Precompact groups and property (T)

TL;DR

This paper explores the dual object of precompact groups, showing how discreteness of the dual varies with metrizability and providing insights into property (T) for such groups.

Contribution

It extends known results about duals of compact groups to precompact groups, especially in non-metrizable cases, and examines property (T) in this context.

Findings

Metrizable precompact groups have discrete duals.

Non-metrizable countable precompact groups have non-discrete duals.

Non-metrizable compact groups contain dense subgroups with non-discrete duals.

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.