Quasiflats in hierarchically hyperbolic spaces

Abstract

The rank of a hierarchically hyperbolic space is the maximal number of unbounded factors in a standard product region. For hierarchically hyperbolic groups, this coincides with the maximal dimension of a quasiflat. Examples for which the rank coincides with familiar quantities include: the dimension of maximal Dehn twist flats for mapping class groups, the maximal rank of a free abelian subgroup for right-angled Coxeter and Artin groups, and, for the Weil--Petersson metric, the rank is the integer part of half the complex dimension of Teichm\"{u}ller space. We prove that any quasiflat of dimension equal to the rank lies within finite distance of a union of standard orthants (under a mild condition satisfied by all natural examples). This resolves outstanding conjectures when applied to various examples. For mapping class group, we verify a conjecture of Farb; for Teichm\"{u}ller space we answer a question of Brock; for CAT(0) cubical groups, we handle special cases including right-angled Coxeter groups. An important ingredient in the proof is that the hull of any finite set in an HHS is quasi-isometric to a CAT(0) cube complex of dimension bounded by the rank. We deduce a number of applications. For instance, we show that any quasi-isometry between HHSs induces a quasi-isometry between certain simpler HHSs. This allows one, for example, to distinguish quasi-isometry classes of right-angled Artin/Coxeter groups. Another application is to quasi-isometric rigidity. Our tools in many cases allow one to reduce the problem of quasi-isometric rigidity for a given hierarchically hyperbolic group to a combinatorial problem. We give a new proof of quasi-isometric rigidity of mapping class groups, which, given our general quasiflats theorem, uses simpler combinatorial arguments than in previous proofs.