Hyperbolic extensions of free groups

TL;DR

This paper establishes conditions under which certain free group extensions are hyperbolic, linking properties of automorphism subgroups to geometric group theory concepts like hyperbolicity and the free factor complex.

Contribution

It provides new criteria for hyperbolicity of free group extensions based on subgroup actions and introduces examples with torsion that are hyperbolic.

Findings

Extensions are hyperbolic if all infinite order elements are atoroidal and the action on the free factor complex has a quasi-isometric orbit map.

Constructs examples of hyperbolic extensions where the subgroup has torsion and is not virtually cyclic.

Develops a detailed analysis of quasigeodesics in Outer space related to the free factor complex.

Abstract

Given a finitely generated subgroup $\Gamma \le \mathrm{Out}(\mathbb{F})$ of the outer automorphism group of the rank $r$ free group $\mathbb{F} = F_r$, there is a corresponding free group extension $1 \to \mathbb{F} \to E_{\Gamma} \to \Gamma \to 1$. We give sufficient conditions for when the extension $E_{\Gamma}$ is hyperbolic. In particular, we show that if all infinite order elements of $\Gamma$ are atoroidal and the action of $\Gamma$ on the free factor complex of $\mathbb{F}$ has a quasi-isometric orbit map, then $E_{\Gamma}$ is hyperbolic. As an application, we produce examples of hyperbolic $\mathbb{F}$-extensions $E_{\Gamma}$ for which $\Gamma$ has torsion and is not virtually cyclic. The proof of our main theorem involves a detailed study of quasigeodesics in Outer space that make progress in the free factor complex. This may be of independent interest.