Hyperbolic extensions of free groups

Dowdall, Spencer¹ ; Taylor, Samuel²

¹ Department of Mathematics, Vanderbilt University, Nashville, TN, United States
² Department of Mathematics, Temple University, Philadelphia, PA, United States

Geometry & topology, Tome 22 (2018) no. 1, pp. 517-570.

Résumé

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.

Détail

DOI : 10.2140/gt.2018.22.517

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

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