Generic representations of countable groups

The fact that a Polish group G has a comeager conjugacy class can be expressed in the language of representations: it is equivalent to the existence of a generic representation of the group of integers Z in G. We will discuss generic representations of other discrete countable groups in automorphism groups of various structures such as the random triangle-free graph, the random tournament, the Urysohn space or the Gurarii space. It turns out that there is a connection between generic representations and certain generalizations of the Hrushovski property, frequently studied in model theory, as well as properties related to the profinite structure of discrete groups (e.g. the Ribes-Zalesskii property). This is joint work with Michal Doucha.