### Abstract

In the context of CAT(0) cubical groups, we develop an analogue of the theory of curve complexes and subsurface projections. The role of the subsurfaces is played by a collection of convex subcomplexes called a factor system, and the role of the curve graph is played by the contact graph. There are a number of close parallels between the contact graph and the curve graph, including hyperbolicity, acylindricity of the action, the existence of hierarchy paths, and a Masur–Minsky-style distance formula. We then define a hierarchically hyperbolic space; the class of such spaces includes a wide class of cubical groups (including all virtually compact special groups) as well as mapping class groups and Teichmüller space with any of the standard metrics. We deduce a number of results about these spaces, all of which are new for cubical or mapping class groups, and most of which are new for both. We show that the quasi-Lipschitz image from a ball in a nilpotent Lie group into a hierarchically hyperbolic space lies close to a product of hierarchy geodesics. We also prove a rank theorem for hierarchically hyperbolic spaces; this generalizes results of Behrstock and Minsky, of Eskin, Masur and Rafi, of Hamenstädt, and of Kleiner. We finally prove that each hierarchically hyperbolic group admits an acylindrical action on a hyperbolic space. This acylindricity result is new for cubical groups, in which case the hyperbolic space admitting the action is the contact graph; in the case of the mapping class group, this provides a new proof of a theorem of Bowditch.

Original language | English |
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Pages (from-to) | 1731-1804 |

Number of pages | 74 |

Journal | Geometry and Topology |

Volume | 21 |

Issue number | 3 |

DOIs | |

Publication status | Published - 10 May 2017 |

### Keywords

- Acylindrical
- Cube complexes
- Curve complex
- Hierarchically hyperbolic
- Teichmüller space

### ASJC Scopus subject areas

- Geometry and Topology

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## Profiles

## Alessandro Sisto

- School of Mathematical & Computer Sciences - Assistant Professor
- School of Mathematical & Computer Sciences, Mathematics - Assistant Professor

Person: Academic (Research & Teaching)

## Cite this

*Geometry and Topology*,

*21*(3), 1731-1804. https://doi.org/10.2140/gt.2017.21.1731