Stallings automata for free-times-abelian groups: intersections and index
We extend the classical Stallings theory (describing subgroups of free groups as automata) to direct products of free and abelian groups: after introducing enriched automata (i.e., automata with extra abelian labels), we obtain an explicit bijection between subgroups and a certain type of such enric...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2022 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/375459 |
| Acceso en línea: | https://hdl.handle.net/2117/375459 https://dx.doi.org/10.5565/PUBLMAT6622209 |
| Access Level: | acceso abierto |
| Palabra clave: | Group theory Automata Direct product Free group Free-abelian group Intersection Stallings Subgroup Grups infinits Grups finits 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 |
| Sumario: | We extend the classical Stallings theory (describing subgroups of free groups as automata) to direct products of free and abelian groups: after introducing enriched automata (i.e., automata with extra abelian labels), we obtain an explicit bijection between subgroups and a certain type of such enriched automata, which—as it happens in the free group—is computable in the finitely generated case. This approach provides a neat geometric description of (even non-(finitely generated)) intersections of finitely generated subgroups within this non-Howson family. In particular, we give a geometric solution to the subgroup intersection problem and the finite index problem, providing recursive bases and transversals, respectively. |
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