Graphs of groups with cyclic edges and GBS
Together with Montserrat Casals-Ruiz and Dario Ascari we study graphs of groups with cyclic edge groups, their group of automorsphims as well as the Isomorphism Problem including the Isomorphism Problem for GBSs.
You can see the slides of our GAGTA-2025 talks “On the isomorphism problem for generalised Baumslag-Solitar groups” and “On automorphisms of graphs of groups with cyclic edge groups” and watch the talks here and here.
Here are our preprints:
Automorphisms of graphs of groups with cyclic edge groups
Abstract
We describe the outer automorphism group of a one-ended fundamental group of a graph of groups, when edge groups are cyclic, and vertex groups are torsion-free with cyclic centralizers. We show that in this case the outer automorphism group is virtually built from the outer automorphisms of the vertex groups (fixing some elements), the outer automorphisms of some associated generalized Baumslag-Solitar groups, and generalized twists - partial conjugations by elements in the centralizers of some elliptic elements in the group.On the isomorphism problem for cyclic JSJ decompositions: vertex elimination
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.Abstract
We prove that the isomorphism problem is decidable for generalized Baumslag-Solitar (GBS) groups with one quasi-conjugacy class and full support gaps. In order to do so we introduce a family of invariants that fully characterize the isomorphism within this class of GBSs.On the isomorphism problem for generalized Baumslag-Solitar groups: angles