Characterization of topological groups by discrete cocompact subgroups
The goal of the article by Kadri is to characterize locally compact groups \(G\) that are: non-compact, contain a non-trivial discrete subgroup, and every non-trivial discrete subgroup is cocompact in \(G\). Such group is called a DC-group. The main strategy is to split the study of DC-groups \(G\) in two cases, depending on whether the identity component of \(G\) is either compact or not, and to use a result of Cornulier. The latter provides an equivalence for a compactly generated locally compact group \(G\) between admitting a discrete cocompact infinite cyclic subgroup and having an open subgroup of index at most 2 which admits a continuous homomorphism with compact kernel onto \(\mathbb{Z}\) or \(\mathbb{R}\). The main theorem of the article proves that a locally compact group is a DC-group if and only if it is torsion-free and has a subgroup of index at most 2 topologically isomorphic either to \(\mathbb{R}\times K\) or \(\mathbb{Z}\ltimes K\), where \(K\) is a compact group.
scientific article; zbMATH DE number 7357041
scientific article