Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Minimal Følner foliations are amenable

Pages: 685 - 707, Volume 31, Issue 3, November 2011      doi:10.3934/dcds.2011.31.685

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Fernando Alcalde Cuesta - Departamento de Xeometría e Topoloxía, Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Spain (email)
Ana Rechtman - Department of Mathematics, Northwestern University 2033 Sheridan Road, Evanston, IL 60208-2730, United States (email)

Abstract: For finitely generated groups, amenability and Følner properties are equivalent. However, contrary to a widespread idea, Kaimanovich showed that Følner 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 Følner 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 Følner condition has to be replaced by $\eta$-Følner (where the usual volume is modified by the modular form $\eta$ of the measure).

Keywords:  Foliations, measurable equivalence relations, local and global means.
Mathematics Subject Classification:  37A20, 43A07, 57R30.

Received: February 2010;      Revised: July 2011;      Available Online: August 2011.