Groups, Geometry, and Dynamics

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Volume 8, Issue 2, 2014, pp. 391–414
DOI: 10.4171/GGD/231

Published online: 2014-07-08

On hyperbolicity of free splitting and free factor complexes

Ilya Kapovich[1] and Kasra Rafi[2]

(1) University of Illinois at Urbana-Champaign, USA
(2) University of Oklahoma, Norman, USA

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

Keywords: Free group, curve complex, outer automorphism group of the free group

Kapovich Ilya, Rafi Kasra: On hyperbolicity of free splitting and free factor complexes. Groups Geom. Dyn. 8 (2014), 391-414. doi: 10.4171/GGD/231