Metric Geometry and Geometric Group Theory Lior Silberman
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Metric Geometry in a Tame Setting
University of California Los Angeles Metric Geometry in a Tame Setting A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Erik Walsberg 2015 c Copyright by Erik Walsberg 2015 Abstract of the Dissertation Metric Geometry in a Tame Setting by Erik Walsberg Doctor of Philosophy in Mathematics University of California, Los Angeles, 2015 Professor Matthias J. Aschenbrenner, Chair We prove basic results about the topology and metric geometry of metric spaces which are definable in o-minimal expansions of ordered fields. ii The dissertation of Erik Walsberg is approved. Yiannis N. Moschovakis Chandrashekhar Khare David Kaplan Matthias J. Aschenbrenner, Committee Chair University of California, Los Angeles 2015 iii To Sam. iv Table of Contents 1 Introduction :::::::::::::::::::::::::::::::::::::: 1 2 Conventions :::::::::::::::::::::::::::::::::::::: 5 3 Metric Geometry ::::::::::::::::::::::::::::::::::: 7 3.1 Metric Spaces . 7 3.2 Maps Between Metric Spaces . 8 3.3 Covers and Packing Inequalities . 9 3.3.1 The 5r-covering Lemma . 9 3.3.2 Doubling Metrics . 10 3.4 Hausdorff Measures and Dimension . 11 3.4.1 Hausdorff Measures . 11 3.4.2 Hausdorff Dimension . 13 3.5 Topological Dimension . 15 3.6 Left-Invariant Metrics on Groups . 15 3.7 Reductions, Ultralimits and Limits of Metric Spaces . 16 3.7.1 Reductions of Λ-valued Metric Spaces . 16 3.7.2 Ultralimits . 17 3.7.3 GH-Convergence and GH-Ultralimits . 18 3.7.4 Asymptotic Cones . 19 3.7.5 Tangent Cones . 22 3.7.6 Conical Metric Spaces . 22 3.8 Normed Spaces . 23 4 T-Convexity :::::::::::::::::::::::::::::::::::::: 24 4.1 T-convex Structures . -
EMBEDDINGS of GROMOV HYPERBOLIC SPACES 1 Introduction
GAFA, Geom. funct. anal. © Birkhiiuser Verlag, Basel 2000 Vol. 10 (2000) 266 - 306 1016-443X/00/020266-41 $ 1.50+0.20/0 I GAFA Geometric And Functional Analysis EMBEDDINGS OF GROMOV HYPERBOLIC SPACES M. BONK AND O. SCHRAMM Abstract It is shown that a Gromov hyperbolic geodesic metric space X with bounded growth at some scale is roughly quasi-isometric to a convex subset of hyperbolic space. If one is allowed to rescale the metric of X by some positive constant, then there is an embedding where distances are distorted by at most an additive constant. Another embedding theorem states that any 8-hyperbolic met ric space embeds isometrically into a complete geodesic 8-hyperbolic space. The relation of a Gromov hyperbolic space to its boundary is fur ther investigated. One of the applications is a characterization of the hyperbolic plane up to rough quasi-isometries. 1 Introduction The study of Gromov hyperbolic spaces has been largely motivated and dominated by questions about Gromov hyperbolic groups. This paper studies the geometry of Gromov hyperbolic spaces without reference to any group or group action. One of our main theorems is Embedding Theorem 1.1. Let X be a Gromov hyperbolic geodesic metric space with bounded growth at some scale. Then there exists an integer n such that X is roughly similar to a convex subset of hyperbolic n-space lHIn. The precise definitions appear in the body of the paper, but now we briefly discuss the meaning of the theorem. The condition of bounded growth at some scale is satisfied, for example, if X is a bounded valence graph, or a Riemannian manifold with bounded local geometry. -
Arxiv:1612.03497V2 [Math.GR] 18 Dec 2017 2-Sphere Is Virtually a Kleinian Group
