Perfect kernel of generalized Baumslag-Solitar groups

In this article, we study the space of subgroups of generalized Baumslag-Solitar groups (GBS groups), that is, groups acting cocompactly on an oriented tree without inversion and with infinite cyclic vertex and edge stabilizers. Our results generalize the study of Baumslag-Solitar groups in [CGLMS22]. Given a GBS group G defined by a graph of groups whose existence is given by Bass-Serre theory, we associate to any subgroup of G an integer, which is a generalization of the phenotype defined in [CGLMS22]. This quantity is invariant under conjugation and allows us to decompose the perfect kernel of G into pieces which are invariant under conjugation and on which G acts highly topologically transitively. To achieve this, we interpret graphs of subgroups of G as "blown up and shrunk" Schreier graphs of transitive actions of G. We also describe the topology of the pieces which appear in the decomposition.