Geometry & Topology

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

Michael Handel and Lee Mosher

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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(Fn1) 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

Subjects
Primary: 20F28: Automorphism groups of groups [See also 20E36]
Secondary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 20E05: Free nonabelian groups 57M07: Topological methods in group theory

Keywords
Lipschitz retraction distortion subgroups of Out(F_n)

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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