# Hierarchically hyperbolic spaces II: Combination theorems and the distance formula

@article{Behrstock2019HierarchicallyHS,
title={Hierarchically hyperbolic spaces II: Combination theorems and the distance formula},
author={Jason A. Behrstock and Mark F. Hagen and Alessandro Sisto},
journal={Pacific Journal of Mathematics},
year={2019}
}
• Published 2 September 2015
• Mathematics
• Pacific Journal of Mathematics
We introduce a number of tools for finding and studying \emph{hierarchically hyperbolic spaces (HHS)}, a rich class of spaces including mapping class groups of surfaces, Teichmuller space with either the Teichmuller or Weil-Petersson metrics, right-angled Artin groups, and the universal cover of any compact special cube complex. We begin by introducing a streamlined set of axioms defining an HHS. We prove that all HHSs satisfy a Masur-Minsky-style distance formula, thereby obtaining a new proof…

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## References

SHOWING 1-10 OF 105 REFERENCES
Hierarchically hyperbolic spaces I: curve complexes for cubical groups
• Mathematics
• 2014
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
Boundaries and automorphisms of hierarchically hyperbolic spaces
• Mathematics
• 2016
Hierarchically hyperbolic spaces provide a common framework for studying mapping class groups of finite type surfaces, Teichmuller space, right-angled Artin groups, and many other cubical groups.
Quasiflats in hierarchically hyperbolic spaces
• Mathematics
• 2017
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
Embedding median algebras in products of trees
We show that a metric median algebra satisfying certain conditions admits a bilipschitz embedding into a finite product of $$\mathbb{R }$$R-trees. This gives rise to a characterisation of closed
Geometry of the complex of curves I: Hyperbolicity
• Mathematics
• 1999
The Complex of Curves on a Surface is a simplicial complex whose vertices are homotopy classes of simple closed curves, and whose simplices are sets of homotopy classes which can be realized
Hyperbolic HHS I:Factor Systems and Quasi-convex subgroups
In this paper we provide a procedure to obtain a non-trivial HHS structure on a hyperbolic space. In particular, we prove that given a finite collection $\mathcal{F}$ of quasi-convex subgroups of a
Coarse median spaces and groups
We introduce the notion of a coarse median on a metric space. This satisfies the axioms of a median algebra up to bounded distance. The existence of such a median on a geodesic space is
Cubulated groups: thickness, relative hyperbolicity, and simplicial boundaries
• Mathematics
• 2016
Let G be a group acting geometrically on a CAT(0) cube complex X. We prove first that G is hyperbolic relative to the collection P of subgroups if and only if the simplicial boundary of X is the
Geometry and rigidity of mapping class groups
• Mathematics
• 2008
We study the large scale geometry of mapping class groups MCG(S), using hyperbolicity properties of curve complexes. We show that any self quasi-isometry of MCG(S) (outside a few sporadic cases) is a
Median structures on asymptotic cones and homomorphisms into mapping class groups
• Mathematics
• 2008
The main goal of this paper is a detailed study of asymptotic cones of the mapping class groups. In particular, we prove that every asymptotic cone of a mapping class group has a bi‐Lipschitz