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Loops with Large Twist Get Short Along Quasi-geodesics in $Out(Fn)$

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Yulan Qing (Toronto)

Abstract: We study the behaviour of quasi-geodesics in $Out(Fn)$. Given an element $\phi$
in $Out(Fn)$ there are several natural paths connecting the origin
to $\phi$ in $Out(Fn)$, for example, a path associate to sequence of Stalling folds
and a path associated to standard geodesics in Outer space. We show that neither
of these paths is, in general, a quasi-geodesic in $Out(Fn)$. In fact, we construct
examples where any quasi-geodesic connecting $\phi$ to the origin will have to
back-track in some free factor of $Fn$.