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## 2. Group von Neumann Algebras

1. ### Property (T)

#### Problem 2.1.

[S. Vaes] Provide an example of an infinite icc property (T) group that is W$^*$ super-rigid.
• #### Problem 2.2.

[C. Houdayer] Take $BS(m,n)=\langle a,t : ta^nt^{-1}=a^n \rangle$ with $n>|m|>2$ and define $$G(m,n)=BS(m,n)//\langle a \rangle.$$ $L(G(m,n))$ is known to be a type III$_{|m|/n}$.
1. Does the isomorphism class depend only on $|m|/n$?
2. Is $L(G(m,n))$ isomorphic to the free Araki-Woods factor?
• #### Problem 2.3.

[J. Peterson] Let $\Gamma$ be an icc property (T) group. Show that $\operatorname{Out}(L(\Gamma))$ is finite.
1. Remark. I believe the question was with the extra assumption that the group is hyperbolic. The motivation is the result of F. Paulin from 1997, which implies that hyperbolic property (T) groups have finite outer automorphism group.
• #### Problem 2.4.

[S. Popa] Describe the groups $\Gamma$ which $W^*$ embed into free groups, i.e. $L(\Gamma)\hookrightarrow L(\mathbb{F}_n)$.
• ### The Class of $\mathcal{V}\mathcal{C}$

#### Problem 2.5.

[S. Popa] Characterize the class of groups $\Gamma$ such that for all II$_1$ factors $N$ and all co-cycle actions $\Gamma \curvearrowright N$ can be perturbed to a genuine action.
1. Remark. [S. Popa] The class $\mathcal{V}\mathcal{C}$ contains all amenable groups and some AFP groups.
• #### Problem 2.6.

[J. Peterson] Let $\Gamma$ be a non-amenable icc group with $\beta_1^{(2)}(\Gamma)>0$. Then $L(\Gamma)$ cannot be generated by two property (T) subfactors with diffuse intersection.

Cite this as: AimPL: Classification of group von Neumann algebras, available at http://aimpl.org/groupvonneumann.