Hyperbolic extensions of free groups

Abstract

Given a finitely generated subgroup $\Gamma \le Out\left(\mathbb{F}\right)$ of the outer automorphism group of the rank-$r$ free group $\mathbb{F}=F_r$, there is a corresponding free group extension $1\to \mathbb{F}\to E_{\Gamma }\to \Gamma \to 1$. We give sufficient conditions for when the extension $E_{\Gamma }$ is hyperbolic. In particular, we show that if all infinite-order elements of $\Gamma$ are atoroidal and the action of $\Gamma$ on the free factor complex of $\mathbb{F}$ has a quasi-isometric orbit map, then $E_{\Gamma }$ is hyperbolic. As an application, we produce examples of hyperbolic $\mathbb{F}$–extensions $E_{\Gamma }$ for which $\Gamma$ 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.