Hierarchically hyperbolic spaces I: curve complexes for cubical groups

By Jason A. Behrstock, Mark F. Hagen, Alessandro Sisto. 2014

204 citations recorded by the source corpus.

In the context of CAT(0) cubical groups, we develop an analogue of the theory of curve complexes and subsurface projections. The role of the subsurfaces is played by a collection of convex subcomplexes called a \emph{factor system}, and the role of the curve graph is played by the \emph{contact graph}. There are a number of close parallels between the contact graph and the curve graph, including hyperbolicity, acylindricity of the action, the existence of hierarchy paths, and a Masur--Minsky-style distance formula. We then define a \emph{hierarchically hyperbolic space}; the class of such spaces includes a wide class of cubical groups (including all virtually compact special groups) as well as mapping class groups and Teichmuller space with any of the standard metrics. We deduce a number of results about these spaces, all of which are new for cubical or mapping class groups, and most of which are new for both. We show that the quasi-Lipschitz image from a ball in a nilpotent Lie group into a hierarchically hyperbolic space lies close to a product of hierarchy geodesics. We also prove a rank theorem for hierarchically hyperbolic spaces; this generalizes results of Behrstock--Minsky, Eskin--Masur--Rafi, Hamenstadt, and Kleiner. We finally prove that each hierarchically hyperbolic group admits an acylindrical action on a hyperbolic space. This acylindricity result is new for cubical groups, in which case the hyperbolic space admitting the action is the contact graph; in the case of the mapping class group, this provides a new proof of a theorem of Bowditch.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/47a5b6294d2db46ad1f12a0912dafdccbea57784

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Citations in the corpus

Quasiflats in hierarchically hyperbolic spaces (2017, methodology): “For completeness, we remark that a result from [BHS17b] implies that the rank is a quasi-isometry invariant of asymphoric HHSs:”
On strongly quasiconvex subgroups (2017, methodology): “ons for convex cocompact subgroups of mapping class groups. ON STRONGLY QUASICONVEX SUBGROUPS 8 1.6. Some questions. Hierarchically hyperbolic groups were recently introduced by Behrstock-Hagen-Sisto [BHS17] to provide a uniform framework in which to study many important families of groups, includin”
Hierarchically hyperbolic spaces II: Combination theorems and the distance formula (2015, methodology): “The existence of such hierarchy paths was hypothesized as part of the de nition of a hierarchically hyperbolic space in [BHS14], but now it is proven as a consequence of the other axioms.”
Boundaries and automorphisms of hierarchically hyperbolic spaces (2016, methodology): “However, G is known to be C ̊–simple under these circumstances, since G is acylindrically hyperbolic [BHS14] and has no finite normal subgroup [DGO11].”
Asymptotic dimension and small‐cancellation for hierarchically hyperbolic spaces and groups (2015, methodology): “This result can be interpreted as stating that hierarchically hyperbolic structures always arise from a coarse version of the factor system construction used in [BHS14] to endow CAT(0) cube complexes with hierarchically hyperbolic structures.”