Limits of conjugacy classes under iterates of hyperbolic elements of Out($\mathbb{F}$)

Abstract

For a free group $\mathbb{F}$ of finite rank such that rank $(\mathbb{F})\ge 3$, we prove that the set of weak limits of a conjugacy class in $\mathbb{F}$ under iterates of some hyperbolic $\varphi \in \mathrm{Out}(\mathbb{F})$ is equal to the collection of generic leaves and lines with endpoints in attracting fixed points of $\varphi$.

As an application we describe the ending lamination set for a hyperbolic extension of $\mathbb{F}$ by a hyperbolic element of 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 in [23] and [19] and use different techniques.