Local Connectedness of Bowditch Boundary of Relatively Hyperbolic Groups
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Hyperbolic Manifolds: an Introduction in 2 and 3 Dimensions Albert Marden Frontmatter More Information
Cambridge University Press 978-1-107-11674-0 - Hyperbolic Manifolds: An Introduction in 2 and 3 Dimensions Albert Marden Frontmatter More information HYPERBOLIC MANIFOLDS An Introduction in 2 and 3 Dimensions Over the past three decades there has been a total revolution in the classic branch of mathematics called 3-dimensional topology, namely the discovery that most solid 3-dimensional shapes are hyperbolic 3-manifolds. This book introduces and explains hyperbolic geometry and hyperbolic 3- and 2-dimensional manifolds in the first two chapters, and then goes on to develop the subject. The author discusses the profound discoveries of the astonishing features of these 3-manifolds, helping the reader to understand them without going into long, detailed formal proofs. The book is heav- ily illustrated with pictures, mostly in color, that help explain the manifold properties described in the text. Each chapter ends with a set of Exercises and Explorations that both challenge the reader to prove assertions made in the text, and suggest fur- ther topics to explore that bring additional insight. There is an extensive index and bibliography. © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-11674-0 - Hyperbolic Manifolds: An Introduction in 2 and 3 Dimensions Albert Marden Frontmatter More information [Thurston’s Jewel (JB)(DD)] Thurston’s Jewel: Illustrated is the convex hull of the limit set of a kleinian group G associated with a hyperbolic manifold M(G) with a single, incompressible boundary component. The translucent convex hull is pictured lying over p. 8.43 of Thurston [1979a] where the theory behind the construction of such convex hulls was first formulated. -
Acylindrical Accessibility for Groups Acting on R-Trees
ACYLINDRICAL ACCESSIBILITY FOR GROUPS ACTING ON R-TREES ILYA KAPOVICH AND RICHARD WEIDMANN Abstract. We prove an acylindrical accessibility theorem for finitely generated groups acting on R-trees. Namely, we show that if G is a freely indecomposable non-cyclic k-generated group acting minimally and M- acylindrically on an R-tree X then there is a finite subtree Y ⊆ X of measure at most 2M(k − 1) + such that GY = X. This generalizes theorems of Z.Sela and T.Delzant about actions on simplicial trees. 1. Introduction An isometric action of a group G on an R-tree X is said to be M- acylindrical (where M ≥ 0) if for any g ∈ G, g 6= 1 we have diam Fix(g) ≤ M, that is any segment fixed point-wise by g has length at most M. For example the action of an amalgamated free product G = A ∗C B on the corresponding Bass-Serre tree is 2-acylindrical if C is malnormal in A and 1-acylindrical if C is malnormal in both A and B. In fact the notion of acylindricity seems to have first appeared in this context in the work of Karras and Solitar [32], who termed it being r-malnormal. Sela [37] proved an important acylindrical accessibility result for finitely generated groups which, when applied to one-ended groups, can be restated as follows: for any one-ended finitely generated group G and any M ≥ 0 there is a constant c(G, M) > 0 such that for any minimal M-acylindrical action of G on a simplicial tree X the quotient graph X/G has at most c(G, M) edges. -
A Brief Survey of the Deformation Theory of Kleinian Groups 1
ISSN 1464-8997 (on line) 1464-8989 (printed) 23 Geometry & Topology Monographs Volume 1: The Epstein birthday schrift Pages 23–49 A brief survey of the deformation theory of Kleinian groups James W Anderson Abstract We give a brief overview of the current state of the study of the deformation theory of Kleinian groups. The topics covered include the definition of the deformation space of a Kleinian group and of several im- portant subspaces; a discussion of the parametrization by topological data of the components of the closure of the deformation space; the relationship between algebraic and geometric limits of sequences of Kleinian groups; and the behavior of several geometrically and analytically interesting functions on the deformation space. AMS Classification 30F40; 57M50 Keywords Kleinian group, deformation space, hyperbolic manifold, alge- braic limits, geometric limits, strong limits Dedicated to David Epstein on the occasion of his 60th birthday 1 Introduction Kleinian groups, which are the discrete groups of orientation preserving isome- tries of hyperbolic space, have been studied for a number of years, and have been of particular interest since the work of Thurston in the late 1970s on the geometrization of compact 3–manifolds. A Kleinian group can be viewed either as an isolated, single group, or as one of a member of a family or continuum of groups. In this note, we concentrate our attention on the latter scenario, which is the deformation theory of the title, and attempt to give a description of various of the more common families of Kleinian groups which are considered when doing deformation theory. -
