Hyperbolic extensions of free groups from atoroidal ping-pong

Abstract

We prove that all atoroidal automorphisms of $Out\left(F_N\right)$ act on the space of projectivized geodesic currents with generalized north–south dynamics. As an application, we produce new examples of nonvirtually cyclic, free and purely atoroidal subgroups of $Out\left(F_N\right)$ such that the corresponding free group extension is hyperbolic. Moreover, these subgroups are not necessarily convex cocompact.