The Teichmüller space of the Hirsch foliation

Alvarez, Sébastien; Lessa, Pablo

Annales de l'Institut Fourier, Tome 68 (2018) no. 1, pp. 1-51.

Nous prouvons que l’espace de Teichmüller du feuilletage de Hirsch (un feuilletage minimal d’une 3-variété fermée par surfaces hyperboliques non compactes) est homéomorphe à l’espace des courbes fermées du plan. Cela nous permet de prouver que l’espace des métriques hyperboliques sur le feuilletage est un fibré principal trivial. De plus, le groupe structural de ce fibré, i.e. la composante neutre du groupe des homéomorphismes qui sont lisses le long des feuilles et varient transversalement continûment dans la topologie lisse, est contractile.

We prove that the Teichmüller space of the Hirsch foliation (a minimal foliation of a closed 3-manifold by non-compact hyperbolic surfaces) is homeomorphic to the space of closed curves in the plane. This allows us to show that the space of hyperbolic metrics on the foliation is a trivial principal fiber bundle. And that the structure group of this bundle, the arc-connected component of the identity in the group of homeomorphisms which are smooth on each leaf and vary continuously in the smooth topology in the transverse direction of the foliation, is contractible.

Classification : 57R30, 30F60
Keywords: Teichmüller theory, Riemann surface foliations
Mot clés : Théorie de Teichmülller, feuilletages par surfaces de Riemann

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