** Negative curvature and rigidity for von Neumann algebras **

**1. What type of rigidity? **

Weakly isomorphic structures are shown to be strongly isomorphic.

Coarse structure determines a much finer structure.

Examples : quasi-isometry implies commensurability, homotopy equivalence implies homeomorphism.

In our context, does a group vN algebra determine cross-products associated to ergodic actions of .

**2. Von Neumann algebras **

Associated with the left regular representation of a group is , the weak closure of the group algebra viewed as operatoirs on .

Connes: all amenable ICC groups yield isomorphic algebras .

However, it has been conjectured that might determine (at least in the torsion free case). Thus taking the closure changes many things.

** 2.1. Ozawa’s solidity theorem **

A -factor is a vN algebra with a finite trace. ‘s are factors.

The following is an analogue of the fact that centralizers in hyperbolic groups are amenable.

Ozawa (2003): for hyperbolic groups, is solid, i.e. whenever a subalgebra is not locally trivial, then is amenable.

Corollary: such are prime, i.e. cannot be written as nontrivial tensor products.

** 2.2. Ozawa’s class **

Ozawa’s theorem has a wider scope.

Say belongs to class if the left-right action of on can be compactified to an action by homeomorphisms on a compact space , with the right action trivial and the left action amenable.

Here amenability of an action means that there exists a sequence of maps which are more and more equivariant.

In hyperbolic groups, these maps map boudnary points to uniform measures on linger and longer geodesic segments pointing to .

Ozawa 2011: belongs to class iff is exact and there is a proper map such that…

Quasi-cocycles (coboundary in norm is uniformly bounded) provide such maps. So this relates to Andreas Thom’s work, and Monod-Shalom’s class of groups whose reduced 1-cohomology does not vanish for some representation.

**3. Rigidity theorems **

Ozawa-Popa 2003: unique prime factorization for , . If two products of groups in class have isomorphic vN algebras, then factors have pairwise isomorphic vN algebra (up to taking powers). In particular, the number of factors is an invariant.

Chifan-de Santiago-Sinclair 2015: idem when the right-hand side group is arbitrary, and the factors in the left-hand side are hyperbolic. Unknown if hyperbolic is replaced with class .

**4. Cross-products **

is associated to a probability measure preserving action of on .

Example: Bernoulli actions, automorphisms of tori have bee thoroughly studied.

** 4.1. Cartan subalgebras **

Assume actions are ergodic and essentially free. Then is a *Cartan subalgebra*, meaning that is maximal abelian, and its normalizer generates.

I. Singer: the inclusion contains the same information as the orbit equivalence relation.

So if one can prove uniqueness of the Cartan subalgebra, then the equivalence relation can be recovered form the factor.

Voiculescu 1995 (using free probability theory) proved that has trivial Cartan subalgebras.

Ozawa 2008 showed that the semi-direct product belongs to class . So class is not appropriate.

Ozawa-Popa 2007 : Assume that is nonamenable, admits a proper cocycle (Haagerup property) and has the complete metric approximation property. Then has no Cartan subalgebra.

Here are definition. The key notion is that of a Fourier multiplier. Weak amenability means that identity is a limit of finitely supported Fourier multipliers with uniformly bounded norms. This holds for hyperbolic groups.

Complete metric approximation property means that identity is a limit of finitely supported Fourier multipliers with norm 1. This fails for lattices in . This does not follow from Haagerup property (Cornulier-Stalder-Valette). Unclear wether it implies Haagerup property (probably not).

** 4.2. Profinite actions **

These are limits of finite actions. Complete metric approximation property passes from to cross-product for profinite actions.

Existence of invariant means on (for weakly amenable groups) has a vN algebra counterpart.

Chifan-Sinclair: class , weakly amenable, nonamenable implies has no Cartan subalgebra. This implies uniqueness of Cartan subalgebra for cross-products of such groups over profinite action.

** 4.3. Arbitrary actions **

Popa-Vaes: idem for arbitrary actions.

Corollary: for free groups, rank can be read from cross products (Gaboriau helps).

Still open wether rank can be read from .

Say a group is Cartan rigid if Cartan subalgebra is unique for all actions.

Examples : above. All free products. Potentially, all groups far from having normal abelian subgroups. What about ?

**5. Locally compact groups **

Definitions (weak amenability, class ) extend.

Brothier-Deprez-Vaes 2017: for class locally compact groups, uniqueness of Cartan subalgebra holds unless…

Beware that is not a Cartan subalgebra in . Need to take a Borel section, in order to get a countable equivalence relation. Nevertheless independent on the choice of section.

Corollary: from the cross-product of a separately ergodic action of a product of groups of class , one can recover everything: the factors and the action up to conjugacy.