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arXiv · 1910.05855

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

Jordi Delgado·Enric Ventura
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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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Jordi Delgado, Enric Ventura. 2019-10-13. Stallings automata for free-times-abelian groups: intersections and index. https://arxiv.org/abs/1910.05855

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