Strong conciseness and equationally Noetherian groups

A word w is said to be concise in a class of groups if, for every G in that class such that the set of w-values w{G} is finite, the verbal subgroup w(G) is also finite. In the context of profinite groups, the notion of strong conciseness imposes a more demanding condition on w, requiring that w(G) i...

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Detalhes bibliográficos
Autores: Heras Kerejeta, Iker de las, Zozaya, Andoni
Formato: artículo
Estado:Versión publicada
Fecha de publicación:2026
País:España
Recursos:Universidad Pública de Navarra
Repositorio:Academica-e. Repositorio Institucional de la Universidad Pública de Navarra
OAI Identifier:oai:dnet:academicae__::45006de82e7f6f523e434fcacda61923
Acesso em linha:https://hdl.handle.net/2454/57012
Access Level:acceso abierto
Palavra-chave:Conciseness
Strong conciseness
Equationally Noetherian groups
Linear groups
Abelian-by-polycyclic groups
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spelling Strong conciseness and equationally Noetherian groupsHeras Kerejeta, Iker de lasZozaya, AndoniConcisenessStrong concisenessEquationally Noetherian groupsLinear groupsAbelian-by-polycyclic groupsA word w is said to be concise in a class of groups if, for every G in that class such that the set of w-values w{G} is finite, the verbal subgroup w(G) is also finite. In the context of profinite groups, the notion of strong conciseness imposes a more demanding condition on w, requiring that w(G) is finite whenever |w{G}| < 2ℵ0 . We investigate the relation between these two properties and the notion of equationally Noetherian groups, by proving that in a profinite group G with a dense equationally Noetherian subgroup, w{G} is finite whenever |w{G}| < 2ℵ0 . Consequently, we conclude that every word is strongly concise in the classes of profinite linear groups, pro-C completions of residually C linear groups and pro-C completions of virtually abelian-by-polycyclic groups, thereby extending wellknown conciseness properties of these classes of groups.Both authors are supported by the Spanish Government, grant PID2020-117281GB-I00 (partly with ERDF). The first author is supported as well by the Basque Government, grant IT483-22. During a large part of this research, he was also supported by the European Union via the Marie Skłodowska-Curie Actions (MSCA), grant HE-MSCA-PF-GF22/02 101067088. The second author is supported by the Slovenian Research Agency, programs P1-0222, P1-0294 and grants J1-50001, J1-4351, N1-0216. Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.SpringerEstadística, Informática y MatemáticasEstatistika, Informatika eta Matematika2026info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfhttps://hdl.handle.net/2454/57012reponame:Academica-e. Repositorio Institucional de la Universidad Pública de Navarrainstname:Universidad Pública de NavarraInglésinfo:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020/PID2020-117281GB-I00© The Author(s) 2025. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.https://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccessoai:dnet:academicae__::45006de82e7f6f523e434fcacda619232026-06-17T12:41:47Z
dc.title.none.fl_str_mv Strong conciseness and equationally Noetherian groups
title Strong conciseness and equationally Noetherian groups
spellingShingle Strong conciseness and equationally Noetherian groups
Heras Kerejeta, Iker de las
Conciseness
Strong conciseness
Equationally Noetherian groups
Linear groups
Abelian-by-polycyclic groups
title_short Strong conciseness and equationally Noetherian groups
title_full Strong conciseness and equationally Noetherian groups
title_fullStr Strong conciseness and equationally Noetherian groups
title_full_unstemmed Strong conciseness and equationally Noetherian groups
title_sort Strong conciseness and equationally Noetherian groups
dc.creator.none.fl_str_mv Heras Kerejeta, Iker de las
Zozaya, Andoni
author Heras Kerejeta, Iker de las
author_facet Heras Kerejeta, Iker de las
Zozaya, Andoni
author_role author
author2 Zozaya, Andoni
author2_role author
dc.contributor.none.fl_str_mv Estadística, Informática y Matemáticas
Estatistika, Informatika eta Matematika
dc.subject.none.fl_str_mv Conciseness
Strong conciseness
Equationally Noetherian groups
Linear groups
Abelian-by-polycyclic groups
topic Conciseness
Strong conciseness
Equationally Noetherian groups
Linear groups
Abelian-by-polycyclic groups
description A word w is said to be concise in a class of groups if, for every G in that class such that the set of w-values w{G} is finite, the verbal subgroup w(G) is also finite. In the context of profinite groups, the notion of strong conciseness imposes a more demanding condition on w, requiring that w(G) is finite whenever |w{G}| < 2ℵ0 . We investigate the relation between these two properties and the notion of equationally Noetherian groups, by proving that in a profinite group G with a dense equationally Noetherian subgroup, w{G} is finite whenever |w{G}| < 2ℵ0 . Consequently, we conclude that every word is strongly concise in the classes of profinite linear groups, pro-C completions of residually C linear groups and pro-C completions of virtually abelian-by-polycyclic groups, thereby extending wellknown conciseness properties of these classes of groups.
publishDate 2026
dc.date.none.fl_str_mv 2026
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/2454/57012
url https://hdl.handle.net/2454/57012
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020/PID2020-117281GB-I00
dc.rights.none.fl_str_mv https://creativecommons.org/licenses/by/4.0/
info:eu-repo/semantics/openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by/4.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Springer
publisher.none.fl_str_mv Springer
dc.source.none.fl_str_mv reponame:Academica-e. Repositorio Institucional de la Universidad Pública de Navarra
instname:Universidad Pública de Navarra
instname_str Universidad Pública de Navarra
reponame_str Academica-e. Repositorio Institucional de la Universidad Pública de Navarra
collection Academica-e. Repositorio Institucional de la Universidad Pública de Navarra
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