Geometry & Topology

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

Michael Handel and Lee Mosher

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Lipschitz retraction distortion subgroups of Out(F_n)


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.

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