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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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