Infinite Group Actions on Polyhedra (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 77)

In the past fifteen years, the theory of right-angled Artin groups and special cube complexes has emerged as a central topic in geometric group theory. This monograph provides an account of this theory, along with other modern techniques in geometric group theory. Structured around the theme of group actions on contractible polyhedra, this book explores two prominent methods for constructing such actions: utilizing the group of deck transformations of the universal cover of a nonpositively curved polyhedron and leveraging the theory of simple complexes of groups. The book presents various approaches to obtaining cubical examples through CAT(0) cube complexes, including the polyhedral product construction, hyperbolization procedures, and the Sageev construction. Moreover, it offers a unified presentation of important non-cubical examples, such as Coxeter groups, Artin groups, and groups that act on buildings. Designed as a resource for graduate students and researchers specializing in geometric group theory, this book should also be of high interest to mathematicians in related areas, such as 3-manifolds. Preface Contents Acronyms Part I Introduction 1 Overview Part II Nonpositively Curved Cube Complexes 2 Polyhedral Preliminaries 2.1 Cell Complexes, Links 2.1.1 The CAT(κ)-inequality 2.1.2 Piecewise Hyperbolic Polyhedra 2.1.3 Quasi-Isometries, Word Hyperbolic Groups 2.2 Flag Complexes and Gromov's Lemma 2.3 Group Actions on CAT(0) Spaces 2.4 Examples of NPC Square Complexes and Polygon Complexes 2.4.1 Trees 2.4.2 Products 2.4.3 Higman Groups 2.4.4 The Burger–Mozes Examples 2.4.5 Nonpositively Curved Polygons of Groups 3 Right-Angled Spaces and Groups 3.1 Polyhedral Products 3.1.1 The Cube Complexes KL , PL , ZL , and TL 3.1.2 Graph Products of Groups 3.1.3 Wreath-Graph Products 3.2 Right-Angled Reflection Groups on Manifolds 3.2.1 Some Basic Notions Concerning Manifolds 3.2.2 Aspherical Manifolds Not Covered by Euclidean Space 3.2.3 NPC Manifolds Not Covered by Euclidean Space 3.2.4 Real Toric Manifolds 3.2.5 Haken Manifolds 3.2.6 The Reflection Group Trick 3.3 Blowing Up Zonotopal Cell Complexes 3.3.1 Zonotopes 3.3.2 Blowing Up Zonotopes to Cube Complexes 3.3.3 Mock Reflection Groups Part III Coxeter Groups, Artin Groups, Buildings 4 Coxeter Groups, Artin Groups, Buildings 4.1 Some Simple Complexes of Groups 4.1.1 Quick Summary 4.1.2 The Basic Examples 4.1.3 The K(π,1)-Question for Simple Complexes of Groups 4.2 Coxeter Groups Outline of This Section 4.2.1 Spherical Coxeter Groups 4.2.2 Coxeter Zonotopes 4.2.3 The Davis–Moussong Complex 4.2.4 Moussong's Lemma 4.2.5 Relatively Hyperbolic Coxeter Groups 4.2.6 A Different Piecewise Euclidean Metric on the Fundamental Chamber 4.2.7 The Tits Representation 4.2.8 Groups of Isometries of Hn Versus Word Hyperbolic Groups 4.2.9 Convex Cocompact Projective Representations 4.3 Artin Groups Outline of This Section 4.3.1 The Deligne Complex 4.3.2 The Complement of a Reflection Arrangement 4.3.3 The Salvetti Complex and Its Universal Cover 4.3.4 The K(π,1)-Conjecture for Artin Groups 4.3.5 Bestvina's Normal Form Complex 4.4 Buildings Outline of This Section 4.4.1 The Combinatorial Theory of Buildings 4.4.2 Geometric Realizations of Buildings 4.4.3 Regular Right-Angled Buildings 4.4.4 A Local Approach to Buildings 4.4.5 Pullbacks 4.4.6 Using Pullbacks to Construct Examples 4.4.7 Generalized Polygons 4.4.8 The Basic Algebraic Examples Part IV More on NPC Cube Complexes 5 General Theory of Cube Complexes 5.1 Abstract CAT(0) Cube Complexes: Half-Space Systems 5.1.1 Hyperplanes in Cube Complexes 5.1.2 Half-Space Systems and Sageev's Construction 5.1.3 Geometric Actions on the Associated CAT(0) Cube Complexes 5.1.4 Cubulating Hyperbolic Manifolds 5.2 Special Cube Complexes 5.2.1 The Square Complexes COX(Γ) and ART(Γ) 5.2.2 Definitions of Specialness 5.2.3 Virtual Specialness 5.3 Separability 5.3.1 RAAGs Are Commensurable with RACGs 5.3.2 Convex Subsets of Coxeter Groups and CubeComplexes 5.3.3 Convex Cocompact Subgroups of RACGs 5.3.4 Separability of Word-Quasiconvex Subgroups 5.3.5 Applications to 3-Manifolds 6 Hyperbolization 6.1 Properties of Hyperbolization 6.1.1 Axioms for Hyperbolization 6.1.2 A Relative Version of Hyperbolization 6.2 Applications to Aspherical Manifolds 6.2.1 Bordisms of Aspherical Manifolds 6.2.2 Nontriangulable Aspherical Manifolds 6.2.3 More on NPC Manifolds not Coveredby Euclidean Space 6.3 The Product with Interval Hyperbolization Procedure 6.3.1 The Reflection Group Trick with Nonpositive Curvature 6.4 Gromov's Hyperbolization Procedure 6.4.1 A General Technique for Hyperbolization 6.4.2 The Space Ω(A,∂A) 6.4.3 Gromov's Construction 6.5 Strict Hyperbolization 7 Morse Theory and Bestvina–Brady Groups 7.1 Morse Theory on Convex Cell Complexes 7.1.1 Functions that Restrict to an Affine Map on Each Cell 7.1.2 Distance Functions 7.1.3 Topology at Infinity 7.2 Finiteness Properties of Groups 7.2.1 Finiteness Properties for Proper Actions 7.3 Level Sets and Bestvina–Brady Groups 7.4 Some Applications 7.4.1 Poincaré Duality Groups that Are Not FinitelyPresentable 7.4.2 Symmetries of RAAGs and Bestvina–Brady Groups 7.4.3 Uncountably Many Groups of Type FP A Complexes of Groups A.1 Simple Complexes of Groups: Definitions A.2 The Basic Construction for a Simple Complex of Groups A.2.1 The Universal Cover When |Q| Is Not SimplyConnected A.3 General Complexes of Groups A.3.1 Overview A.3.2 The Category CGR A.3.3 The Fundamental Group and Universal Cover of a Complex of Groups A.4 Aspherical Realizations and the K(π,1)-Question A.4.1 Complexes of Spaces and van Kampen's Theorem A.4.2 Aspherical Realizations References Index