Preface -- Contents -- Chapter 1 Preliminaries on Coxeter groups -- 1.1 Historical background and motivation -- 1.2 Coxeter systems and Coxeter groups -- 1.2.1 A combinatorial definition -- 1.2.2 A geometric definition -- 1.2.3 Examples -- 1.3 Basic properties of Coxeter groups -- 1.3.1 Reflections -- 1.3.2 Lengths of words and geodesics -- 1.3.3 Parabolic subgroups -- 1.3.4 The Deletion Condition -- 1.4 Van Kampen diagrams -- 1.5 Other geometric viewpoints: the Coxeter complex and the Davis complex -- 1.5.1 Simplicial complexes -- 1.5.2 The Coxeter complex -- 1.5.3 The Davis complex -- 1.5.4 The metric on -- 1.6 Exercises -- Chapter 2 Further properties of Coxeter groups -- 2.1 The (co)homology of a Coxeter group -- 2.1.1 Group (co)homology -- 2.1.2 Poincaré duality -- 2.2 Normalizers and centralizers -- 2.2.1 The normalizer of a parabolic subgroup -- 2.2.2 Centralizers in Coxeter groups -- 2.3 Visual decompositions of Coxeter groups -- 2.3.1 Graph-of-groups decompositions -- 2.3.2 Visual decompositions -- 2.4 The Word and Conjugacy Problems -- 2.5 Exercises -- Chapter 3 Rigidity -- 3.1 Rigidity and related conditions -- 3.1.1 Rigidity and strong rigidity -- 3.1.2 Reflection independence -- 3.1.3 Reflection rigidity and strong reflection rigidity -- 3.1.4 Classes of Coxeter groups -- 3.2 Some first results -- 3.2.1 Right-angled Coxeter groups -- 3.2.2 The role of reflections -- 3.3 Even Coxeter groups -- 3.4 Large-type and two-dimensional Coxeter groups -- 3.5 Related issues -- 3.5.1 Automorphisms of right-angled Coxeter groups -- 3.5.2 Automorphisms of more general Coxeter groups -- 3.5.3 Artin groups -- 3.6 Exercises -- Chapter 4 In the beginning: some early results -- 4.1 Rigidity of right-angled Coxeter groups -- 4.2 A slight generalization -- 4.3 Strong rigidity through homological conditions -- 4.3.1 The conditions HMn and PMn.

4.3.2 Fixed point sets and the nerve of (W, S) -- 4.3.3 The action of (W, S') on -- 4.3.4 Completing the proof -- 4.3.5 Aut(W) -- 4.4 Two-dimensional groups -- 4.5 Exercises -- Chapter 5 Even Coxeter groups -- 5.1 The structure of even Coxeter groups -- 5.2 Conjugate simplices in even systems -- 5.3 Reflection rigidity in even groups -- 5.4 Reflection preservation and reflection independence -- 5.5 The Isomorphism Problem for even Coxeter groups -- 5.5.1 The algorithm -- 5.5.2 Matching edges between even and non-even diagrams -- 5.5.3 Finding u" -- 5.6 Exercises -- Chapter 6 More general groups -- 6.1 Large-type Coxeter systems -- 6.1.1 The chamber system associated to a Coxeter system -- 6.1.2 The chamber system Ch(W, S) and reflection subgroups of W -- 6.1.3 Universal root systems -- 6.1.4 The first case: V is edge-connected -- 6.1.5 The general case: applying visual decompositions -- 6.2 Circuits and centralizers -- 6.2.1 Walking around a circuit -- 6.2.2 Centralizer chasing in even Coxeter systems -- 6.2.3 The first case: V is a simple circuit -- 6.2.4 The general case -- 6.3 Exercises -- Chapter 7 Refinements and generalizations: automorphisms and Artin groups -- 7.1 Automorphisms of right-angled groups -- 7.1.1 The group Spe(W) -- 7.1.2 What does Spe(W) look like? -- 7.2 Coxeter systems with V complete -- 7.2.1 A chain of subgroups of Aut(W) -- 7.2.2 The first step -- 7.2.3 The second step -- 7.2.4 A quick aside: properties of the Tits cone -- 7.2.5 The proof of Theorem 7.4 -- 7.3 Dehn twists and decompositions: Aut(W) for some large-type groups -- 7.3.1 Centralizer chasing and visual decompositions: reprise -- 7.3.2 The ratios of conjugating elements -- 7.3.3 The unit graph and the structure of an automorphism -- 7.4 Artin groups -- 7.4.1 Rigidity of Artin groups -- 7.4.2 Automorphisms of Artin groups -- 7.5 Exercises -- Bibliography -- Index.