Stallings automata for free-times-abelian groupsintersections and index

  1. Delgado, Jordi 1
  2. Ventura, Enric 2
  1. 1 Universidad del País Vasco. Departamento de Matemáticas
  2. 2 Universitat Politècnica de Catalunya. Departament de Matemàtiques
Journal:
Publicacions matematiques

ISSN: 0214-1493

Year of publication: 2022

Volume: 66

Issue: 2

Pages: 789-830

Type: Article

More publications in: Publicacions matematiques

Abstract

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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