Commensurations of Out(F_n)

Details Commensurations of Out(F_n) Commensurations of Out(F_n)

2006-07-21

arxiv.org/abs/math/0607556v2

Mathematics

Group Theory

Revised version, 43 pages. To appear in Publ. Math. IHES

Scientific paper

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 $\Out(F_n)$.

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