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Hierarchically hyperbolic
spaces, I: Curve complexes for cubical groups
Jason Behrstock, Mark F Hagen and Alessandro Sisto |
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Geometry & Topology 21 (2017) 1731–1804
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Abstract |
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In the context of 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. |
Keywords
hierarchically hyperbolic, cube complexes, acylindrical,
Teichmüller space, curve complex
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Mathematical Subject Classification 2010
Primary: 20F36, 20F55, 20F65
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References |
Publication
Received: 22 April 2015
Revised: 8 October 2016
Accepted: 10 November 2016
Published: 10 May 2017
Proposed: Walter Neumann
Seconded: Bruce Kleiner, Dmitri Burago
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Authors |
| Jason Behrstock | |
| The Graduate Center and Lehman
College CUNY 365 5th Avenue New York, NY 10036 United States |
Barnard College Columbia University New York NY USA |
| Mark F Hagen | |
| Department of Pure Mathematics and
Mathematical Statistics University of Cambridge Wilberforce Road Cambridge CB3 0WB United Kingdom |
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| Alessandro Sisto | |
| Departement Mathematik HG G 28 ETH Rämistrasse 101 CH-8092 Zürich Switzerland |
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