On hyperbolicity of free splitting and free factor complexes

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 $[\tau (x),\tau (y)]$.