Equivalence Relations with Amenable Leaves Need Not Be Amenable

There are two notions of amenability for discrete equivalence relations. The \global" amenability (which is usually referred to just as \amenability") is the property of existence of leafwise invariant means, which, by a theorem of Connes{Feldman{Weiss, is equivalent to hyperrniteness, or, to being the orbit equivalence relation of a Z-action. The notion of \local" amenability applies to equivalence relations endowed with an additional leafwise graph structure and means that a.e. leafwise graph is amenable (or, FFlner) in the sense that it has subsets A with arbitrary small isoperimetric ratio j@Aj=jAj (equivalently, that 0 belongs to the spectrum of leafwise Laplacians). In the present article we exhibit examples showing that local amenability does not imply global amenability contrary to a widespread opinion expressed in a number of earlier papers. We construct these examples both in the measure-theoretical (for discrete equivalence relations) and in the smooth (for foliations of compact manifolds) categories. We also formulate a general criterion of global amenability in isoperimetric terms. 1. Amenability We begin with recalling the deenition of amenable groups. Denote by l 1 1 (G) the space of probability measures on a countable group G, and by (l 1) 1 (G) the space of normalized positive linear functionals on l 1 (G), i.e., the space of means ((nitely additive probability measures) on G. Obviously, niteness of G is equivalent to existence of a nite invariant measure on G: There are two natural ways of generalizing property (1): either to look for xed points in the larger space (l 1) or to replace precise invariance with approximative invariance in the same space l