The co-surface graph and the geometry of hyperbolic free group extensions

We introduce the co-surface graph CS of a finitely generated free group F and use it to study the geometry of hyperbolic group extensions of F. Among other things, we show that the Gromov boundary of the co-surface graph is equivariantly homeomorphic to the space of free arational F–trees and use this to prove that a finitely generated subgroup of Out(F) quasi-isometrically embeds into the co-surface graph if and only if it is purely atoroidal and quasi-isometrically embeds into the free factor complex. This answers a question of I. Kapovich. Our earlier work [DT] shows that every such group gives rise to a hyperbolic extension of F, and here we prove a converse to this result that characterizes the hyperbolic extensions of F arising in this manner. As an application of our techniques, we additionally obtain a Scott–Swarup type theorem for this class of extensions.