On [formula omitted] factors arising from 2-cocycles of w-rigid groups

Abstract

We consider II 1 factors L μ ( G ) arising from 2-cocyles μ ∈ H 2 ( 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 II 1 factor M, the set of 2-cocycles μ | H ∈ H 2 ( H , T ) with the property that L μ ( G ) is embeddable into M is at most countable. We use this result, the relative property (T) of Z 2 ⊂ Z 2 ⋊ Γ for Γ ⊂ SL ( 2 , Z ) non-amenable and the fact that every cocycle μ α ∈ H 2 ( Z 2 , T ) ≃ T extends to a cocycle on Z 2 ⋊ SL ( 2 , Z ) , to show that the one parameter family of II 1 factors M α ( Γ ) = L μ α ( Z 2 ⋊ Γ ) , α ∈ 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.