The Geometry and Topology of Coxeter Groups 1st Edition by Michael W Davis ISBN 1400845947 9781400845941

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ISBN 10: 1400845947
ISBN 13: 9781400845941
Author: Michael W Davis

The Geometry and Topology of Coxeter Groups 1st Table of contents:

Chapter 1: Introduction and Preview

• 1.1 Introduction

• 1.2 A Preview of the Right-Angled Case

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Chapter 2: Some Basic Notions in Geometric Group Theory

• 2.1 Cayley Graphs and Word Metrics

• 2.2 Cayley 2-Complexes

• 2.3 Background on Aspherical Spaces

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Chapter 3: Coxeter Groups

• 3.1 Dihedral Groups

• 3.2 Reflection Systems

• 3.3 Coxeter Systems

• 3.4 The Word Problem

• 3.5 Coxeter Diagrams

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Chapter 4: More Combinatorial Theory of Coxeter Groups

• 4.1 Special Subgroups in Coxeter Groups

• 4.2 Reflections

• 4.3 The Shortest Element in a Special Coset

• 4.4 Another Characterization of Coxeter Groups

• 4.5 Convex Subsets of W

• 4.6 The Element of Longest Length

• 4.7 The Letters with Which a Reduced Expression Can End

• 4.8 A Lemma of Tits

• 4.9 Subgroups Generated by Reflections

• 4.10 Normalizers of Special Subgroups

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Chapter 5: The Basic Construction

• 5.1 The Space U

• 5.2 The Case of a Pre-Coxeter System

• 5.3 Sectors in U

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Chapter 6: Geometric Reflection Groups

• 6.1 Linear Reflections

• 6.2 Spaces of Constant Curvature

• 6.3 Polytopes with Nonobtuse Dihedral Angles

• 6.4 The Developing Map

• 6.5 Polygon Groups

• 6.6 Finite Linear Groups Generated by Reflections

• 6.7 Examples of Finite Reflection Groups

• 6.8 Geometric Simplices: The Gram Matrix and the Cosine Matrix

• 6.9 Simplicial Coxeter Groups: Lanner’s Theorem

• 6.10 Three-dimensional Hyperbolic Reflection Groups: Andreev’s Theorem

• 6.11 Higher-dimensional Hyperbolic Reflection Groups: Vinberg’s Theorem

• 6.12 The Canonical Representation

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Chapter 7: The Complex Σ

• 7.1 The Nerve of a Coxeter System

• 7.2 Geometric Realizations

• 7.3 A Cell Structure on Σ

• 7.4 Examples

• 7.5 Fixed Posets and Fixed Subspaces

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Chapter 8: The Algebraic Topology of U and of Σ

• 8.1 The Homology of U

• 8.2 Acyclicity Conditions

• 8.3 Cohomology with Compact Supports

• 8.4 The Case Where X Is a General Space

• 8.5 Cohomology with Group Ring Coefficients

• 8.6 Background on the Ends of a Group

• 8.7 The Ends of W

• 8.8 Splittings of Coxeter Groups

• 8.9 Cohomology of Normalizers of Spherical Special Subgroups

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Chapter 9: The Fundamental Group and the Fundamental Group at Infinity

• 9.1 The Fundamental Group of U

• 9.2 What Is Simply Connected at Infinity?

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Chapter 10: Actions on Manifolds

• 10.1 Reflection Groups on Manifolds

• 10.2 The Tangent Bundle

• 10.3 Background on Contractible Manifolds

• 10.4 Background on Homology Manifolds

• 10.5 Aspherical Manifolds Not Covered by Euclidean Space

• 10.6 When Is a Manifold? (?)

• 10.7 Reflection Groups on Homology Manifolds

• 10.8 Generalized Homology Spheres and Polytopes

• 10.9 Virtual Poincaré Duality Groups

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Chapter 11: The Reflection Group Trick

• 11.1 The First Version of the Trick

• 11.2 Examples of Fundamental Groups of Closed Aspherical Manifolds

• 11.3 Nonsmoothable Aspherical Manifolds

• 11.4 The Borel Conjecture and the PDn-Group Conjecture

• 11.5 The Second Version of the Trick

• 11.6 The Bestvina–Brady Examples

• 11.7 The Equivariant Reflection Group Trick

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Chapter 12: Is CAT(0): Theorems of Gromov and Moussong

• 12.1 A Piecewise Euclidean Cell Structure on Σ

• 12.2 The Right-Angled Case

• 12.3 The General Case

• 12.4 The Visual Boundary of Σ

• 12.5 Background on Word Hyperbolic Groups

• 12.6 When Is CAT(−1)?

• 12.7 Free Abelian Subgroups of Coxeter Groups

• 12.8 Relative Hyperbolization

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Chapter 13: Rigidity

• 13.1 Definitions, Examples, Counterexamples

• 13.2 Spherical Parabolic Subgroups and Their Fixed Subspaces

• 13.3 Coxeter Groups of Type PM

• 13.4 Strong Rigidity for Groups of Type PM

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Chapter 14: Free Quotients and Surface Subgroups

• 14.1 Largeness

• 14.2 Surface Subgroups

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Chapter 15: Another Look at (Co)homology

• 15.1 Cohomology with Constant Coefficients

• 15.2 Decompositions of Coefficient Systems

• 15.3 The W-Module Structure on (Co)homology

• 15.4 The Case Where W Is Finite

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Chapter 16: The Euler Characteristic

• 16.1 Background on Euler Characteristics

• 16.2 The Euler Characteristic Conjecture

• 16.3 The Flag Complex Conjecture

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Chapter 17: Growth Series

• 17.1 Rationality of the Growth Series

• 17.2 Exponential versus Polynomial Growth

• 17.3 Reciprocity

• 17.4 Relationship with the h-Polynomial

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Chapter 18: Buildings

• 18.1 The Combinatorial Theory of Buildings

• 18.2 The Geometric Realization of a Building

• 18.3 Buildings Are CAT(0)

• 18.4 Euler–Poincaré Measure

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Chapter 19: Hecke–von Neumann Algebras

• 19.1 Hecke Algebras

• 19.2 Hecke–von Neumann Algebras

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