Groups with the Haagerup Property

A locally compact group has the Haagerup property, or is a-T-menable in the sense of Gromov, if it admits a proper isometric action on some affine Hilbert space. As Gromov's pun is trying to indicate, this definition is designed as a strong negation to Kazhdan's property (T), characterized by the fact that every isometric action on some affine Hilbert space has a fixed point. This book is to covers various aspects of the Haagerup property. It gives several new examples.

1 Introduction.- 1.1 Basic definitions.- 1.2 Examples.- 1.3 What is the Haagerup property good for?.- 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.- 4.2 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.- 7.3 Group constructions.- 7.4 Geometric characterizations.- 7.5 Other dynamical characterizations.