Lipschitz retraction and distortion for subgroups of Out(F_n)

TL;DR

This paper investigates the distortion properties of certain subgroups within Out(F_n) and Aut(F_n), providing characterizations of when these subgroups are distorted or undistorted, and establishing Lipschitz retraction results with applications to Dehn functions and automaticity.

Contribution

It offers new characterizations of subgroup distortion in Out(F_n), proves Lipschitz retraction for nondistorted subgroups, and applies these results to Dehn functions and automaticity.

Findings

Characterization of distortion for stabilizers of free factors

Proof that certain embeddings are nondistorted and Lipschitz retracts

Determination of Dehn functions and automaticity for Out(F_n) and Aut(F_n)

Abstract

Given a free factor A of the rank n free group F_n, we characterize when the subgroup of Out(F_n) that stabilizes the conjugacy class of A is distorted in Out(F_n). We also prove that the image of the natural embedding of Aut(F_{n-1}) in Aut(F_n) is nondistorted, that the stabilizer in Out(F_n) 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 Out(F_n) and Aut(F_n).