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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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