Stallings' Foldings and Subgroups of Amalgams of Finite Groups

TL;DR

This paper extends Stallings' folding techniques from free groups to amalgams of finite groups, developing algorithms to represent and analyze finitely generated subgroups using finite automata.

Contribution

It introduces a modified Stallings folding algorithm for amalgams of finite groups, enabling the representation and algorithmic analysis of their finitely generated subgroups.

Findings

Developed an algorithm to compute finite automata for subgroups of amalgams of finite groups.

Enabled solutions to various algorithmic problems in these groups.

Extended automata-based subgroup analysis beyond free groups.

Abstract

In the 1980's Stallings showed that every finitely generated subgroup of a free group is canonically represented by a finite minimal immersion of a bouquet of circles. In terms of the theory of automata, this is a minimal finite inverse automaton. This allows for the deep algorithmic theory of finite automata and finite inverse monoids to be used to answer questions about finitely generated subgroups of free groups. In this paper we attempt to apply the same methods to other classes of groups. A fundamental new problem is that the Stallings folding algorithm must be modified to allow for ``sewing'' on relations of non-free groups. We look at the class of groups that are amalgams of finite groups. It is known that these groups are locally quasiconvex and thus all finitely generated subgroups are represented by finite automata. We present an algorithm to compute such a finite automaton and use it to solve various algorithmic problems.