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.

The aim of this book is to cover, for the first time in book form, various aspects of the Haagerup property. New characterizations are brought in, using ergodic theory or operator algebras. Several new examples are given and new approaches to previously known examples are proposed. Connected Lie groups with the Haagerup property are completely characterized.

*The book is extremely interesting, stimulating and well written (...) and it is strongly recommended to graduate students and researchers in the fields of geometry, group theory, harmonic analysis, ergodic theory and operator algebras.
The first chapter, by Valette, is a stimulating introduction to the whole book.

• (Mathematical Reviews)

  *This book constitutes a collective volume due to five authors, featuring important breakthroughs in an intensively studied subject.

• (Zentralblatt MATH)

  The aim of this book is to cover, for the first time in book form, various aspects of the Haagerup property New characterizations are brought in New approaches to previously known examples are proposed Includes supplementary material: sn.pub/extras

  Inhalt
  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 BaumConnes 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 the Gromov 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.- 4.1.2A 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.