On hyperbolicity of free splitting and free factor complexes

TL;DR

This paper provides an alternative proof of the hyperbolicity of the free factor complex of a free group by deriving it from the hyperbolicity of the free splitting complex, and analyzes the geometric relationship between these complexes.

Contribution

It offers a new proof of a known hyperbolicity result and studies the geometric correspondence between free splitting and free factor complexes.

Findings

Hyperbolicity of the free factor complex derived from free splitting complex

Geodesics in free splitting complex map close to geodesics in free factor complex

Provides an alternative proof of a theorem of Bestvina-Feighn

Abstract

We show how to derive hyperbolicity of the free factor complex of $F_N$ from the Handel-Mosher proof of hyperbolicity of the free splitting complex of $F_N$, thus obtaining an alternative proof of a theorem of Bestvina-Feighn. We also show that under the natural map $\tau$ from the free splitting complex to free factor complex, a geodesic $[x,y]$ maps to a path that is uniformly Hausdorff-close to a geodesic $[\tau(x),\tau(y)]$ .