For a free group $\mathbb{ F}$ of finite rank such that $\text{ rank}(\mathbb{ F})\geq 3$, we prove that the set of weak limits of a conjugacy class in $\mathbb{ F}$ under iterates of some hyperbolic $\phi\in\mathsf{ Out(\mathbb{ F})}$ is equal to the collection of generic leaves and singular lines of $\phi$. As an application we describe the ending lamination set for a hyperbolic extension of $\mathbb{ F}$ by a hyperbolic subgroup of $\mathsf{ Out(\mathbb{ F})}$ in a new way and use it to prove results about Cannon-Thurston maps for such extensions. We also use it to derive conditions for quasiconvexity of finitely generated, infinite index subgroups of $\mathbb{ F}$ in the extension group. These results generalize similar results obtained by Mahan Mj, Kapovich-Lustig and use different techniques.
Categories: math.GR