Detecting surface bundles in finite covers of hyperbolic closed 3-manifolds

TL;DR

This paper establishes explicit criteria involving volume, injectivity radius, and other parameters for finite covers of hyperbolic 3-manifolds to contain surface bundles or fibrations, advancing understanding of their topological structures.

Contribution

It provides new explicit inequalities that guarantee the existence of surface bundles and fibrations in finite covers of hyperbolic 3-manifolds, linking geometric and topological properties.

Findings

Conditions for a finite cover to be a surface bundle.

Lower bounds for genus of Heegaard splittings.

Criteria for homology classes to correspond to fibrations.

Abstract

The main theorem of this article provides sufficient conditions for a degree $d$ finite cover $M'$ of a hyperbolic 3-manifold $M$ to be a surface-bundle. Let $F$ be an embedded, closed and orientable surface of genus $g$, close to a minimal surface in the cover $M'$, splitting $M'$ into a disjoint union of $q$ handlebodies and compression bodies. We show that there exists a fiber in the complement of $F$ provided that $d$, $q$ and $g$ satisfy some inequality involving an explicit constant $k$ depending only on the volume and the injectivity radius of $M$. In particular, this theorem applies to a Heegaard splitting of a finite covering $M'$, giving an explicit lower bound for the genus of a strongly irreducible Heegaard splitting of $M'$. Applying the main theorem to the setting of a circular decomposition associated to a non trivial homology class of $M$ gives sufficient conditions for this homology class to correspond to a fibration over the circle. Similar methods lead also to a sufficient condition for an incompressible embedded surface in $M$ to be a fiber.