Generic representations of abelian groups and extreme amenability

Abstract

If $G$ is a Polish group and $Γ$ is a countable group, denote by $\Hom(Γ, G)$ the space of all homomorphisms $Γ\to G$. We study properties of the group $\cl{π(Γ)}$ for the generic $π\in \Hom(Γ, G)$, 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 $\cl{π(Γ)}$; in the other two, we show that the generic $\cl{π(Γ)}$ 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 King from ergodic theory.