Generic representations of abelian groups and extreme amenability

Abstract

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

Version 2

References in corpus (3)

• Turbulence, representations, and trace-preserving actions
• Representations of residually finite groups by isometries of the Urysohn space
• Non-genericity phenomenon in some ordered Fraisse classes

Cited by in corpus (11)

• Closed subgroups generated by generic measure automorphisms
• Decompositions and measures on countable Borel equivalence relations
• Notes on quasi-Polish spaces
• From isolated subgroups to generic permutation representations
• Applications of amenable semigroups in operator theory
• Generic Behavior of a Measure Preserving Transformation
• Higher dimensional obstructions for star reductions
• Amenability and Ramsey theory in the metric setting
• Logic Blog 2023-2024
• Topological isomorphism for rank-1 systems
• Structural, point-free, non-Hausdorff topological realization of Borel groupoid actions