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 WeilPetersson metrics, rightangled 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 MasurMinskystyle distance formula, thereby obtaining a new proof of the distance formula in the mapping class group without relying on the MasurMinsky 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 Gromovhyperbolic spaces.
 Publication:

arXiv eprints
 Pub Date:
 September 2015
 arXiv:
 arXiv:1509.00632
 Bibcode:
 2015arXiv150900632B
 Keywords:

 Mathematics  Group Theory
 EPrint:
 Revised in view of various referee and reader comments. Accepted in Pacific J. Math