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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

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).

Lipschitz retraction, distortion, subgroups of Out(F_n)
Mathematical Subject Classification 2010
Primary: 20F28
Secondary: 20F65, 20E05, 57M07
Received: 16 April 2011
Revised: 8 September 2012
Accepted: 30 November 2012
Published: 15 June 2013
Proposed: Benson Farb
Seconded: Walter Neumann, Martin R. Bridson
Michael Handel
Mathematics & Computer Science Department
Herbert H Lehman College (CUNY)
Bronx, NY 10468-1589
Lee Mosher
Department of Mathematics and Computer Science
Rutgers University Newark
Newark, NJ 07102