The Notion of Commensurability in Group Theory and Geometry
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Dynamics for Discrete Subgroups of Sl 2(C)
DYNAMICS FOR DISCRETE SUBGROUPS OF SL2(C) HEE OH Dedicated to Gregory Margulis with affection and admiration Abstract. Margulis wrote in the preface of his book Discrete subgroups of semisimple Lie groups [30]: \A number of important topics have been omitted. The most significant of these is the theory of Kleinian groups and Thurston's theory of 3-dimensional manifolds: these two theories can be united under the common title Theory of discrete subgroups of SL2(C)". In this article, we will discuss a few recent advances regarding this missing topic from his book, which were influenced by his earlier works. Contents 1. Introduction 1 2. Kleinian groups 2 3. Mixing and classification of N-orbit closures 10 4. Almost all results on orbit closures 13 5. Unipotent blowup and renormalizations 18 6. Interior frames and boundary frames 25 7. Rigid acylindrical groups and circular slices of Λ 27 8. Geometrically finite acylindrical hyperbolic 3-manifolds 32 9. Unipotent flows in higher dimensional hyperbolic manifolds 35 References 44 1. Introduction A discrete subgroup of PSL2(C) is called a Kleinian group. In this article, we discuss dynamics of unipotent flows on the homogeneous space Γn PSL2(C) for a Kleinian group Γ which is not necessarily a lattice of PSL2(C). Unlike the lattice case, the geometry and topology of the associated hyperbolic 3-manifold M = ΓnH3 influence both topological and measure theoretic rigidity properties of unipotent flows. Around 1984-6, Margulis settled the Oppenheim conjecture by proving that every bounded SO(2; 1)-orbit in the space SL3(Z)n SL3(R) is compact ([28], [27]). -
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. -
Colimit Theorems for Coarse Coherence with Applications
COLIMIT THEOREMS FOR COARSE COHERENCE WITH APPLICATIONS BORIS GOLDFARB AND JONATHAN L. GROSSMAN Abstract. We establish two versions of a central theorem, the Family Colimit Theorem, for the coarse coherence property of metric spaces. This is a coarse geometric property and so is well-defined for finitely generated groups with word metrics. It is known that coarse coherence of the fundamental group has important implications for classification of high-dimensional manifolds. The Family Colimit Theorem is one of the permanence theorems that give struc- ture to the class of coarsely coherent groups. In fact, all known permanence theorems follow from the Family Colimit Theorem. We also use this theorem to construct new groups from this class. 1. Introduction Let Γ be a finitely generated group. Coarse coherence is a property of the group viewed as a metric space with a word metric, defined in terms of algebraic proper- ties invariant under quasi-isometries. It has emerged in [1, 6] as a condition that guarantees an equivalence between the K-theory of a group ring K(R[Γ]) and its G-theory G(R[Γ]). The K-theory is the central home for obstructions to a num- ber of constructions in geometric topology and number theory. We view K(R[Γ]) as a non-connective spectrum associated to a commutative ring R and the group Γ whose stable homotopy groups are the Quillen K-groups in non-negative dimen- sions and the negative K-groups of Bass in negative dimensions. The corresponding G-theory, introduced in this generality in [2], is well-known as a theory with better computational tools available compared to K-theory. -
ON BI-INVARIANT WORD METRICS 1. Introduction 1.1. the Result Let OV
June 28, 2011 11:19 WSPC/243-JTA S1793525311000556 Journal of Topology and Analysis Vol. 3, No. 2 (2011) 161–175 c World Scientific Publishing Company DOI: 10.1142/S1793525311000556 ON BI-INVARIANT WORD METRICS SWIATOS´ LAW R. GAL∗ and JAREK KEDRA† ∗University of Wroclaw, Poland and Universit¨at Wien, Austria †University of Aberdeen, UK and University of Szczecin ∗[email protected] †[email protected] Received 7 February 2011 We prove that bi-invariant word metrics are bounded on certain Chevalley groups. As an application we provide restrictions on Hamiltonian actions of such groups. Keywords: Bi-invariant metric; Hamiltonian actions; Chevalley group. 