Computably locally compact groups and their closed subgroups

TL;DR

This paper explores the computability properties of locally compact groups and their closed subgroups, establishing new results on the effective structure of the Chabauty space and providing algorithmic characterizations.

Contribution

It introduces a computable framework for locally compact groups, analyzes the effective structure of their closed subgroups, and characterizes the Chabauty space in the totally disconnected case.

Findings

The 1-point compactification of a computably locally compact space is computably compact.

The Chabauty space of closed subgroups has a canonical effectively-closed presentation.

There exists a computably locally compact abelian group with a non-computably closed space of closed subgroups.

Abstract

Given a computably locally compact Polish space $M$, we show that its 1-point compactification $M^*$ is computably compact. Then, for a computably locally compact group $G$, we show that the Chabauty space $\mathcal S(G)$ of closed subgroups of $G$ has a canonical effectively-closed (i.e., $\Pi^0_1$) presentation as a subspace of the hyperspace $\mathcal K(G^*)$ of closed sets of $G^*$. We construct a computable discrete abelian group $H$ such that $\mathcal S(H)$ is not computably closed in $\mathcal K(H^*)$; in fact, the only computable points of $\mathcal S(H)$ are the trivial group and $H$ itself, while $\mathcal S(H)$ is uncountable. In the case that a computably locally compact group $G$ is also totally disconnected, we provide a further algorithmic characterization of $\mathcal S(G)$ in terms of the countable meet groupoid of $G$ introduced recently by the authors (arXiv: 2204.09878). We apply our results and techniques to show that the index set of the computable locally compact abelian groups that contain a closed subgroup isomorphic to $(\mathbb{R},+)$ is arithmetical.