Invariants L 2 de relations d'équivalence et de groupes

D. Gaboriau

Article publié: Publ. math., Inst. Hautes Étud. Sci.,  95 no. 1  (2002), 93-150.
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We introduce the notion of L2-Betti numbers for probability measure preserving equivalence relations and prove their main properties.
We deduce that ℓ2-Betti numbers of groups are invariant of Orbit Equivalence (OE) for free actions and of Measure Equivalence modulo a multiplicative constant.
We derive a lot of OE rigidity results, as well as some results about the ℓ2-Betti numbers of various types of groups.
We obtain a proportionality principle: If Γ and Λ are lattices in a locally compact second countable group G, then for every j≥0, their j-th ℓ2-Betti numbers normalized by their covolume are equal:
βj(2)(Γ) / Haar(G/Γ)=βj(2)(Λ)/ Haar(G/Λ).
This quantity can be taken as a definition of the j-th ℓ2-Betti numbers of G.