DOCSLIB.ORG

Quasi-Isometries of Graph Manifolds Do Not Preserve Non-Positive Curvature

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Quasi-Isometries of Graph Manifolds Do Not Preserve Non-Positive Curvature QUASI-ISOMETRIES OF GRAPH MANIFOLDS DO NOT PRESERVE NON-POSITIVE CURVATURE DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Andrew Nicol, MS Graduate Program in Mathematics The Ohio State University 2014 Dissertation Committee: Jean-Fran¸coisLafont, Advisor Brian Ahmer Nathan Broaddus Michael Davis c Copyright by Andrew Nicol 2014 ABSTRACT Continuing the work of Frigerio, Lafont, and Sisto [8], we recall the definition of high dimensional graph manifolds. They are compact, smooth manifolds which decompose into finitely many pieces, each of which is a hyperbolic, non-compact, finite volume manifold of some dimension with toric cusps which has been truncated at the cusps and crossed with an appropriate dimensional torus. This class of manifolds is a generalization of two of the geometries described in Thurston's geometrization conjecture: H3 and H2 × R. In their monograph, Frigerio, Lafont, and Sisto describe various rigidity results and prove the existence of infinitely many graph manifolds not supporting a locally CAT(0) metric. Their proof relies on the existence of a cochain having infinite order. They leave open the question of whether or not there exists pairs of graph manifolds with quasi-isometric fundamental group but where one supports a locally CAT(0) metric while the other cannot. Using a number of facts about bounded cohomology and relative hyperbolicity, I extend their result, showing that there exists a cochain that is not only of infinite order but is also bounded. I also show that this is sufficient to construct two graph manifolds with the desired properties. ii ACKNOWLEDGMENTS I wish to acknowledge all of the hard work put forth by my advisor, Jean-Fran¸cois Lafont. His wisdom, insights, and patience were greatly appreciated. iii VITA 2007 . B.A. in Mathematics, The College of Wooster 2011 . MA in Mathematics, The Ohio State University 2007-Present . Graduate Teaching Associate, The Ohio State University FIELDS OF STUDY Major Field: Mathematics Specialization: Geometric Group Theory iv TABLE OF CONTENTS Abstract . ii Acknowledgments . iii Vita . iv List of Figures . vii CHAPTER PAGE 1 Introduction . 1 2 Background . 5 2.1 Principle S1-Bundles and the Euler Class . 5 2.1.1 Universal Bundles . 7 2.2 Hyperbolic Manifolds . 9 2.3 Quasi-Isometries . 10 2.4 Graph Manifolds . 12 2.5 Various Cohomology Theories . 15 2.5.1 Group Cohomology . 16 2.5.2 Bounded Cohomology . 18 2.6 A proof of lemma 1.3 . 22 3 Relative Hyperbolicity and Isolated Flats . 25 4 Exploring some Bounded Cohomology . 34 5 Condition (b) . 42 6 Completing the Proof . 48 APPENDICES A Various Definitions of Relatively Hyperbolic Groups . 50 v Bibliography . 56 vi LIST OF FIGURES FIGURE PAGE 3.1 One chamber of DAg ........................... 30 vii CHAPTER 1 INTRODUCTION In what follows, we will show the following Theorem 1.1. There exists infinitely many examples of pairs of graph manifolds with quasi-isometric fundamental groups, with the property that one of them supports a locally CAT(0) metric, but the other one cannot support any locally CAT(0) metric. Furthermore, these examples arise in every dimension n ≥ 4. This result was inspired by the following question, posed by Roberto Frigerio, Jean-Francois Lafont, and Alessandro Sisto in [8]: Problem 1.2. Is there a pair of irreducible graph manifolds with quasi-isometric fundamental groups, with the property that one of them supports a locally CAT(0) metric, but the other one cannot support any locally CAT(0) metric? There is already some progress on this question for the classical definition of graph manifold following the work of Gersten (see definition 2.15). He looks at the unit tangent bundle X, of a surface Σ of genus g ≥ 2. This is a principal S1-bundle. Using the Milnor-Wood inequality following [18] and [26], the S1-bundle yields a bounded characteristic class for the central extension Z ! π1(X) ! π1(Σ) from the long exact sequence of fiber bundles. Another result of Gersten [9] tells us that fundamental group of the unit tangent bundle is quasi-isometric to the fundamental 1 group of product Σ × S1. Further arguments can conclude that this example answers the above question. However, this example is restricted to dimension 3. Theorem 1.1 extends this to higher dimensions. To work towards proving the theorem, we will construct a locally CAT(0) space, N, with certain properties, and a principle S1-bundle, 1 S ! M2 ! N with Euler class, β, of infinite order and represented by a bounded cocycle. Then, we 1 q:i: will define M1 = N × S . By theorem 2.14, M1 ∼ M2. Additionally, M1 will be the product of two locally CAT(0) spaces, hence is locally CAT(0). Meanwhile, since ' has infinite order, M2 cannot support a locally CAT(0) metric (lemma 2.18). The result, along with the conditions required to get such manifolds, is stated as follows: Lemma 1.3. Suppose we can find a locally CAT(0) n-manifold N which decomposes as N = A [f B, where A and B are both truncated, complete, finite-volume, non- compact, hyperbolic n-manifolds with toric cusps, glued together via f : @A ! @B so that A \ B = [T n−1. Suppose also that there exists α 2 H1(A \ B) with ∗ 1 1 a) hαi \ i (H (A; Z) ⊕ H (B; Z)) = 0 2 2 b) δ(hαi) \ im(Hb (N; Z) ! H (N; Z)) 6= ; 1 δ 2 where δ : H (A \ B; Z) ! H (N; Z) is the map from the Mayer Vietoris sequence and i∗ is the map induced by the inclusions A \ B,! A and A \ B,! B. Then we can find a pair of graph manifolds with quasi-isometric fundamental groups with the property that one of them supports a locally CAT(0) metric, but the other one cannot support any locally CAT(0) metric. 2 The manifold N will come from the double of a suitably chosen manifold A. The existence of this A along with α 2 H2(@A) satisfying property (a) from lemma 1.3 is a combination of results from [17] and [8]. All that remains is the fact that the image of the subgroup generated by α in H2(DA) is bounded. This uses a number of diagram 2 2 chases and primarily uses the surjectivity of the map Hb (DA; @A) ! H (DA; @A). This fact follows from the fact that π1(DA) is hyperbolic relative to π1(@A), which is a consequence of their action on DAg, a space with isolated flats. In chapter 2 we will provide the necessary definitions. These will include the back- ground material on graph manifolds, group cohomology, and bounded cohomology of spaces and groups. We will also recall some results from Gersten (2.14) and Frigerio, Lafont, and Sisto (2.18). Then, we will be able to prove the above lemma. In chapter 3, we define what is means for a metric space to have isolated flats and we verify that the universal covers of graph manifolds have isolated flats. Then, we use this fact along with some results from [13] and [6] to show that for DA, the double of A, a finite volume, non-compact, hyperbolic manifold with toric cusps, π1(DA) is hyperbolic relative to π1(@A). In chapter 4, we take a detour and explore part of a paper of Bieri and Eckmann. We will define relative bounded cohomology and explore their proof of various iso- morphisms between cohomology involving a space X and a (possibly disconnected) subspace Y , and the cohomology involving π1(X) and a subgroup (or possibly union of subgroups) π1(Y ) (Theorem 4.2). Working through the proof, we will see that these isomorphisms also hold for bounded cohomology (Theorem 4.3). In chapter 5, we return to lemma 1.3 and specialize to the case where A = B, A is formed by truncating A at the cusps, and N = DA, the double of A. We will show that in this setting, if condition (a) is satisfied, then condition (b) in the lemma is automatically satisfied. As a result, we can restate lemma 1.3 as follows: 3 Lemma 1.4. Let A be a finite volume non-compact hyperbolic manifold, with all cusps diffeomorphic to the product of a torus and [0; 1). Let A be the compact manifold obtained by truncating the cusps. Suppose we can find a non-trivial cohomology class 1 ∗ 1 1 α 2 H (A \ A) with hαi \ i (H (A; Z) ⊕ H (A; Z)) = 0. Then we can find a pair of graph manifolds with quasi-isometric fundamental groups with the property that one of them supports a locally CAT(0) metric, but the other one cannot support any locally CAT(0) metric. Finally, in chapter 6, we answer the above question posed by Frigerio, Lafont, and Sisto, combining some of their results with our results in section 4, to show that there exist infinitely many examples of such manifolds satisfying condition (a) from lemma 1.3. 4 CHAPTER 2 BACKGROUND 2.1 Principle S1-Bundles and the Euler Class Definition 2.1. A fiber bundle consists of a (connected) base space B, a total space E, a projection π : E ! B, a space F called the fiber such that for any x 2 B, π−1(x) is homeomorphic to F . Furthermore, for each x 2 B, there exists Ux ⊂ B containing −1 x such that π (Ux) is homeomorphic to Ux × F and both of their projections onto Ux agree. In other words, the following diagram commutes: −1 ≈ π (Ux) / Ux × F y Ux For a topological group G, a principal G-bundle has the additional structure of a right G action on the total space E.
