# 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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Mathématiques de l'IHÉS

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We introduce the notion of L^{2}-Betti
numbers for probability measure preserving equivalence relations and
prove their main properties.

We deduce that
ℓ^{2}-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 ℓ^{2}-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 ℓ^{2}-Betti
numbers normalized by their covolume are equal:

β^{j}^{(2)}(Γ) / Haar(G/Γ)=β^{j}^{(2)}(Λ)/ Haar(G/Λ).

This quantity can be taken as a definition of the j-th ℓ^{2}-Betti numbers of G.