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

We consider ${\text{II}}_1$ factors $L_{\mu }(G)$ arising from 2-cocyles $\mu \in $\text{H}$^2(G,\Bbb T)$ongroups$G$ containing infinite normal subgroups $H\subset G$ with the relative property $\text{(T)}$ (i.e. $G$ {\it w-rigid}). We prove that given any separable ${\text{II}}_1$ factor $M$, the set of 2-cocycles ${\mu }_{\mathrm{\mid }H}\in {\text{H}}^2(H,\mathbb{T})$ with the property that $L_{\mu }(G)$ is embeddable into $M$ is at most countable. We use this result, the relative property (T) of ${\mathbb{Z}}^2\subset {\mathbb{Z}}^2⋊\mathrm{\Gamma }$ for $\Gamma \subset SL(2,\Bbb Z)$non-amenableandthefactthateverycocycle$\mu_\alpha \in {$\text{H}$}^2(\Bbb Z^2,\Bbb T)\simeq \Bbb T$extendstoacocycleon$\Bbb Z^2 \rtimes SL(2,\Bbb Z)$, to show that the one parameter family of II${}_1$ factors $M_{\alpha }(\mathrm{\Gamma })=L_{{\mu }_{\alpha }}({\mathbb{Z}}^2⋊\mathrm{\Gamma })$, $\alpha \in \mathbb{T}$, 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.