Mikaël Pichot

Turbulence, representations, and trace-preserving actions

We establish criteria for turbulence in certain spaces of C -algebra repre- sentations and apply this to the problem of nonclassiability by countable structures for group actions on a standard atomless probability space (X; ) and on the hypernite II1 factor R. We also prove that the conjugacy action on the space of free actions of a countably innite amenable group on R is turbulent, and that the conjugacy action on the space of ergodic measure-preserving ows on ( X; ) is generically turbulent.

Closed manifolds with transcendental L2-Betti numbers

In this paper, we show how to construct examples of closed manifolds with explicitly computed irrational, even transcendental L 2 Betti numbers, defined via the universal covering. We show that every non-negative real number shows up as an L 2 Betti number of some covering of a compact manifold, and that many computable real numbers appear as an L 2 -Betti number of a universal covering of a compact manifold (with a precise meaning of computable given below). In algebraic terms, for many given computable real numbers (in particular for many transcendental numbers) we show how to construct a finitely presented group and an element in the integral group ring such that the L 2 -dimension of the kernel is the given number.

Sofic dimension for discrete measured groupoids

For discrete measured groupoids preserving a probability measure we introduce a notion of sofic dimension that measures the asymptotic growth of the number of sofic approximations on larger and larger finite sets. In the case of groups we give a formula for free products with amalgamation over an amenable subgroup. We also prove a free product formula for measure-preserving actions.

ASYMPTOTIC ABELIANNESS, WEAK MIXING, AND PROPERTY T

Abstract Let G be a second countable locally compact group and H a closed subgroup. We characterize the lack of Kazhdans property T for the pair (G, H) by the genericity of G-actions on the hyperfinite II1 factor with a certain asymptotic Abelianness property relative to H, as well as by the genericity of measure-preserving G-actions on a nonatomic standard probability space that are weakly mixing for H. The latter furnishes a definitive generalization of a classical theorem of Halmos for single automorphisms and strengthens a recent result of Glasner, Thouvenot, and Weiss on generic ergodicity. We also establish a weak mixing version of Glasner and Weisss characterization of property T for discrete G in terms of the invariant state space of a Bernoulli shift and show that on the CAR algebra a type of norm asymptotic Abelianness is generic for G-actions when G is discrete and admits a nontorsion Abelian quotient.

Uniform non-amenability, cost, and the first ℓ2-Betti number

It is shown that 2β1() � h() for any countable group , where β1() is the first l 2 -Betti number and h() the uniform isoperimetric constant. In particular, a countable group with non-vanishing first l 2 -Betti number is uniformly non-amenable. We then define isoperimetric constants in the framework of measured equivalence re- lations. For an ergodic measured equivalence relation R of type II1, the uniform isoperi- metric constant h(R) of R is invariant under orbit equivalence and satisfies

Le coût est un invariant isopérimétrique

For a type II_1 ergodic measured equivalence relation R on a probability space without atom, we prove that h(R)=2C(R)-2, where C(R) is the cost, and h(R) the isoperimetric constant. This follows recent result by Lyons and the authors.

Conditions simpliciales de rigidité pour les relations de type II1

We give a geometric criterion for a probability measure preserving equivalence relation with countable classes to have Kazhdans property (T). This generalizes a similar theorem in geometric group theory. To cite this article: M. Pichot, C. R. Acad. Sci. Paris, Ser. I 337 (2003).