Comeagre conjugacy classes and free products with amalgamation

The main theorem is that if G is a Polish group with a comeagre conjugacy class, and G acts without inversions on some tree T, then for every g∈G there is a vertex of T fixed by g. In particular, such a group cannot be written non-trivially as a free product with amalgamation. The same conclusion holds if G is the automorphism group of an ω-categorical structure and some open subgroup of G has a comeagre conjugacy class.