Minimal Følner foliations are amenable - Mathematics > Dynamical SystemsReportar como inadecuado

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Abstract: For finitely generated groups, amenability and F{\o}lner properties areequivalent. However, contrary to a widespread idea, Kaimanovich showed thatF{\o}lner condition does not imply amenability for discrete measuredequivalence relations. In this paper, we exhibit two examples of $C^\infty$foliations of closed manifolds that are F{\o}lner and non amenable with respectto a finite transverse invariant measure and a transverse invariant volume,respectively. We also prove the equivalence between the two notions when thefoliation is minimal, that is all the leaves are dense, giving a positiveanswer to a question of Kaimanovich. The equivalence is stated with respect totransverse invarian measures or some tangentially smooth measures. The latterinclude harmonic measures, and in this case the F{\o}lner condition has to bereplaced by $\eta$-F{\o}lner where the usual volume is modified by the modularform $\eta$ of the measure.

Autor: Fernando Alcalde Cuesta, Ana Rechtman


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