Multiple conjugacy problem in graphs of free abelian groups

TL;DR

This paper investigates the multiple conjugacy problem in graphs of free abelian groups, providing solutions under certain conditions and fully solving it for specific cases like Generalized Baumslag-Solitar groups.

Contribution

It proves the solvability of the multiple conjugacy problem in vGBS groups when elements do not generate elliptic subgroups, and fully solves it for infinite cyclic cases.

Findings

Solvability of the multiple conjugacy problem in vGBS groups under specific conditions

Complete solution for the multiple conjugacy problem in GBS groups

Extension of conjugacy problem solutions to broader classes of free abelian groups

Abstract

A group G is a vGBS group if it admits a decomposition as a finite graph of groups with all edge and vertex groups finitely generated and free abelian. We prove that the multiple conjugacy problem is solvable between two n-tuples A and B of elements of G whenever the elements of A does not generate an elliptic subgroup. When the edge and vertex groups are infinite cyclic, i.e. G is a Generalized Baumslag-Solitar group, we prove that the multiple conjugacy problem is fully solvable.