Abstract
For finitely generated groups, amenability and Folner properties are equivalent. However, contrary to a widespread idea, Kaimanovich showed that Folner condition does not imply amenability for discrete measured equivalence relations. In this paper, we exhibit two examples of $C^\infty$ foliations of closed manifolds that are Folner and non amenable with respect to a finite transverse invariant measure and a transverse invariant volume, respectively. We also prove the equivalence between the two notions when the foliation is minimal, that is all the leaves are dense, giving a positive answer to a question of Kaimanovich. The equivalence is stated with respect to transverse invariant measures or some tangentially smooth measures. The latter include harmonic measures, and in this case the Folner condition has to be replaced by $\eta$-Folner (where the usual volume is modified by the modular form $\eta$ of the measure).
Highlights
Discrete equivalence relations provide a natural approach to the study of foliations on compact manifolds: the leaves induce a discrete equivalence relation on any total transversal
A group satisfying the latter condition is said to be Følner. Both definitions can be stated for a compact foliated manifold equipped with a transverse invariant measure, using the induced equivalence relation
The previous proposition implies that a foliation F of a compact manifold M that is amenable with respect to a quasi-invariant measure ν induces a δ-Følner equivalence relation on any total transversal
Summary
Discrete equivalence relations provide a natural approach to the study of foliations on compact manifolds: the leaves induce a discrete equivalence relation on any total transversal. A group satisfying the latter condition is said to be Følner Both definitions can be stated for a compact foliated manifold equipped with a transverse invariant measure, using the induced equivalence relation. In this paper we will study the amenability and Følner properties with respect to either transverse invariant measures, or tangentially smooth measures The implication in the second theorem stating that an amenable foliation is Følner is a generalization of Carriere and Ghys’ result They proved this implication for transverse invariant measures. The paper is divided in three sections: the first one contains the definitions and concepts we will use; in the second one we describe the two examples of non-amenable Følner foliations; and the third one contains the proofs of the main theorems.
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