Whitehead algorithm for automorphisms of generalized Baumslag-Solitar groups

TL;DR

This paper introduces an algorithm for GBS groups that determines whether an element belongs to a special factor, extending Whitehead's algorithm from free groups to a broader class of groups.

Contribution

It develops a Whitehead-like algorithm for GBS groups to decide element membership in special factors and proves the uniqueness of minimal special factors containing a given element.

Findings

Algorithm successfully decides element membership in special factors.

Proves existence and uniqueness of minimal special factors for any element.

Extends Whitehead's algorithm to generalized Baumslag-Solitar groups.

Abstract

In analogy with the free factors of a free group we define special factors of Generalized Baumslag-Solitar (GBS) groups as non-cyclic subgroups which appear in splittings over infinite cyclic groups. We give an algorithm which, given a GBS group $G$ and an element $g \in G$, decides whether there exists a special factor $H < G$ such that $g \in H$. This algorithm is analogous to an algorithm by Whitehead for free groups. Furthermore we prove that given $g \in G$ there exists a unique minimal special factor containing $g$ and give an algorithm which finds it.