Hyperbolic extensions of free groups

Given a finitely generated subgroup $\mathrm{\Gamma }\le \mathrm{O}\mathrm{u}\mathrm{t}(\mathbb{F})$ of the outer automorphism group of the rank $r$ free group $\mathbb{F}=F_r$, there is a corresponding free group extension $1 \to $\mathbb{F}$ \to E_{\Gamma} \to \Gamma \to 1$\mathrm{.}Wegivesufficientconditionsforwhentheextension$E_{\Gamma}$ is hyperbolic. In particular, we show that if all infinite order elements of $\mathrm{\Gamma }$ are atoroidal and the action of $\mathrm{\Gamma }$ on the free factor complex of $\mathbb{F}$ has a quasi-isometric orbit map, then $E_{\mathrm{\Gamma }}$ is hyperbolic. As an application, we produce examples of hyperbolic $\mathbb{F}$-extensions $E_{\mathrm{\Gamma }}$ for which $\mathrm{\Gamma }$ has torsion and is not virtually cyclic. The proof of our main theorem involves a detailed study of quasigeodesics in Outer space that make progress in the free factor complex. This may be of independent interest.