Load more

Recommended publications
  • Arxiv:2008.00328V3 [Math.DS] 28 Apr 2021 Uniformly Hyperbolic and Topologically Mixing
    ERGODICITY AND EQUIDISTRIBUTION IN STRICTLY CONVEX HILBERT GEOMETRY FENG ZHU Abstract. We show that dynamical and counting results characteristic of negatively-curved Riemannian geometry, or more generally CAT(-1) or rank- one CAT(0) spaces, also hold for geometrically-finite strictly convex projective structures equipped with their Hilbert metric. More specifically, such structures admit a finite Sullivan measure; with respect to this measure, the Hilbert geodesic flow is strongly mixing, and orbits and primitive closed geodesics equidistribute, allowing us to asymptotically enumerate these objects. In [Mar69], Margulis established counting results for uniform lattices in constant negative curvature, or equivalently for closed hyperbolic manifolds, by means of ergodicity and equidistribution results for the geodesic flows on these manifolds with respect to suitable measures. Thomas Roblin, in [Rob03], obtained analogous ergodicity and equidistribution results in the more general setting of CAT(−1) spaces. These results include ergodicity of the horospherical foliations, mixing of the geodesic flow, orbital equidistribution of the group, equidistribution of primitive closed geodesics, and, in the geometrically finite case, asymptotic counting estimates. Gabriele Link later adapted similar techniques to prove similar results in the even more general setting of rank-one isometry groups of Hadamard spaces in [Lin20]. These results do not directly apply to manifolds endowed with strictly convex projective structures, considered with their Hilbert metrics and associated Bowen- Margulis measures, since strictly convex Hilbert geometries are in general not CAT(−1) or even CAT(0) (see e.g. [Egl97, App. B].) Nevertheless, these Hilbert geometries exhibit substantial similarities to Riemannian geometries of pinched negative curvature. In particular, there is a good theory of Busemann functions and of Patterson-Sullivan measures on these geometries.
    [Show full text]
  • [Math.GT] 9 Oct 2020 Symmetries of Hyperbolic 4-Manifolds
    Symmetries of hyperbolic 4-manifolds Alexander Kolpakov & Leone Slavich R´esum´e Pour chaque groupe G fini, nous construisons des premiers exemples explicites de 4- vari´et´es non-compactes compl`etes arithm´etiques hyperboliques M, `avolume fini, telles M G + M G que Isom ∼= , ou Isom ∼= . Pour y parvenir, nous utilisons essentiellement la g´eom´etrie de poly`edres de Coxeter dans l’espace hyperbolique en dimension quatre, et aussi la combinatoire de complexes simpliciaux. C¸a nous permet d’obtenir une borne sup´erieure universelle pour le volume minimal d’une 4-vari´et´ehyperbolique ayant le groupe G comme son groupe d’isom´etries, par rap- port de l’ordre du groupe. Nous obtenons aussi des bornes asymptotiques pour le taux de croissance, par rapport du volume, du nombre de 4-vari´et´es hyperboliques ayant G comme le groupe d’isom´etries. Abstract In this paper, for each finite group G, we construct the first explicit examples of non- M M compact complete finite-volume arithmetic hyperbolic 4-manifolds such that Isom ∼= G + M G , or Isom ∼= . In order to do so, we use essentially the geometry of Coxeter polytopes in the hyperbolic 4-space, on one hand, and the combinatorics of simplicial complexes, on the other. This allows us to obtain a universal upper bound on the minimal volume of a hyperbolic 4-manifold realising a given finite group G as its isometry group in terms of the order of the group. We also obtain asymptotic bounds for the growth rate, with respect to volume, of the number of hyperbolic 4-manifolds having a finite group G as their isometry group.
