Generic representations of countable groups

The paper is devoted to a study of generic representations (homomorphisms) of discrete countable groups $\mathrm{\Gamma }$ in Polish groups $G$, i.e. those elements in the Polish space $\mathrm{R}\mathrm{e}\mathrm{p}(\mathrm{\Gamma },G)$ of all representations of $\mathrm{\Gamma }$ in $G$, whose orbit under the conjugation action of $G$ on $\mathrm{R}\mathrm{e}\mathrm{p}(\mathrm{\Gamma },G)$ 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 $\mathrm{\Gamma }$ in automorphism groups of metric structures such as the isometry group $\mathrm{I}\mathrm{s}\mathrm{o}(\mathbb{U})$ of the Urysohn space, isometry group $\mathrm{I}\mathrm{s}\mathrm{o}({\mathbb{U}}_1)$ of the Urysohn sphere, or the linear isometry group $LIso(\mathbb{G})$ of the Gurarii space. We show that the conjugation action of $\mathrm{I}\mathrm{s}\mathrm{o}(\mathbb{U})$ on $\mathrm{R}\mathrm{e}\mathrm{p}(\mathrm{\Gamma },\mathrm{I}\mathrm{s}\mathrm{o}(\mathbb{U}))$ is generically turbulent answering a question of Kechris and Rosendal.