On II1 factors arising from 2-cocycles of w-rigid groups

We consider II1 factors Lμ(G) arising from 2-cocyles μ∈H2(G,T) on groups G containing infinite normal subgroups H⊂G with the relative property (T) (i.e., G w-rigid). We prove that given any separable II1 factor M, the set of 2-cocycles μ|H∈H2(H,T) with the property that Lμ(G) is embeddable into M is at most countable. We use this result, the relative property (T) of Z2⊂Z2⋊Γ for Γ⊂SL(2,Z) non-amenable and the fact that every cocycle μα∈H2(Z2,T)≃T extends to a cocycle on Z2⋊SL(2,Z), to show that the one parameter family of II1 factors Mα(Γ)=Lμα(Z2⋊Γ), α∈T, are mutually non-isomorphic, modulo countable sets, and cannot all be embedded into the same separable II1 factor. Other examples and applications are discussed.