Cubulating Drilled bundles over graphs

Abstract

We start with a Gromov-hyperbolic surface bundle $E$ over a graph, and drill out essential simple closed curves from fibers to obtain a drilled bundle $F$. We prove that for such drilled bundles $F$, the fundamental group ${\pi }_1(F)$ is relatively hyperbolic with $(\mathbb{Z}\oplus \mathbb{Z})$ peripheral groups. Combining the relative hyperbolicity of ${\pi }_1(F)$ thus obtained with a theorem of Wise, we establish virtually special cubulability of ${\pi }_1(F)$ provided that the maximal undrilled subbundles of $F$ are cubulable.

arXiv comment37 pages, no figures

References in corpus (7)

• Convex cocompact subgroups of mapping class groups
• Residually finite rationally solvable groups and virtual fibring
• Relative Hyperbolicity, Trees of Spaces and Cannon-Thurston Maps
• A Combination Theorem for Strong Relative Hyperbolicity
• A Combination Theorem for Metric Bundles
• Algebraic Ending Laminations and Quasiconvexity
• Tight trees and model geometries of surface bundles over graphs