Unbounded domains in hierarchically hyperbolic groups

TL;DR

This paper explores the structure of hierarchically hyperbolic groups (HHGs), characterizing virtually abelian cases, showing non-closure under finite extensions, and establishing that infinite torsion groups are not HHGs, thereby clarifying invariance properties.

Contribution

It provides new constraints on unbounded domains in HHGs, characterizes virtually abelian HHGs, and demonstrates that HHGs are not closed under finite extensions, answering key invariance questions.

Findings

Virtually abelian HHGs are characterized.

HHGs are not closed under finite extensions.

Infinite torsion groups are not HHGs.

Abstract

We investigate unbounded domains in hierarchically hyperbolic groups and obtain constraints on the possible hierarchical structures. Using these insights, we characterise the structures of virtually abelian HHGs and show that the class of HHGs is not closed under finite extensions. This provides a strong answer to the question of whether being an HHG is invariant under quasiisometries. Along the way, we show that infinite torsion groups are not HHGs. By ruling out pathological behaviours, we are able to give simpler, direct proofs of the rank-rigidity and omnibus subgroup theorems for HHGs. This involves extending our techniques so that they apply to all subgroups of HHGs.