#### Vol. 299, No. 2, 2019

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Hierarchically hyperbolic spaces II: Combination theorems and the distance formula

### Jason Behrstock, Mark Hagen and Alessandro Sisto

Vol. 299 (2019), No. 2, 257–338
DOI: 10.2140/pjm.2019.299.257
##### Abstract

We introduce a number of tools for finding and studying 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 HHS 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 HHS; 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 $\mathit{Nil}$ nor $\mathit{Sol}$. We establish this by proving a general combination theorem for trees of HHS (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.

##### Keywords
geometric group theory, hierarchically hyperbolic, mapping class group
##### Mathematical Subject Classification 2010
Primary: 20F36, 20F65, 20F67