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

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.