Abstract
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.
Citation
Spencer Dowdall. Samuel Taylor. "Hyperbolic extensions of free groups." Geom. Topol. 22 (1) 517 - 570, 2018. https://doi.org/10.2140/gt.2018.22.517
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