On II$_1$ factors arising from 2-cocycles of w-rigid groups

Abstract

We consider $\text{\rm II}_1$ factors $L_\mu(G)$ arising from 2-cocyles $\mu \in \text{\rm H}^2(G,\Bbb T)$ on groups $G$ containing infinite normal subgroups $H \subset G$ with the relative property $\text{\rm(T)}$ (i.e. $G$ {\it w-rigid}). We prove that given any separable $\text{\rm II}_1$ factor $M$, the set of 2-cocycles $\mu_{|H}\in \text{\rm H}^2(H,\Bbb T)$ with the property that $L_\mu(G)$ is embeddable into $M$ is at most countable. We use this result, the relative property (T) of $\Bbb Z^2 \subset \Bbb Z^2 \rtimes \Gamma$ for $\Gamma \subset SL(2,\Bbb Z)$ non-amenable and the fact that every cocycle $\mu_\alpha \in {\text{\rm H}}^2(\Bbb Z^2,\Bbb T)\simeq \Bbb T$ extends to a cocycle on $\Bbb Z^2 \rtimes SL(2,\Bbb Z)$, to show that the one parameter family of II$_1$ factors $M_\alpha(\Gamma)=L_{\mu_{\alpha}}(\Bbb Z^2 \rtimes \Gamma)$, $\alpha \in \Bbb T$, are mutually non-isomorphic, modulo countable sets, and cannot all be embedded into the same separable II$_1$ factor. Other examples and applications are discussed.