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

Abstract

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.

An introduction of the modular homomorphism at the beginning of the paper, and the statement of the second theorem in the introduction. Some comments about the differences and challenges that arise when passing from rank 1 to higher rank in the introduction. Some definitions added to avoid ambiguities in Section 3.1. Claim 5.3 added in the proof of Theorem 5.2. Some typos corrected