1. Introduction 1.1. The result Let OV ⊂ K be a ring of V-integers in a number field K,whereV is a set of valuations containing all Archimedean ones. Let Gπ(Φ, OV) be the Chevalley group associated with a faithful representation π : g → gl(V ) of a simple complex Lie algebra of rank at least two (see Sec. 4.1 for details). Theorem 1.1. Let Γ be a finite extension or a supergroup of finite index of the Chevalley group Gπ(Φ, OV). Then any bi-invariant metric on Γ is bounded. The examples to which Theorem 1.1 applies include the following groups. (1) SL(n; Z);√ it is a non-uniform lattice in SL(n; R). (2) SL(n; Z[ 2]); the image of its diagonal embedding into the product SL(n; R) × SL(n; R) is a non-uniform lattice. (3) SO(n; Z):={A ∈ SL(n, Z) | AJAT = J}, where J is the matrix with ones on the anti-diagonal and zeros elsewhere. -
Quasi-Isometry Rigidity of Groups
Quasi-isometry rigidity of groups Cornelia DRUT¸U Universit´e de Lille I, [email protected] Contents 1 Preliminaries on quasi-isometries 2 1.1 Basicdefinitions .................................... 2 1.2 Examplesofquasi-isometries ............................. 4 2 Rigidity of non-uniform rank one lattices 6 2.1 TheoremsofRichardSchwartz . .. .. .. .. .. .. .. .. 6 2.2 Finite volume real hyperbolic manifolds . 8 2.3 ProofofTheorem2.3.................................. 9 2.4 ProofofTheorem2.1.................................. 13 3 Classes of groups complete with respect to quasi-isometries 14 3.1 Listofclassesofgroupsq.i. complete. 14 3.2 Relatively hyperbolic groups: preliminaries . 15 4 Asymptotic cones of a metric space 18 4.1 Definition,preliminaries . .. .. .. .. .. .. .. .. .. 18 4.2 Asampleofwhatonecandousingasymptoticcones . 21 4.3 Examplesofasymptoticconesofgroups . 22 5 Relatively hyperbolic groups: image from infinitely far away and rigidity 23 5.1 Tree-graded spaces and cut-points . 23 5.2 The characterization of relatively hyperbolic groups in terms of asymptotic cones 25 5.3 Rigidity of relatively hyperbolic groups . 27 5.4 More rigidity of relatively hyperbolic groups: outer automorphisms . 29 6 Groups asymptotically with(out) cut-points 30 6.1 Groups with elements of infinite order in the center, not virtually cyclic . 31 6.2 Groups satisfying an identity, not virtually cyclic . 31 6.3 Existence of cut-points in asymptotic cones and relative hyperbolicity . 33 7 Open questions 34 8 Dictionary 35 1 These notes represent a slightly modified version of the lectures given at the summer school “G´eom´etriesa ` courbure n´egative ou nulle, groupes discrets et rigidit´es” held from the 14-th of June till the 2-nd of July 2004 in Grenoble. -
Hyperbolic Groups
Hyperbolic groups Giang Le OSU,10/18/2010 Outline • Quasi-isometry • Hyperbolic metric spaces • Hyperbolic groups • Isoperimetric inequaliBes • Dehn’s problems for groups • Rips complex and Rips theorem Quasi-isometry (X, d) and (X’,d’) two metric spaces. F: X X’ F is an isometry iF d’(F(x),F(y)) = d(x,y) For all x,y. Note. If f is an isometry then f is con2nuous and injecve. If f is surjecve then X and X’ are called isometric. Quasi-isometry is a weaker equivalence relaBon. f is (C,k)-quasi-isometry if 1/C d(x,y) – k ≤ d’(F(x),F(y)) ≤ C d(x,y) + k (X,d) and (X’,d’) are quasi-isometric if • There is a quasi-isometry F: X X’ • Every point oF X’ is in a bounded distance From the image oF F Examples oF quasi-isometric metrics spaces • Every bounded metric space is quasi-isometric to a point • (Z,d) and (R,d) • A finitely generated group G=<S|R> with the word metric and its Cayley graph Γ=Γ(G,S) with the natural metric • If S and T are two finite generaBng sets oF G then (G, ) and (G, ) are quasi-isometric Hyperbolic metric spaces MoBvaBon: in hyperbolic plane H all triangles are thin. Want to extend the concept For a general metric space? What is a triangle in a metric space? (X,d) is a metric space. A geodesic segment oF length l in (X,d) is the image oF an isometric embedding i: [0,l] -> X A (geodesic) triangle in X with verBces x, y, z is the union oF three geodesic segments, From x to y, y to z, and z to x respecBvely. -
Growth of Quotients of Groups Acting by Isometries on Gromov Hyperbolic
