Grothendieck rigidity and virtual retraction of higher-rank GBS groups
Daxun Wang
TL;DR
The paper studies Grothendieck rigidity and virtual retract properties for higher rank $GBS_n$ groups, i.e., groups that split as finite graphs of groups with vertex/edge groups $\mathbb{Z}^n$. It proves that every residually finite $GBS_n$ group is Grothendieck rigid by establishing rigidity for strictly ascending HNN-extensions of $\mathbb{Z}^n$ and applying it to the residually finite case. It also fully characterizes property (VRC) for $GBS_n$ groups: (VRC) holds if and only if the monodromy is trivial, equivalently when $G$ is virtually $\mathbb{Z}^n\times F_r$, with $F_r$ a free group. These results link profinite completion behavior to the graph-of-groups structure and monodromy, clarifying when cyclic subgroups are virtual retracts in higher rank GBS groups.
Abstract
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 trivial.