#### 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
##### Milestones
Received: 25 January 2018
Revised: 15 November 2018
Accepted: 27 November 2018
Published: 21 May 2019
##### Authors
 Jason Behrstock The Graduate Center and Lehman College CUNY New York, NY United States Mark Hagen School of Mathematics University of Bristol Bristol United Kingdom Alessandro Sisto Departement Mathematik ETH Zürich Zürich Switzerland