GROWTH OF QUOTIENTS OF GROUPS ACTING BY ISOMETRIES ON GROMOV HYPERBOLIC SPACES STEPHANE´ SABOURAU Abstract. We show that every non-elementary group G acting prop- erly and cocompactly by isometries on a proper geodesic Gromov hyper- bolic space X is growth tight. In other words, the exponential growth rate of G for the geometric (pseudo)-distance induced by X is greater than the exponential growth rate of any of its quotients by an infinite normal subgroup. This result generalizes from a unified framework pre- vious works of Arzhantseva-Lysenok and Sambusetti, and provides an answer to a question of the latter. 1. Introduction In this article, we investigate the asymptotic geometry of some discrete groups (G, d) endowed with a left-invariant metric through their exponential growth rate. Recall that the exponential growth rate of (G, d) is defined as log cardB (R) ω(G, d) = limsup G (1.1) R→+∞ R where BG(R) is the ball of radius R formed of the elements of G at distance at most R from the neutral element e. The quotient group G¯ = G/N of G by a normal subgroup N inherits the quotient distance d¯ given by the least distance between representatives. The distance d¯ is also left-invariant. Clearly, we have ω(G,¯ d¯) ≤ ω(G, d). The metric group (G, d) is said to be growth tight if ω(G,¯ d¯) <ω(G, d) for any quotient G¯ of G by an infinite normal subgroup N ✁ G. In other words, (G, d) is growth tight if it can be characterized by its exponential growth rate among its quotients by an infinite normal subgroup. -
Local Rigidity of Group Actions: Past, Present, Future
Recent Progress in Dynamics MSRI Publications Volume 54, 2007 Local rigidity of group actions: past, present, future DAVID FISHER To Anatole Katok on the occasion of his 60th birthday. ABSTRACT. This survey aims to cover the motivation for and history of the study of local rigidity of group actions. There is a particularly detailed dis- cussion of recent results, including outlines of some proofs. The article ends with a large number of conjectures and open questions and aims to point to interesting directions for future research. 1. Prologue Let be a finitely generated group, D a topological group, and D a W ! homomorphism. We wish to study the space of deformations or perturbations of . Certain trivial perturbations are always possible as soon as D is not discrete, namely we can take dd 1 where d is a small element of D. This motivates the following definition: DEFINITION 1.1. Given a homomorphism D, we say is locally rigid 0 W ! if any other homomorphism which is close to is conjugate to by a small element of D. We topologize Hom.; D/ with the compact open topology which means that two homomorphisms are close if and only if they are close on a generating set for . If D is path connected, then we can define deformation rigidity instead, meaning that any continuous path of representations t starting at is conjugate to the trivial path t by a continuous path dt in D with d0 being D Author partially supported by NSF grant DMS-0226121 and a PSC-CUNY grant. 45 46 DAVID FISHER the identity in D. -
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 certain structures: • Polynomials (field extensions) | Galois groups. • Vector spaces, possibly equipped with a bilinear form | Matrix groups. -
4 Aug 2016 Metric Geometry of Locally Compact Groups
Metric geometry of locally compact groups Yves Cornulier and Pierre de la Harpe July 31, 2016 arXiv:1403.3796v4 [math.GR] 4 Aug 2016 2 Abstract This book offers to study locally compact groups from the point of view of appropriate metrics that can be defined on them, in other words to study “Infinite groups as geometric objects”, as Gromov writes it in the title of a famous article. The theme has often been restricted to finitely generated groups, but it can favourably be played for locally compact groups. The development of the theory is illustrated by numerous examples, including matrix groups with entries in the the field of real or complex numbers, or other locally compact fields such as p-adic fields, isometry groups of various metric spaces, and, last but not least, discrete group themselves. Word metrics for compactly generated groups play a major role. In the particular case of finitely