We study topological properties of conjugacy classes in Polish groups, with emphasis on
automorphism groups of homogeneous countable structures. We first consider the existence
of dense conjugacy classes (the topological Rokhlin property). We then characterize when
an automorphism group admits a comeager conjugacy class (answering a question of
Truss) and apply this to show that the homeomorphism group of the Cantor space has a
comeager conjugacy class (answering a question of Akin, Hurley and Kennedy). Finally, we�…

Let G be a permutation group on the set Ω. Usually we take| Ω|= ℵ0. We seek a notion of
genericity for members of G. The idea is that a member of G should be generic if it is in some
sense 'typical'. We argue that the following is the correct definition. Suppose that G is
endowed with a metric so that it becomes a complete metric space. It follows that the Baire
category theorem holds, and we may use the notions of 'meagre'and 'comeagre'sets. An
element of G is then said to be generic if it lies in a comeagre conjugacy class. The definition�…