English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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.

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