Notes of Stefan Vaes’s Cambridge lecture 09-05-2017

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 {L(\Gamma)} determine cross-products associated to ergodic actions of {\Gamma}.

2. Von Neumann algebras

Associated with the left regular representation of a group {\Gamma} is {L(\Gamma)}, the weak closure of the group algebra viewed as operatoirs on {\ll^2(\Gamma)}.

Connes: all amenable ICC groups yield isomorphic algebras {L(\Gamma)}.

However, it has been conjectured that {{\mathbb C}[\Gamma]} might determine {\Gamma} (at least in the torsion free case). Thus taking the closure changes many things.

2.1. Ozawa’s solidity theorem

A {II_1}-factor is a vN algebra with a finite trace. {L(\Gamma)}‘s are {II_1} factors.

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

Ozawa (2003): for hyperbolic groups, {L(\Gamma)} is solid, i.e. whenever a subalgebra {A\subset M} is not locally trivial, then {A'\cap M} is amenable.

Corollary: such {L(\Gamma)} are prime, i.e. cannot be written as nontrivial tensor products.

2.2. Ozawa’s class {S}

Ozawa’s theorem has a wider scope.

Say {\Gamma} belongs to class {S} if the left-right action of {\Gamma\times\Gamma} on {\Gamma} can be compactified to an action by homeomorphisms on a compact space {K}, with the right action trivial and the left action amenable.

Here amenability of an action means that there exists a sequence of maps {\xi_n:K\rightarrow Prob(\Gamma)} which are more and more equivariant.

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

Ozawa 2011: {\Gamma} belongs to class {S} iff {\Gamma} is exact and there is a proper map {c:\Gamma\rightarrow\ell^2(\Gamma)} such that…

Quasi-cocycles (coboundary in {\ell^2} 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 {L(\Gamma)}, {\Gamma\in S}. If two products of groups in class {S} 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 {S}.

4. Cross-products

{M=L^\infty(X)\times \Gamma} is associated to a probability measure preserving action of {\Gamma} on {X}.

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

4.1. Cartan subalgebras

Assume actions are ergodic and essentially free. Then {A=L^\infty(X)\subset A\times\Gamma} is a Cartan subalgebra, meaning that {A} is maximal abelian, and its normalizer generates.

I. Singer: the inclusion {A\subset A\times\Gamma} 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 {II_1} factor.

Voiculescu 1995 (using free probability theory) proved that {L(F_n)} has trivial Cartan subalgebras.

Ozawa 2008 showed that the semi-direct product {{\mathbb Z}^2\times Sl(2,{\mathbb Z})} belongs to class {S}. So class {S} is not appropriate.

Ozawa-Popa 2007 : Assume that {\Gamma} is nonamenable, admits a proper cocycle {\Gamma\rightarrow \ell^2(\Gamma)} (Haagerup property) and has the complete metric approximation property. Then {L(\Gamma)} 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 {Sp(n,1)}. 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 {\Gamma} to cross-product for profinite actions.

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

Chifan-Sinclair: class {S}, weakly amenable, nonamenable implies {L(\Gamma)} 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 {L(\Gamma)}.

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 {Sl(n,{\mathbb Z})} ?

5. Locally compact groups

Definitions (weak amenability, class {S}) extend.

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

Beware that {L^\infty(X)} is not a Cartan subalgebra in {L^\infty(X)\times G}. 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 {S}, one can recover everything: the factors and the action up to conjugacy.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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