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Computably locally compact groups and their closed subgroups.

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Title: Computably locally compact groups and their closed subgroups.
Authors: Melnikov, Alexander G.1 (AUTHOR) alexander.g.melnikov@gmail.com, Nies, André2 (AUTHOR) andre@cs.auckland.ac.nz
Source: International Journal of Algebra & Computation. Feb2026, Vol. 36 Issue 1, p81-114. 34p.
Subjects: Compactification (Mathematics), Abelian groups, Compact groups, Computable functions, Topological spaces
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 (G) of closed subgroups of G has a canonical effectively-closed (i.e., Π 1 0 ) presentation as a subspace of the hyperspace (G ∗) of closed sets of G ∗ . We construct a computable discrete abelian group H such that (H) is not computably closed in (H ∗) ; in fact, the only computable points of (H) are the trivial group and H itself, while (H) is uncountable. In the case that a computably locally compact group G is also totally disconnected, we provide a further algorithmic characterization of (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. [ABSTRACT FROM AUTHOR]
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Database: Engineering Source
Description
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 (G) of closed subgroups of G has a canonical effectively-closed (i.e., Π 1 0 ) presentation as a subspace of the hyperspace (G ∗) of closed sets of G ∗ . We construct a computable discrete abelian group H such that (H) is not computably closed in (H ∗) ; in fact, the only computable points of (H) are the trivial group and H itself, while (H) is uncountable. In the case that a computably locally compact group G is also totally disconnected, we provide a further algorithmic characterization of (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. [ABSTRACT FROM AUTHOR]
ISSN:02181967
DOI:10.1142/S0218196725500432