Locally compact groups approximable by subgroups isomorphic to $\mathbb Z$ or $\mathbb R$

Abstract

Let $G$ be a locally compact topological group, $G_0$ the connected component of its identity element, and comp(G) the union of all compact subgroups. A topological group will be called inductively monothetic if any subgroup generated (as a topological group) by finitely many elements is generated (as a topological group) by a single element. The space SUB(G) of all closed subgroups of $G$ carries a compact Hausdorff topology called the Chabauty topology. Let $F_1(G)$, respectively, $R_1(G)$, denote the subspace of all discrete subgroups isomorphic to $\mathbb Z$, respectively, all subgroups isomorphic to $\mathbb R$. It is shown that a necessary and sufficient condition for $G\in\overline{F_1(G)}$ to hold is that $G$ is abelian, and either that $G\cong \mathbb R\times {\rm comp}(G)$ and $G/G_0$ is inductively monothetic, or else that $G$ is discrete and isomorphic to a subgroup of $\mathbb Q$. It is further shown that a necessary and sufficient condition for $G\in\overline{R_1(G)}$ to hold is that $G\cong\mathbb R\times C$ for a compact connected abelian group $C$.