Hyperbolic extensions of free groups

TLDR: In particular, if all infinite order elements of the outer automorphism group are atoroidal and the action of a subgroup on the free factor complex of the rank r free group F = Fr has a quasi-isometric orbit map, then the subgroup is hyperbolic as mentioned in this paper.

Abstract: Given a finitely generated subgroup Γ≤ Out(F) of the outer automorphism group of the rank r free group F = Fr, there is a corresponding free group extension 1→ F→ EΓ→ Γ→ 1. We give sufficient conditions for when the extension EΓ 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 F has a quasi-isometric orbit map, then EΓ is hyperbolic. As an application, we produce examples of hyperbolic F–extensions EΓ 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.