Property (QT) of relatively hierarchically hyperbolic groups

TL;DR

This paper establishes a general condition under which relatively hierarchically hyperbolic groups possess property (QT), extending previous results and applying to various classes of groups.

Contribution

It unifies and generalizes prior work by providing a sufficient condition for property (QT) in relatively hierarchically hyperbolic groups.

Findings

Groups with residual finiteness and certain classifications have property (QT)

Property (QT') is invariant under graph products

Introduces a stronger version of property (QT) and proves its invariance

Abstract

Using the projection complex machinery, Bestvina-Bromberg-Fujiwara, Hagen-Petyt and Han-Nguyen-Yang prove that several classes of nonpositively-curved groups admit equivariant quasi-isometric embeddings into finite products of quasi-trees, i.e. having property (QT). In this paper, we unify and generalize the above results by establishing a sufficient condition for relatively hierarchically hyperbolic groups to have property (QT). As applications, we show that a group has property (QT) if it is residually finite and belongs to one of the following classes of groups: admissible groups, hyperbolic--$2$--decomposable groups with no distorted elements, Artin groups of large and hyperbolic type. We also introduce a slightly stronger version of property (QT), called property (QT'), and show the invariance of property (QT') under graph products.