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

TL;DR

This paper extends Stallings automata theory to free-times-abelian groups, providing a geometric framework for subgroup intersections and index problems, with computable bijections and solutions for finitely generated subgroups.

Contribution

It introduces enriched automata with abelian labels to represent subgroups in free-times-abelian groups, enabling explicit, computable descriptions and solutions for intersection and index problems.

Findings

Provides a geometric description of subgroup intersections.

Offers recursive bases and transversals for subgroup problems.

Extends Stallings automata to a broader class of groups.

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.