The geometry and topology of coxeter groups / Michael W. Davis.

Format:

Book

Author/Creator:

Davis, Michael, 1949 April 26-

Series:

London Mathematical Society monographs.

London Mathematical Society monographs series

Language:

English

Subjects (All):

Coxeter groups.

Geometric group theory.

Physical Description:

1 online resource (601 p.)

Edition:

Course Book

Place of Publication:

Princeton : Princeton University Press, c2008.

Language Note:

English

Summary:

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.

Contents:

Frontmatter

Contents

Preface

Chapter One. INTRODUCTION AND PREVIEW

Chapter Two. SOME BASIC NOTIONS IN GEOMETRIC GROUP THEORY

Chapter Three. COXETER GROUPS

Chapter Four. MORE COMBINATORIAL THEORY OF COXETER GROUPS

Chapter Five. THE BASIC CONSTRUCTION

Chapter Six. GEOMETRIC REFLECTION GROUPS

Chapter Seven. THE COMPLEX Σ

Chapter Eight. THE ALGEBRAIC TOPOLOGY OF U AND OF Σ

Chapter Nine. THE FUNDAMENTAL GROUP AND THE FUNDAMENTAL GROUP AT INFINITY

Chapter Ten. ACTIONS ON MANIFOLDS

Chapter Eleven. THE REFLECTION GROUP TRICK

Chapter Twelve. Σ IS CAT(O): THEOREMS OF GROMOV AND MOUSSONG

Chapter Thirteen. RIGIDITY

Chapter Fourteen. FREE QUOTIENTS AND SURFACE SUBGROUPS

Chapter Fifteen. ANOTHER LOOK AT (CO)HOMOLOGY

Chapter Sixteen. THE EULER CHARACTERISTIC

Chapter Seventeen. GROWTH SERIES

Chapter Eighteen. BUILDINGS

Chapter Nineteen. HECKE-VON NEUMANN ALGEBRAS

Chapter Twenty. WEIGHTED L2-(CO)HOMOLOGY

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: L2-(CO)HOMOLOGY

Bibliography

Index

Notes:

Description based upon print version of record.

Includes bibliographical references (p. [555]-572) and index.

ISBN:

9781283851282

1283851288

9781400845941

1400845947

OCLC:

823283891