Precompact groups and property (T)

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