Groups with the Haagerup Property

1 Introduction.- 1.1 Basic definitions.- 1.1.1 The Haagerup property, or a-T-menability.- 1.1.2 Kazhdan's property (T).- 1.2 Examples.- 1.2.1 Compact groups.- 1.2.2 SO(n, 1) and SU(n, 1).- 1.2.3 Groups acting properly on trees.- 1.2.4 Groups acting properly on R-trees.- 1.2.5 Coxeter groups.- 1.2.6 Amenable groups.- 1.2.7 Groups acting on spaces with walls.- 1.3 What is the Haagerup property good for?.- 1.3.1 Harmonic analysis: weak amenability.- 1.3.2 K-amenability.- 1.3.3 The Baum-Connes conjecture.- 1.4 What this book is about.- 2 Dynamical Characterizations.- 2.1 Definitions and statements of results.- 2.2 Actions on measure spaces.- 2.3 Actions on factors.- 3 Simple Lie Groups of Rank One.- 3.1 The Busemann cocycle and theGromov scalar product.- 3.2 Construction of a quadratic form.- 3.3 Positivity.- 3.4 The link with complementary series.- 4 Classification of Lie Groups with the Haagerup Property.- 4.0 Introduction.- 4.1 Step one.- 4.1.1 The fine structure of Lie groups.- 4.1.2 A criterion for relative property (T).- 4.1.3 Conclusion of step one.- 4.2 Step two.- 4.2.1 The generalized Haagerup property.- 4.2.2 Amenable groups.- 4.2.3 Simple Lie groups.- 4.2.4 A covering group.- 4.2.5 Spherical functions.- 4.2.6 The group SU(n,1).- 4.2.7 The groups SO(n, 1) and SU(n,1)..- 4.2.8 Conclusion of step two.- 5 The Radial Haagerup Property.- 5.0 Introduction.- 5.1 The geometry of harmonic NA groups.- 5.2 Harmonic analysis on H-type groups.- 5.3 Analysis on harmonic NA groups.- 5.4 Positive definite spherical functions.- 5.5 Appendix on special functions.- 6 Discrete Groups.- 6.1 Some hereditary results.- 6.2 Groups acting on trees.- 6.3 Group presentations.- 6.4 Appendix: Completely positive mapson amalgamated products,by Paul Jolissaint.- 7 Open Questions and Partial Results.- 7.1 Obstructions to the Haagerup property.- 7.2 Classes of groups.- 7.2.1 One-relator groups.- 7.2.2 Three-manifold groups.- 7.2.3 Braid groups.- 7.3 Group constructions.- 7.3.1 Semi-direct products.- 7.3.2 Actions on trees.- 7.3.3 Central extensions.- 7.4 Geometric characterizations.- 7.4.1 Chasles' relation.- 7.4.2 Some cute and sexy spaces.- 7.5 Other dynamical characterizations.- 7.5.1 Actions on infinite measure spaces.- 7.5.2 Invariant probability measures.