Simplicial embeddings between multicurve graphs

We study some graphs associated to a surface, called k-multicurve graphs, which interpolate between the curve complex and the pants graph. Our main result is that, under certain conditions, simplicial embeddings between multicurve graphs are induced by $\pi_1$-injective embeddings of the corresponding surfaces. We also prove the rigidity of the multicurve graphs.Comment: New introduction and some changes in Section 2, main results unchanged. References added. 18 pages, 5 figure

Lipschitz retraction and distortion for subgroups of Out(F_n)

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).Comment: Version 3: 35 pages. Revised for publication. Changes from previous versions: significant economies in exposition. Added an explicit description of the stabilizer of a free splitting, in Lemma 1