Projection complexes and quasimedian maps

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.

Mapping class groups are quasicubical

“…This gives a complete answer to Question 1 of [HP19]. It can also be viewed as a "globalisation" of powerful results on approximations of finite subsets by CAT(0) cube complexes [BHS21,Bow18] in the setting of colourable hierarchically hyperbolic groups.…”

“…This is a hyperbolic graph that is built by taking the disjoint union of those curve graphs and adding edges between certain pairs of them; roughly, a pair gets an edge when their subsurface projections to all other subsurfaces of that colour almost coincide. It was shown in [BBF15] that MCGpSq quasiisometrically embeds in the product of these hyperbolic graphs, and the embedding was shown to be quasimedian in [HP19]. However, this is not the end of the story, because the hyperbolic graphs contain isometrically embedded curve graphs, and so are not all quasitrees.…”