Hyperbolic extensions of free groups

Given a finitely generated subgroup Gamma <= Out (F) of the outer automorphism group of the rank- r free group F = F-r, there is a corresponding free group extension 1 -> F -> E (Gamma) -> Gamma -> 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 F has a quasi-isometric orbit map, then E-Gamma is hyperbolic. As an application, we produce examples of hyperbolic 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.