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

This is a comprehensive--nearly encyclopedic--survey of results concerning Coxeter groups. No other book covers the more recent important results, many of which are due to Michael Davis himself. This is an excellent, thoughtful, and well-written book, and it should have a wide readership among pure mathematicians in geometry, topology, representation theory, and group theory. -- Graham A. Niblo, University of Southampton Davis's book is a significant addition to the mathematics literature and it provides an important access point for geometric group theory. Although the book is a focused research monograph, it does such a nice job of presenting important material that it will also serve as a reference for quite some time. In fact, for years to come mathematicians will be writing 'terminology and notation follow Davis' in the introductions to papers on the geometry and topology of infinite Coxeter groups. -- John Meier, Lafayette College

Presents a comprehensive treatment of Coxeter groups from the viewpoint of geometric group theory. This 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.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.