Perfect kernel of generalized Baumslag-Solitar groups

Source abstract

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 performed by Carderi, Gaboriau, Le Maître and Stalder. Given a GBS group Γ \Gamma defined by a graph of groups whose existence is given by Bass-Serre theory, we associate to any subgroup of Γ \Gamma an integer, which is a generalization of the phenotype introduced by the aforementioned authors. This quantity is invariant under conjugation and allows us to decompose the perfect kernel of Γ \Gamma into pieces which are invariant under conjugation and on which Γ \Gamma acts highly topologically transitively. To achieve this, we interpret graphs of subgroups of Γ \Gamma as “blown up and shrunk” Schreier graphs of transitive actions of Γ \Gamma . We also describe the topology of the pieces which appear in the decomposition.