Conjugacy class conditions in locally compact second countable groups

Abstract

Abstract. Many non-locally compact second countable groups admit a comeagre conjugacy class. For example, this is the case for S∞, Aut(Q, <), and, less trivially, Aut(R) for R the random graph [Truss]. A. Kechris and C. Rosendal ask if a non-trivial locally compact second countable group can admit a comeagre conjugacy class. We answer the question in the negative via an analysis of locally compact second countable groups with topological conditions on a conjugacy class.

The Polish topology of the isometry group of the infinite dimensional hyperbolic space

“…For matrix groups, the spectrum is a continuous conjugacy invariant and this fact essentially proves that there is no dense conjugacy class. Actually, Wesolek proved that no locally compact second countable group has a dense conjugacy class [Wes16].…”

Generic elements in isometry groups of Polish ultrametric spaces

“…On the other hand, P.Wesolek [7] has recently proved that locally compact Polish groups cannot have comeager conjugacy classes. We present an example of a locally compact group G such that every diagonal action of G on G n+1 , n ∈ AE, by conjugation has a dense orbit (Corollary 6.4.)…”