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Hyperbolic extensions of free groups

Spencer Dowdall and Samuel J Taylor

Geometry & Topology 22 (2018) 517–570

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 1 F EΓ Γ 1. 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.

hyperbolic group extensions, $\mathrm{Out}(\mathbb{F}_n)$, Outer space, free factor complex
Mathematical Subject Classification 2010
Primary: 20F28, 20F67
Secondary: 20E06, 57M07
Received: 9 March 2016
Revised: 20 October 2016
Accepted: 23 November 2016
Published: 31 October 2017
Proposed: Walter Neumann
Seconded: Benson Farb, Bruce Kleiner
Spencer Dowdall
Department of Mathematics
Vanderbilt University
Nashville, TN
United States
Samuel J Taylor
Department of Mathematics
Temple University
Philadelphia, PA
United States