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

TL;DR

This paper investigates the limits of conjugacy classes in free groups under hyperbolic automorphisms, revealing their connection to generic leaves and singular lines, and applies these findings to hyperbolic extensions and subgroup properties.

Contribution

It characterizes the weak limits of conjugacy classes under hyperbolic automorphisms in free groups and applies this to describe ending laminations and subgroup quasiconvexity in hyperbolic extensions.

Findings

Weak limits are generic leaves and singular lines of the automorphism.

New description of ending lamination sets for hyperbolic extensions.

Conditions for quasiconvexity of subgroups in the extension group.

Abstract

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.