Hierarchically hyperbolic spaces II: Combination theorems and the distance formula

Abstract

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, Teichmüller space with either the Teichmüller 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 of the distance formula in the mapping class group without relying on the Masur-Minsky hierarchy machinery. We then study examples of HHSs; for instance, we prove that when $M$ is a closed irreducible $3$--manifold then ${\pi }_{1}M$ is an HHS if and only if it is neither $Nil$ nor $Sol$. We establish this by proving a general combination theorem for trees of HHSs (and graphs of HH groups). We also introduce a notion of "hierarchical quasiconvexity", which in the study of HHS is analogous to the role played by quasiconvexity in the study of Gromov-hyperbolic spaces.

Revised in view of various referee and reader comments. Accepted in Pacific J. Math

References in corpus (4)

• Boundaries and automorphisms of hierarchically hyperbolic spaces
• Asymptotic dimension and small-cancellation for hierarchically hyperbolic spaces and groups
• Higher rank lattices are not coarse median
• Non-existence of boundary maps for some hierarchically hyperbolic spaces

Cited by in corpus (22)

• On hierarchical hyperbolicity of cubical groups
• Hierarchical hyperbolicity of graphs of multicurves
• Stable cubulations, bicombings, and barycenters
• Stable Subgroups of the Genus Two Handlebody Group
• Dehn filling Dehn twists
• Lattices in a product of trees, hierarchically hyperbolic groups, and virtual torsion-freeness
• Coarse-median preserving automorphisms
• Superrigidity of actions on finite rank median spaces
• Automorphisms of contact graphs of $\mathrm{CAT}(0)$ cube complexes
• Random walks and quasi-convexity in acylindrically hyperbolic groups
• An embedding of the Morse boundary in the Martin boundary
• Conjugacy languages in virtual graph products
• A cubulation with no factor system
• Relatively hyperbolic groups with strongly shortcut parabolics are strongly shortcut
• Short hierarchically hyperbolic groups II: quotients and the Hopf property for Artin groups
• Counting pseudo-Anosovs as weakly contracting isometries
• Rigidity of mapping class groups mod powers of twists
• Extensions of multicurve stabilizers are hierarchically hyperbolic
• Curve graphs for Artin-Tits groups of type $B$, $\tilde{A}$ and $\tilde{C}$ are hyperbolic
• Problems on handlebody groups
• Asymptotic cones of snowflake groups and the strong shortcut property
• A Combinatorial Structure for Many Hierarchically Hyperbolic Spaces