Le coût est un invariant isopérimétrique

Abstract

Let $R$ be an ergodic measured equivalence relation of type II1 on a probability space $(X,\mu )$ without atom. We show that

where $C(R)$ is the cost of $R$, assumed to be finite, and $h(R)$ its isoperimetric constant. This is a continuation of recent work of Russell Lyons and the authors on the subject.