The Dehn functions of $Out(F_n)$ and $Aut(F_n)$

Bridson, Martin R.; Vogtmann, Karen

doi:10.5802/aif.2736

The Dehn functions of

O u t (F_{n})

and

A u t (F_{n})

[Les fonctions de Dehn de

O u t (F_{n})

A u t (F_{n})

]

Bridson, Martin R.¹ ; Vogtmann, Karen ²

¹ Mathematical Institute 24-29 St Giles’ Oxford OX1 3LB (U.K.)
² Cornell University Department of Mathematics Ithaca NY 14853 (USA)

Annales de l'Institut Fourier, Tome 62 (2012) no. 5, pp. 1811-1817.

Résumé
Abstract

Pour $n$ au moins 3, les fonctions de Dehn de $O u t (F_{n})$ et $A u t (F_{n})$ 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 $O u t (F_{n})$ and $A u t (F_{n})$ 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.

MR Zbl

DOI : 10.5802/aif.2736

Classification : 20F65, 20F28, 53C24, 57S25
Keywords: Automorphism groups of free groups, Dehn functions
Mot clés : groupes des automorphismes des groupes libres, Fonctions de Dehn

Affiliations des auteurs :

Bridson, Martin R. ¹ ; Vogtmann, Karen ²

¹ Mathematical Institute 24-29 St Giles’ Oxford OX1 3LB (U.K.)
² Cornell University Department of Mathematics Ithaca NY 14853 (USA)

@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},
     zbl = {1259.20048},
     mrnumber = {3025154},
     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
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/
DO  - 10.5802/aif.2736
LA  - en
ID  - AIF_2012__62_5_1811_0
ER  -

%0 Journal Article
%A Bridson, Martin R.
%A Vogtmann, Karen
%T The Dehn functions of $Out(F_n)$ and $Aut(F_n)$
%J Annales de l'Institut Fourier
%D 2012
%P 1811-1817
%V 62
%N 5
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2736/
%R 10.5802/aif.2736
%G en
%F AIF_2012__62_5_1811_0

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/

Bibliographie
Cité par

[1] Alibegovic, Emina Translation lengths in $Out (F_{n})$ , Geom. Dedicata, Volume 92 (2002), pp. 87-93 | DOI | MR | Zbl

[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 | Zbl

[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 | DOI | MR | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

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

Cité par Sources :