Generic representations of countable groups

Abstract:

The paper is devoted to a study of generic representations (homomorphisms) of discrete countable groups $\Gamma$ in Polish groups $G$, i.e., elements in the Polish space $\mathrm {Rep}(\Gamma ,G)$ of all representations of $\Gamma$ in $G$ whose orbits under the conjugation action of $G$ on $\mathrm {Rep}(\Gamma ,G)$ are 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–Zalesskii-like properties of the acting groups. We prove that $\mathbb {Z}$ 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 $\mbox {Iso}(\mathbb {U})$ of the Urysohn space, isometry group $\mbox {Iso}(\mathbb {U}_1)$ of the Urysohn sphere, or the linear isometry group $\mbox {LIso}(\mathbb {G})$ of the Gurarii space. We show that the conjugation action of $\mbox {Iso}(\mathbb {U})$ on $\mathrm {Rep}(\Gamma ,\mbox {Iso}(\mathbb {U}))$ is generically turbulent, answering a question of Kechris and Rosendal.