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Coût des relations d'équivalence et des groupes

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D. Gaboriau

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Article publié :

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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 l^{2 }Betti number and the free entropy
dimension (of D. Voiculescu).