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

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 ΓΓ 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://archive.numdam.org/articles/10.1007/s10240-007-0007-7/}
}
TY  - JOUR
AU  - Farb, Benson
AU  - Handel, Michael
TI  - Commensurations of Out$(F_n)$
JO  - Publications Mathématiques de l'IHÉS
PY  - 2007
SP  - 1
EP  - 48
VL  - 105
PB  - Springer
UR  - http://archive.numdam.org/articles/10.1007/s10240-007-0007-7/
DO  - 10.1007/s10240-007-0007-7
LA  - en
ID  - PMIHES_2007__105__1_0
ER  - 
%0 Journal Article
%A Farb, Benson
%A Handel, Michael
%T Commensurations of Out$(F_n)$
%J Publications Mathématiques de l'IHÉS
%D 2007
%P 1-48
%V 105
%I Springer
%U http://archive.numdam.org/articles/10.1007/s10240-007-0007-7/
%R 10.1007/s10240-007-0007-7
%G en
%F PMIHES_2007__105__1_0
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. http://archive.numdam.org/articles/10.1007/s10240-007-0007-7/

1. M. Bestvina, M. Feighn, M. Handel, The Tits alternative for Out(Fn ), I: Dynamics of exponentially-growing automorphisms, Ann. Math. (2), 151 (2000), 517-623 | MR | Zbl

2. M. Bestvina, M. Feighn, M. Handel, The Tits alternative for Out(Fn ) II: A Kolchin type theorem, Ann. Math. (2), 161 (2005), 1-59 | MR

3. M. Bestvina, M. Feighn, M. Handel, Solvable subgroups of Out(Fn ) are virtually Abelian, Geom. Dedicata, 104 (2004), 71-96 | MR | Zbl

4. M. Bestvina, M. Handel, Train tracks and automorphisms of free groups, Ann. Math. (2), 135 (1992), 1-51 | MR | Zbl

5. M. Burger, P. Harpe, Constructing irreducible representations of discrete groups, Proc. Indian Acad. Sci., Math. Sci., 107 (1997), 223-235 | MR | Zbl

6. M. Bridson, K. Vogtmann, Automorphisms of automorphism groups of free groups, J. Algebra, 229 (2000), 785-792 | MR | Zbl

7. J. Dyer, E. Formanek, The automorphism group of a free group is complete, J. Lond. Math. Soc., II. Ser., 11 (1975), 181-190 | MR | Zbl

8. M. Feighn and M. Handel, Abelian subgroups of Out(Fn ), preprint, December 2006.

9. E. Formanek, C. Procesi, The automorphism group of a free group is not linear, J. Algebra, 149 (1992), 494-499 | MR | Zbl

10. N. Ivanov, J. Mccarthy, On injective homomorphisms between Teichmüller modular groups, I, Invent. Math., 135 (1999), 425-486 | MR | Zbl

11. N. Ivanov, Mapping class groups, Handbook of Geometric Topology, North Holland, Amsterdam (2002), pp. 523-633 | MR | Zbl

12. N. Ivanov, Automorphisms of complexes of curves and of Teichmüller spaces, Int. Math. Res. Not., 1997 (1997), 651-666 | MR | Zbl

13. D.G. Khramtsov, Completeness of groups of outer automorphisms of free groups, Group-Theoretic Investigations (Russian), Akad. Nauk SSSR Ural. Otdel., Sverdlovsk (1990), pp. 128-143 | MR | Zbl

14. G.A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Springer, Berlin (1990) | MR | Zbl

15. G. Prasad, Discrete subgroups isomorphic to lattices in semisimple Lie groups, Am. J. Math., 98 (1976), 241-261 | MR | Zbl

16. K. Vogtmann, Automorphisms of free groups and outer space, Geom. Dedicata, 94 (2002), 1-31 | MR | Zbl

17. R. Zimmer, Ergodic Theory and Semisimple Groups, Birkhäuser, Basel (1984) | MR | Zbl

Cité par Sources :