | A ∈ Γ }. (2) This is a direct consequence of Lemma 4.1.1(1). (3) We show that for every x ∈ H3 and ǫ > 0, the compact set B(x, ǫ) satisﬁes the conclusion. To this end, put r = d(x, Γ H∞ ). By Lemma A.1.1(2), B(x, r + 2ǫ) intersects only ﬁnitely many horoballs A1 (H∞ ), A2 (H∞ ), · · · , An (H∞ ) in Γ H∞ . Suppose A(P h(Γ )) ∩ B(x, ǫ) = ∅ for some A ∈ Γ . Pick a point y from the intersection. Then d(x, A−1 (H∞ )) ≤ ǫ + d(y, A−1 (H∞ )) = ǫ + d(y, Γ H∞ ) ≤ 2ǫ + d(x, Γ H∞ ) = 2ǫ + r. Here the identity in the above follows from the assumption that y ∈ A(P h(Γ )). Hence B(x, r + 2ǫ) ∩ A(H∞ ) = ∅ and therefore A−1 (H∞ ) = Aj (H∞ ) for some j. This implies A−1 (P h(Γ )) = Aj (P h(Γ )). Hence the desired result holds for the compact set B(x, ǫ). (4) Let K be a compact subset of C, and consider the isometric hemispheres Ih(A) (A ∈ Γ − Γ∞ ) which intersect K. Then as in the proof of Lemma A.1.1(2), we see that their Euclidean radii are bounded below, their centers are contained in a compact subset of C, and that the radii of the corresponding horoballs A(H∞ ) are bounded below. Hence we see that there are only ﬁnitely many such isometric hemispheres. (We can also deduce this by using Lemma 2.5.2(2)). Hence P h(Γ ) satisﬁes the condition for a convex polyhedron in Deﬁnition 3.4.1(2). The following lemma is proved by imitating the argument of [55, Proof of Lemma 2.13] for the Dirichlet domain (cf. [72, Lemma 5.37]). Lemma A.1.3. For every point ξ in the Ford polygon P (Γ ), there is a horoball Hξ centered at ξ such that Hξ ∩ H∞ is a singleton and (int Hξ ) ∩ Γ H∞ = ∅. In particular, any point of P (Γ ) ∩ Λ(Γ ) is not a horospherical limit point of Γ ([55, Deﬁnition in p.51]).

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Proof. Let ξ be a point in P (Γ ). Then the vertical geodesic (ξ, ∞) is contained in P h(Γ ). Hence, for each x ∈ (ξ, ∞) − H∞ , int B(x, d(x, H∞ )) is disjoint from Γ H∞ . Thus the open horoball ∪{int B(x, d(x, H∞ )) | x ∈ (ξ, ∞) − H∞ }, centered at ξ, is disjoint from Γ H∞ . Moreover its closure, Hξ , intersect H∞ precisely at the point ∂H∞ ∩ (ξ, ∞). Thus we obtain the ﬁrst assertion. To see the second assertion, pick a point x from int H∞ . Then its orbit Γ x is disjoint from Hξ . Hence ξ is not a horospherical limit point. Corollary A.1.4. If Γ is geometrically ﬁnite, then any point of P (Γ ) ∩ Λ(Γ ) is the (parabolic) ﬁxed point of a parabolic element of Γ which is not conjugate to an element of Γ∞ . In particular, if Γ is a quasifuchsian punctured torus group, then P (Γ ) ⊂ Ω(Γ ). Proof. Suppose Γ is geometrically ﬁnite and let ξ be a point in P (Γ ) ∩ Λ(Γ ). Then ξ is not a horospherical limit point and hence it is a bounded parabolic ﬁxed point (see [55, Theorem 3.7]). On the other hand, we see that the orbit Γ ∞ is disjoint from P (Γ ) as follows. Suppose to the contrary that A(∞) belongs to P (Γ ) = P h(Γ ) ∩ C. Then A(int H∞ ) ∩ int P h(Γ ) = ∅ and hence A ∈ Γ∞ by Proposition A.1.2(2). Thus ∞ = A(∞) ∈ P (Γ ), a contradiction. Hence ξ does not belong to the orbit Γ ∞, and we obtain the ﬁrst assertion. The second assertion follows from the fact that every parabolic transformation of a quasifuchsian punctured torus group is conjugate to an element of Γ∞ . The following lemma is a reﬁnement of Lemma A.1.3. Lemma A.1.5. A point ξ ∈ C belongs to P (Γ ), if and only if there is a horoball Hξ centered at ξ such that Hξ ∩ Γ H∞ = ∅ and d(Hξ , H∞ ) ≤ d(Hξ , A(H∞ )) for every A ∈ Γ . Proof. Suppose that ξ belongs to P (Γ ). Then by Lemma A.1.3, there is a horoball Hξ centered at ξ such that Hξ ∩ H∞ is a singleton and (int Hξ ) ∩ Γ H∞ = ∅. Reset Hξ to be a horoball contained in the interior of this horoball. Then we see that this new horoball Hξ satisﬁes the desired conditions. Conversely, suppose that the latter condition is satisﬁed. We show that (ξ, ∞) ∩ Hξ ⊂ P h(Γ ). To this end, pick a point x ∈ (ξ, ∞) ∩ Hξ and A ∈ Γ . Let γ be the shortest geodesic segment joining x to A(H∞ ). Then d(x, A(H∞ )) = length(γ) = length(γ ∩ Hξ ) + length(γ ∩ (H3 − int Hξ )) ≥ d(x, ∂Hξ ) + d(Hξ , A(H∞ )) ≥ d(x, ∂Hξ ) + d(Hξ , H∞ )

