Conjugacy class conditions in locally compact second countable groups

Abstract

Many non-locally compact second countable groups admit a co- meagre conjugacy class. For example, this is the case for S∞, Aut(ℚ, <), and, less trivially, Aut(R) for R the random graph. 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