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Article Dans Une Revue Israel Journal of Mathematics Année : 2013

Generic representations of abelian groups and extreme amenability

Julien Melleray
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Résumé

If $G$ is a Polish group and $\Gamma$ is a countable group, denote by $\Hom(\Gamma, G)$ the space of all homomorphisms $\Gamma \to G$. We study properties of the group $\cl{\pi(\Gamma)}$ for the generic $\pi \in \Hom(\Gamma, G)$, when $\Gamma$ is abelian and $G$ is one of the following three groups: the unitary group of an infinite-dimensional Hilbert space, the automorphism group of a standard probability space, and the isometry group of the Urysohn metric space. Under mild assumptions on $\Gamma$, we prove that in the first case, there is (up to isomorphism of topological groups) a unique generic $\cl{\pi(\Gamma)}$; in the other two, we show that the generic $\cl{\pi(\Gamma)}$ is extremely amenable. We also show that if $\Gamma$ is torsion-free, the centralizer of the generic $\pi$ is as small as possible, extending a result of King from ergodic theory.
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Dates et versions

hal-00607406 , version 1 (08-07-2011)
hal-00607406 , version 2 (23-11-2012)

Identifiants

Citer

Julien Melleray, Todor Tsankov. Generic representations of abelian groups and extreme amenability. Israel Journal of Mathematics, 2013, 198 (1), http://link.springer.com/article/10.1007/s11856-013-0036-5. ⟨10.1007/s11856-013-0036-5⟩. ⟨hal-00607406v2⟩
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