Projection complexes and quasimedian maps

TL;DR

This paper explores the structure of hierarchically hyperbolic groups using projection complexes, establishing conditions under which such groups are quasiisometric to CAT(0) cube complexes and analyzing properties like the Helly property.

Contribution

It introduces new conditions linking projection complexes and hyperbolic spaces to CAT(0) cube complexes, expanding understanding of hierarchically hyperbolic groups.

Findings

Groups with BBF coloring and tree-like hyperbolic spaces are quasiisometric to CAT(0) cube complexes.

Establishes the Helly property for hierarchically quasiconvex subsets.

Provides new insights into the geometric structure of hierarchically hyperbolic groups.

Abstract

We use the projection complex machinery of Bestvina--Bromberg--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.