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(Fn-1) 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).
All Science Journal Classification (ASJC) codes
- Geometry and Topology
- Lipschitz retraction
- Subgroups of Out(F-n)