Hyperbolic extensions of free groups

Abstract

© 2018, Mathematical Sciences Publishers. All rights reserved. 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.