= d(x, H∞ )

(This inequality is easily seen by performing a coordinate change so that Hξ is centered at ∞.) This implies x ∈ P h(Γ ). Thus we have (ξ, ∞) ∩ Hξ ⊂ P h(Γ ). Hence ξ ∈ P (Γ ).

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The above lemma immediately implies the following corollary. Corollary A.1.6. For each A0 ∈ Γ , a point ξ ∈ C belongs to A0 (P (Γ )), if and only if there is a horoball Hξ centered at ξ such that Hξ ∩ Γ H∞ = ∅ and d(Hξ , A0 (H∞ )) ≤ d(Hξ , A(H∞ )) for every A ∈ Γ . We now prove the following proposition. Proposition A.1.7. The intersection, P (Γ ) ∩ Ω(Γ ), of the Ford polygon and the domain of discontinuity is a “fundamental polygon of Γ modulo Γ∞ ”, for the action of Γ on Ω(Γ ) in the following sense. 1. Ω(Γ ) = ∪{A(P (Γ ) ∩ Ω(Γ )) | A ∈ Γ }. 2. int P (Γ ) = int(P (Γ ) ∩ Ω(Γ )) is precisely (Γ, Γ∞ )-invariant. 3. For any compact set K of Ω(Γ ), only ﬁnitely many images A(P (Γ )) (A ∈ Γ ) can intersect K, namely the set {AΓ∞ ∈ Γ/Γ∞ | A(P (Γ )) ∩ K = ∅} is ﬁnite. Proof. (1) Let ξ be a point in Ω(Γ ). Then some neighborhood of ξ in C is disjoint from the centers of the horoballs in Γ H∞ , because they are contained in Λ(Γ ). Since the Euclidean radii of the horoballs in Γ H∞ − {H∞ } are 3 bounded above, we can ﬁnd a closed neighborhood D of ξ in H such that D ∩ Γ H∞ = ∅. Claim A.1.8. Any horoball Hξ centered at ξ ∈ Ω(Γ ) can intersect only ﬁnitely many horoballs in Γ H∞ . Proof. Let Hξ be a horoball centered at ξ. Then the closure of Hξ − D in H3 is compact and a horoball in Γ H∞ intersects Hξ if and only if it intersects the relatively compact set Hξ − D. Hence we have the claim by Lemma A.1.1(2). Now pick a small horoball Hξ centered at ξ contained in D. Then by applying the above claim to a closed r-neighborhood of Hξ in H3 , which is again a horoball centered at ξ, for suﬃciently large r, we see that d(Hξ , Γ H∞ ) := min{d(Hξ , AH∞ ) | A ∈ Γ } is a well-deﬁned positive number, i.e., there is an element Aξ ∈ Γ such that d(Hξ , Aξ (H∞ )) ≤ d(Hξ , A(H∞ )) for every A ∈ Γ . Thus we see ξ ∈ Aξ (P (Γ )) by Corollary A.1.6, and hence A−1 ξ (ξ) ∈ P (Γ ) ∩ Ω(Γ ). So we have Ω(Γ ) = ∪{A(P (Γ ) ∩ Ω(Γ )) | A ∈ Γ }. (2) This is a direct consequence of Lemma 4.1.1(1). (3) Let K be a compact subset of Ω(Γ ). Then by the argument in the 3 proof of (1), there is a compact neighborhood D of K in H such that D ∩ Γ H∞ = ∅. Pick a constant c > 0 such that each horoball, Hξ , with Euclidean radius c centered at a point ξ ∈ K is contained in int(D ∩ H3 ). Throughout the proof we reserve the symbol Hξ to denote these horoballs, and set HK = ∪{Hξ | ξ ∈ K}. By using the compactness of K, we can