BOUNDARIES OF DEHN FILLINGS DANIEL GROVES, JASON FOX MANNING, AND ALESSANDRO SISTO Abstract. We begin an investigation into the behavior of Bowditch and Gro- mov boundaries under the operation of Dehn filling. In particular we show many Dehn fillings of a toral relatively hyperbolic group with 2{sphere bound- ary are hyperbolic with 2{sphere boundary. As an application, we show that the Cannon conjecture implies a relatively hyperbolic version of the Cannon conjecture. Contents 1. Introduction 1 2. Preliminaries 5 3. Weak Gromov–Hausdorff convergence 12 4. Spiderwebs 15 5. Approximating the boundary of a Dehn filling 20 6. Proofs of approximation theorems for hyperbolic fillings 23 7. Approximating boundaries are spheres 39 8. Ruling out the Sierpinski carpet 42 9. Proof of Theorem 1.2 44 10. Proof of Corollary 1.4 45 Appendix A. δ{hyperbolic technicalities 46 References 50 1. Introduction One of the central problems in geometric group theory and low-dimensional topology is the Cannon Conjecture (see [Can91, Conjecture 11.34], [CS98, Con- jecture 5.1]), which states that a hyperbolic group whose (Gromov) boundary is a arXiv:1612.03497v2 [math.GR] 18 Dec 2017 2-sphere is virtually a Kleinian group. By a result of Bowditch [Bow98] hyperbolic groups can be characterized in terms of topological properties of their action on the boundary. The Cannon Conjecture is that (in case the boundary is S2) this topological action is in fact conjugate to an action by M¨obiustransformations. Rel- atively hyperbolic groups are a natural generalization of hyperbolic groups which are intended (among other things) to generalize the situation of the fundamental group of a finite-volume hyperbolic n-manifold acting on Hn. -
Nonpositive Curvature and the Ptolemy Inequality
Zurich Open Repository and Archive University of Zurich Main Library Strickhofstrasse 39 CH-8057 Zurich www.zora.uzh.ch Year: 2007 Nonpositive curvature and the Ptolemy inequality Foertsch, T ; Lytchak, A ; Schroeder, Viktor Abstract: We provide examples of nonlocally, compact, geodesic Ptolemy metric spaces which are not uniquely geodesic. On the other hand, we show that locally, compact, geodesic Ptolemy metric spaces are uniquely geodesic. Moreover, we prove that a metric space is CAT(0) if and only if it is Busemann convex and Ptolemy. DOI: https://doi.org/10.1093/imrn/rnm100 Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-21536 Journal Article Published Version Originally published at: Foertsch, T; Lytchak, A; Schroeder, Viktor (2007). Nonpositive curvature and the Ptolemy inequality. International Mathematics Research Notices, 2007(rnm100):online. DOI: https://doi.org/10.1093/imrn/rnm100 Foertsch, T., A. Lytchak, and V. Schroeder. (2007) “Nonpositive Curvature and the Ptolemy Inequality,” International Mathematics Research Notices, Vol. 2007, Article ID rnm100, 15 pages. doi:10.1093/imrn/rnm100 Nonpositive Curvature and the Ptolemy Inequality Thomas Foertsch1, Alexander Lytchak1, Viktor Schroeder2 1Universitat¨ Bonn, Mathematisches Institut, Beringstr. 1, 53115 Bonn, Germany, and 2Universitat¨ Zurich¨ , Institut fur¨ Mathematik, Winterthurerstr. 190, 8057 Zurich¨ , Switzerland Correspondence to be sent to: Thomas Foertsch, Universitat¨ Bonn, Mathematisches Institut, Beringstr. 1, 53115 Bonn, Germany. e-mail: [email protected] We provide examples of nonlocally, compact, geodesic Ptolemy metric spaces which are not uniquely geodesic. On the other hand, we show that locally, compact, geodesic Ptolemy metric spaces are uniquely geodesic. -
LECTURE 6: FIBER BUNDLES in This Section We Will Introduce The
LECTURE 6: FIBER BUNDLES In this section we will introduce the interesting class of fibrations given by fiber bundles. Fiber bundles play an important role in many geometric contexts. For example, the Grassmaniann varieties and certain fiber bundles associated to Stiefel varieties are central in the classification of vector bundles over (nice) spaces. The fact that fiber bundles are examples of Serre fibrations follows from Theorem ?? which states that being a Serre fibration is a local property. 1. Fiber bundles and principal bundles Definition 6.1. A fiber bundle with fiber F is a map p: E ! X with the following property: every ∼ −1 point x 2 X has a neighborhood U ⊆ X for which there is a homeomorphism φU : U × F = p (U) such that the following diagram commutes in which π1 : U × F ! U is the projection on the first factor: φ U × F U / p−1(U) ∼= π1 p * U t Remark 6.2. The projection X × F ! X is an example of a fiber bundle: it is called the trivial bundle over X with fiber F . By definition, a fiber bundle is a map which is `locally' homeomorphic to a trivial bundle. The homeomorphism φU in the definition is a local trivialization of the bundle, or a trivialization over U. Let us begin with an interesting subclass. A fiber bundle whose fiber F is a discrete space is (by definition) a covering projection (with fiber F ). For example, the exponential map R ! S1 is a covering projection with fiber Z. Suppose X is a space which is path-connected and locally simply connected (in fact, the weaker condition of being semi-locally simply connected would be enough for the following construction). -
