# Coût des relations d’équivalence et des groupes

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DOI: 10.1007/s002229900019

- Cite this article as:
- Gaboriau, D. Invent. math. (2000) 139: 41. doi:10.1007/s002229900019

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## Abstract.

We study a new dynamical invariant for dicrete groups: the cost. It is a real number in {1−1/*n*}∪[1,∞], bounded by the number of generators of the group, and it is well behaved with respect to finite index subgroups. Namely, the quantities 1 minus the cost are related by multiplying by the index. The cost of every infinite amenable group equals 1. We compute it in some other situations, including free products, free products with amalgamation and HNN-extensions over amenable groups and for direct product situations. For instance, the cost of the free group on *n* generators equals *n*. We prove that each possible finite value of the cost is achieved by a finitely generated group. It is dynamical because it relies on measure preserving free actions on probability Borel spaces. In most cases, groups have *fixed price*, which implies that two freely acting groups which define the same orbit partition must have the same cost. It enables us to distinguish the orbit partitions of probability-preserving free actions of free groups of different ranks. At the end of the paper, we give a *mercuriale*, i.e. a list of costs of different groups. The cost is in fact an invariant of ergodic measure-preserving equivalence relations and is defined using *graphings*. A *treeing* is a measurable way to provide every equivalence class (=orbit) with the structure of a simplicial tree, this an example of graphing. Not every relation admits a treeing: we prove that every free action of a cost 1 non-amenable group is not treeable, but we prove that subrelations of treeable relations are treeable. We give examples of relations which cannot be produced by an action of any finitely generated group. The cost of a relation which can be decomposed as a direct product is shown to be 1. We define the notion for a relation to be a free product or an HNN-extension and compute the cost for the resulting relation from the costs of the building blocks. The cost is also an invariant of the pairs von Neumann algebra/Cartan subalgebra.