Asymptotique des nombres de Betti, invariants l2 et laminations

N. Bergeron et D. Gaboriau

Article publié :

Comment. Math. Helv., 79 (2004) 2, pages 362 - 395
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Let K be a finite simplicial complex. We are interested in the asymptotic behavior of the Betti numbers of a sequence of finite sheeted covers of K, when normalized by the index of the covers. W. Lück, has proved that for regular coverings, these sequences of numbers converge to the l2 Betti numbers of the associated (in general infinite) limit regular cover of K.

In this article we investigate the non regular case. We show that the sequences of normalized Betti numbers still converge. But this time the ``good'' limit object is no longer the associated limit cover of K, but a lamination by simplicial complexes. We prove that the limits of sequences of normalized Betti numbers are equal to the l2 Betti numbers of this lamination.

Even if the associated limit cover of K is contractible, its l2 Betti numbers are in general different from those of the lamination. We construct such examples. We also give a dynamical condition for these numbers to be equal. It turns out that this condition is equivalent to a former criterion due to M. Farber. We hope that our results clarify its meaning and show to which extent it is optimal.

In a second part of this paper we study non free measure-preserving ergodic actions of a countable group G on a standard Borel probability space. Extending group-theoretic similar results of the second author, we obtain relations between the l2 Betti numbers of G and those of the generic stabilizers. For example, if b1(2) (G) is different from 0, then either almost each stabilizer is finite or almost each stabilizer has an infinite first l2 Betti number.