Alcalde Cuesta, Fernando; Rechtman, Ana Minimal Følner foliations are amenable. (English) Zbl 1251.37002 Discrete Contin. Dyn. Syst. 31, No. 3, 685-707 (2011). The concepts of amenability and Følner condition are originally defined for locally compact groups and are equivalent. For a compact foliated space with an invariant measure, the foliation defines an equivalence relation on a total transversal (two points are equivalent if they belong to the same leaf), amenable foliations and Følner leaves can be defined in terms of this equivalence relation. Contrary to a widespread belief, V. Kaimanovich [Transl., Ser. 2, Am. Math. Soc. 202(50), 151–166 (2001; Zbl 0990.28013)] constructed a smooth non-amenable foliation with almost all leaves satisfying the Følner condition. In the example of Kaimanovich any transverse invariant measure is not locally finite. The paper under review has two goals, the first of them is to show that amenability for foliations cannot be deduced from the condition of the foliation having Følner leaves. This is done by exhibiting two examples of non-amenable foliations whose leaves are Følner. Theses examples are constructed by using plugs, and their importance lies in the measures they present: a finite transverse invariant measure and transverse invariant volume. The second goal is to answer the following question (due to Kaimanovich): does the assumption of minimality on the foliation assure the equivalence between the two concepts? Here ‘minimality’ means that the ambient manifold is a minimal set for the foliation. A positive answer is given by two theorems. Under the assumptions that almost all leaves have trivial holonomy and that the foliation is minimal, the authors prove, in the cases of transverse invariant measures and tangentially smooth measures, that the foliation is amenable if and only if almost all leaves are Følner. Reviewer: Albetã Costa Mafra (Rio de Janeiro) Cited in 2 Documents MSC: 37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations 43A07 Means on groups, semigroups, etc.; amenable groups 57R30 Foliations in differential topology; geometric theory Keywords:foliations; amenable foliations; Følner leaves; measurable equivalence relations Citations:Zbl 0990.28013 PDFBibTeX XMLCite \textit{F. Alcalde Cuesta} and \textit{A. Rechtman}, Discrete Contin. Dyn. Syst. 31, No. 3, 685--707 (2011; Zbl 1251.37002) Full Text: DOI arXiv