    [Show full text]
  • Apollonian Circle Packings: Dynamics and Number Theory
    APOLLONIAN CIRCLE PACKINGS: DYNAMICS AND NUMBER THEORY HEE OH Abstract. We give an overview of various counting problems for Apol- lonian circle packings, which turn out to be related to problems in dy- namics and number theory for thin groups. This survey article is an expanded version of my lecture notes prepared for the 13th Takagi lec- tures given at RIMS, Kyoto in the fall of 2013. Contents 1. Counting problems for Apollonian circle packings 1 2. Hidden symmetries and Orbital counting problem 7 3. Counting, Mixing, and the Bowen-Margulis-Sullivan measure 9 4. Integral Apollonian circle packings 15 5. Expanders and Sieve 19 References 25 1. Counting problems for Apollonian circle packings An Apollonian circle packing is one of the most of beautiful circle packings whose construction can be described in a very simple manner based on an old theorem of Apollonius of Perga: Theorem 1.1 (Apollonius of Perga, 262-190 BC). Given 3 mutually tangent circles in the plane, there exist exactly two circles tangent to all three. Figure 1. Pictorial proof of the Apollonius theorem 1 2 HEE OH Figure 2. Possible configurations of four mutually tangent circles Proof. We give a modern proof, using the linear fractional transformations ^ of PSL2(C) on the extended complex plane C = C [ f1g, known as M¨obius transformations: a b az + b (z) = ; c d cz + d where a; b; c; d 2 C with ad − bc = 1 and z 2 C [ f1g. As is well known, a M¨obiustransformation maps circles in C^ to circles in C^, preserving angles between them.
    [Show full text]
  • On Eigenvalues of Geometrically Finite Hyperbolic Manifolds With
    On Eigenvalues of Geometrically Finite Hyperbolic Manifolds with Infinite Volume Xiaolong Hans Han April 6, 2020 Abstract Let M be an oriented geometrically finite hyperbolic manifold of dimension n ≥ 3 with infinite volume. Then for all k ≥ 0, we provide a lower bound on the k-th eigenvalue of the Laplacian operator acting on functions on M by a constant and the k-th eigenvalue of some neighborhood Mf of the thick part of the convex core. 1. Introduction As analytic data of a manifold, the spectrum of the Laplacian operator acting on func- tions contains rich information about the geometry and topology of a manifold. For example, by results of Cheeger and Buser, with a lower bound on Ricci curvature, the first eigenvalue is equivalent to the Cheeger constant of a manifold, which depends on the geometry of the manifold. The rigidity and richness of hyperbolic geometry make the connection even more intriguing: by Canary [4], for infinite volume, topologically tame hyperbolic 3-manifolds, the 2 bottom of the L -spectrum of −∆, denoted by λ0, is zero if and only if the manifold is not geometrically finite. If λ0 of a geometrically finite, infinite volume hyperbolic manifold is 1, by Canary and Burger [2], the manifold is either the topological interior of a handlebody or an R-bundle over a closed surface, which is a very strong topological restriction. Conversely, if we have data about the geometry of a manifold, we can convert it into data about its spectrum, like lower bounds on eigenvalues, dimensions of eigenspaces, existence of embedded eigenvalues, etc.
    [Show full text]
  • ISOMETRY TRANSFORMATIONS of HYPERBOLIC 3-MANIFOLDS Sadayoshi KOJIMA 0. Introduction by a Hyperbolic Manifold, We Mean a Riemanni
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 29 (1988) 297-307 297 North-Holland ISOMETRY TRANSFORMATIONS OF HYPERBOLIC 3-MANIFOLDS Sadayoshi KOJIMA Department of Information Science, Tokyo Institute of Technology, Ohokayama, Meguro, Tokyo, Japan Received 1 December 1986 Revised 10 June 1987 The isometry group of a compact hyperbolic manifold is known to be finite. We show that every finite group is realized as the full isometry group of some compact hyperbolic 3-manifold. AMS (MOS) Subj. Class.: 57S17, 53C25, 57M99 3-manifold hyperbolic manifold isometry 0. Introduction By a hyperbolic manifold, we mean a Riemannian manifold with constant sectional curvature -1. It is known by several ways that the isometry group of a hyperbolic manifold of finite volume is finite. On the other hand, Greenberg has shown in [3] that every finite group is realized as the full conformal automorphism group of some Riemann surface of hyperbolic type. Since such a Riemann surface, which is a 2-manifold with a conformal structure, inherits a unique conformally equivalent hyperbolic structure through the uniformization theorem, there are no limitations on the group structure of the full isometry group of a hyperbolic manifold in dimension 2. In this paper, we first give a quick proof of the theorem of Greenberg quoted above, and then along the same line we show, raising the dimension by one. Theorem. Every jinite group is realized as the full isometry group of some closed hyperbolic 3 -manifold. An immediate corollary to this is the following.
