Stallings automata for free-times-abelian groups: intersections and index

• Delgado, Jordi ^[1] ; Ventura, Enric ^[2]
  1. [1] Universidad del País Vasco/Euskal Herriko Unibertsitatea

     Universidad del País Vasco/Euskal Herriko Unibertsitatea

     Leioa, España

     
  2. [2] Universitat Politècnica de Catalunya

     Universitat Politècnica de Catalunya

     Barcelona, España

     
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 66, Nº 2, 2022, págs. 789-830
• Idioma: inglés
• Enlaces
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• Resumen

  • 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.