Grothendieck rigidity and virtual retraction of higher-rank GBS groups

A rank $n$ generalized Baumslag-Solitar group ($GBS_n$ group) is a group that splits as a finite graph of groups such that all vertex and edge groups are isomorphic to $\mathbb{Z}^n$. This paper investigates Grothendieck rigidity and virtual retraction properties of $GBS_n$ groups. We show that every residually finite $GBS_n$ group is Grothendieck rigid. Further, we characterize when a $GBS_n$ group satisfies property (VRC), showing that it holds precisely when the monodromy is finite.