Drilled hyperbolic surface bundles over graphs are CAT(0) cubulable

Abstract

By works of Agol, Bergeron–Wise, Dufour, and Kahn–Markovic, among others, it is known that fundamental groups of fibred hyperbolic 3-manifolds (and those of closed hyperbolic 3-manifolds in general) act geometrically on CAT(0) cube complexes and thus these manifolds possess strong group theoretic and topological properties. However, it is unknown if these results extend to the fundamental groups of all hyperbolic surface bundles over finite graphs (in other words, arbitrary hyperbolic surface-by-free groups). In this talk, we will describe how, after modifying these bundles by drilling (that is, removing small tubular neighbourhoods of) enough simple closed curves in specific fibres, the corresponding fundamental groups become hyperbolic relative to the introduced tori subgroups, and as a consequence, act geometrically on CAT(0) cube complexes. This is joint work with Mahan Mj.