Lipschitz retraction and distortion for subgroups of Out(Fn)

Handel, Michael¹ ; Mosher, Lee²

¹ Mathematics & Computer Science Department, Herbert H Lehman College (CUNY), Bronx, NY 10468-1589, USA
² Department of Mathematics and Computer Science, Rutgers University Newark, Newark, NJ 07102, USA

Geometry & topology, Tome 17 (2013) no. 3, pp. 1535-1579.

Résumé

Given a free factor $A$ of the rank $n$ free group $F_n$, we characterize when the subgroup of $\mathsf{Out}\left(F_n\right)$ that stabilizes the conjugacy class of $A$ is distorted in $\mathsf{Out}\left(F_n\right)$. We also prove that the image of the natural embedding of $\mathsf{Aut}\left(F_{n-1}\right)$ in $\mathsf{Aut}\left(F_n\right)$ is nondistorted, that the stabilizer in $\mathsf{Out}\left(F_n\right)$ of the conjugacy class of any free splitting of $F_n$ is nondistorted and we characterize when the stabilizer of the conjugacy class of an arbitrary free factor system of $F_n$ is distorted. In all proofs of nondistortion, we prove the stronger statement that the subgroup in question is a Lipschitz retract. As applications we determine Dehn functions and automaticity for $\mathsf{Out}\left(F_n\right)$ and $\mathsf{Aut}\left(F_n\right)$.

Détail

DOI : 10.2140/gt.2013.17.1535

Classification : 20F28, 20F65, 20E05, 57M07
Keywords: Lipschitz retraction, distortion, subgroups of Out(F_n)

Bibliographie

[1] E Alibegović, Translation lengths in Out(Fn), Geom. Dedicata 92 (2002) 87

[2] M Bestvina, M Feighn, M Handel, The Tits alternative for Out(Fn), I : Dynamics of exponentially-growing automorphisms, Ann. of Math. 151 (2000) 517

[3] M R Bridson, K Vogtmann, On the geometry of the automorphism group of a free group, Bull. London Math. Soc. 27 (1995) 544

[4] M R Bridson, K Vogtmann, The Dehn functions of Out(Fn) and Aut(Fn), Ann. Inst. Fourier 62 (2012) 1811

[5] M Culler, K Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986) 91

[6] D B A Epstein, J W Cannon, D F Holt, S V F Levy, M S Paterson, W P Thurston, Word processing in groups, Jones and Bartlett (1992)

[7] S M Gersten, H B Short, Rational subgroups of biautomatic groups, Ann. of Math. 134 (1991) 125

[8] A Hatcher, Homological stability for automorphism groups of free groups, Comment. Math. Helv. 70 (1995) 39

[9] A Hatcher, K Vogtmann, Isoperimetric inequalities for automorphism groups of free groups, Pacific J. Math. 173 (1996) 425

[10] T Hawkins, Continued fractions and the origins of the Perron–Frobenius theorem, Arch. Hist. Exact Sci. 62 (2008) 655

[11] G Levitt, Automorphisms of hyperbolic groups and graphs of groups, Geom. Dedicata 114 (2005) 49

[12] H A Masur, Y N Minsky, Geometry of the complex of curves, II : Hierarchical structure, Geom. Funct. Anal. 10 (2000) 902

[13] J Milnor, A note on curvature and fundamental group, J. Differential Geometry 2 (1968) 1

[14] E H Spanier, Algebraic topology, McGraw-Hill (1966)

[15] A S Švarc, A volume invariant of coverings, Dokl. Akad. Nauk SSSR 105 (1955) 32

[16] K Vogtmann, Automorphisms of free groups and outer space, Geom. Dedicata 94 (2002) 1