Geodesic
Commensurations of Out(F n )
Publications Mathématiques de l'IHÉS, Tome 105 (2007), pp. 1-48

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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 Γ<Out(F n ), the group Aut(Γ) is isomorphic to the normalizer of Γ in Out(F n ). We prove that Γ is co-Hopfian: every injective homomorphism Γ→Γ is surjective. Finally, we prove that the abstract commensurator Comm(Out(F n )) is isomorphic to Out(F n ).

@article{PMIHES_2007__105__1_0,
     author = {Farb, Benson and Handel, Michael},
     title = {Commensurations of {Out}$(F_n)$},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {1--48},
     publisher = {Springer},
     volume = {105},
     year = {2007},
     doi = {10.1007/s10240-007-0007-7},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10240-007-0007-7/}
}
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Farb, Benson; Handel, Michael. Commensurations of Out$(F_n)$. Publications Mathématiques de l'IHÉS, Tome 105 (2007), pp. 1-48. doi: 10.1007/s10240-007-0007-7

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