Many non-locally compact second countable groups admit a comeagre conjugacy
class. For example, this is the case for the automorphism group of the rational
order and the automorphism group of 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.Comment: Referee's suggestions incorporate
