Detecting surface bundles in finite covers of hyperbolic closed 3-manifolds.
Pré-print du CMLA 2012-07 - version du 18 mai 2012
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Auteur : Claire Renard
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.
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.
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