{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,12]],"date-time":"2026-02-12T15:14:19Z","timestamp":1770909259752,"version":"3.50.1"},"reference-count":24,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2023,12,22]],"date-time":"2023-12-22T00:00:00Z","timestamp":1703203200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["J. Aust. Math. Soc."],"published-print":{"date-parts":[[2024,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We prove that, given a finitely generated subgroup <jats:italic>H<\/jats:italic> of a free group <jats:italic>F<\/jats:italic>, the following questions are decidable: is <jats:italic>H<\/jats:italic> closed (dense) in <jats:italic>F<\/jats:italic> for the pro-(met)abelian topology? Is the closure of <jats:italic>H<\/jats:italic> in <jats:italic>F<\/jats:italic> for the pro-(met)abelian topology finitely generated? We show also that if the latter question has a positive answer, then we can effectively construct a basis for the closure, and the closure has decidable membership problem in any case. Moreover, it is decidable whether <jats:italic>H<\/jats:italic> is closed for the pro-<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788723000162_inline2.png\"\/><jats:tex-math>\n$\\mathbf {V}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> topology when <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788723000162_inline3.png\"\/><jats:tex-math>\n$\\mathbf {V}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is an equational pseudovariety of finite groups, such as the pseudovariety <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788723000162_inline4.png\"\/><jats:tex-math>\n$\\mathbf {S}_k$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> of all finite solvable groups with derived length <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788723000162_inline5.png\"\/><jats:tex-math>\n$\\leq k$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. We also connect the pro-abelian topology with the topologies defined by abelian groups of bounded exponent.<\/jats:p>","DOI":"10.1017\/s1446788723000162","type":"journal-article","created":{"date-parts":[[2023,12,22]],"date-time":"2023-12-22T08:27:08Z","timestamp":1703233628000},"page":"363-383","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":2,"title":["THE PRO--SOLVABLE TOPOLOGY ON A FREE GROUP"],"prefix":"10.1017","volume":"116","author":[{"given":"CLAUDE","family":"MARION","sequence":"first","affiliation":[]},{"given":"PEDRO V.","family":"SILVA","sequence":"additional","affiliation":[]},{"given":"GARETH","family":"TRACEY","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2023,12,22]]},"reference":[{"key":"S1446788723000162_r5","volume-title":"Handbook of Automata Theory","author":"Bartholdi","year":"2021"},{"key":"S1446788723000162_r4","doi-asserted-by":"crossref","first-page":"853","DOI":"10.1007\/s00208-006-0767-2","article-title":"Varieties of finite supersolvable groups with the M. 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