Intersection configurations in free and free times free-abelian groups
In this paper, we study intersection configurations – which describe the behaviour of multiple (finite) intersections of subgroups with respect to finite generability – in the realm of free and free times free-abelian (FTFA) groups. We say that a configuration is realizable in a group G if there exi...
| Autores: | , , |
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| Formato: | artículo |
| Fecha de publicación: | 2023 |
| País: | España |
| Recursos: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/394696 |
| Acesso em linha: | https://hdl.handle.net/2117/394696 https://dx.doi.org/10.1017/prm.2023.79 |
| Access Level: | acceso abierto |
| Palavra-chave: | Free groups Free group Free-abelian group Direct product Subgroup Multiple intersection Intersection configuration Grups lliures Classificació AMS::20 Group theory and generalizations::20E Structure and classification of infinite or finite groups Classificació AMS::20 Group theory and generalizations::20F Special aspects of infinite or finite groups Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra::Teoria de grups |
| Resumo: | In this paper, we study intersection configurations – which describe the behaviour of multiple (finite) intersections of subgroups with respect to finite generability – in the realm of free and free times free-abelian (FTFA) groups. We say that a configuration is realizable in a group G if there exist subgroups H1,…,Hk¿G realizing it. It is well known that free groups Fn satisfy the Howson property: the intersection of any two finitely generated subgroups is again finitely generated. We show that the Howson property is indeed the only obstruction for multiple intersection configurations to be realizable within nonabelian free groups. On the contrary, FTFA groups Fn×Zm are well known to be non-Howson. We also study multiple intersections within FTFA groups, providing an algorithm to decide, given k=2 finitely generated subgroups, whether their intersection is again finitely generated and, in the affirmative case, compute a ‘basis’ for it. We finally prove that any intersection configuration is realizable in an FTFA group Fn×Zm , for n=2 and large enough m . As a consequence, we exhibit finitely presented groups where every intersection configuration is realizable. |
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