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Lipschitz retraction and distortion for subgroups of \(\mathrm{Out}(F_n)\). (English) Zbl 1285.20033

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

MSC:

20F28 Automorphism groups of groups
20E05 Free nonabelian groups
20F65 Geometric group theory
57M07 Topological methods in group theory
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
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