A.1 Basic facts concerning the Ford domain

237

ﬁnd a constant r > 0 such that d(Hξ , Γ H∞ ), which is well-deﬁned by (the proof of) (1), is at most r for every ξ ∈ K. Consider the closed r-neighborhood B(HK , r) of HK in H3 . Then B(HK , r) − D is relatively compact in H3 . Thus we see, by Lemma A.1.1(2), that B(HK , r) intersects only ﬁnitely many horoballs, A1 (H∞ ), A2 (H∞ ), · · · , An (H∞ ), in Γ H∞ . Now suppose K∩A(P (Γ )) = ∅. Pick a point ξ from the intersection. Then, by Corollary A.1.6, d(Hξ , A(H∞ )) = d(Hξ , Γ H∞ ) ≤ r. Hence B(HK , r)∩A(H∞ ) = ∅. Thus A(H∞ ) = Aj (H∞ ) for some j. This implies A(P (Γ )) = Aj (P (Γ )). Hence we obtain the desired result. Remark A.1.9. (1) Since two “edges” of fr P (Γ ) may be tangent, fr P (Γ ) is not necessarily a 1-dimensional manifold. (2) fr P (Γ )∩Ω(Γ ) is locally ﬁnite in the sense that any point x ∈ fr P (Γ )∩ Ω(Γ ), there is a neighborhood U of x in Ω(Γ ) such that fr P (Γ ) ∩ U is a ﬁnite union of circular arcs. However, fr P (Γ ) is not necessarily locally ﬁnite around points in fr P (Γ ) ∩ Λ(Γ ). At the end of this appendix, we prove the following ﬁniteness property for the Ford domain. Lemma A.1.10. For a point p in P h(Γ ), let [p] be the set of points in P h(Γ ) which are Γ -equivalent to p, namely [p] = Γ p ∩ P h(Γ ) = {x ∈ P h(Γ ) | x = A(p) for some A ∈ Γ }. Then the quotient set [p]/Γ∞ is ﬁnite provided that p ∈ H3 ∪ Ω(Γ ) or p is a bounded parabolic ﬁxed point of Γ . In particular, if Γ is geometrically ﬁnite, then [p]/Γ∞ is a ﬁnite set for every p ∈ P h(Γ ). Proof. We prove the lemma only for the case when p is a point in C (and hence in P (Γ )). (The proof for the case p ∈ H3 is parallel to this case and is much simpler.) Let ξ be a point in P (Γ ), and let Hξ be a horoball centered at ξ satisfying the condition in Lemma A.1.5. Then r := d(Hξ , Γ H∞ ) is welldeﬁned and is equal to d(Hξ , H∞ ). By Corollary A.1.6, for each A ∈ Γ , we have A(ξ) ∈ P (Γ ) if and only if d(Hξ , A−1 (H∞ )) = r. The latter condition holds if and only if A−1 (H∞ ) intersects the horoball B(Hξ , r). Hence we see [p] = {A(p) | B(Hξ , r) ∩ A−1 (H∞ ) = ∅}. Now suppose that ξ ∈ Ω(Γ ). Then by Claim A.1.8, only ﬁnitely many horoballs in Γ H∞ can intersect B(Hξ , r). Moreover, for two elements A1 and −1 −1 −1 A2 of Γ , A−1 1 (H∞ ) = A2 (H∞ ) if and only if A1 Γ∞ = A2 Γ∞ ∈ Γ/Γ∞ . Hence the set {A−1 Γ∞ ∈ Γ/Γ∞ | B(Hξ , r) ∩ A−1 (H∞ ) = ∅} is ﬁnite. By the observation in the previous paragraph, the correspondence A−1 Γ∞ → Γ∞ A(ξ) determines a surjective map from the above ﬁnite set to

