Dynamics on the perfect kernel of higher rank generalized Baumslag-Solitar groups

In this article, we study the space of subgroups of non-amenable generalized Baumslag-Solitar groups (GBS groups) of rank $d$, that is, groups acting cocompactly on an oriented tree with vertex and edge stabilizers isomorphic to ${\mathbb{Z}}^d$. Our results generalize the study of Baumslag-Solitar groups, and of GBS groups of rank $1$. We give an explicit description of the perfect kernel of a non-amenable GBS group $G$ of rank $d$ and show the existence of a partition of the perfect kernel into a countably infinite set of pieces which are invariant under the action by conjugation of $G$, and such that each piece contains a dense orbit.