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

*(English)*Zbl 1454.20087Summary: 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 [Geom. Topol. 22, No. 1, 517–570 (2018; Zbl 1439.20034)] 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.

Reviewer: Reviewer (Berlin)