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Hyperbolic extensions of free groups. (English) Zbl 1439.20034
Summary: Given a finitely generated subgroup $$\Gamma \leq \operatorname{Out}(\mathbb{F})$$ of the outer automorphism group of the rank-$$r$$ free group $$\mathbb{F}=F_r$$, there is a corresponding free group extension $$1 \rightarrow \mathbb{F} \rightarrow E_{\Gamma}\rightarrow\Gamma\rightarrow 1$$. 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.
Reviewer: Reviewer (Berlin)

##### MSC:
 20F28 Automorphism groups of groups 20F67 Hyperbolic groups and nonpositively curved groups 20E05 Free nonabelian groups 20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 57M07 Topological methods in group theory
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