TY - JOUR
T1 - Quasiflats in hierarchically hyperbolic spaces
AU - Behrstock, Jason
AU - Hagen, Mark F.
AU - Sisto, Alessandro
N1 - 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.
PY - 2021/4/1
Y1 - 2021/4/1
N2 - 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 Teichmöller 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 Teichmöller space we answer a question of Brock. In the context of certain 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 CAT(0) cube complex of dimension bounded by the rank. (If the HHS is a 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 quasiisometry 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.
AB - 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 Teichmöller 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 Teichmöller space we answer a question of Brock. In the context of certain 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 CAT(0) cube complex of dimension bounded by the rank. (If the HHS is a 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 quasiisometry 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.
UR - http://www.scopus.com/inward/record.url?scp=85104547812&partnerID=8YFLogxK
U2 - 10.1215/00127094-2020-0056
DO - 10.1215/00127094-2020-0056
M3 - Article
AN - SCOPUS:85104547812
SN - 0012-7094
VL - 170
SP - 909
EP - 996
JO - Duke Mathematical Journal
JF - Duke Mathematical Journal
IS - 5
ER -