The Dehn functions of Out(F n ) and Aut(F n )
[Les fonctions de Dehn de Out(F n ) et Aut(F n )]
Annales de l'Institut Fourier, Tome 62 (2012) no. 5, pp. 1811-1817.

Pour n au moins 3, les fonctions de Dehn de Out(F n ) et Aut(F n ) sont exponentielles. Hatcher et Vogtmann ont montré qu’elles étaient au plus exponentielles, et la borne inférieure a été établie par Bridson et Vogtmann dans le cas n=3. Handel et Mosher ont complété la démonstration en ramenant la preuve de la borne inférieure pour n au moins 4 au cas n=3. Dans cet article, nous donnons un argument plus direct permettant de passer du cas n=3 au cas général.

For n at least 3, the Dehn functions of Out(F n ) and Aut(F n ) are exponential. Hatcher and Vogtmann proved that they are at most exponential, and the complementary lower bound in the case n=3 was established by Bridson and Vogtmann. Handel and Mosher completed the proof by reducing the lower bound for n bigger than 3 to the case n=3. In this note we give a shorter, more direct proof of this last reduction.

DOI : https://doi.org/10.5802/aif.2736
Classification : 20F65,  20F28,  53C24,  57S25
Mots clés : groupes des automorphismes des groupes libres, Fonctions de Dehn
@article{AIF_2012__62_5_1811_0,
     author = {Bridson, Martin R. and Vogtmann, Karen},
     title = {The {Dehn} functions of $Out(F_n)$ and $Aut(F_n)$},
     journal = {Annales de l'Institut Fourier},
     pages = {1811--1817},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {62},
     number = {5},
     year = {2012},
     doi = {10.5802/aif.2736},
     mrnumber = {3025154},
     zbl = {1259.20048},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2736/}
}
TY  - JOUR
AU  - Bridson, Martin R.
AU  - Vogtmann, Karen
TI  - The Dehn functions of $Out(F_n)$ and $Aut(F_n)$
JO  - Annales de l'Institut Fourier
PY  - 2012
DA  - 2012///
SP  - 1811
EP  - 1817
VL  - 62
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2736/
UR  - https://www.ams.org/mathscinet-getitem?mr=3025154
UR  - https://zbmath.org/?q=an%3A1259.20048
UR  - https://doi.org/10.5802/aif.2736
DO  - 10.5802/aif.2736
LA  - en
ID  - AIF_2012__62_5_1811_0
ER  - 
Bridson, Martin R.; Vogtmann, Karen. The Dehn functions of $Out(F_n)$ and $Aut(F_n)$. Annales de l'Institut Fourier, Tome 62 (2012) no. 5, pp. 1811-1817. doi : 10.5802/aif.2736. http://www.numdam.org/articles/10.5802/aif.2736/

[1] Alibegovic, Emina Translation lengths in Out (F n ), Geom. Dedicata, Volume 92 (2002), pp. 87-93 | Article | MR 1934012 | Zbl 1041.20024

[2] Bridson, M. R. The geometry of the word problem, Invitations to geometry and topology (Oxf. Grad. Texts Math.), Volume 7, Oxford Univ. Press, Oxford, 2002, pp. 29-91 | MR 1967744 | Zbl 0996.54507

[3] Bridson, M. R.; Vogtmann, Karen On the geometry of the automorphism group of a free group, Bull. Math Londres. Soc., Volume 27 (1995), pp. 544-552 | Article | MR 1348708 | Zbl 0836.20045

[4] Bridson, M. R.; Vogtmann, Karen Automorphism groups of free groups, surface groups and free abelian groups, Problems on mapping class groups and related topics (Proc. Sympos. Pure Math.), Volume 74, Amer. Math. Soc., Providence, RI, 2006, pp. 301-316 | MR 2264548 | Zbl 1184.20034

[5] Culler, Marc; Vogtmann, Karen Moduli of graphs and automorphisms of free groups, Invent. Math., Volume 84 (1986) no. 1, pp. 91-119 | Article | MR 830040 | Zbl 0589.20022

[6] Epstein, David B. A.; Cannon, James W.; Holt, Derek F.; Levy, Silvio V. F.; Paterson, Michael S.; Thurston, William P. Word processing in groups, Jones and Bartlett Publishers, Boston, MA, 1992 | MR 1161694 | Zbl 0764.20017

[7] Handel, Michael; Mosher, Lee Lipschitz retraction and distortion for subgroups of Out (F n ), arXiv:1009.5018, 2010

[8] Hatcher, Allen; Vogtmann, Karen Isoperimetric inequalities for automorphism groups of free groups, Pacific J. Math., Volume 173 (1996) no. 2, pp. 425-441 | MR 1394399 | Zbl 0862.20030

[9] Mosher, Lee Mapping class groups are automatic, Ann. of Math. (2), Volume 142 (1995) no. 2, pp. 303-384 | Article | MR 1343324 | Zbl 0867.57004

[10] Young, Robert The Dehn function of SL(n; ) (2009) (arXiv:0912.2697v1)

Cité par Sources :