Gradients de Heegaard sous-logarithmiques d’une variété hyperbolique de dimension trois et fibres virtuelles

Renard, Claire

Séminaire de théorie spectrale et géométrie, Tome 29 (2010-2011), pp. 97-131.

J. Maher a montré qu’une variété hyperbolique de dimension $3$ compacte sans bord, connexe et orientable fibre virtuellement sur le cercle si et seulement si elle admet une famille infinie de revêtements finis de genre de Heegaard borné. En s’appuyant sur la démonstration de Maher, cet article présente un théorème donnant une condition suffisante pour qu’un revêtement fini d’une variété hyperbolique compacte de dimension $3$ contienne une fibre virtuelle, qui s’exprime en fonction du degré $d$ du revêtement et de son genre de Heegaard. On introduit des versions sous-logarithmiques des gradients de Heegaard de Lackenby. Dans ce contexte, on propose des analogues aux conjectures du gradient de Heegaard et du gradient de Heegaard fort de Lackenby.

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.

Classification : 57M27, 57M10, 57M50, 20F67
Mots clés : variété de dimension trois, fibration, revêtement fini, géométrie hyperbolique, corps en anses.

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