Compatibility JSJ decomposition of graphs of free abelian groups

TL;DR

This paper investigates the compatibility JSJ decomposition of vGBS groups, showing that while such decompositions can be described, they are not always computable algorithmically.

Contribution

It provides a description of the compatibility JSJ decomposition over abelian groups for vGBS groups and proves the non-computability of this decomposition in general.

Findings

Compatibility JSJ decompositions can be explicitly described.

Such decompositions are not always algorithmically computable.

The paper advances understanding of the structure of vGBS 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 describe the compatibility JSJ decomposition over abelian groups. We prove that in general this decomposition is not algorithmically computable.