We study the behaviour of quasi-geodesics in $Out({\mathbb{F}}_n)$. Given an element $\varphi$ in $Out({\mathbb{F}}_n)$, there are several natural paths connecting the origin to $\varphi$ in $Out({\mathbb{F}}_n)$; for example, paths given by Stallings' folding algorithm 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 $Out({\mathbb{F}}_n)$. In fact, in contrast with the mapping class group setting, we construct examples where any quasi-geodesic in $Out({\mathbb{F}}_n)$ connecting $\varphi$ to the origin will have to backtrack in some free factor of ${\mathbb{F}}_n$.

中文翻译：

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Out(Fn) 中的准测地线及其在子因子中的阴影

我们研究准测地线的行为 $出去({\mathbb{F}}_n)$. 给定一个元素$\phi$ 在 $出去({\mathbb{F}}_n)$，有几条自然路径将原点连接到 $\phi$ 在 $出去({\mathbb{F}}_n)$; 例如，由 Stallings 折叠算法给出的路径和由外层空间中贪婪折叠路径的阴影引起的路径。我们表明，这些路径中没有一条通常是$出去({\mathbb{F}}_n)$. 事实上，与映射类组设置相反，我们构造了示例，其中任何准测地线在$出去({\mathbb{F}}_n)$ 连接 $\phi$ 到原点将不得不在某些自由因素中回溯 ${\mathbb{F}}_n$.