The hyperbolicity of the sphere complex via surgery paths
Abstract
Handel and Mosher have proved that the free splitting complex FS for the free group is Gromov hyperbolic. This is a deep and much sought-after result, since it establishes FS 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 some other complexes such as the free factor complex or the arc complex (of a surface with boundary).
Keywords
Cite
@article{arxiv.1210.6183,
title = {The hyperbolicity of the sphere complex via surgery paths},
author = {Arnaud Hilion and Camille Horbez},
journal= {arXiv preprint arXiv:1210.6183},
year = {2013}
}
Comments
Old sections 8.1 and 8.2 have been modified and merged in new section 8.1. Old section 8.3 has become new section 8.2. The end of the introduction has been modified accordingly. References have been updated