Commensurations of Out ( F n )

Benson Farb; Michael Handel

Publications Mathématiques de l'IHÉS (2007)

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

Abstract

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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

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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

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  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
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  14. 14. G.A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Springer, Berlin (1990) Zbl0732.22008MR1090825
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