Hyperbolic extensions of free groups

Abstract

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