Generic representations of countable groups

TL;DR

This paper studies generic representations of countable groups in Polish groups, exploring their properties, conditions for finite approximability, and turbulence, with applications to automorphism groups of various metric structures.

Contribution

It introduces new conditions for finite approximability, proves the existence of generic representations in certain automorphism groups, and clarifies turbulence phenomena.

Findings

Existence of a generic representation of  in the automorphism group of the random tournament.

A Ribes-Zalesskii-like condition guarantees finite approximability of group actions on tournaments.

The conjugation action of () on representations in () is generically turbulent.

Abstract

The paper is devoted to a study of generic representations (homomorphisms) of discrete countable groups $\Gamma$ in Polish groups $G$, i.e. those elements in the Polish space $\mathrm{Rep}(\Gamma,G)$ of all representations of $\Gamma$ in $G$, whose orbit under the conjugation action of $G$ on $\mathrm{Rep}(\Gamma,G)$ is comeager. We investigate a closely related notion of finite approximability of actions on countable structures such as tournaments or $K_n$-free graphs, and we show its connections with Ribes-Zalesski-like properties of the acting groups. We prove that $\mathbb{N}$ has a generic representation in the automorphism group of the random tournament (i.e., there is a comeager conjugacy class in this group). We formulate a Ribes-Zalesskii-like condition on a group that guarantees finite approximability of its actions on tournaments. We also provide a simpler proof of a result of Glasner, Kitroser and Melleray characterizing groups with a generic permutation representation. We also investigate representations of infinite groups $\Gamma$ in automorphism groups of metric structures such as the isometry group $\mathrm{Iso}(\mathbb{U})$ of the Urysohn space, isometry group $\mathrm{Iso}(\mathbb{U}_1)$ of the Urysohn sphere, or the linear isometry group $\mbox{LIso}(\mathbb{G})$ of the Gurarii space. We show that the conjugation action of $\mathrm{Iso}(\mathbb{U})$ on $\mathrm{Rep}(\Gamma,\mathrm{Iso}(\mathbb{U}))$ is generically turbulent answering a question of Kechris and Rosendal.