Nouvelles approches de la propriété (T) de Kazhdan

Valette, Alain

Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Exposé no. 913, pp. 97-124.

Un groupe localement compact $G$ a la propriété (T) de Kazhdan si la $1$-cohomologie de tout $G$-module hilbertien est nulle. Cette propriété de rigidité de la théorie des représentations de $G$ a trouvé des applications qui vont de la théorie ergodique à la théorie des graphes. Pendant près de 30 ans, les seuls exemples connus de groupes avec la propriété (T), provenaient des groupes algébriques simples sur les corps locaux, ou de leurs réseaux. La situation a radicalement changé ces dernières années : nouvelles caractérisations (Y. Shalom), nouveaux exemples (M. Gromov, Y. Shalom, A. Zuk), de sorte qu’on peut même parler de “généricité” des groupes discrets ayant la propriété (T).

A locally compact group $G$ has Kazhdan’s property (T) if the 1-cohomology of any unitary $G$-module is zero. This rigidity property of the representation theory of $G$ found applications ranging from ergodic theory to graph theory. For nearly 30 years, the only known examples of groups with property (T) came from simple algebraic groups over local fields, and their lattices. Situation dramatically changed during the last years: new characterizations (Y. Shalom), new examples (M. Gromov, Y. Shalom, A. Zuk), so that one may talk of “genericity” of discrete groups with property (T).

Classification : 22D10, 22E40, 22E41, 05C50, 53C43
Mot clés : représentations unitaires, 1-cohomologie, groupes algébriques simples, réseaux, applications harmoniques, spectres de graphes
Keywords: unitary representations, 1-cohomology, simple algebraic groups, lattices, harmonic maps, graph spectra

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