Quasi-geodesics in Out(F_n) and their shadows in sub-factors

Abstract

We study the behaviour of quasi-geodesics in Out(F_n). Given an element f in Out(F_n) there are several natural paths connecting the origin to f in Out(F_n); for example, paths associated to sequences of Stallings folds and paths induced by the shadow of greedy folding paths in Outer Space. We show that none of these paths is, in general, a quasi-geodesic in Out(F_n). In fact, in contrast with the mapping class group setting, we construct examples where any quasi-geodesic in Out(F_n) connecting f to the origin will have to back-track in some free factor of F_n.

24 pages, 5 figure. Final version accepted to Journal of the London Mathematical Society