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

Description

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 Poincare 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.

Michael W. Davis is professor of mathematics at Ohio State University.

"This book is one of those that grows with the reader: A graduate student can learn many properties, details and examples of Coxeter groups, while an expert can read about aspects of recent results in the theory of Coxeter groups and use the book as a guide to the literature. I strongly recommend this book to anybody who has any interest in geometric group theory. Anybody who reads (parts of) this book with an open mind will get a lot out of it."--Ralf Gramlich, Mathematical Reviews "The Geometry and Topology of Coxeter Groups is a comprehensive and authoritative treatment of Coxeter groups from the viewpoint of geometric group theory."--L'Enseignement Mathematique "[An] excellent introduction to other, important aspects of the study of geometric and topological approaches to group theory. Davis's exposition gives a delightful treatment of infinite Coxeter groups that illustrates their continued utility to the field."--John Meier, Bulletin of the AMS