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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.<\/jats:p>","DOI":"10.1142\/s0218196725500432","type":"journal-article","created":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T02:32:09Z","timestamp":1760063529000},"page":"81-114","source":"Crossref","is-referenced-by-count":0,"title":["Computably locally compact groups and their closed subgroups"],"prefix":"10.1142","volume":"36","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8781-7432","authenticated-orcid":false,"given":"Alexander G.","family":"Melnikov","sequence":"first","affiliation":[{"name":"Victoria University of Wellington, School of Physical and Mathematical Sciences, Wellington, New Zealand"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0666-5180","authenticated-orcid":false,"given":"Andr\u00e9","family":"Nies","sequence":"additional","affiliation":[{"name":"The University of Auckland, School of Computer Science, Auckland, New Zealand"}]}],"member":"219","published-online":{"date-parts":[[2025,10,23]]},"reference":[{"key":"S0218196725500432BIB001","series-title":"Studies in Logic and the Foundations of Mathematics","volume-title":"Computable Structures and the Hyperarithmetical Hierarchy","volume":"144","author":"Ash C.","year":"2000"},{"key":"S0218196725500432BIB002","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2023.103304"},{"key":"S0218196725500432BIB003","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/jdu035"},{"key":"S0218196725500432BIB004","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2011.12.020"},{"key":"S0218196725500432BIB005","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2019.09.017"},{"key":"S0218196725500432BIB006","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(98)00286-2"},{"key":"S0218196725500432BIB007","doi-asserted-by":"publisher","DOI":"10.2140\/agt.2011.11.2007"},{"key":"S0218196725500432BIB008","unstructured":"P. de la Harpe, Spaces of closed subgroups of locally compact groups, preprint (2008), arXiv:0807.2030 [math.GR]."},{"issue":"3","key":"S0218196725500432BIB009","first-page":"208","volume":"22","author":"Dobrica V.","year":"1981","journal-title":"Sibirsk. 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