The hyperbolicity of the sphere complex via surgery paths
-
Arnaud Hilion
and
Camille Horbez
Abstract
In [10], Handel and Mosher have proved that the free splitting complex
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.
Funding source: Agence Nationale de la Recherche
Award Identifier / Grant number: ANR-10-JCJC 01010
Funding statement: First author supported by the grant ANR-10-JCJC 01010 of the Agence Nationale de la Recherche.
Acknowledgements
We would like to thank the organizers of the Park City Mathematics Institute 2012 Summer Session. The present work has benefited from discussions we had there: we are grateful to Vincent Guirardel, Lee Mosher, Saul Schleimer and Karen Vogtmann for helpful remarks, and in particular to Ursula Hamenstädt for sharing with us her own insight regarding surgery paths and the hyperbolicity of the sphere complex. Finally, we thank the referee for useful suggestions that improved readability of the paper.
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© 2017 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Universal eigenvarieties, trianguline Galois representations, and p-adic Langlands functoriality
- Commuting-liftable subgroups of Galois groups II
- The hyperbolicity of the sphere complex via surgery paths
- Lefschetz properties for noncompact arithmetic ball quotients
- Plünnecke inequalities for countable abelian groups
- Minimal resolutions, Chow forms and Ulrich bundles on K3 surfaces
- On deformations of ℚ-Fano threefolds II
- K-theoretic duality for hyperbolic dynamical systems
Articles in the same Issue
- Frontmatter
- Universal eigenvarieties, trianguline Galois representations, and p-adic Langlands functoriality
- Commuting-liftable subgroups of Galois groups II
- The hyperbolicity of the sphere complex via surgery paths
- Lefschetz properties for noncompact arithmetic ball quotients
- Plünnecke inequalities for countable abelian groups
- Minimal resolutions, Chow forms and Ulrich bundles on K3 surfaces
- On deformations of ℚ-Fano threefolds II
- K-theoretic duality for hyperbolic dynamical systems
