Se p 20 04 Turbulence , amalgamation and generic automorphisms of homogeneous structures

(A) We study in this paper topological properties of conjugacy classes in Polish groups. There are two questions which we are particularly interested in. First, does a Polish group G have a dense conjugacy class? This is equivalent (see, e.g., Kechris [95, 8.47]) to the following generic ergodicity property of G: Every conjugacy invariant subset A ⊆ G with the Baire property (e.g., a Borel set) is either meager or comeager. There is an extensive list of Polish groups that have dense conjugacy classes, like, e.g., the automorphism group Aut(X,μ) of a standard measure space (X,μ), i.e., a standard Borel space X with a non-atomic Borel probability measure μ (see, e.g., Halmos [56]), the unitary group U(H) of separable infinite-dimensional Hilbert space H (see, e.g., Choksi-Nadkarni [90]) and the homeomorphism groups H(X) of various compact metric spaces X with the uniform convergence topology, including X = [0, 1] (the Hilbert cube), X = 2 (the Cantor space), X = S (even dimensional spheres), etc. (see Glasner-Weiss [01] and Akin-Hurley-Kennedy [03]). In Glasner-Weiss [01] groups that have dense conjugacy classes are said to have the topological Rohlin property, motivated by the existence of dense conjugacy classes in Aut(X,μ), which is usually seen as a consequence of the well-known Rohlin Lemma in ergodic theory. Recently, Glasner and Weiss also found examples of locally compact Polish groups with dense conjugacy class. Any such group must be non-compact and totally disconnected. (We would like to thank Karl Hofmann for supplying a proof of this.) The second question we consider is whether G has a (necessarily unique) dense Gδ conjugacy class (which is well-known to be equivalent to whether it has a dense non-meager class, see, e.g., Becker-Kechris [96]). Following Truss [92], we call any element of G whose conjugacy class is dense Gδ a generic element of G. This is a much stronger property which fails in very