#### Volume 19, issue 3 (2019)

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Hyperbolic extensions of free groups from atoroidal ping-pong

### Caglar Uyanik

Algebraic & Geometric Topology 19 (2019) 1385–1411
##### Abstract

We prove that all atoroidal automorphisms of $Out\left({F}_{N}\right)$ act on the space of projectivized geodesic currents with generalized north–south dynamics. As an application, we produce new examples of nonvirtually cyclic, free and purely atoroidal subgroups of $Out\left({F}_{N}\right)$ such that the corresponding free group extension is hyperbolic. Moreover, these subgroups are not necessarily convex cocompact.

##### Keywords
free groups, hyperbolic extensions, $\mathrm{Out}(F_N)$, geodesic currents
##### Mathematical Subject Classification 2010
Primary: 20F28, 20F67
Secondary: 37D40