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Revêtements finis d'une variété hyperbolique de dimension trois et fibres virtuelles.

Abstract : In the setting of hyperbolic 3-manifolds, Thurston conjectured that every connected, orientable, complete hyperbolic 3-manifold of finite volume has a finite cover fibered over the circle. Having this conjecture in mind, the main result of this thesis provides sufficient conditions for a finite cover of a hyperbolic 3-manifold M to fiber over the circle, or at least to contain a virtual fiber. Let F be an embedded, closed and orientable surface close to a minimal surface, in a finite cover M' of M, such that M' cut along F is a disjoint union of handlebodies and compression bodies. The condition to show that there exists a virtual fiber in the complement of F is given by an inequality involving the degree d of the cover, the genus g of the surface, the number q of compression bodies and a constant k depending only on the volume and the injectivity radius of M. Applying this theorem to a minimal genus Heegaard splitting of the finite cover M' leads to a sub-logarithmic version of Lackenby's conjectures of the Heegaard gradient and the strong Heegaard gradient. The main theorem also applies to the setting of a circular decomposition associated to a non trivial homology class. For example, we obtain sufficient conditions for a non trivial homology class of M 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 virtual fiber. Eventually, we give a criterion to show that the first Betti number in a tower of finite covers tends to infinity.
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Contributor : Claire Renard Connect in order to contact the contributor
Submitted on : Tuesday, March 20, 2012 - 10:10:04 AM
Last modification on : Monday, July 4, 2022 - 9:04:15 AM
Long-term archiving on: : Thursday, June 21, 2012 - 2:35:12 AM


  • HAL Id : tel-00680760, version 1


Claire Renard. Revêtements finis d'une variété hyperbolique de dimension trois et fibres virtuelles.. Topologie géométrique [math.GT]. Université Paul Sabatier - Toulouse III, 2011. Français. ⟨tel-00680760⟩



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