Geometry & Topology

Lipschitz retraction and distortion for subgroups of $\mathsf{Out}(F_n)$

Abstract

Given a free factor $A$ of the rank $n$ free group $Fn$, we characterize when the subgroup of $Out(Fn)$ that stabilizes the conjugacy class of $A$ is distorted in $Out(Fn)$. We also prove that the image of the natural embedding of $Aut(Fn−1)$ in $Aut(Fn)$ is nondistorted, that the stabilizer in $Out(Fn)$ of the conjugacy class of any free splitting of $Fn$ is nondistorted and we characterize when the stabilizer of the conjugacy class of an arbitrary free factor system of $Fn$ 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 $Out(Fn)$ and $Aut(Fn)$.

Article information

Source
Geom. Topol., Volume 17, Number 3 (2013), 1535-1579.

Dates
Received: 16 April 2011
Revised: 8 September 2012
Accepted: 30 November 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732613

Digital Object Identifier
doi:10.2140/gt.2013.17.1535

Mathematical Reviews number (MathSciNet)
MR3073930

Zentralblatt MATH identifier
1285.20033

Citation

Handel, Michael; Mosher, Lee. Lipschitz retraction and distortion for subgroups of $\mathsf{Out}(F_n)$. Geom. Topol. 17 (2013), no. 3, 1535--1579. doi:10.2140/gt.2013.17.1535. https://projecteuclid.org/euclid.gt/1513732613

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