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. Several noteworthy 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 groups and right-angled Artin groups (in the latter this can also be observed as the clique number of the defining graph); and, for the Weil–Petersson metric, the rank is the integer part of half the complex dimension of Teichmüller space.

We prove that, in a hierarchically hyperbolic space (HHS), any quasiflat of dimension equal to the rank lies within finite distance of a union of standard orthants (under a very mild condition on the HHS satisfied by all natural examples). This resolves outstanding conjectures when applied to a number of different groups and spaces.

In the case of the mapping class group, we verify a conjecture of Farb. For Teichmüller space we answer a question of Brock. In the context of certain $\mathrm{CAT}(0)$ cubical groups, our result handles novel special cases, including right-angled Coxeter groups.

An important ingredient in the proof, which we expect will have other applications, is that the hull of any finite set in an HHS is quasi-isometric to a $\mathrm{CAT}(0)$ cube complex of dimension bounded by the rank. (If the HHS is a $\mathrm{CAT}(0)$ cube complex, then the rank can be lower than the dimension of the space.)

We deduce a number of applications of these results. For instance, we show that any quasi-isometry between HHSs induces a quasi-isometry between certain factored spaces, which are simpler HHSs. This allows one, for example, to distinguish quasi-isometry classes of right-angled Artin/Coxeter groups.

Another application of our results 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. As a template, we give a new proof of quasi-isometric rigidity of mapping class groups, which, once we have established our general quasiflats theorem, uses simpler combinatorial arguments than in previous proofs.