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.,Gw-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) nonamenable 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 nonisomorphic, modulo countable sets, and cannot all be embedded into the same separable II1 factor. Other examples and applications are discussed. © 2006 Elsevier Inc. All rights reserved