The book is the first to give a comprehensive overview of the techniques and tools currently being used in the study of combinatorial problems in Coxeter groups. It is self-contained, and accessible even to advanced undergraduate students of mathematics.The primary purpose of the book is to highlight approximations to the difficult isomorphism problem in Coxeter groups. A number of theorems relating to this problem are stated and proven. Most of the results addressed here concern conditions which can be seen as varying degrees of uniqueness of representations of Coxeter groups. Throughout the investigation, the readers are introduced to a large number of tools in the theory of Coxeter groups, drawn from dozens of recent articles by prominent researchers in geometric and combinatorial group theory, among other fields. As the central problem of the book may in fact be solved soon, the book aims to go further, providing the readers with many techniques that can be used to answer more general questions. The readers are challenged to practice those techniques by solving exercises, a list of which concludes each chapter.
Preliminaries on Coxeter Groups; Further Properties of Coxeter Groups; Rigidity; In the Beginning: Some Early Results; Even Coxeter Groups; More General Groups; Refinements and Generalizations: Automorphisms and Artin Groups