The ultrapower $\mathcal R^ω$ is not a group factor

Abstract

Let $(\mathcal M_n)_{n\geqslant 1}$ be a sequence of type $\mathrm{II}_1$ factors with property $Γ$ and let $ω$ be a free ultrafilter on $\mathbb N$. We prove that every subfactor of the ultraproduct $\prod_ω\mathcal M_n$ of finite Jones index is not isomorphic to the group von Neumann algebra of any discrete group. In particular, the ultrapower $\mathcal R^ω$ of the hyperfinite type $\mathrm{II}_1$ factor $\mathcal R$ is not a group factor.