    [Show full text]
  • Hyperbolic Geometry
    Flavors of Geometry MSRI Publications Volume 31,1997 Hyperbolic Geometry JAMES W. CANNON, WILLIAM J. FLOYD, RICHARD KENYON, AND WALTER R. PARRY Contents 1. Introduction 59 2. The Origins of Hyperbolic Geometry 60 3. Why Call it Hyperbolic Geometry? 63 4. Understanding the One-Dimensional Case 65 5. Generalizing to Higher Dimensions 67 6. Rudiments of Riemannian Geometry 68 7. Five Models of Hyperbolic Space 69 8. Stereographic Projection 72 9. Geodesics 77 10. Isometries and Distances in the Hyperboloid Model 80 11. The Space at Infinity 84 12. The Geometric Classification of Isometries 84 13. Curious Facts about Hyperbolic Space 86 14. The Sixth Model 95 15. Why Study Hyperbolic Geometry? 98 16. When Does a Manifold Have a Hyperbolic Structure? 103 17. How to Get Analytic Coordinates at Infinity? 106 References 108 Index 110 1. Introduction Hyperbolic geometry was created in the first half of the nineteenth century in the midst of attempts to understand Euclid’s axiomatic basis for geometry. It is one type of non-Euclidean geometry, that is, a geometry that discards one of Euclid’s axioms. Einstein and Minkowski found in non-Euclidean geometry a This work was supported in part by The Geometry Center, University of Minnesota, an STC funded by NSF, DOE, and Minnesota Technology, Inc., by the Mathematical Sciences Research Institute, and by NSF research grants. 59 60 J. W. CANNON, W. J. FLOYD, R. KENYON, AND W. R. PARRY geometric basis for the understanding of physical time and space. In the early part of the twentieth century every serious student of mathematics and physics studied non-Euclidean geometry.
    [Show full text]
  • Hyperbolic Geometry, Surfaces, and 3-Manifolds Bruno Martelli
    Hyperbolic geometry, surfaces, and 3-manifolds Bruno Martelli Dipartimento di Matematica \Tonelli", Largo Pontecorvo 5, 56127 Pisa, Italy E-mail address: martelli at dm dot unipi dot it version: december 18, 2013 Contents Introduction 1 Copyright notices 1 Chapter 1. Preliminaries 3 1. Differential topology 3 1.1. Differentiable manifolds 3 1.2. Tangent space 4 1.3. Differentiable submanifolds 6 1.4. Fiber bundles 6 1.5. Tangent and normal bundle 6 1.6. Immersion and embedding 7 1.7. Homotopy and isotopy 7 1.8. Tubolar neighborhood 7 1.9. Manifolds with boundary 8 1.10. Cut and paste 8 1.11. Transversality 9 2. Riemannian geometry 9 2.1. Metric tensor 9 2.2. Distance, geodesics, volume. 10 2.3. Exponential map 11 2.4. Injectivity radius 12 2.5. Completeness 13 2.6. Curvature 13 2.7. Isometries 15 2.8. Isometry group 15 2.9. Riemannian manifolds with boundary 16 2.10. Local isometries 16 3. Measure theory 17 3.1. Borel measure 17 3.2. Topology on the measure space 18 3.3. Lie groups 19 3.4. Haar measures 19 4. Algebraic topology 19 4.1. Group actions 19 4.2. Coverings 20 4.3. Discrete groups of isometries 20 v vi CONTENTS 4.4. Cell complexes 21 4.5. Aspherical cell-complexes 22 Chapter 2. Hyperbolic space 25 1. The models of hyperbolic space 25 1.1. Hyperboloid 25 1.2. Isometries of the hyperboloid 26 1.3. Subspaces 27 1.4. The Poincar´edisc 29 1.5. The half-space model 31 1.6.