Ergodic Currents Dual to a Real Tree Thierry Coulbois, Arnaud Hilion
Ergodic currents dual to a real tree Thierry Coulbois, Arnaud Hilion To cite this version: Thierry Coulbois, Arnaud Hilion. Ergodic currents dual to a real tree. Ergodic Theory and Dynamical Systems, Cambridge University Press (CUP), 2016, 36 (3), pp.745-766. 10.1017/etds.2014.78. hal- 01481866 HAL Id: hal-01481866 https://hal.archives-ouvertes.fr/hal-01481866 Submitted on 3 Mar 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ERGODIC CURRENTS DUAL TO A REAL TREE THIERRY COULBOIS, ARNAUD HILION Abstract. Let T be an R-tree with dense orbits in the boundary of Outer space. When the free group FN acts freely on T , we prove that the number of projective classes of ergodic currents dual to T is bounded above by 3N − 5. We combine Rips induction and splitting induction to define unfolding induction for such an R-tree T . Given a current µ dual to T , the unfolding induction produces a sequence of approximations converging towards µ. We also give a unique ergodicity criterion. 1. Introduction 1.1. Main results. Let FN be the free group with N generators. -
Beyond Serre's" Trees" in Two Directions: $\Lambda $--Trees And
Beyond Serre’s “Trees” in two directions: Λ–trees and products of trees Olga Kharlampovich,∗ Alina Vdovina† October 31, 2017 Abstract Serre [125] laid down the fundamentals of the theory of groups acting on simplicial trees. In particular, Bass-Serre theory makes it possible to extract information about the structure of a group from its action on a simplicial tree. Serre’s original motivation was to understand the structure of certain algebraic groups whose Bruhat–Tits buildings are trees. In this survey we will discuss the following generalizations of ideas from [125]: the theory of isometric group actions on Λ-trees and the theory of lattices in the product of trees where we describe in more detail results on arithmetic groups acting on a product of trees. 1 Introduction Serre [125] laid down the fundamentals of the theory of groups acting on sim- plicial trees. The book [125] consists of two parts. The first part describes the basics of what is now called Bass-Serre theory. This theory makes it possi- ble to extract information about the structure of a group from its action on a simplicial tree. Serre’s original motivation was to understand the structure of certain algebraic groups whose Bruhat–Tits buildings are trees. These groups are considered in the second part of the book. Bass-Serre theory states that a group acting on a tree can be decomposed arXiv:1710.10306v1 [math.GR] 27 Oct 2017 (splits) as a free product with amalgamation or an HNN extension. Such a group can be represented as a fundamental group of a graph of groups. -
Curriculum Vitae
Curriculum vitae Kate VOKES Adresse E-mail [email protected] Institut des Hautes Études Scientifiques 35 route de Chartres 91440 Bures-sur-Yvette Site Internet www.ihes.fr/~vokes/ France Postes occupées • octobre 2020–juin 2021: Postdoctorante (HUAWEI Young Talents Programme), Institut des Hautes Études Scientifiques, Université Paris-Saclay, France • janvier 2019–octobre 2020: Postdoctorante, Institut des Hautes Études Scien- tifiques, Université Paris-Saclay, France • juillet 2018–decembre 2018: Fields Postdoctoral Fellow, Thematic Program on Teichmüller Theory and its Connections to Geometry, Topology and Dynamics, Fields Institute for Research in Mathematical Sciences, Toronto, Canada Formation • octobre 2014 – juin 2018: Thèse de Mathématiques University of Warwick, Royaume-Uni Titre de thèse: Large scale geometry of curve complexes Directeur de thèse: Brian Bowditch • octobre 2010 – juillet 2014: MMath(≈Licence+M1) en Mathématiques Durham University, Royaume-Uni, Class I (Hons) Domaine de recherche Topologie de basse dimension et géométrie des groupes, en particulier le groupe mod- ulaire d’une surface, l’espace de Teichmüller, le complexe des courbes et des complexes similaires. Publications et prépublications • (avec Jacob Russell) Thickness and relative hyperbolicity for graphs of multicurves, prépublication (2020) : arXiv:2010.06464 • (avec Jacob Russell) The (non)-relative hyperbolicity of the separating curve graph, prépublication (2019) : arXiv:1910.01051 • Hierarchical hyperbolicity of graphs of multicurves, accepté dans Algebr. -
Arxiv:Math/9701212V1
