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Projection complexes and quasimedian maps

Mark F Hagen and Harry Petyt

Algebraic & Geometric Topology 22 (2022) 3277–3304
Abstract

We use the projection complex machinery of Bestvina, Bromberg, and Fujiwara to study hierarchically hyperbolic groups. In particular, we show that if the group has a BBF colouring and its associated hyperbolic spaces are quasiisometric to trees, then the group is quasiisometric to a finite-dimensional CAT(0) cube complex. We deduce various properties, including the Helly property for hierarchically quasiconvex subsets.

Keywords
projection complexes, hierarchical hyperbolicity, quasitrees, quasiisometries, CAT(0) cube complexes
Mathematical Subject Classification 2010
Primary: 20F65, 20F67
References
Publication
Received: 31 October 2019
Revised: 2 August 2021
Accepted: 28 August 2021
Published: 30 January 2023
Authors
Mark F Hagen
School of Mathematics
University of Bristol
Bristol
United Kingdom
Harry Petyt
School of Mathematics
University of Bristol
Bristol
United Kingdom