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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.
List of contents
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.
Summary
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.
