Grothendieck rigidity and virtual retraction of higher-rank GBS groups

TL;DR

This paper studies the properties of higher-rank GBS groups, proving residual finiteness implies Grothendieck rigidity and characterizing when they satisfy virtual retraction based on monodromy finiteness.

Contribution

It establishes that all residually finite GBS_n groups are Grothendieck rigid and provides a complete characterization of virtual retraction in terms of monodromy.

Findings

Residually finite GBS_n groups are Grothendieck rigid.

Virtual retraction holds iff the monodromy is finite.

Characterization of property (VRC) in GBS_n 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 finite.