GEOMETRY AND TOPOLOGY OF COMPLEX HYPERBOLIC AND CR-MANIFOLDS Boris Apanasov ABSTRACT. We study geometry, topology and deformation spaces of noncompact complex hyperbolic manifolds (geometrically finite, with variable negative curvature), whose properties make them surprisingly different from real hyperbolic manifolds with constant negative curvature. This study uses an interaction between K¨ahler geometry of the complex hyperbolic space and the contact structure at its infinity (the one-point compactification of the Heisenberg group), in particular an established structural theorem for discrete group actions on nilpotent Lie groups. 1. Introduction This paper presents recent progress in studying topology and geometry of com- plex hyperbolic manifolds M with variable negative curvature and spherical Cauchy- Riemannian manifolds with Carnot-Caratheodory structure at infinity M∞. Among negatively curved manifolds, the class of complex hyperbolic manifolds occupies a distinguished niche due to several reasons. First, such manifolds fur- nish the simplest examples of negatively curved K¨ahler manifolds, and due to their complex analytic nature, a broad spectrum of techniques can contribute to the study. Simultaneously, the infinity of such manifolds, that is the spherical Cauchy- Riemannian manifolds furnish the simplest examples of manifolds with contact structures. Second, such manifolds provide simplest examples of negatively curved arXiv:math/9701212v1 [math.DG] 26 Jan 1997 manifolds not having constant sectional curvature, and already obtained -
Lecture 4, Geometric and Asymptotic Group Theory
Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Quasi-isometry Limit groups Free actions Lecture 4, Geometric and asymptotic group theory Olga Kharlampovich NYC, Sep 16 1 / 32 Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Homomorphisms of groups Quasi-isometry Limit groups Free actions The universal property of free groups allows one to describe arbitrary groups in terms of generators and relators. Let G be a group with a generating set S. By the universal property of free groups there exists a homomorphism ': F (S) ! G such that '(s) = s for s 2 S. It follows that ' is onto, so by the first isomorphism theorem G ' F (S)=ker('): In this event ker(') is viewed as the set of relators of G, and a group word w 2 ker(') is called a relator of G in generators S. If a subset R ⊂ ker(') generates ker(') as a normal subgroup of F (S) then it is termed a set of defining relations of G relative to S. 2 / 32 Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Homomorphisms of groups Quasi-isometry Limit groups Free actions The pair hS j Ri is called a presentation of G, it determines G uniquely up to isomorphism. The presentation hS j Ri is finite if both sets S and R are finite. A group is finitely presented if it has at least one finite presentation. Presentations provide a universal method to describe groups. Example of finite presentations 1 G = hs1;:::; sn j [si ; sj ]; 81 ≤ i < j ≤ ni is the free abelian group of rank n. -
Fractal Geometry and Applications in Forest Science
ACKNOWLEDGMENTS Egolfs V. Bakuzis, Professor Emeritus at the University of Minnesota, College of Natural Resources, collected most of the information upon which this review is based. We express our sincere appreciation for his investment of time and energy in collecting these articles and books, in organizing the diverse material collected, and in sacrificing his personal research time to have weekly meetings with one of us (N.L.) to discuss the relevance and importance of each refer- enced paper and many not included here. Besides his interdisciplinary ap- proach to the scientific literature, his extensive knowledge of forest ecosystems and his early interest in nonlinear dynamics have helped us greatly. We express appreciation to Kevin Nimerfro for generating Diagrams 1, 3, 4, 5, and the cover using the programming package Mathematica. Craig Loehle and Boris Zeide provided review comments that significantly improved the paper. Funded by cooperative agreement #23-91-21, USDA Forest Service, North Central Forest Experiment Station, St. Paul, Minnesota. Yg._. t NAVE A THREE--PART QUE_.gTION,, F_-ACHPARToF:WHICH HA# "THREEPAP,T_.<.,EACFi PART" Of:: F_.AC.HPART oF wHIct4 HA.5 __ "1t4REE MORE PARTS... t_! c_4a EL o. EP-.ACTAL G EOPAgTI_YCoh_FERENCE I G;:_.4-A.-Ti_E AT THB Reprinted courtesy of Omni magazine, June 1994. VoL 16, No. 9. CONTENTS i_ Introduction ....................................................................................................... I 2° Description of Fractals .................................................................................... -
Lec 11, Limit Groups, R-Trees
