The isomorphism problem for small-rose generalized Baumslag-Solitar groups (Q6632087)

A \textit{generalized Baumslag-Solitar group} (a GBS group) is a finitely generated group acting on a tree with infinite cyclic vertex and edge stabilizers. Thus these groups can be realized as fundamental groups of finite graphs of groups where all the vertex and edge groups are infinite cyclic. An \(n\)-rose GBS group is a fundamental group of such a graph of groups such that the graph consists of single vertex and \(n\) loops. Each loop is labelled with two numbers \(p\) and \(q\) such that the two associated homomorphisms \(\mathbb{Z}\to \mathbb{Z}\) from the edge group to the vertex group are multiplications by \(p\) and \(q\), respectively. The group \(G\) is said to be \textit{non-ascending} if for every such graph describing \(G\) there is no edge such that one of the labels is \(\pm 1\) and the other one is neither equal to \(-1\) nor \(1\). The main result of the paper says that the isomorphisms problem for non-ascending \(n\)-rose GBS groups is solvable. More precisely, if \(\Gamma\) and \(\Gamma'\) are graphs of groups such that the fundamental groups are GBS groups and \(G\) is a non-ascending \(n\) rose group and \(\Gamma\) is a graph of groups as described above then there is an algorithm that determines if \(G\) and \(G'\) are isomorphic. The study of the isomorphism problem for GBS groups was started in [\textit{M. Forester}, Geom. Dedic. 121, 43--59 (2006; Zbl 1117.20023)] and [\textit{M. Clay} and \textit{M. Forester}, Algebraic Geom. Topol. 8, 2289--2322 (2008; Zbl 1191.20021)] and in these two papers the main tool used in the present work, the theory of deformation spaces, is developed in the context of GBS groups.