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Transactions of the American Mathematical Society

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

Author: Claire Renard
Journal: Trans. Amer. Math. Soc. 366 (2014), 979-1027
MSC (2010): Primary 57M27, 57M10, 57M50, 20F67
Published electronically: July 3, 2013
MathSciNet review: 3130323
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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 also lead to a sufficient condition for an incompressible embedded surface in $ M$ to be a fiber.

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Claire Renard
Affiliation: Centre de Mathématiques et de Leurs Applications, École Normale Supérieure de Cachan, 61 avenue du président Wilson, F-94235 Cachan Cedex, France

Received by editor(s): April 9, 2012
Received by editor(s) in revised form: July 7, 2012
Published electronically: July 3, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.