Precompact groups and property (T)

Abstract

For any topological group $G$ the dual object $\hat G$ is defined as the set of equivalence classes of irreducible unitary representations of $G$ equipped with the Fell topology. If $G$ is compact, $\hat G$ is discrete, and we investigate to what extent this remains true for precompact groups, i.e. for dense subgroups of compact groups. We find that: (a) if $G$ is a metrizable precompact group, then $\hat G$ is discrete; (b) if $G$ is a countable non-metrizable precompact group, then $\hat G$ is not discrete; (c) every non-metrizable compact group contains a dense subgroup $G$ for which $\hat G$ is not discrete. This generalizes to the non-Abelian case what was known for Abelian groups. Kazhdan's property (T) can be defined in similar terms, but we must consider representations without non-zero invariant vectors rather than irreducible representations. If $G$ is any countable Abelian precompact group, then $G$ does not have property (T), although $\hat G$ is discrete if $G$ is metrizable.