Minimal F{\o}lner foliations are amenable

TL;DR

This paper explores the relationship between F{46}lner properties and amenability in foliations, providing examples and proving equivalence under minimality, thus clarifying longstanding questions in the theory.

Contribution

It constructs examples of F{46}lner but non-amenable foliations and establishes the equivalence of F{46}lner and amenability for minimal foliations, answering a key open question.

Findings

Examples of F{46}lner non-amenable foliations are provided.

Proves equivalence of F{46}lner and amenability for minimal foliations.

Shows the F{46}lner condition's relation to measures and harmonic measures.

Abstract

For finitely generated groups, amenability and F{\o}lner properties are equivalent. However, contrary to a widespread idea, Kaimanovich showed that F{\o}lner 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 F{\o}lner 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 invarian measures or some tangentially smooth measures. The latter include harmonic measures, and in this case the F{\o}lner condition has to be replaced by $\eta$-F{\o}lner (where the usual volume is modified by the modular form $\eta$ of the measure).