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

TL;DR

This paper investigates the properties of quasi-geodesics in Out(F_n), revealing that natural paths often fail to be quasi-geodesics and demonstrating the necessity of back-tracking in free factors, contrasting with the mapping class group case.

Contribution

It provides the first examples showing that natural paths in Out(F_n) are not quasi-geodesics and highlights differences from the mapping class group setting.

Findings

Natural paths are not quasi-geodesics in Out(F_n)

Any quasi-geodesic must back-track in some free factor

Contrasts with the behavior in the mapping class group

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.