2016/01/31 by Spencer Dowdall, Samuel J. Taylor · 2 citations
Mathematics · #Advanced Combinatorial Mathematics #Combinatorics #Converse #Differential geometry #Discrete mathematics #Finitely generated group #Finitely-generated abelian group #Free group #Geometric and Algebraic Topology #Geometry #Graph #Group (periodic table) #Homotopy and Cohomology in Algebraic Topology #Hyperbolic 3-manifold #Hyperbolic function #Hyperbolic geometry #Hyperbolic group #Hyperbolic manifold #Hyperbolic space #Mapping class group #Mathematics #Pure mathematics #Relatively hyperbolic group #Surface (topology) #math.GR #math.GT
paper · pdf · doi:10.1112/topo.12013
33 pages. Minor changes and other updates to incorporate referee comments. Final version; accepted for publication in the Journal of Topology
openalex created_date 2016/06/24 · arxiv created 2017/02/21 · openalex publication_date 2017/04/21 · arxiv updated 2017/05/04 · openalex updated_date 2026/08/06
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.