Submission date: 28 July 2024

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 𝒮(G) of closed subgroups of G has a canonical effectively-closed (i.e., Π^0_1) 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.

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Full text arXiv 2407.19440: pdf, ps.