Separability properties of higher-rank GBS groups

A rank $n$ generalized Baumslag-Solitar group is a group that splits as a finite graph of groups such that all vertex and edge groups are isomorphic to ${\mathbb{Z}}^n$. In this paper we classify these groups in terms of their separability properties. Specifically, we determine when they are residually finite, subgroup separable and cyclic subgroup separable.