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

We introduce the co-surface graph $\mathcal{CS}$ of a finitely generated free group $\mathbb{F}$ and use it to study the geometry of hyperbolic group extensions of $\mathbb{F}$. Among other things, we show that the Gromov boundary of the co-surface graph is equivariantly homeomorphic to the space of free arational $\mathbb{F}$-trees and use this to prove that a finitely generated subgroup of $\mathrm{Out}(\mathbb{F})$ 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 [Hyperbolic extensions of free groups, arXiv:1406.2567] shows that every such group gives rise to a hyperbolic extension of $\mathbb{F}$, and here we prove a converse to this result that characterizes the hyperbolic extensions of $\mathbb{F}$ arising in this manner. As an application of our techniques, we additionally obtain a Scott--Swarup type theorem for this class of extensions.