Characterization of topological groups by discrete cocompact subgroups

Abstract

A subgroup H of a topological group G is called cocompact (or uniform) if the quotient space $$G/{\overline{H}}$$ is compact, where $${\overline{H}}$$ denotes the closure of H in G. The aim of this note is to determine all topological groups with the property that every non-trivial discrete subgroup is cocompact.