Overview

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 class of a-T-menable groups is remarkably large, containing amenable groups, free groups (more generally groups acting properly on trees), Coxeter groups, closed subgroups of isometries of real or complex hyperbolic spaces, and much more. Although the Haagerup property was originally introduced in functional and harmonic analysis as an approximation property of certain operator algebras, it was gradually realised that this property was actually of geometric nature. So its study leads into various areas of mathematics, such as geometric group theory, ergodic theory, representation theory of Lie groups, and operator algebras. (The latter enter in particular via the celebrated Baum-Connes conjecture; it is a remarkable result by Higson and Kasparov that the class of a-T-menable groups satisfy the Baum-Connes conjecture.) The aim of this book is to cover, for the first time in book form, various aspects of the Haagerup property. New characterisations 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.

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