Relative property (T) and linear groups

Fernós, Talia

Annales de l'Institut Fourier, Tome 56 (2006) no. 6, pp. 1767-1804.

La propriété (T) relative a récemment été utilisée pour démontrer l’existence de divers nouveaux phénomènes de rigidité, par exemple dans la théorie des algèbres de von Neumann et dans l’étude des relations d’équivalence définies par les orbites d’un groupe. Cependant, jusqu’à récemment, il n’y avait pas beaucoup d’exemples dans la littérature de paires de groupes qui jouissent de la propriété (T) relative. Cette situation a motivé le théorème suivant : Un groupe $\Gamma$ de type fini admet une représentation dans $S{L}_n\left(R\right)$ dont la fermeture de Zariski n’est pas moyennable si et seulement si $\Gamma$ agit par automorphismes sur un groupe $A$ abélien de rang rationnel fini et non nul, de telle façon que la paire $(\Gamma ⋉A,A)$ ait la propriété (T) relative.

La preuve de ce théorème est constructive. Les ingrédients principaux sont le lemme de Furstenberg sur les mesures invariantes sur l’espace projectif et le théorème spectral pour la décomposition des représentations unitaires de groupes abéliens. Des méthodes provenant de la théorie des groupes algébriques, telles que la restriction des scalaires, sont également employées.

Relative property (T) has recently been used to show the existence of a variety of new rigidity phenomena, for example in von Neumann algebras and the study of orbit-equivalence relations. However, until recently there were few examples of group pairs with relative property (T) available through the literature. This motivated the following result: A finitely generated group $\Gamma$ admits a special linear representation with non-amenable $R$-Zariski closure if and only if it acts on an Abelian group $A$ (of finite nonzero $Q$-rank) so that the corresponding group pair $(\Gamma ⋉A,A)$ has relative property (T).

The proof is constructive. The main ingredients are Furstenberg’s celebrated lemma about invariant measures on projective spaces and the spectral theorem for the decomposition of unitary representations of Abelian groups. Methods from algebraic group theory, such as the restriction of scalars functor, are also employed.

Classification : 20F99, 20E22, 20G25, 46G99
Keywords: Relative property (T), group extensions, linear algebraic groups
Mot clés : propriété (T) relative, extension de groupes, groupes algébriques linéaires

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