Computably locally compact groups and their closed subgroups

Given a computably locally compact Polish space M, we show that its 1-point compactification [Formula: see text] is computably compact. Then, for a computably locally compact group G, we show that the Chabauty space [Formula: see text] of closed subgroups of G has a canonical effectively-closed (i.e., [Formula: see text]) presentation as a subspace of the hyperspace [Formula: see text] of closed sets of [Formula: see text]. We construct a computable discrete abelian group H such that [Formula: see text] is not computably closed in [Formula: see text]; in fact, the only computable points of [Formula: see text] are the trivial group and H itself, while [Formula: see text] is uncountable. In the case that a computably locally compact group G is also totally disconnected, we provide a further algorithmic characterization of [Formula: see text] 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 [Formula: see text] is arithmetical.