Volume 17, issue 3 (2013)

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

Michael Handel and Lee Mosher

Geometry & Topology 17 (2013) 1535–1579
Abstract

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 $\mathsf{Aut}\left({F}_{n}\right)$.

Keywords
Lipschitz retraction, distortion, subgroups of Out(F_n)
Mathematical Subject Classification 2010
Primary: 20F28
Secondary: 20F65, 20E05, 57M07