Le co\^ut est un invariant isop\'erim\'etrique

TL;DR

This paper establishes a precise relationship between the isoperimetric constant and the cost of a type II_1 ergodic measured equivalence relation, extending recent findings in the field.

Contribution

It proves that for such relations, the isoperimetric constant equals twice the cost minus two, providing a new exact formula connecting these invariants.

Findings

h(R)=2C(R)-2 for type II_1 ergodic measured equivalence relations

Extends recent results by Lyons and the authors

Provides a new invariant relationship in measured equivalence relations

Abstract

For a type II_1 ergodic measured equivalence relation R on a probability space without atom, we prove that h(R)=2C(R)-2, where C(R) is the cost, and h(R) the isoperimetric constant. This follows recent result by Lyons and the authors.