Generic representations of abelian groups and extreme amenability |
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Authors: | Julien Melleray Todor Tsankov |
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Affiliation: | 1. CNRS UMR 5208 Université Lyon 1, Institut Camille Jordan, Université de Lyon, 43 blvd. du 11 novembre 1918, F-69622, Villeurbanne Cedex, France 2. UFR de Mathématiques, case 7012 Institut de Mathématiques de Jussieu, Université Paris 7, 75205, Paris cedex 13, France
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Abstract: | If G is a Polish group and Γ is a countable group, denote by Hom(Γ, G) the space of all homomorphisms Γ → G. We study properties of the group $overline {pi (Gamma )} $ for the generic π ∈ Hom(Γ, G), when Γ 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 Γ, we prove that in the first case, there is (up to isomorphism of topological groups) a unique generic $overline {pi (Gamma )} $ ; in the other two, we show that the generic $overline {pi (Gamma )} $ is extremely amenable. We also show that if Γ is torsionfree, the centralizer of the generic π is as small as possible, extending a result of Chacon and Schwartzbauer from ergodic theory. |
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