Generalised Baumslag-Solitar groups and Hierarchically Hyperbolic Groups

TL;DR

This paper investigates the properties of generalized Baumslag-Solitar groups acting on hyperbolic spaces, revealing that hierarchical hyperbolicity and property (QT) are not preserved under quasi-isometry.

Contribution

It demonstrates that being virtually hierarchically hyperbolic or having property (QT) is not invariant under quasi-isometry for these groups.

Findings

Hierarchically hyperbolic groups are not quasi-isometry invariants.

Virtually being hierarchically hyperbolic is not preserved under quasi-isometry.

Property (QT) is not invariant under quasi-isometry for generalized Baumslag-Solitar groups.

Abstract

We look at isometric actions on arbitrary hyperbolic spaces of generalised Baumslag - Solitar groups of arbitrary dimension (the rank of the free abelian vertex and edge subgroups). It is known that being a hierarchically hyperbolic group is not a quasi-isometric invariant. We show that virtually being a hierarchically hyperbolic group is not invariant under quasi-isometry either, and nor is property (QT).