Dynamics on the Perfect Kernel of Non-Amenable 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 Z d Zᵈ double struck upper Z Superscript d . Our results generalize the study of Baumslag-Solitar groups, and of GBS groups of rank 1 $1$ 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 that are invariant under the action by conjugation of G , and such that each piece contains a dense orbit.