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.
