# Asymptotique des nombres de Betti, invariants l^{2} et laminations

## N. Bergeron et D. Gaboriau

### Article publié :

**Comment. Math. Helv.****, 79 (2004) 2, pages 362 - 395**

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## Abstract

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 l^{2}
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 l^{2} Betti numbers of this lamination.

Even if the associated limit cover of K is contractible, its l^{2}
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 l^{2} Betti numbers of G and those of the generic stabilizers. For example,
if b_{1}^{(2)} (G) is different
from 0, then either almost each stabilizer is finite or almost each stabilizer
has an infinite first l^{2} Betti number.