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

By Spencer Dowdall, Samuel J. Taylor. 2016

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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 [S. Dowdall and S. J. Taylor, ‘Hyperbolic extensions of free groups’, to appear in Geom. Topol.] 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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/88a064ec8780395703aa4a57015ad6e72ea1b39f