### Abstract

The rank of a hierarchically hyperbolic space is the maximal number of unbounded factors of standard product regions; this coincides with the maximal dimension of a quasiflat for hierarchically hyperbolic groups. Noteworthy examples where 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 coincides with the clique number of the defining graph), and, for the Weil-Petersson metric the rank is half the complex dimension of Teichmuller space.

We prove that in a HHS, any quasiflat of dimension equal to the rank lies within finite distance of a union of standard orthants (under a very mild condition satisfied by all natural examples). This resolves outstanding conjectures when applied to a number of different groups and spaces. The mapping class group case resolves a conjecture of Farb, in Teichmuller space this resolves a question of Brock, and in the context of CAT(0) cubical groups it strengthens previous results (so as to handle, for example, the right-angled Coxeter case).

An important ingredient, is our proof that the hull of any finite set in an HHS is quasi-isometric to a cube complex of dimension equal to the rank.

We deduce a number of applications; for instance we show that any quasi-isometry between HHS induces a quasi-isometry between certain simpler HHS. This allows one, for example, to distinguish quasi-isometry classes of right-angled Artin/Coxeter groups.

Another application is that our tools, in many cases, allow one to reduce the problem of quasi-isometric rigidity for a given HHG to a combinatorial problem. As a template, we give a new proof of quasi-isometric rigidity of mapping class groups, using simpler combinatorial arguments than in previous proofs.

We prove that in a HHS, any quasiflat of dimension equal to the rank lies within finite distance of a union of standard orthants (under a very mild condition satisfied by all natural examples). This resolves outstanding conjectures when applied to a number of different groups and spaces. The mapping class group case resolves a conjecture of Farb, in Teichmuller space this resolves a question of Brock, and in the context of CAT(0) cubical groups it strengthens previous results (so as to handle, for example, the right-angled Coxeter case).

An important ingredient, is our proof that the hull of any finite set in an HHS is quasi-isometric to a cube complex of dimension equal to the rank.

We deduce a number of applications; for instance we show that any quasi-isometry between HHS induces a quasi-isometry between certain simpler HHS. This allows one, for example, to distinguish quasi-isometry classes of right-angled Artin/Coxeter groups.

Another application is that our tools, in many cases, allow one to reduce the problem of quasi-isometric rigidity for a given HHG to a combinatorial problem. As a template, we give a new proof of quasi-isometric rigidity of mapping class groups, using simpler combinatorial arguments than in previous proofs.

Original language | English |
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Number of pages | 58 |

Journal | Duke Mathematical Journal |

Publication status | Accepted/In press - 21 Aug 2020 |

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## Cite this

Hagen, M., Behrstock, J., & Sisto, A. (Accepted/In press). Quasiflats in hierarchically hyperbolic spaces.

*Duke Mathematical Journal*. https://arxiv.org/abs/1704.04271