Lec 11, Limit Groups, R-trees Theorem (Kharl, Mias,razzzz). Suppose H is not free. Then there is a finite set S = fs : H Hsg of proper epimorphisms such that: • for all h 2 Hom(H; F); there exists a 2 Aut(H) such that h ◦ a factors through S. a .... H ................................................................................................................... H . s . h . ....... ....... ... ... ... Hs ................................................................................................................ F • We will refine this statement and discuss a proof. Real trees • Areal tree ( T; dT )is a metric space such that between any two points t; t0 2 T , there is a unique arc∗ from t to t0 and this arc is the image of an isometric embedding of an interval. Example. A finite simplicial real tree is a finite tree with each edge identified with an interval. Example. A countable increasing union of finite simplicial real trees. Example. 0-hyperbolic spaces embed into real trees. ∗the image of an embedding σ : [x; x0] ! T with σ(x) = t and σ(x0) = t0 1 Isometries of real trees • An isometry η of a real tree T either elliptic or hyperbolic. • Elliptic η fixes a point of T . The axis of η is F ix(η). • Hyperbolic η leaves invariant an isometrically embedded R (its axis Aη). Points on Aη are translated by `T (η):= min fdT (t; η(t)) j t 2 T g: • We will be interested in isometric actions of a fg group H on T . • The H-tree T is minimal if T contains no proper invariant H-subtrees. Lemma. If H is fg and T is a minimal H-tree, then T is either a point or the union of the axes of the hyperbolic elements of H. -
A Metric Between Quasi-Isometric Trees 1
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 140, Number 1, January 2012, Pages 325–335 S 0002-9939(2011)11286-3 Article electronically published on August 11, 2011 A METRIC BETWEEN QUASI-ISOMETRIC TREES ALVARO´ MART´INEZ-PEREZ´ (Communicated by Alexander N. Dranishnikov) Abstract. It is known that PQ-symmetric maps on the boundary charac- terize the quasi-isometry type of visual hyperbolic spaces, in particular, of geodesically complete R-trees. We define a map on pairs of PQ-symmetric ultrametric spaces which characterizes the branching of the space. We also show that when the ultrametric spaces are the corresponding end spaces, this map defines a metric between rooted geodesically complete simplicial trees with minimal vertex degree 3 in the same quasi-isometry class. Moreover, this metric measures how far the trees are from being rooted isometric. 1. Introduction The study of quasi-isometries between trees and the induced maps on their end spaces has a voluminous literature. This is often set in the more general context of hyperbolic metric spaces and their boundaries. See Bonk and Schramm [2], Buyalo and Schroeder [4], Ghys and de la Harpe [6], and Paulin [14] to name a few. For a quasi-isometry f : X → Y between Gromov hyperbolic, almost geodesic metric spaces, Bonk and Schramm define [2, Proposition 6.3] the induced map ∂f : ∂X → ∂Y between the boundaries at infinity and prove [2, Theorem 6.5] that ∂f is PQ-symmetric with respect to any metrics on ∂X and ∂Y in their canonical gauges. Moreover, they prove that a PQ-symmetric map between bounded metric spaces can be extended to a map between their hyperbolic cones and obtain that a PQ-symmetric map between the boundaries at infinity of Gromov hyperbolic, almost geodesic metric spaces, implies a quasi-isometry equivalence between the spaces. -
Curriculum Vitae
Curriculum vitae Kate Vokes Address Email [email protected] Institut des Hautes Études Scientifiques 35 route de Chartres 91440 Bures-sur-Yvette Webpage www.ihes.fr/~vokes/ France Positions • October 2020–June 2021: Postdoctoral Researcher (HUAWEI Young Tal- ents Programme), Institut des Hautes Études Scientifiques, Université Paris- Saclay, France • January 2019–September 2020: Postdoctoral Researcher, Institut des Hautes Études Scientifiques, Université Paris-Saclay, France • July 2018–December 2018: Fields Postdoctoral Fellow, Thematic Pro- gram on Teichmüller Theory and its Connections to Geometry, Topology and Dynamics, Fields Institute for Research in Mathematical Sciences, Toronto, Canada Education • October 2014 – June 2018: PhD in Mathematics University of Warwick, United Kingdom Thesis title: Large scale geometry of curve complexes Supervisor: Professor Brian Bowditch • October 2010 – July 2014: MMath in Mathematics (Class I (Hons)) Durham University, United Kingdom Research interests Low-dimensional topology and geometric group theory, particularly mapping class groups, Teichmüller spaces, curve complexes and related complexes. Papers • (with Jacob Russell) Thickness and relative hyperbolicity for graphs of multic- urves, preprint (2020); available at arXiv:2010.06464 • (with Jacob Russell) The (non)-relative hyperbolicity of the separating curve graph, preprint (2019); available at arXiv:1910.01051 • Hierarchical hyperbolicity of graphs of multicurves, accepted in Algebr. Geom. Topol.; available at arXiv:1711.03080 • Uniform quasiconvexity