Hyperbolic extensions of free groups

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.

50 pages. Minor changes and other updates to incorporate referee comments. Final version; accepted for publication in Geometry & Topology

References in corpus (3)

• The co-surface graph and the geometry of hyperbolic free group extensions
• Knots, sutures and excision
• Right-angled Artin groups and Out(F_n) I: quasi-isometric embeddings

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• Quasi-geodesics in Out(F_n) and their shadows in sub-factors
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