Generic representations of abelian groups and extreme amenability

If G is a Polish group and Γ is a countable group, denote byHom(Γ,G) the space of all homomorphisms Γ → G. We study properties of the group π(Γ) 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 π(Γ); in the other two, we show that the generic π(Γ) is extremely amenable. We also show that if Γ is torsion-free, the centralizer of the generic π is as small as possible, extending a result of Chacon–Schwartzbauer from ergodic theory.