Arxiv:0803.2592V3 [Math.GR] 29 Jan 2010 F)I N Nyi Vr Isometric Every If Only and If (FH) Taeyfrpoigsao’ Eutadterm
FIXED POINT PROPERTIES IN THE SPACE OF MARKED GROUPS YVES STALDER Abstract. We explain, following Gromov, how to produce uniform isometric actions of groups starting from isometric actions without fixed point, using common ultralimits techniques. This gives in particular a simple proof of a result by Shalom: Kazhdan’s property (T) defines an open subset in the space of marked finitely generated groups. 1. Introduction In this expository note, we are interested in groups whose actions on some particular kind of spaces always have (global) fixed points. Definition 1.1. Let G be a (discrete) group. We say that G has: – Serre’s Property (FH), if any isometric G-action on an affine Hilbert space has a fixed point [HV89, Chap 4]; – Serre’s Property (FA), if any G-action on a simplicial tree (by automorphisms and without inversion) has a fixed point [Ser77, Chap I.6]; – Property (FRA), if if any isometric G-action on a complete R-tree has a fixed point [HV89, Chap 6.b]. These definitions extend to topological groups: one has then to require the actions to be continuous. Such properties give information about the structure of the group G. Serre proved that a countable group has Property (FA) if and only if (i) it is finitely generated, (ii) it has no infinite cyclic quotient, and (iii) it is not an amalgam [Ser77, Thm I.15]. Among locally compact, second countable groups, Guichardet and Delorme proved that Property (FH) is equivalent to Kazhdan’s Property (T) [Gui77, Del77]. Kazhdan groups are known to be compactly generated and to have a compact abelianization; see e.g. -
MTH 304: General Topology Semester 2, 2017-2018
MTH 304: General Topology Semester 2, 2017-2018 Dr. Prahlad Vaidyanathan Contents I. Continuous Functions3 1. First Definitions................................3 2. Open Sets...................................4 3. Continuity by Open Sets...........................6 II. Topological Spaces8 1. Definition and Examples...........................8 2. Metric Spaces................................. 11 3. Basis for a topology.............................. 16 4. The Product Topology on X × Y ...................... 18 Q 5. The Product Topology on Xα ....................... 20 6. Closed Sets.................................. 22 7. Continuous Functions............................. 27 8. The Quotient Topology............................ 30 III.Properties of Topological Spaces 36 1. The Hausdorff property............................ 36 2. Connectedness................................. 37 3. Path Connectedness............................. 41 4. Local Connectedness............................. 44 5. Compactness................................. 46 6. Compact Subsets of Rn ............................ 50 7. Continuous Functions on Compact Sets................... 52 8. Compactness in Metric Spaces........................ 56 9. Local Compactness.............................. 59 IV.Separation Axioms 62 1. Regular Spaces................................ 62 2. Normal Spaces................................ 64 3. Tietze's extension Theorem......................... 67 4. Urysohn Metrization Theorem........................ 71 5. Imbedding of Manifolds.......................... -
Amenable Actions, Free Products and a Fixed Point Property
