Generic representations of abelian groups and extreme amenability

TL;DR

This paper investigates the typical properties of subgroups generated by homomorphisms from abelian groups into certain large Polish groups, revealing unique or extremely amenable structures in the generic case.

Contribution

It establishes the nature of the closure of the image of generic homomorphisms for abelian groups into specific Polish groups, showing uniqueness or extreme amenability.

Findings

Unique generic subgroup in the unitary group case

Generic subgroup is extremely amenable in automorphism and isometry groups

Centralizer of the generic homomorphism is minimal for torsion-free groups

Abstract

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.