Mapping class groups are quasicubical

Abstract

It is proved in two distinct ways that the mapping class group of any closed surface with finitely many marked points is quasiisometric to a CAT(0) cube complex. Contents 1. Introduction 1 2. Preliminaries 4 3. Proof using quasitrees of quasigeodesics 10 4. Proof using hyperbolic cones 16 5. Median-quasiconvexity 21 References 23

Non-colorable hierarchically hyperbolic groups

“…In [HP22,Pet21], this is used to produce quasi-isometries from various hierarchically hyperbolic groups to CAT(0) cube complexes. In [DMS20], it is used to apply cubical geometry to hierarchically hyperbolic groups, for example to prove semihyperbolicity of the mapping class group.…”