Submitted exclusively to the London Mathematical Society doi:10.1112/S0000000000000000 AMENABLE ACTIONS, FREE PRODUCTS AND A FIXED POINT PROPERTY Y. GLASNER and N. MONOD Abstract We investigate the class of groups admitting an action on a set with an invariant mean. It turns out that many free products admit interesting actions of that kind. A complete characterization of such free products is given in terms of a fixed point property. 1. Introduction 1.A. In the early 20th century, the construction of Lebesgue’s measure was followed by the discovery of the Banach-Hausdorff-Tarski paradoxes ([20], [16], [4], [30]; see also [18]). This prompted von Neumann [32] to study the following general question: Given a group G acting on a set X, when is there an invariant mean on X ? Definition 1.1. An invariant mean is a G-invariant map µ from the collection of subsets of X to [0, 1] such that (i) µ(A ∪ B) = µ(A) + µ(B) when A ∩ B = ∅ and (ii) µ(X) = 1. If such a mean exists, the action is called amenable. Remarks 1.2. (1) For the study of the classical paradoxes, one also considers normalisations other than (ii). (2) The notion of amenability later introduced by Zimmer [33] and its variants [3] are different, being in a sense dual to the above. 1.B. The thrust of von Neumann’s article was to show that the paradoxes, or lack thereof, originate in the structure of the group rather than the set X. He therefore proposed the study of amenable groups (then “meßbare Gruppen”), i.e. -
Abstract Commensurability and Quasi-Isometry Classification of Hyperbolic Surface Group Amalgams
ABSTRACT COMMENSURABILITY AND QUASI-ISOMETRY CLASSIFICATION OF HYPERBOLIC SURFACE GROUP AMALGAMS EMILY STARK Abstract. Let XS denote the class of spaces homeomorphic to two closed orientable surfaces of genus greater than one identified to each other along an essential simple closed curve in each surface. Let CS denote the set of fundamental groups of spaces in XS . In this paper, we characterize the abstract commensurability classes within CS in terms of the ratio of the Euler characteristic of the surfaces identified and the topological type of the curves identified. We prove that all groups in CS are quasi-isometric by exhibiting a bilipschitz map between the universal covers of two spaces in XS . In particular, we prove that the universal covers of any two such spaces may be realized as isomorphic cell complexes with finitely many isometry types of hyperbolic polygons as cells. We analyze the abstract commensurability classes within CS : we characterize which classes contain a maximal element within CS ; we prove each abstract commensurability class contains a right-angled Coxeter group; and, we construct a common CAT(0) cubical model geometry for each abstract commensurability class. 1. Introduction Finitely generated infinite groups carry both an algebraic and a geometric structure, and to study such groups, one may study both algebraic and geometric classifications. Abstract commensurability defines an algebraic equivalence relation on the class of groups, where two groups are said to be abstractly commensurable if they contain isomorphic subgroups of finite-index. Finitely generated groups may also be viewed as geometric objects, since a finitely generated group has a natural word metric which is well-defined up to quasi-isometric equivalence. -
Lectures on Quasi-Isometric Rigidity Michael Kapovich 1 Lectures on Quasi-Isometric Rigidity 3 Introduction: What Is Geometric Group Theory? 3 Lecture 1
Contents Lectures on quasi-isometric rigidity Michael Kapovich 1 Lectures on quasi-isometric rigidity 3 Introduction: What is Geometric Group Theory? 3 Lecture 1. Groups and Spaces 5 1. Cayley graphs and other metric spaces 5 2. Quasi-isometries 6 3. Virtual isomorphisms and QI rigidity problem 9 4. Examples and non-examples of QI rigidity 10 Lecture 2. Ultralimits and Morse Lemma 13 1. Ultralimits of sequences in topological spaces. 