    [Show full text]
  • Hyperbolic Deep Learning for Chinese Natural Language Understanding
    Hyperbolic Deep Learning for Chinese Natural Language Understanding Marko Valentin Micic1 and Hugo Chu2 Kami Intelligence Ltd, HK SAR [email protected] [email protected] Abstract Recently hyperbolic geometry has proven to be effective in building embeddings that encode hierarchical and entailment information. This makes it particularly suited to modelling the complex asymmetrical relationships between Chinese characters and words. In this paper we first train a large scale hyperboloid skip-gram model on a Chinese corpus, then apply the character embeddings to a downstream hyperbolic Transformer model derived from the principles of gyrovector space for Poincare disk model. In our experiments the character-based Transformer outperformed its word-based Euclidean equivalent. To the best of our knowledge, this is the first time in Chinese NLP that a character-based model outperformed its word- based counterpart, allowing the circumvention of the challenging and domain-dependent task of Chinese Word Segmentation (CWS). 1 Introduction Recent advances in deep learning have seen tremendous improvements in accuracy and scalability in all fields of application. Natural language processing (NLP), however, still faces many challenges, particularly in Chinese, where even word-segmentation proves to be a difficult and as-yet unscalable task. Traditional neural network architectures employed to tackle NLP tasks are generally cast in the standard Euclidean geometry. While this setting is adequate for most deep learning tasks, where it has often allowed for the achievement of superhuman performance in certain areas (e.g. AlphaGo [35], ImageNet [36]), its performance in NLP has yet to reach the same levels. Recent research arXiv:1812.10408v1 [cs.CL] 11 Dec 2018 indicates that some of the problems NLP faces may be much more fundamental and mathematical in nature.
    [Show full text]
  • Hyperbolic Geometry
    Flavors of Geometry MSRI Publications Volume 31, 1997 Hyperbolic Geometry JAMES W. CANNON, WILLIAM J. FLOYD, RICHARD KENYON, AND WALTER R. PARRY Contents 1. Introduction 59 2. The Origins of Hyperbolic Geometry 60 3. Why Call it Hyperbolic Geometry? 63 4. Understanding the One-Dimensional Case 65 5. Generalizing to Higher Dimensions 67 6. Rudiments of Riemannian Geometry 68 7. Five Models of Hyperbolic Space 69 8. Stereographic Projection 72 9. Geodesics 77 10. Isometries and Distances in the Hyperboloid Model 80 11. The Space at Infinity 84 12. The Geometric Classification of Isometries 84 13. Curious Facts about Hyperbolic Space 86 14. The Sixth Model 95 15. Why Study Hyperbolic Geometry? 98 16. When Does a Manifold Have a Hyperbolic Structure? 103 17. How to Get Analytic Coordinates at Infinity? 106 References 108 Index 110 1. Introduction Hyperbolic geometry was created in the first half of the nineteenth century in the midst of attempts to understand Euclid’s axiomatic basis for geometry. It is one type of non-Euclidean geometry, that is, a geometry that discards one of Euclid’s axioms. Einstein and Minkowski found in non-Euclidean geometry a ThisworkwassupportedinpartbyTheGeometryCenter,UniversityofMinnesota,anSTC funded by NSF, DOE, and Minnesota Technology, Inc., by the Mathematical Sciences Research Institute, and by NSF research grants. 59 60 J. W. CANNON, W. J. FLOYD, R. KENYON, AND W. R. PARRY geometric basis for the understanding of physical time and space. In the early part of the twentieth century every serious student of mathematics and physics studied non-Euclidean geometry. This has not been true of the mathematicians and physicists of our generation.
