The hyperbolicity of the sphere complex via surgery paths

TL;DR

This paper provides a shorter proof that the sphere complex related to free groups is Gromov hyperbolic, using surgery paths, and extends the hyperbolicity results to other related complexes.

Contribution

It introduces a new proof technique using surgery paths in the sphere complex, simplifying the understanding of hyperbolicity for these complexes.

Findings

Surgery paths are unparameterized quasi-geodesics in the sphere complex.

The hyperbolicity of the free splitting complex is established via an alternative approach.

Hyperbolicity results extend to the free factor and arc complexes.

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).