13 2. Ultralimits of sequences of metric spaces 14 3. Ultralimits and CAT(0) metric spaces 14 4. Asymptotic Cones 15 5. Quasi-isometries and asymptotic cones 15 6. Morse Lemma 16 Lecture 3. Boundary extension and quasi-conformal maps 19 1. Boundary extension of QI maps of hyperbolic spaces 19 2. Quasi-actions 20 3. Conical limit points of quasi-actions 21 4. Quasiconformality of the boundary extension 21 Lecture 4. Quasiconformal groups and Tukia's rigidity theorem 27 1. Quasiconformal groups 27 2. Invariant measurable conformal structure for qc groups 28 3. Proof of Tukia's theorem 29 4. QI rigidity for surface groups 31 Lecture 5. Appendix 33 1. Appendix 1: Hyperbolic space 33 2. Appendix 2: Least volume ellipsoids 35 3. Appendix 3: Different measures of quasiconformality 35 Bibliography 37 i Lectures on quasi-isometric rigidity Michael Kapovich IAS/Park City Mathematics Series Volume XX, XXXX Lectures on quasi-isometric rigidity Michael Kapovich Introduction: What is Geometric Group Theory? Historically (in the 19th century), groups appeared as automorphism groups of some structures: • Polynomials (field extensions) | Galois groups. • Vector spaces, possibly equipped with a bilinear form| Matrix groups. -
Inducing Maps Between Gromov Boundaries
INDUCING MAPS BETWEEN GROMOV BOUNDARIES J. DYDAK AND Z.ˇ VIRK Abstract. It is well known that quasi-isometric embeddings of Gromov hy- perbolic spaces induce topological embeddings of their Gromov boundaries. A more general question is to detect classes of functions between Gromov hyperbolic spaces that induce continuous maps between their Gromov bound- aries. In this paper we introduce the class of visual functions f that do induce continuous maps f˜ between Gromov boundaries. Its subclass, the class of ra- dial functions, induces H¨older maps between Gromov boundaries. Conversely, every H¨older map between Gromov boundaries of visual hyperbolic spaces induces a radial function. We study the relationship between large scale prop- erties of f and small scale properties of f˜, especially related to the dimension theory. In particular, we prove a form of the dimension raising theorem. We give a natural example of a radial dimension raising map and we also give a general class of radial functions that raise asymptotic dimension. arXiv:1506.08280v1 [math.MG] 27 Jun 2015 Date: May 11, 2018. 2010 Mathematics Subject Classification. Primary 53C23; Secondary 20F67, 20F65, 20F69. Key words and phrases. Gromov boundary, hyperbolic space, dimension raising map, visual metric, coarse geometry. This research was partially supported by the Slovenian Research Agency grants P1-0292-0101. 1 2 1. Introduction Gromov boundary (as defined by Gromov) is one of the central objects in the geometric group theory and plays a crucial role in the Cannon’s conjecture [7]. It is a compact metric space that represents an image at infinity of a hyperbolic metric space. -
3-Manifold Groups
3-Manifold Groups Matthias Aschenbrenner Stefan Friedl Henry Wilton University of California, Los Angeles, California, USA E-mail address: [email protected] Fakultat¨ fur¨ Mathematik, Universitat¨ Regensburg, Germany E-mail address: [email protected] Department of Pure Mathematics and Mathematical Statistics, Cam- bridge University, United Kingdom E-mail address: [email protected] Abstract. We summarize properties of 3-manifold groups, with a particular focus on the consequences of the recent results of Ian Agol, Jeremy Kahn, Vladimir Markovic and Dani Wise. Contents Introduction 1 Chapter 1. Decomposition Theorems 7 1.1. Topological and smooth 3-manifolds 7 1.2. The Prime Decomposition Theorem 8 1.3. The Loop Theorem and the Sphere Theorem 9 1.4. Preliminary observations about 3-manifold groups 10 1.5. Seifert fibered manifolds 11 1.6. The JSJ-Decomposition Theorem 14 1.7. The Geometrization Theorem 16 1.8. Geometric 3-manifolds 20 1.9. The Geometric Decomposition Theorem 21 1.10. The Geometrization Theorem for fibered 3-manifolds 24 1.11. 3-manifolds with (virtually) solvable fundamental group 26 Chapter 2. The Classification of 3-Manifolds by their Fundamental Groups 29 2.1. Closed 3-manifolds and fundamental groups 29 2.2. Peripheral structures and 3-manifolds with boundary 31 2.3. Submanifolds and subgroups 32 2.4. Properties of 3-manifolds and their fundamental groups 32 2.5. Centralizers 35 Chapter 3. 3-manifold groups after Geometrization 41 3.1. Definitions and conventions 42 3.2. Justifications 45 3.3. Additional results and implications 59 Chapter 4. The Work of Agol, Kahn{Markovic, and Wise 63 4.1.