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

Abstract

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