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

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 $\mathsf{Aut}\left(F_n\right)$.