#### Volume 22, issue 1 (2018)

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Hyperbolic extensions of free groups

### Spencer Dowdall and Samuel J Taylor

Geometry & Topology 22 (2018) 517–570
##### Abstract

Given a finitely generated subgroup $\Gamma \le Out\left(\mathbb{F}\right)$ 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.

##### Keywords
hyperbolic group extensions, $\mathrm{Out}(\mathbb{F}_n)$, Outer space, free factor complex
##### Mathematical Subject Classification 2010
Primary: 20F28, 20F67
Secondary: 20E06, 57M07