Coût des relations d'équivalence et des groupes

D. Gaboriau

Article publié :

Invent. Math. 139 (2000) 1, pages 41-98.

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Abstract :

The cost is a dynamical invariant of discrete groups and equivalence relations. Establishing a list of prices (mercuriale), we show that it is not trivial for some non amenable groups. It enables us to prove that measure preserving free actions of free groups of different ranks on probability spaces cannot be orbit equivalent. For a simplified quite complete proof of that particular result for free groups of finite rank, see also the paper [D. Gaboriau, Mercuriale de groupes et de relations, C. R. Acad. Sci. Paris Sér. I Math. 326 no. 2 (1998) 219-222.]. Other groups are also considered : Free products and HNN-extensions with amalgamation over amenable subgroups, direct products, some lattices in semi-simple Lie groups.

"Most" of the discrete groups have cost 1. We show that non-amenable cost 1 groups do not admit treeable actions.

Various links are established with previous notions in the classification of II_1 ergodic equivalence relations.

Later on, various unexplained numerical coincidences have been noticed between the cost,the first l2 Betti number and the free entropy dimension (of D. Voiculescu).