Hyperbolic extensions of free groups from atoroidal ping-pong

TL;DR

This paper demonstrates that atoroidal automorphisms of outer automorphism groups of free groups exhibit specific dynamic behaviors and constructs new hyperbolic extensions, expanding understanding of free group automorphisms and their geometric properties.

Contribution

It establishes generalized north-south dynamics for atoroidal automorphisms and constructs new hyperbolic free group extensions with non-virtually cyclic, purely atoroidal subgroups.

Findings

Atoroidal automorphisms act with north-south dynamics on geodesic currents.

New examples of hyperbolic free group extensions are provided.

Subgroups constructed are not necessarily convex cocompact.

Abstract

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