## 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.