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.

37 pages, no figures