# The Dehn functions of $Out\left({F}_{n}\right)$ and $Aut\left({F}_{n}\right)$

• [1] Mathematical Institute 24-29 St Giles’ Oxford OX1 3LB (U.K.)
• [2] Cornell University Department of Mathematics Ithaca NY 14853 (USA)
• Volume: 62, Issue: 5, page 1811-1817
• ISSN: 0373-0956

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## Abstract

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For $n$ at least 3, the Dehn functions of $Out\left({F}_{n}\right)$ and $Aut\left({F}_{n}\right)$ 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.

## How to cite

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Bridson, Martin R., and Vogtmann, Karen. "The Dehn functions of $Out(F_n)$ and $Aut(F_n)$." Annales de l’institut Fourier 62.5 (2012): 1811-1817. <http://eudml.org/doc/251062>.

@article{Bridson2012,
abstract = {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.},
affiliation = {Mathematical Institute 24-29 St Giles’ Oxford OX1 3LB (U.K.); Cornell University Department of Mathematics Ithaca NY 14853 (USA)},
author = {Bridson, Martin R., Vogtmann, Karen},
journal = {Annales de l’institut Fourier},
keywords = {Automorphism groups of free groups; Dehn functions; automorphism groups of free groups; outer automorphisms},
language = {eng},
number = {5},
pages = {1811-1817},
publisher = {Association des Annales de l’institut Fourier},
title = {The Dehn functions of $Out(F_n)$ and $Aut(F_n)$},
url = {http://eudml.org/doc/251062},
volume = {62},
year = {2012},
}

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
PB - Association des Annales de l’institut Fourier
VL - 62
IS - 5
SP - 1811
EP - 1817
AB - 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.
LA - eng
KW - Automorphism groups of free groups; Dehn functions; automorphism groups of free groups; outer automorphisms
UR - http://eudml.org/doc/251062
ER -

## References

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1. Emina Alibegovic, Translation lengths in $\mathrm{Out}\left({F}_{n}\right)$, Geom. Dedicata 92 (2002), 87-93 Zbl1041.20024MR1934012
2. M. R. Bridson, The geometry of the word problem, Invitations to geometry and topology 7 (2002), 29-91, Oxford Univ. Press, Oxford Zbl0996.54507MR1967744
3. M. R. Bridson, Karen Vogtmann, On the geometry of the automorphism group of a free group, Bull. Math Londres. Soc. 27 (1995), 544-552 Zbl0836.20045MR1348708
4. M. R. Bridson, Karen Vogtmann, Automorphism groups of free groups, surface groups and free abelian groups, Problems on mapping class groups and related topics 74 (2006), 301-316, Amer. Math. Soc., Providence, RI Zbl1184.20034MR2264548
5. Marc Culler, Karen Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986), 91-119 Zbl0589.20022MR830040
6. David B. A. Epstein, James W. Cannon, Derek F. Holt, Silvio V. F. Levy, Michael S. Paterson, William P. Thurston, Word processing in groups, (1992), Jones and Bartlett Publishers, Boston, MA Zbl0764.20017MR1161694
7. Michael Handel, Lee Mosher, Lipschitz retraction and distortion for subgroups of $\mathrm{Out}\left({F}_{n}\right)$, arXiv:1009.5018 (2010) Zbl1285.20033
8. Allen Hatcher, Karen Vogtmann, Isoperimetric inequalities for automorphism groups of free groups, Pacific J. Math. 173 (1996), 425-441 Zbl0862.20030MR1394399
9. Lee Mosher, Mapping class groups are automatic, Ann. of Math. (2) 142 (1995), 303-384 Zbl0867.57004MR1343324
10. Robert Young, The Dehn function of SL(n;$ℤ$), (2009)

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