Minimal Følner foliations are amenable

Fernando Alcalde Cuesta; Ana Rechtman

doi:10.3934/dcds.2011.31.685

Article Contents

2011, Volume 31, Issue 3: 685-707. Doi: 10.3934/dcds.2011.31.685

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Minimal Følner foliations are amenable

Fernando Alcalde Cuesta^1, and
Ana Rechtman^2,

1.
Departamento de Xeometría e Topoloxía, Universidade de Santiago de Compostela, E-15782 Santiago de Compostela
2.
Department of Mathematics, Northwestern University 2033 Sheridan Road, Evanston, IL 60208-2730

Received: February 2010

Revised: July 2011

Published: September 2011

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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).

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Mathematics Subject Classification: 37A20, 43A07, 57R30.

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