238

A Appendix

the quotient set [p]/Γ∞ . (To be precise, the symbol [p]/Γ∞ should be denoted by Γ∞ \[p], because the action of Γ∞ on [p] is a left action.) Hence [p]/Γ∞ is ﬁnite. Next suppose that ξ is a bounded parabolic ﬁxed point. Then we can easily see that, modulo the action of the parabolic stabilizer Γξ of ξ, only ﬁnitely many horoballs in Γ H∞ can intersect B(Hξ , r). Thus the set {Γp A−1 Γ∞ ∈ Γp \Γ/Γ∞ | B(Hξ , r) ∩ A−1 (H∞ ) = ∅} is ﬁnite. Hence we obtain the ﬁniteness of [p]/Γ∞ as in the previous case.

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Notation

•

•

•

•

•

•

General topology – int X: the interior of X – fr X: the frontier of X – X: the closure of X – π1 (X): the fundamental group of X Cellular complex – C (k) : the k-skeleton of a complex C – lk(ξ, L): the link of ξ in L – st0 (ξ, L): the subcomplex of L spanned by ξ and lk(ξ, L) Abstract group – [X, Y ] := XY X −1 Y −1 – X Y := Y XY −1 – X: the normal closure of X Numbers – N: the set of natural numbers – Z: the set of integral numbers – Q: the set of rational numbers – R: the set of real numbers – C: the set of complex numbers ˆ the union Q ∪ {∞} – Q:

– C: the Riemann sphere C ∪ {∞} – [j]: the integer in {0, 1, 2} such that [j] ≡ j (mod 3) – ǫ: a sign − or + Surfaces – T : the once-punctured torus – S: the four-times punctured sphere – O: the (2, 2, 2, ∞)-orbifold over S 2 – Teich(X): the Teichm¨ uller space of a surface X Model spaces – Hn : the hyperbolic n-space

246

• •

•

•

Notation

– Hn : the closure, Hn ∪ ∂Hn , of Hn 3 – H : the closure, H3 ∪ C, of the upper half space model of H3 – E1,n : the (n + 1)-dimensional Minkowski space – P(H3 ): the space of convex polyhedra in H3 3 3 – P(H ): the space of convex polyhedra in H – E: the symbol to mean something which is Euclidean – C(X): the closed convex hull of X Matrix groups: for R = Z, R or C – SL(2, R): the set of all 2 × 2-matrices with determinant 1 – P SL(2, R) := SL(2, R)/{±1} Farey triangulation – D: the Farey triangulation (the modular diagram) – v0 , . . . , vk : the k-simplex in D spanned by vertices v0 , . . . , vk ∈ D(0) – s(X) ∈ D(0) : the slope of X, where X is a generator of π1 (T ) or an elliptic generator of π1 (O) – T : the binary tree dual to D − → – E (T ): the set of directed edges of T → – − e : a directed edge of T – EG: the set of elliptic generators – Σ = (σ1 , . . . , σm ): a chain of triangles – σ ǫ : the ǫ-terminal triangle of a chain – Σ(ν): the chain of triangles determined by a label ν – σ ǫ (ν): the ǫ-terminal triangle of Σ(ν) Ideal triangulation of T × [−1, 1] – trg(σ): the topological ideal triangulation of T determined by σ – trg(σ): the topological ideal triangulation of R2 − Z2 determined by σ – Trg(ν): the layer of topological ideal triangulations of T determined by ν – Trg(ν): the layer of topological ideal triangulations of R2 − Z2 determined by ν – spine(σ): the spine of T determined by σ – Spine(δ − , δ + ): the “trace of spines” of T in T × [−1, 1] For an element X of SL(2, C) or P SL(2, C) – tr X: the trace of X – Axis X: the axis of X ⊂ H3 3 – AxisX: the closure of Axis X in H – Fix X: the ﬁxed point of X in C – I(X): the isometric circle of X – E(X): the exterior of I(X) – Ih(X): the isometric hemisphere of X – Eh(X): the exterior of Ih(X) 3 – Ih(X): the closure of Ih(X) in H 3 – Eh(X): the closure of Eh(X) in H 3 – Dh(X): the closure of Dh(X) in H

