The Geometry and Topology of Coxeter Groups. (LMS-32)

Michael W. Davis

• Publisher: Princeton University Press
• Year: 2012
• Print ISBN:
  9780691131382
• eBook ISBN:
  9781400845941
• Topic: Mathematics
• Subtopic: Geometry
• Index: Mathematics

The Geometry and Topology of Coxeter Groups is a comprehensive and authoritative treatment of Coxeter groups from the viewpoint of geometric group theory. Groups generated by reflections are ubiquitous in mathematics, and there are classical examples of reflection groups in spherical, Euclidean, and hyperbolic geometry. Any Coxeter group can be realized as a group generated by reflection on a certain contractible cell complex, and this complex is the principal subject of this book. The book explains a theorem of Moussong that demonstrates that a polyhedral metric on this cell complex is nonpositively curved, meaning that Coxeter groups are "CAT(0) groups." The book describes the reflection group trick, one of the most potent sources of examples of aspherical manifolds. And the book discusses many important topics in geometric group theory and topology, including Hopf's theory of ends; contractible manifolds and homology spheres; the Poincaré Conjecture; and Gromov's theory of CAT(0) spaces and groups. Finally, the book examines connections between Coxeter groups and some of topology's most famous open problems concerning aspherical manifolds, such as the Euler Characteristic Conjecture and the Borel and Singer conjectures.

Table of contents

• Cover
• Contents
• Preface
• Chapter 1 INTRODUCTION AND PREVIEW
• Chapter 2 SOME BASIC NOTIONS IN GEOMETRIC GROUP THEORY
• Chapter 3 COXETER GROUPS
• Chapter 4 MORE COMBINATORIAL THEORY OF COXETER GROUPS
• Chapter 5 THE BASIC CONSTRUCTION
• Chapter 6 GEOMETRIC REFLECTION GROUPS
• Chapter 7 THE COMPLEX Σ
• Chapter 8 THE ALGEBRAIC TOPOLOGY OF U AND OF Σ
• Chapter 9 THE FUNDAMENTAL GROUP AND THE FUNDAMENTAL GROUP AT INFINITY
• Chapter 10 ACTIONS ON MANIFOLDS
• Chapter 11 THE REFLECTION GROUP TRICK
• Chapter 12 Σ IS CAT(O): THEOREMS OF GROMOV AND MOUSSONG
• Chapter 13 RIGIDITY
• Chapter 14 FREE QUOTIENTS AND SURFACE SUBGROUPS
• Chapter 15 ANOTHER LOOK AT (CO)HOMOLOGY
• Chapter 16 THE EULER CHARACTERISTIC
• Chapter 17 GROWTH SERIES
• Chapter 18 BUILDINGS
• Chapter 19 HECKE–VON NEUMANN ALGEBRAS
• Chapter 20 WEIGHTED L[sup(2)]-(CO)HOMOLOGY
• 20.4 Weighted Poincaré Duality
• Appendix A: CELL COMPLEXES
• Appendix B: REGULAR POLYTOPES
• Appendix C: THE CLASSIFICATION OF SPHERICAL AND EUCLIDEAN COXETER GROUPS
• Appendix D: THE GEOMETRIC REPRESENTATION
• Appendix E: COMPLEXES OF GROUPS
• Appendix F: HOMOLOGY AND COHOMOLOGY OF GROUPS
• Appendix G: ALGEBRAIC TOPOLOGY AT INFINITY
• Appendix H: THE NOVIKOV AND BOREL CONJECTURES
• Appendix I: NONPOSITIVE CURVATURE
• Appendix J: L[sup(2)]-(CO)HOMOLOGY
• Bibliography
• Index