Title: Quasiflats in hierarchically hyperbolic spaces,
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.