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The co-surface graph and the geometry of hyperbolic free group extensions. (English) Zbl 1454.20087
Summary: 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)

20F67 Hyperbolic groups and nonpositively curved groups
20F28 Automorphism groups of groups
20E05 Free nonabelian groups
20E22 Extensions, wreath products, and other compositions of groups
20F65 Geometric group theory
57M07 Topological methods in group theory
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