Geometry & Topology
- Geom. Topol.
- Volume 22, Number 1 (2018), 517-570.
Hyperbolic extensions of free groups
Given a finitely generated subgroup of the outer automorphism group of the rank- free group , there is a corresponding free group extension . We give sufficient conditions for when the extension is hyperbolic. In particular, we show that if all infinite-order elements of are atoroidal and the action of on the free factor complex of has a quasi-isometric orbit map, then is hyperbolic. As an application, we produce examples of hyperbolic –extensions for which 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.
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
Permanent link to this document
Digital Object Identifier
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
Dowdall, Spencer; Taylor, Samuel. Hyperbolic extensions of free groups. Geom. Topol. 22 (2018), no. 1, 517--570. doi:10.2140/gt.2018.22.517. https://projecteuclid.org/euclid.gt/1513774918