    [Show full text]
  • Classifying Hyperbolic Manifolds— All’S Well That Ends Well Danamackenzie
    Hall of Mirrors. Spectacular images result from plotting the action of a Kleinian group in 3-dimensional space. To a topologist, this image represents the outside, or convex hull, of a 3-dimensional “hall of mir- rors.” Each individual room in the hall of mirrors corresponds to a 3-dimensional manifold. In a case like this one, where the walls of the hall of mirrors form visible spheres, the manifold is said to be “geomet- rically finite,” and its ends are tame. Until recently topologists did not know if “geometrically infinite” manifolds, where the spheres shrink down to an infinitely fine froth, also had tame ends. (c JosLeys.) 14 What’s Happening in the Mathematical Sciences Classifying Hyperbolic Manifolds— All’s Well That Ends Well DanaMackenzie hile the status of one famous unsolved problem in topology remained uncertain, topologists conquered Wfour other major problems in 2004. The four theorems that went into this unique sundae—the Ending Laminations Conjecture, the Tame Ends Conjecture, the Ahlfors Measure Conjecture, and the Density Conjecture—were not household names, even among mathematicians. Yet to topologists who study three-dimensional spaces, they were as familiar (and went together as well) as chocolate, vanilla, hot fudge sauce and a maraschino cherry. The Poincaré Conjecture (see “First of Seven Millennium Danny Calegari. (Photo courtesy Problems Nears Completion,” p. 2) can be compared to the of Danny Calegari.) quest for the Loch Ness Monster: Whether you discover it or prove it doesn’t exist, fame and glory are sure to follow. On the other hand, if topologists spent all their time hunting Loch Ness monsters, the subject would never advance.
    [Show full text]
  • A Short Proof of Unique Ergodicity of Horospherical Foliations on Infinite Volume Hyperbolic Manifolds Tome 8, No 1 (2016), P
    CONFLUENTES MATHEMATICI Barbara SCHAPIRA A short proof of unique ergodicity of horospherical foliations on infinite volume hyperbolic manifolds Tome 8, no 1 (2016), p. 165-174. <http://cml.cedram.org/item?id=CML_2016__8_1_165_0> © Les auteurs et Confluentes Mathematici, 2016. Tous droits réservés. L’accès aux articles de la revue « Confluentes Mathematici » (http://cml.cedram.org/), implique l’accord avec les condi- tions générales d’utilisation (http://cml.cedram.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation á fin strictement personnelle du copiste est constitutive d’une infrac- tion pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Confluentes Math. 8, 1 (2016) 165-174 A SHORT PROOF OF UNIQUE ERGODICITY OF HOROSPHERICAL FOLIATIONS ON INFINITE VOLUME HYPERBOLIC MANIFOLDS BARBARA SCHAPIRA Abstract. We provide a new proof of the fact that the horospherical group N < G = SOo(d, 1) acting on the frame bundle Γ\G of a hyperbolic manifold admits a unique invariant ergodic measure (up to multiplicative constants) supported on the set of frames whose orbit under the geodesic flow comes back infinitely often in a compact set. This result is known, but our proof is more direct and much shorter. 1. Introduction The (unstable) horocycle flow on the unit tangent bundle of compact hyperbolic surfaces is uniquely ergodic. Furstenberg [6] proved that the Liouville measure is the unique invariant measure under this flow.
    [Show full text]
  • Commensurability of Hyperbolic Manifolds with Geodesic Boundary
    Commensurability of hyperbolic manifolds with geodesic boundary Roberto Frigerio∗ September 24, 2018 Abstract Suppose n > 3, let M1,M2 be n-dimensional connected complete finite- volume hyperbolic manifolds with non-empty geodesic boundary, and suppose that π1(M1) is quasi-isometric to π1(M2) (with respect to the word metric). Also suppose that if n = 3, then ∂M1 and ∂M2 are compact. We show that M1 is commensurable with M2. Moreover, we show that there exist homotopically equivalent hyperbolic 3-manifolds with non-compact geodesic boundary which are not commensurable with each other. We also prove that if M is as M1 above and G is a finitely generated group which is quasi-isometric to π1(M), then there exists a hyperbolic manifold with geodesic boundary M ′ with the following properties: M ′ is commensurable with ′ M, and G is a finite extension of a group which contains π1(M ) as a finite-index subgroup. MSC (2000): 20F65 (primary), 30C65, 57N16 (secondary). Keywords: Fundamental group, Cayley graph, Quasi-isometry, Quasi-conformal homeomorphism, Hyperbolic manifold. A quasi-isometry between metric spaces is a (not necessarily continuous) map which looks biLipschitz from afar (see Section 1 for a precise definition). Even if they do arXiv:math/0502209v1 [math.GT] 10 Feb 2005 not give information about the local structure of metric spaces, quasi-isometries usually capture the most important properties of their large-scale geometry. Let Γ be any group with a finite set S of generators. Then a locally finite graph is defined which depends on Γ and S (such a graph is called a Cayley graph of Γ, see Section 1).
    [Show full text]