Hyperbolic extensions of free groups

By Spencer Dowdall, Samuel J. Taylor. 2014

61 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f9a563698456b92a5d6a0bc2e64583e60c1ff394

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In Geometry and Topology

1. Spencer Dowdall
2. Samuel J. Taylor
3. Jason A. Behrstock
4. Carolyn R. Abbott

In Mathematical Physics

1. Spencer Dowdall
2. Samuel J. Taylor
3. Jason A. Behrstock
4. Carolyn R. Abbott

Citations in the corpus

• The co‐surface graph and the geometry of hyperbolic free group extensions (2016, methodology): “In this section, we develop a converse to the main theorem of [15], which established hyperbolicity of EΓ.”