Lipschitz retraction and distortion for subgroups of Out(Fn)

Handel, Michael; Mosher, Lee

doi:10.2140/gt.2013.17.1535

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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

DOI: 10.2140/gt.2013.17.1535

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 $Aut (F_{n})$ .

Keywords

Lipschitz retraction, distortion, subgroups of Out(F_n)

Mathematical Subject Classification 2010

Primary: 20F28

Secondary: 20F65, 20E05, 57M07

References

Publication

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

Authors

Michael Handel
Mathematics & Computer Science Department Herbert H Lehman College (CUNY) Bronx, NY 10468-1589 USA

Lee Mosher
Department of Mathematics and Computer Science Rutgers University Newark Newark, NJ 07102 USA

Volume 17, issue 3 (2013)