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...
| Autores: | , |
|---|---|
| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2026 |
| País: | España |
| Institución: | Universidad Pública de Navarra |
| Repositorio: | Academica-e. Repositorio Institucional de la Universidad Pública de Navarra |
| OAI Identifier: | oai:dnet:academicae__::45006de82e7f6f523e434fcacda61923 |
| Acceso en línea: | https://hdl.handle.net/2454/57012 |
| Access Level: | acceso abierto |
| Palabra clave: | Conciseness Strong conciseness Equationally Noetherian groups Linear groups Abelian-by-polycyclic groups |
| Sumario: | 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. |
|---|