JSJ decompositions of Quadratic Baumslag-Solitar groups

TL;DR

This paper extends the understanding of JSJ decompositions to a broader class of generalized Baumslag-Solitar groups, including those with quadratically hanging surface groups, building on Forester's foundational work.

Contribution

It generalizes Forester's results to graphs of groups with vertex groups that are either infinite cyclic or quadratically hanging surface groups.

Findings

Extended JSJ decomposition results to new classes of groups

Identified conditions for cyclic JSJ decompositions in these groups

Provided a framework for analyzing more complex group structures

Abstract

Generalized Baumslag-Solitar groups are defined as fundamental groups of graphs of groups with infinite cyclic vertex and edge groups. Forester proved (in "On uniqueness of JSJ decompositions of finitely generated groups", Comment. Math. Helv. 78 (2003) pp 740-751) that in most cases the defining graphs are cyclic JSJ decompositions, in the sense of Rips and Sela. Here we extend Forester's results to graphs of groups with vertex groups that can be either infinite cyclic or quadratically hanging surface groups.