### 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 orWeil-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 _{π1}M 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 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.

Original language | English |
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Pages (from-to) | 257-338 |

Number of pages | 82 |

Journal | Pacific Journal of Mathematics |

Volume | 299 |

Issue number | 2 |

DOIs | |

Publication status | Published - 21 May 2019 |

### Keywords

- Geometric group theory
- Hierarchically hyperbolic
- Mapping class group

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Pacific Journal of Mathematics*,

*299*(2), 257-338. https://doi.org/10.2140/pjm.2019.299.257