Use and disuse of plugs in foliations

Abstract : In this text we deal with two main questions. The first one is to know if geodesible vector fields on closed 3-manifolds have periodic orbits. The second one studies the relation between the concepts of amenability and having Folner leaves in the context of foliations. The common point is the use of plugs. Plugs are a useful tool for changing a foliation inside a foliated chart.

The first chapter is dedicated to the first question. A non singular vector field is geodesible if there is a Riemannian metric of the ambient manifold making the orbits of the vector field geodesics. Let X be a geodesible vector field on a closed oriented 3-manifold, and assume that the 3-manifold is either diffeomorphic to the sphere or has non trivial second homotopy group. The main theorems of this chapter said that under this assumptions X has a periodic orbit if it is real analytic or if it preserves a volume.

In the second chapter we talk about the second question. In 1983, R. Brooks stated that a foliation with all its leaves Folner is amenable, with respect to an invariant measure. Using a plug, we will construct a counter-example of this statement, that is a non-amenable foliation whose leaves are Folner. We will then show that if we assume that the foliation is minimal, that is that all the leaves are dense, the fact that the leaves are F{\o}lner implies that the foliation is amenable. This hypothesis was suggested by V. A. Kaimanovich. The proof uses a theorem by D. Cass that describes minimal leaves.
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Ana Rechtman. Use and disuse of plugs in foliations. Mathematics [math]. Ecole normale supérieure de lyon - ENS LYON, 2009. English. ⟨tel-00361633⟩

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