Equivalence relations with amenable leaves need not be amenable (Q2735117)

A countable equivalence relation \(R\) on a Lebesgue space is `globally' amenable (usually called amenable) if it has leafwise-invariant means, or, equivalently, if it is the orbit equivalence relation of a \(\mathbb Z\)-action by the result of \textit{A. Connes}, \textit{J. Feldman} and \textit{B. Weiss} [Ergodic Theory Dyn. Syst. 1, 431-450 (1981; Zbl 0491.28018)]. On the other hand \(R\) is `locally' amenable if it has an additional leafwise graph structure with respect to which a.e. leafwise graph is amenable as a graph (that is, it has subsets \(A\) with arbitrarily small isoperimetric ratio \(|\delta A|/|A|\)). In this paper the author presents examples showing that local amenability does not imply global amenability; moreover, a general necessary and sufficient condition for global amenability is found in terms of leafwise isoperimetric properties. These conditions do much to clarify the situation. One measure of the importance of these examples is that it has been widely assumed (and frequently explicitly stated) that no such examples could exist.NEWLINENEWLINEFor the entire collection see [Zbl 0961.00011].