Generic representations of countable groups

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 Rep(Gamma, G) of all representations of G in G whose orbits under the conjugation action of G on 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 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 Iso(U) of the Urysohn space, isometry group Iso(U-1) of the Urysohn sphere, or the linear isometry group LIso(G) of the Gurarii space. We show that the conjugation action of Iso(U) on Rep(Gamma, Iso(U)) is generically turbulent, answering a question of Kechris and Rosendal.