Generalised Baumslag-Solitar groups and Hierarchically Hyperbolic Groups
J.O.Button

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).
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Taxonomy
TopicsGeometric and Algebraic Topology · Homotopy and Cohomology in Algebraic Topology · Mathematical Dynamics and Fractals
