The subgroup membership problem in amalgamated products of finitely generated free groups

TL;DR

This paper modifies Stallings folding theory with double coset representatives to solve the subgroup membership problem in amalgamated free groups, providing a decidable algorithm and analyzing its complexity.

Contribution

It introduces a novel approach to subgroup membership in amalgamated free groups using modified Stallings folding and double coset techniques.

Findings

Subgroup membership problem is decidable for these groups.

An explicit algorithm with complexity analysis is provided.

Groups in this class have the Howson property.

Abstract

Stallings folding theory is modified, using double coset representatives, and to applied to the study of subgroups of amalgamated products of finite rank free groups. As a first application the subgroup membership problem for such groups is shown to be decidable. An algorithm for this problem is constructed and its complexity is analysed. Groups in this class are also shown to possess the Howson property.