Hierarchically hyperbolic spaces II: Combination theorems and the distance formula

By Jason A. Behrstock, Mark F. Hagen, Alessandro Sisto. Pacific Journal of Mathematics, 2015

143 citations recorded by the source corpus.

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 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 $\pi_1M$ 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a8b9bcfbaf21df50bd5b17c23dc0ffb1c2b22d42

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Citations in the corpus

Largest acylindrical actions and Stability in hierarchically hyperbolic groups (2017, methodology): “Hierarchically hyperbolic structures for these spaces were constructed in [BHS17b] and [BHS15].”
Quasiflats in hierarchically hyperbolic spaces (2017, methodology): “1 of [BHS19].”