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Sub-logarithmic Heegaard gradients of a hyperbolic 3-manifold and virtual fibers. (French. English summary) Zbl 1377.57020
Actes de Séminaire de Théorie Spectrale et Géométrie. Année 2010–2011. St. Martin d’Hères: Université de Grenoble I, Institut Fourier. Séminaire de Théorie Spectrale et Géométrie 29, 97-131 (2011).
Summary: J. Maher has proven that a closed, connected and orientable hyperbolic 3-manifold $$M$$ virtually fibers over the circle if and only if it admits an infinite family of finite covers with bounded Heegaard genus. Building on Maher’s proof, we present in this article a theorem giving a sufficient condition for a finite cover of a closed hyperbolic 3-manifold to contain a virtual fiber in terms of the covering degree $$d$$ and the Heegaard genus of the cover. We introduce sub-logarithmic versions of Lackenby’s infimal Heegaard gradients. In this setting, we expose the analogues of Lackenby’s Heegaard gradient and strong Heegaard gradient conjectures.
For the entire collection see [Zbl 1356.35008].
##### MSC:
 57M50 General geometric structures on low-dimensional manifolds 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 57M10 Covering spaces and low-dimensional topology 20F67 Hyperbolic groups and nonpositively curved groups
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