Geometry & Topology

Hyperbolic extensions of free groups

Spencer Dowdall and Samuel Taylor

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Given a finitely generated subgroup Γ Out(F) of the outer automorphism group of the rank-r free group F=Fr, there is a corresponding free group extension 1FEΓΓ1. We give sufficient conditions for when the extension EΓ is hyperbolic. In particular, we show that if all infinite-order elements of Γ are atoroidal and the action of Γ on the free factor complex of F has a quasi-isometric orbit map, then EΓ is hyperbolic. As an application, we produce examples of hyperbolic F–extensions EΓ for which Γ 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.

Article information

Geom. Topol., Volume 22, Number 1 (2018), 517-570.

Received: 9 March 2016
Revised: 20 October 2016
Accepted: 23 November 2016
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F28: Automorphism groups of groups [See also 20E36] 20F67: Hyperbolic groups and nonpositively curved groups
Secondary: 20E06: Free products, free products with amalgamation, Higman-Neumann- Neumann extensions, and generalizations 57M07: Topological methods in group theory

hyperbolic group extensions $\mathrm{Out}(\mathbb{F}_n)$ Outer space free factor complex


Dowdall, Spencer; Taylor, Samuel. Hyperbolic extensions of free groups. Geom. Topol. 22 (2018), no. 1, 517--570. doi:10.2140/gt.2018.22.517.

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