## 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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Pages (from-to) | 909 - 996 |

Number of pages | 88 |

Journal | Duke Mathematical Journal |

Volume | 170 |

Issue number | 5 |

Early online date | 22 Aug 2020 |

DOIs | |

Publication status | Published - 1 Apr 2021 |

### Bibliographical note

Funding Information:The authors’ work was partially supported by the National Science Foundation (NSF) under grant DMS-1440140 at the Mathematical Sciences Research Institute in Berkeley during the 2016 program in geometric group theory, and by Engineering and Physical Sciences Research Council (EPSRC) grant EP/K032208/1. Behrstock’s work was partially supported by NSF grant DMS-1710890. Hagen’s work was partially supported by EPRSRC grant EP/L026481/1. Sisto’s work was partially supported by the Swiss National Science Foundation grant 182186.

Publisher Copyright:

© 2021 Duke University Press. All rights reserved.

## Keywords

- cubical groups
- hierarchically hyperbolic spaces
- mapping class groups
- quasi-isometric rigidity