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).Comment: Revised version, 43 pages. To appear in Publ. Math. IHE