Rigidity of the free factor complex

We establish the following non-abelian analogue of the Fundamental Theorem of Projective Geometry: the natural map from $\mathrm{A}\mathrm{u}\mathrm{t}(F_n)$ to the automorphism group of the free-factor complex ${\mathcal{A}\mathcal{F}}_n$ is an isomorphism. We also prove the corresponding theorem for the action of $\mathrm{O}\mathrm{u}\mathrm{t}(F_n)$ on the complex of conjugacy classes of free factors.