Notation

•

•

•

•

247

– c(X): the center of I(X) – r(X): the radius of I(X) For a Kleinian group Γ – StabΓ (x): the stabilizer of x with respect to the action of Γ – Ω(Γ ): the domain of discontinuity of Γ – Ω ǫ (Γ ): the ǫ-component of Ω(Γ ) – Λ(Γ ): the limit set of Γ – M (Γ ): the hyperbolic manifold (or orbifold) H3 /Γ ¯ (Γ ): the quotient manifold (or orbifold) (H3 ∪ Ω(Γ ))/Γ – M

– P (Γ ): the Ford polygon of Γ in C – P h(Γ ): the Ford domain of Γ in H3 – P h(Γ ) := P h(Γ ) ∪ P (Γ ) ¯ (Γ ) – Ford(Γ ): the Ford complex of Γ in M – ∆E (Γ ): the Euclidean decomposition of M (Γ ) Spaces of representations – Homtp (π1 (X), P SL(2, C)): the space of all type-preserving P SL(2, C)representations of π1 (X) for X = T , O and S – X : = Homtp (π1 (X), P SL(2, C))/P SL(2, C) – ρ: a type-preserving representation – QF: the space of quasifuchsian representations ⊂ X – QF: the closure of QF in X – Φ: the space of Markoﬀ maps – φ: a Markoﬀ map – Ψ : the space of complex probabilities ǫ ǫ : the polynomial function Φ → C deﬁned by ζν,σ (φ) = φ(s0 ) + – ζν,σ ǫ ǫ α φ(s1 ) + β φ(s2 ) ǫ – Φǫν,σ : the subvariety (ζν,σ )−1 (0) of Φ – Φǫν : the “geometric” irreducible component of Φǫν,σ – dG (ν): the geometric degree of ν Side parameter – ν: the side parameter – ν ǫ (ρ): the ǫ-component of ν – ν: used to denote a label ν ∈ D – θǫ (ρ, σ): the ǫ-angle invariant of ρ at σ – θǫ (ρ, σ; s[j] ): the s[j] -component of θǫ (ρ, σ) – J [QF] ⊂ X × (H2 × H2 ): the space of good labeled representations – µ1 : J [QF] → X : the natural projection – µ2 : J [QF] → H2 × H2 : the natural projection – ρ = (ρ, ν): a labeled representation For a pair (ρ, σ) – I(j) = I(ρ(Pj )): the isometric circle of ρ(Pj ) – D(j) = D(ρ(Pj )): the disk bounded by I(ρ(Pj )) – E(j) = E(ρ(Pj )): the exterior of I(ρ(Pj )) – Ih(j) = Ih(ρ(Pj )): the isometric hemisphere of ρ(Pj )

248

•

•

•

Notation

– Dh(j) = Dh(ρ(Pj )): the half space of H3 bounded by Ih(ρ(Pj )) whose closure contains D(ρ(Pj )) – Eh(j) = Eh(ρ(Pj )): the half space H3 − int Dh(ρ(Pj )) – c(j) = c(ρ(Pj )): the center of I(ρ(Pj )) → → – − c (j, j + 1) = − c (ρ; Pj , Pj+1 ): the vector or oriented line c(ρ(Pj+1 )) − c(ρ(Pj )) – Fixǫ (j) = Fixǫσ (ρ(Pj )): the ﬁxed point of ρ(Pj ) which lies in the ǫ-side of L(ρ, σ) – Axis(j) = Axis(ρ(Pj )): the axis of ρ(Pj ) −−−−−−−−−−−−−+−−−−−→ − → – f (j): the oriented line Fix− σ (ρ(Pj )) Fixσ (ρ(Pj )). – v ǫ (j, j + 1) = v ǫ (ρ; Pj , Pj+1 ): the point of I(ρ(Pj )) ∩ I(ρ(Pj+1 )) which lies in the ǫ-side of L(ρ, σ) – eǫ (j) = eǫ (ρ, σ; Pj ): the ǫ-ideal edge in I(ρ(Pj )) (Notation 4.3.7) – ∆ǫj = ∆ǫj (ρ, σ): “the j-th triangle” in the ǫ-side of L(ρ, σ) (Deﬁnition 4.2.10) – ∆(ρ, σ): the model triangle for ∆ǫj (ρ, σ)’s – fρǫ (ξ): the ǫ-ideal face determined by ρ and ξ – α(ρ, σ; sj ): the (inner) angle of the triangle at the vertex wj Elliptic generator complex – L(ρ, σ): a bi-inﬁnite broken line in C determined by ρ and σ – L(ρ, Σ): the union of bi-inﬁnite broken lines L(ρ, σ) in C for triangles σ in a chain Σ – L(ρ): the set L(ρ, Σ(ν)) for ρ = (ρ, ν) – L(σ): an abstract bi-inﬁnite broken line determined by σ – L(Σ): the elliptic generator complex associated with Σ – L(ν): the elliptic generator complex associated with Σ(ν) – L∗ (ν): the augmentation of L(ν) ǫ L∗ (ν): the “ǫ-boundary” of L∗ (ν) – ∂aug Dual map from L to ∂Eh – Fρ : L(ν)(≤2) → P(H3 ): the dual map to H3 – Fρ (ξ): the image of ξ by Fρ 3 3 – F ρ : L(ν)(≤2) → P(H ): the dual map to H – F ρ (ξ): the image of ξ by F ρ Quotient by the action of K – Cusp(K): = H3 /ρ(K) 3 – Cusp(K): = H /ρ(K) – ∂Cusp(K): = C/ρ(K) 3 – qK : H → Cusp(K): the projection – EhK (ρ) := qK (Eh(ρ)) ⊂ Cusp(K) – EK (ρ) := qK (E(ρ)) ⊂ ∂Cusp(K) – FK,ρ (ξ) := qK (Fρ (ξ))

