Detecting surface bundles in finite covers of hyperbolic closed 3-manifolds
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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.References
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Additional Information
- 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
- Email: claire.renard@normalesup.org
- Received by editor(s): April 9, 2012
- Received by editor(s) in revised form: July 7, 2012
- Published electronically: July 3, 2013
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 366 (2014), 979-1027
- MSC (2010): Primary 57M27, 57M10, 57M50, 20F67
- DOI: https://doi.org/10.1090/S0002-9947-2013-05914-4
- MathSciNet review: 3130323