Journal of Noncommutative Geometry

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Volume 4, Issue 3, 2010, pp. 381–387
DOI: 10.4171/JNCG/60

Published online: 2010-07-26

Le coût est un invariant isopérimétrique

Mikaël Pichot[1] and Stéphane Vassout[2]

(1) IPMU, Tokyo
(2) Institut de Mathématiques de Jussieu, Paris

Let $R$ be an ergodic measured equivalence relation of type II1 on a probability space $(X, \mu)$ without atom. We show that $$ C(R)=1+\frac{1}{2}h(R) $$ 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.

Keywords: Measured equivalence relations, cost, isoperimetry

Pichot Mikaël, Vassout Stéphane: Le coût est un invariant isopérimétrique. J. Noncommut. Geom. 4 (2010), 381-387. doi: 10.4171/JNCG/60