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., Π⁰₁) 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.