Index

α(ρ, σ; sj ), 60 αr , 16 α ˜ r , 16 Axis(j), 58 → − − c (ρ; Pj , Pj+1 ) = → c (j, j + 1), 38, 58 c(·), 58 Ct , 146 Cusp(K), 143 Cusp(K), 143 ∂Cusp(K), 143 D(·), 3, 58 D, 7 |D|, 12 ∆ǫj , 59 ∆ǫj (ρ, σ), 59 ∆(ρ, σ), 59 ∆E (·), 4, 5, 26 δ(·), 13 dG (·), 227 Dh(·), 3, 58 Dh(·), 3 D(·) , 8 |D(·) |, 12 E(·), 3, 45, 58 E ǫ (·), 46 eǫ (ρ, σ; Pj ) = eǫ (j), 69 → − E (T ), 31 EG, 17 EG(Σ), 20 EG(σ), 20 Eh(·), 3, 45, 58 Eh(·), 3 EhK (·), 143

EK (·), 143 → − E φ (T ), 31 F ρ , 46 Fρ , 46 fρǫ (ξ), 160 → − f (j), 58 ǫ , 217 Fν,σ Fixǫσ (ρ(Pj )) = Fixǫ (j), 57, 58 FK,· , 143 FK,· (·), 143 Ford(·), 4, 26 Ford(·), 4 HL (·), 40 HR (·), 40 3 H ,3 Hn , 45 Homtp (·, P SL(2, C)), 21 Homtp (·, SL(2, C)), 27 I(·), 3, 58 Ih(·), 3, 58 Ih(·), 3 Int(·, ·), 227 [j], 19 J [QF ], 135 K, 16, 42 L(·), 42–44, 160 L(·)(k) , 42 L∗ (ν), 160 L(·) = L(·, ·), 38, 42–44 |L(·, ·)|, 38 L0 , 136 ǫ ∂aug L∗ (ν), 160 ∂ ǫ L(ν), 160

250

Index

L(·)/K, 42, 43 L(·)/K 2 , 42, 43 lk(·, ·), 46 {Lt }t∈(−1,1) , 145 L±1 , 146 MO (ρ), 144 µ1 : J [QF ] → X , 135 µ2 : J [QF ] → H2 × H2 , 135 ν(·), 12 O, 16 P (·), 3 p : St → Tt , 146 P h(·), 3, 26 P h(·), 3 Φ, 27, 215 Φǫν,σ , 216, 217 ˇǫν,σ , 219 Φ ˇǫν , 221 Φ Φ, 228 ǫν ǫ ,σǫ , 228 Φ φ = φρ˜, 27 {Pj }, 20, 21 {Pj′ }, 21 Ψ , 32 ǫ , 219 Ψν,σ ˆ Q, 7 QF , 2 QF , 2 3 qK : H → Cusp(K), 143 r(·), 3 S, 16 s(·), 17 Σ(·), 13, 44 Σ ∗ (·), 160 Σ0 , 136 σ ǫ (·), 44 σ ǫ,∗ (·), 157 Spine(·), 13 Spine(·, ·), 8 spine(·), 8 St , 146 st0 (·, ·), 46 T , 2, 16 T , 31 τ , 21 τ (·), 44 θǫ (ρ, σ; s[j] ), 59 θǫ (ρ, σ), 59

