Precompact groups and property (T)

For a topological group GG, the dual object Ĝ is defined as the set of equivalence classes of irreducible unitary representations of GG equipped with the Fell topology. It is well known that, if GG is compact, Ĝ is discrete. In this paper, we investigate to what extent this remains true for precompact groups, that is, dense subgroups of compact groups. We show that: (a) if GG is a metrizable precompact group, then Ĝ is discrete; (b) if GG is a countable non-metrizable precompact group, then Ĝ is not discrete; (c) every non-metrizable compact group contains a dense subgroup GG for which Ĝ is not discrete. This extends to the non-Abelian case what was known for Abelian groups. We also prove that, if GG is a countable Abelian precompact group, then GG does not have Kazhdan’s property (T), although Ĝ is discrete if GG is metrizable.