Property QT of relatively hierarchically hyperbolic groups

Bingxue Tao

Vol. 343 (2026), No. 1, 231–260

DOI: 10.2140/pjm.2026.343.231

Abstract

Using the projection complex machinery, a number of authors (Bestvina, Bromberg and Fujiwara; Hagen and Petyt; and Han, Nguyen and Yang) have proved that several classes of nonpositively curved groups admit equivariant quasi-isometric embeddings into finite products of quasitrees, i.e., having property QT. Here we unify and generalize those 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, and Artin groups of large and hyperbolic type. We also introduce a slightly stronger version of property QT, called property QT${}_0$, and show the invariance of property QT${}_0$ under graph products.

Keywords

hierarchically hyperbolic, quasitree, projection complex, residually finite

Mathematical Subject Classification

Primary: 20F65, 20F67

Secondary: 20E06, 20E26, 20F36

Milestones

Received: 1 January 2025

Revised: 12 December 2025

Accepted: 6 April 2026

Published: 29 April 2026