Commensurations of Out ( F n )

Benson Farb; Michael Handel

Publications Mathématiques de l'IHÉS (2007)

  • Volume: 105, page 1-48
  • ISSN: 0073-8301

Abstract

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

How to cite

top

Farb, Benson, and Handel, Michael. "Commensurations of Out$(F_n)$." Publications Mathématiques de l'IHÉS 105 (2007): 1-48. <http://eudml.org/doc/104223>.

@article{Farb2007,
abstract = {Let Out(Fn) denote the outer automorphism group of the free group Fn with n&gt;3. We prove that for any finite index subgroup Γ&lt;Out(Fn), the group Aut(Γ) is isomorphic to the normalizer of Γ in Out(Fn). We prove that Γ is co-Hopfian: every injective homomorphism Γ→Γ is surjective. Finally, we prove that the abstract commensurator Comm(Out(Fn)) is isomorphic to Out(Fn).},
author = {Farb, Benson, Handel, Michael},
journal = {Publications Mathématiques de l'IHÉS},
keywords = {outer automorphisms of free groups; groups of outer automorphisms; subgroups of finite index; injective homomorphisms; almost fixed elements; inner automorphisms; normalizations; commensurator subgroup},
language = {eng},
pages = {1-48},
publisher = {Springer},
title = {Commensurations of Out$(F_n)$},
url = {http://eudml.org/doc/104223},
volume = {105},
year = {2007},
}

TY - JOUR
AU - Farb, Benson
AU - Handel, Michael
TI - Commensurations of Out$(F_n)$
JO - Publications Mathématiques de l'IHÉS
PY - 2007
PB - Springer
VL - 105
SP - 1
EP - 48
AB - Let Out(Fn) denote the outer automorphism group of the free group Fn with n&gt;3. We prove that for any finite index subgroup Γ&lt;Out(Fn), the group Aut(Γ) is isomorphic to the normalizer of Γ in Out(Fn). We prove that Γ is co-Hopfian: every injective homomorphism Γ→Γ is surjective. Finally, we prove that the abstract commensurator Comm(Out(Fn)) is isomorphic to Out(Fn).
LA - eng
KW - outer automorphisms of free groups; groups of outer automorphisms; subgroups of finite index; injective homomorphisms; almost fixed elements; inner automorphisms; normalizations; commensurator subgroup
UR - http://eudml.org/doc/104223
ER -

References

top
  1. 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 Zbl0984.20025MR1765705
  2. 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 Zbl1139.20026MR2150382
  3. 3. M. Bestvina, M. Feighn, M. Handel, Solvable subgroups of Out(Fn) are virtually Abelian, Geom. Dedicata, 104 (2004), 71-96 Zbl1052.20027MR2043955
  4. 4. M. Bestvina, M. Handel, Train tracks and automorphisms of free groups, Ann. Math. (2), 135 (1992), 1-51 Zbl0757.57004MR1147956
  5. 5. M. Burger, P. Harpe, Constructing irreducible representations of discrete groups, Proc. Indian Acad. Sci., Math. Sci., 107 (1997), 223-235 Zbl0908.22005MR1467427
  6. 6. M. Bridson, K. Vogtmann, Automorphisms of automorphism groups of free groups, J. Algebra, 229 (2000), 785-792 Zbl0959.20027MR1769698
  7. 7. J. Dyer, E. Formanek, The automorphism group of a free group is complete, J. Lond. Math. Soc., II. Ser., 11 (1975), 181-190 Zbl0313.20021MR379683
  8. 8. M. Feighn and M. Handel, Abelian subgroups of Out(Fn), preprint, December 2006. 
  9. 9. E. Formanek, C. Procesi, The automorphism group of a free group is not linear, J. Algebra, 149 (1992), 494-499 Zbl0780.20023MR1172442
  10. 10. N. Ivanov, J. McCarthy, On injective homomorphisms between Teichmüller modular groups, I, Invent. Math., 135 (1999), 425-486 Zbl0978.57014MR1666775
  11. 11. N. Ivanov, Mapping class groups, Handbook of Geometric Topology, North Holland, Amsterdam (2002), pp. 523-633 Zbl1002.57001MR1886678
  12. 12. N. Ivanov, Automorphisms of complexes of curves and of Teichmüller spaces, Int. Math. Res. Not., 1997 (1997), 651-666 Zbl0890.57018MR1460387
  13. 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 Zbl0808.20036MR1159135
  14. 14. G.A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Springer, Berlin (1990) Zbl0732.22008MR1090825
  15. 15. G. Prasad, Discrete subgroups isomorphic to lattices in semisimple Lie groups, Am. J. Math., 98 (1976), 241-261 Zbl0336.22008MR399351
  16. 16. K. Vogtmann, Automorphisms of free groups and outer space, Geom. Dedicata, 94 (2002), 1-31 Zbl1017.20035MR1950871
  17. 17. R. Zimmer, Ergodic Theory and Semisimple Groups, Birkhäuser, Basel (1984) Zbl0571.58015MR776417

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.