Commensurations of Out(F_n)

Abstract

Let $\Out(F_n)$ denote the outer automorphism group of the free group $F_n$ with $n>3$. We prove that for any finite index subgroup $\Gamma<\Out(F_n)$, the group $\Aut(\Gamma)$ is isomorphic to the normalizer of $\Gamma$ in $\Out(F_n)$. We prove that $\Gamma$ is {\em co-Hopfian} : every injective homomorphism $\Gamma\to \Gamma$ is surjective. Finally, we prove that the abstract commensurator $\Comm(\Out(F_n))$ is isomorphic to $\Out(F_n)$.