The property (\(T\)) of Kazhdan for locally compact groups. (With an appendix by Marc Burger). (La propriété (\(T\)) de Kazhdan pour les groupes localement compacts. (Avec un appendice de Marc Burger).) (French) Zbl 0759.22001

de la Harpe, Pierre; Valette, Alain

Astérisque. 175. Paris: Société Mathématique de France, Centre National de la Recherche Scientifique: 158 p. (1989).

Summary: A locally compact group \(G\) has property (\(T\)) of Kazhdan if the unit representation of \(G\) is an isolated point of the unitary dual \(\widehat G\) with respect to the Fell topology. This property has several consequences on the structure of \(G\), such as that of being compactly generated. Most of the known examples of groups with property (\(T\)), such as \(SL_ 3(\mathbb{R})\) and \(SL_ 3(\mathbb{Z})\), are linked up with simple Lie groups having split rank at least two and with their lattices. These facts are exposed in Chapters 1 to 3. The examples \(Sp(1,n)\) of real rank one (with \(n\geq 2\)), due to Kostant and more delicate to establish, are given in Chapter 9.
Property (\(T\)) is equivalent to a fixed point property for isometric actions on affine Hilbert spaces, as well as to various conditions on functions of positive and negative type, due to Guichardet and Delorme (Chapters 4 and 5). We show several applications of property (\(T\)) concerning actions of groups on various spaces (Chapter 6), problems about finitely additive measures on spheres (Chapter 7, results of Rosenblatt, Margulis and Sullivan), and expanding graphs (Chapter 8, results of Margulis). Finally, Chapter 10 indicates briefly fruitful connections with operator algebras.

MSC:

22-02	Research exposition (monographs, survey articles) pertaining to topological groups
22D10	Unitary representations of locally compact groups
05C99	Graph theory
22E99	Lie groups
28C10	Set functions and measures on topological groups or semigroups, Haar measures, invariant measures

Keywords:

locally compact group; property (\(T\)) of Kazhdan; unit representation; simple Lie groups; split rank; fixed point property; affine Hilbert spaces; actions of groups; finitely additive measures on spheres; expanding graphs; connections with operator algebras