Hyperbolic extensions of free groups

Given a finitely generated subgroup [math] of the outer automorphism group of the rank- [math] free group [math] , there is a corresponding free group extension [math] . We give sufficient conditions for when the extension [math] is hyperbolic. In particular, we show that if all infinite-order elements of [math] are atoroidal and the action of [math] on the free factor complex of [math] has a quasi-isometric orbit map, then [math] is hyperbolic. As an application, we produce examples of hyperbolic [math] –extensions [math] for which [math] 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.