generated groups, they were introduced by Dehn around 1910 in connection with the Word Problem. Some of the results exposed concern general locally compact groups, such as criteria for the existence of compatible metrics (Birkhoff-Kakutani, Kakutani-Kodaira, Struble). Other results concern special classes of groups, for example those mapping onto Z (the Bieri-Strebel splitting theorem, generalized to locally compact groups). Prior to their applications to groups, the basic notions of coarse and large-scale geometry are developed in the general framework of metric spaces. Coarse geometry is that part of geometry concerning properties of metric spaces that can be formulated in terms of large distances only. -
Rigidity Theorems in Kähler Geometry and Fundamental Groups of Varieties
Several Complex Variables MSRI Publications Volume 37, 1999 Rigidity Theorems in K¨ahler Geometry and Fundamental Groups of Varieties DOMINGO TOLEDO Abstract. We review some developments in rigidity theory of compact K¨ahler manifolds and related developments on restrictions on their possible fundamental groups. 1. Introduction This article surveys some developments, which started almost twenty years ago, on the applications of harmonic mappings to the study of topology and geometry of K¨ahler manifolds. The starting point of these developments was the strong rigidity theorem of Siu [1980], which is a generalization of a special case of the strong rigidity theorem of Mostow [1973] for locally symmetric manifolds. Siu’s theorem introduced for the first time an effective way of using, in a broad way, the theory of harmonic mappings to study mappings between manifolds. Many interesting applications of harmonic mappings to the study of mappings of K¨ahler manifolds to nonpositively curved spaces have been developed since then by various authors. More generally the linear representations (and other rep- resentations) of their fundamental groups have also been studied. Our purpose here is to give a general survey of this work. One interesting by-product of this study is that it has produced new results on an old an challenging question: what groups can be fundamental groups of smooth projective varieties (or of compact K¨ahler manifolds)? These groups are called K¨ahler groups for short, and have been intensively studied in the last decade. New restrictions on K¨ahler groups have been obtained by these techniques. On the other hand new examples of K¨ahler groups have also shown the limitations of some of these methods. -
Boundaries of Hyperbolic Groups
Contemporary Mathematics Boundaries of hyperbolic groups Ilya Kapovich and Nadia Benakli Abstract. In this paper we survey the known results about boundaries of word-hyperbolic groups. 1. Introduction In the last fifteen years Geometric Group Theory has enjoyed fast growth and rapidly increasing influence. Much of this progress has been spurred by remarkable work of M.Gromov [95], [96] who has advanced the theory of word-hyperbolic groups (also referred to as Gromov-hyperbolic or negatively curved groups). The basic idea was to explore the connections between abstract algebraic properties of a group and geometric properties of a space on which this group acts nicely by isometries. This connection turns out to be particularly strong when the space in question has some hyperbolic or negative curvature characteristics. This led M.Gromov [95] as well as J.Cannon [48] to the notions of a Gromov-hyperbolic (or "negatively curved") space, a word-hyperbolic group and to the development of rich, beautiful and powerful theory of word-hyperbolic groups. These ideas have caused something of a revolution in the more general field of Geometric and Combinatorial Group Theory. For example it has turned out that most traditional algorithmic problems (such as the word-problem and the conjugacy problem), while unsolvable for finitely presented groups in general, have fast and transparent solutions for hyperbolic groups. Even more amazingly, a result of Z.Sela [167] states that the isomorphism problem is also solvable for torsion-free word-hyperbolic groups. Gromov's theory has also found numerous applications to and connections with many branches of Mathematics.