The automorphism group of the random poset

A number of well-studied properties of Polish groups concern the interactions between the topological and algebraic structure of those groups. Examples of such properties are the small index property, the automatic continuity, and the Bergman property. An important approach for proving them is showing that the group has ample generics. Therefore we are often interested whether a given Polish group has a comeager conjugacy class, i.e a generic element, a generic pair, or more generally, a generic n-tuple. After a survey on this topic, I will discuss a recent result joint with Aristotelis Panagiotopoulos, where we show that the automorphism group of the random poset does not admit a generic pair. This answers a question of Truss and Kuske-Truss.