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Journal für die reine und angewandte Mathematik

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Volume 2017, Issue 730


The hyperbolicity of the sphere complex via surgery paths

Arnaud Hilion
  • Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, Technopôle Château-Gombert, 39 rue F. Joliot Curie, 13453 Marseille Cedex 13, France
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/ Camille Horbez
Published Online: 2015-03-10 | DOI: https://doi.org/10.1515/crelle-2014-0128


In [10], Handel and Mosher have proved that the free splitting complex 𝒮n for the free group Fn is Gromov hyperbolic. This is a deep and much sought-after result, since it establishes 𝒮n as a good analogue of the curve complex for surfaces.

We give a shorter alternative proof of this theorem, using surgery paths in Hatcher’s sphere complex (another model for the free splitting complex), instead of Handel and Mosher’s fold paths. As a byproduct, we get that surgery paths are unparameterized quasi-geodesics in the sphere complex.

We explain how to deduce from our proof the hyperbolicity of the free factor complex and the arc complex of a surface with boundary.


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About the article

Received: 2012-12-19

Revised: 2014-11-03

Published Online: 2015-03-10

Published in Print: 2017-09-01

Funding Source: Agence Nationale de la Recherche

Award identifier / Grant number: ANR-10-JCJC 01010

First author supported by the grant ANR-10-JCJC 01010 of the Agence Nationale de la Recherche.

Citation Information: Journal für die reine und angewandte Mathematik, Volume 2017, Issue 730, Pages 135–161, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crelle-2014-0128.

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