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 $\pi_1M$ 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.
- Publication:
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arXiv e-prints
- Pub Date:
- September 2015
- DOI:
- 10.48550/arXiv.1509.00632
- arXiv:
- arXiv:1509.00632
- Bibcode:
- 2015arXiv150900632B
- Keywords:
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- Mathematics - Group Theory
- E-Print:
- Revised in view of various referee and reader comments. Accepted in Pacific J. Math