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 Zᵈ. 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.
N. Schneider (Tue,) studied this question.