Trg(·), 14 trg(·), 8 ·), 13 Trg(·, t rg(·), 13 Tt , 146 v ǫ (ρ; Pj , Pj+1 ) = v ǫ (j, j + 1), 57, 58 X , 25 X˜ , 26 ǫ , 65, 215 ζν,σ algebraic root, 65, 216 associated (elliptic) generator triple, 17 Brenner (representation), 119 canonical (spine), 6 chain, 13, 41 chain rule (for isometric circles), 51 complete geodesic, 45 complex probability, 29, 31, 33 complex probability map, 31 convex polyhedron, 45 convex to the (+)-side, 40 convex to the (−)-side, 40 convex to the above, 40 convex to the below, 40 cut locus, 4 directed edge, 31 doubly folded (bi-inﬁnite broken line), 95 (geometric) dual, 4 dual (to an oriented simplex), 31 (condition) Duality, 134 elementary transformation, 8 elliptic generator, 17 elliptic generator complex, 42–44 elliptic generator pair, 20 elliptic generator triple, 17 EPH-decomposition, 6 equivalent (punctured torus groups), 2 equivalent (spines), 13 essential (simple arc), 7, 16 essential (simple loop), 7, 16 Euclidean (hyperplane), 6 Euclidean decomposition, 6 extreme edge, 184 Farey triangulation, 7

Index ﬂat (union of bi-inﬁnite broken lines), 78 folded (bi-inﬁnite broken line), 95 Ford complex, 4, 6 Ford domain, 3 Ford polygon, 3 Fricke diagram, 16 Fricke surface, 16 (condition) ǫ-Frontier, 134 (condition) Frontier, 135 (left/right) generator, 17 generator pair, 16 generator triple, 17 generic (leaf), 146 generic edge, 184 generic label, 157 geodesic segment, 45 geometric multiplicity, 227 geometric root, 216 good (labeled representation), 135 half-geodesic, 45 (condition) HausdorﬀConvergence, 182 head (of a directed edge), 31 hidden isometric hemisphere, 155 imaginary (representation), 126 inner (leaf of foliation), 145 isometric circle, 3 isometric hemisphere, 3 isosceles (representation), 107 Knapp (representation), 120 label, 44 label (of a labeled representation), 44 labeled representation, 44 link (of a simplex), 46 marked fuchsian punctured torus group, 2 marked punctured torus group, 2 marked quasifuchsian punctured torus group, 2 Markoﬀ map, 27 modular diagram, 7 monogon, 16 (condition) NonZero, 134

251

NonZero, 44 normalized (representation), 32 OPTi, 10 pivot, 41 precisely invariant, 3 predecessor, 41 quasifuchsian (labeled representation), 134 quasifuchsian punctured torus space, 2 real (representation), 102 SameStratum, 136 sequence of elliptic generators, 20, 29 side parameter, 12, 59 simple (broken line), 38 simple (union of bi-inﬁnite broken lines), 41, 43, 78 folded (bi-inﬁnite broken line), 95 slope, 16, 17 slope (of an edge), 8 slope (of circle/arc), 7 spine, 6 spine (of T ), 8 star (of a simplex), 46 successor, 41 tail (of a directed edge), 31 terminal triangle, 44, 69 thick (label), 44 thick (labeled representation), 44 thick (chain/label), 13 thick chain, 41 thin (label), 44 thin (labeled representation), 44 thin (chain/label), 13 thin chain, 41 topological ideal polygonal decomposition, 8 topological ideal triangulation, 8 triangle inequality (for a representation), 57 trivial Markoﬀ map, 27 type-preserving (representation of π1 (S)), 21 type-preserving (representation of π1 (T )/π1 (O)), 21, 26

252

Index

upward (Markoﬀ map), 38

vertical geodesic, 30 virtual Ford domain, 45

weakly simple, 38 weight (on edge), 12 weight system (on spine), 12 weighted relative spine, 13 weighted spine (of T ), 12

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