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This book, now in a revised and extended second edition, offers an in-depth account of Coxeter groups through the perspective of geometric group theory. It examines the connections between Coxeter groups and major open problems in topology related to aspherical manifolds, such as the Euler Characteristic Conjecture and the Borel and Singer Conjectures. The book also discusses key 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. In addition, this second edition includes new chapters on Artin groups and their Betti numbers. Written by a leading expert, the book is an authoritative reference on the subject.
Michael W. Davis received a PhD in mathematics from Princeton University in 1975. He was a Professor of Mathematics at Ohio State University for thirty-nine years, retiring in 2022 as Professor Emeritus. In 2015, he became a Fellow of the AMS. His research is in geometric group theory and topology. Since 1981, his work has focused on topics related to reflection groups including the construction of new examples of aspherical manifolds and the study of their properties.
Chapter 1. Introduction and preview.- Chapter 2. Some basic notions in geometric group theory.- Chapter 3. Coxeter groups.- Chapter 4. More combinatorics of Coxeter groups.- Chapter 5. The basic construction.- Chapter 6. Geometric reflection groups.- Chapter 7. The complex E.- Chapter 8. The algebraic topology of U and of E.- Chapter 9. The fundamental group and the fundamental group at infinity.- Chapter 10. Actions on manifolds.- Chapter 11. The reflection group trick.- Chapter 12. E is CAT(0).- 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. Artin Groups.- Chapter 19. L2-Betti numbers of Artin groups.- Chapter 20. Buildings.- Chapter 21. Hecke - von Neumann algebras.- Chapter 22. Weighted L2- (co)homology.
