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

TL;DR

This paper characterizes when a locally compact group can be approximated by subgroups isomorphic to integers or real numbers, revealing structural conditions involving abelian groups, connected components, and inductively monothetic properties.

Contribution

It provides necessary and sufficient conditions for such approximations, linking group topology, subgroup structure, and the Chabauty topology in a comprehensive framework.

Findings

A group is in the closure of integer-isomorphic subgroups iff it is abelian with specific structural properties.

A group is in the closure of real-isomorphic subgroups iff it is a product of real numbers and a compact connected abelian group.

Characterization of approximation conditions using the Chabauty topology and inductively monothetic groups.

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 $C$.