Cubulating Drilled bundles over graphs

TL;DR

The paper develops a framework for drilled surface bundles over graphs, proving that drilling essential curves from fibers yields a fundamental group $G=\pi_1(F)$ that is strongly relatively hyperbolic relative to a finite family of $\mathbb{Z}^2$ peripheral subgroups. It combines partial electrification, annulus-flaring arguments, and Bestvina-Feighn theory to establish hyperbolicity of the partially electrified bundle, then uses Sisto’s guessing geodesics criterion to obtain a robust path family that yields relative hyperbolicity and relative quasiconvexity. Building on this, Wise’s quasiconvex hierarchy is employed to obtain virtual specialness (cubulability) of $G$ under suitable undrilled-subbundle hypotheses, with explicit constructions of alternative graph-of-groups decompositions and reductions to avoid accidental parabolics. Further, Kielak’s criterion and Lott-Lück/Fernos-Valette results imply vanishing $l^2$-Betti numbers, hence virtual algebraic fibering of $G$. The work demonstrates a coherent path from drilling in hyperbolic surface bundles to cubulation and algebraic fibering, yielding new families of cubulable groups and insights for 3-manifold group analogs.

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 $π_1(F)$ is relatively hyperbolic with $(\mathbb{Z}\oplus \mathbb{Z})$ peripheral groups. Combining the relative hyperbolicity of $π_1(F)$ thus obtained with a theorem of Wise, we establish virtually special cubulability of $π_1(F)$ provided that the maximal undrilled subbundles of $F$ are cubulable.

Theorems & Definitions (104)

• Definition 1.2
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 2.1
• Lemma 2.2
• proof
• Theorem 3.1
• Definition 3.2
• Lemma 3.3