On the isomorphism problem for cyclic JSJ decompositions: vertex elimination

TL;DR

This paper introduces new graph moves that preserve fundamental groups, simplifying the isomorphism problem for cyclic JSJ decompositions and reducing it to GBS groups, enabling solutions for broad classes of these groups.

Contribution

It presents novel moves on graphs of groups that maintain the fundamental group and reduces the isomorphism problem to simpler cases like GBS groups.

Findings

New moves preserve fundamental groups in graphs of groups.

Isomorphism problem reduces to GBS groups and one-vertex graphs.

Solved the isomorphism problem for a broad class of flexible GBS groups.

Abstract

We introduce two new moves on graphs of groups with cyclic edge groups that preserve the fundamental group. These moves allow us to address the isomorphism problem without the use of expansions, therefore keeping the number of vertices and edges constant along sequences of moves witnessing an isomorphism between two groups. We further show that the isomorphism problem for a large family of cyclic JSJ decompositions reduces to the case of generalized Baumslag-Solitar groups (GBS), and that among GBSs, it suffices to consider one-vertex graphs. As an application of the methods, we solve the isomorphism problem for a broad class of flexible GBSs. Finally, we discuss potential further applications of our techniques.