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

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).

@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 = {https://www.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  - https://www.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 https://www.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. https://www.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

  • Bridson, Martin R.; Wade, Richard D. Commensurations of Aut(FN) and Its Torelli Subgroup, Geometric and Functional Analysis, Volume 34 (2024) no. 5, p. 1310 | DOI:10.1007/s00039-024-00681-y
  • Bridson, Martin R.; Wade, Richard D. Direct Products of Free Groups in Aut(FN), Geometric and Functional Analysis, Volume 34 (2024) no. 5, p. 1337 | DOI:10.1007/s00039-024-00688-5
  • Silva, Eduardo The Poisson boundary of lampshuffler groups, Mathematische Zeitschrift, Volume 308 (2024) no. 3 | DOI:10.1007/s00209-024-03607-4
  • Bradford, Henry; Fournier-Facio, Francesco Hopfian wreath products and the stable finiteness conjecture, Mathematische Zeitschrift, Volume 308 (2024) no. 4 | DOI:10.1007/s00209-024-03589-3
  • Guerch, Yassine Roots of Outer Automorphisms of Free Groups and Centralizers of Abelian Subgroups of Out(FN), Michigan Mathematical Journal, Volume -1 (2024) no. -1 | DOI:10.1307/mmj/20226327
  • Bou-Rabee, Khalid; Studenmund, Daniel Abstract commensurators of surface groups, Journal of Topology and Analysis, Volume 13 (2021) no. 03, p. 607 | DOI:10.1142/s1793525320500235
  • Horbez, Camille; Wade, Richard Commensurations of subgroups of 𝑂𝑢𝑡(𝐹_𝑁), Transactions of the American Mathematical Society, Volume 373 (2020) no. 4, p. 2699 | DOI:10.1090/tran/7991
  • Huang, Jingyin Quasi-isometric classification of right-angled Artin groups, I: The finite out case, Geometry Topology, Volume 21 (2017) no. 6, p. 3467 | DOI:10.2140/gt.2017.21.3467
  • Bregman, Corey; Fullarton, Neil Infinite groups acting faithfully on the outer automorphism group of a right-angled Artin group, Michigan Mathematical Journal, Volume 66 (2017) no. 3 | DOI:10.1307/mmj/1496995334
  • Dudkin, F. A. The abstract commensurator of Baumslag–Solitar groups, Algebra and Logic, Volume 52 (2013) no. 1, p. 46 | DOI:10.1007/s10469-013-9218-9
  • Merenkov, Sergei A Sierpiński carpet with the co-Hopfian property, Inventiones mathematicae, Volume 180 (2010) no. 2, p. 361 | DOI:10.1007/s00222-010-0231-5
  • Bartholdi, L.; Bogopolski, O. On abstract commensurators of groups, Journal of Group Theory, Volume 13 (2010) no. 6 | DOI:10.1515/jgt.2010.021
  • Feighn, Mark; Handel, Michael Abelian subgroups of Out(Fn), Geometry Topology, Volume 13 (2009) no. 3, p. 1657 | DOI:10.2140/gt.2009.13.1657

Cité par 13 documents. Sources : Crossref