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OverviewA 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. Full Product DetailsAuthor: Pierre-Alain Cherix , Michael Cowling , Paul Jolissaint , Pierre JulgPublisher: Birkhauser Verlag AG Imprint: Birkhauser Verlag AG Edition: 2001 ed. Volume: v. 197 Dimensions: Width: 15.60cm , Height: 0.90cm , Length: 23.40cm Weight: 0.830kg ISBN: 9783764365981ISBN 10: 3764365986 Pages: 126 Publication Date: 01 August 2001 Audience: Professional and scholarly , Professional and scholarly , Professional & Vocational , Professional & Vocational Format: Hardback Publisher's Status: Active Availability: In Print This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us. Table of Contents1 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.ReviewsAuthor InformationTab Content 6Author Website:Countries AvailableAll regions |