Computably locally compact groups and their closed subgroups

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 S (G) of closed subgroups of G has a canonical effectively-closed (i. e. , ⁰₁) presentation as a subspace of the hyperspace K (G^*) of closed sets of G^*. We construct a computable discrete abelian group H such that S (H) is not computably closed in K (H^*) ; in fact, the only computable points of S (H) are the trivial group and H itself, while 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 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 (R, +) is arithmetical.