Hyperbolic extensions of free groups
Spencer Dowdall, Samuel J. Taylor

TL;DR
This paper establishes conditions under which certain free group extensions are hyperbolic, linking properties of automorphism subgroups to geometric group theory concepts like hyperbolicity and the free factor complex.
Contribution
It provides new criteria for hyperbolicity of free group extensions based on subgroup actions and introduces examples with torsion that are hyperbolic.
Findings
Extensions are hyperbolic if all infinite order elements are atoroidal and the action on the free factor complex has a quasi-isometric orbit map.
Constructs examples of hyperbolic extensions where the subgroup has torsion and is not virtually cyclic.
Develops a detailed analysis of quasigeodesics in Outer space related to the free factor complex.
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.
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