A remark on TT-valued cohomology groups of algebraic group actions

We prove that for a weakly mixing algebraic action σ:G↷(X,ν)σ:G↷(X,ν), the n  th cohomology group Hn(G↷X;T)Hn(G↷X;T), after quotienting out the natural subgroup Hn(G,T)Hn(G,T), contains Hn(G,Xˆ) as a natural subgroup for n=1n=1. If we further assume the diagonal actions σ2σ2, σ4σ4 are TT-cocycle superrigid and H2(G,Xˆ) is torsion free as an abelian group, then the above also holds true for n=2n=2. Applying it for principal algebraic actions when n=1n=1, we show that H2(G,ZG)H2(G,ZG) is torsion free as an abelian group when G   has property (T) as a direct corollary of Sorin Popa's cocycle superrigidity theorem; we also use it (when n=2n=2) to answer, negatively, a question of Sorin Popa on the 2nd cohomology group of Bernoulli shift actions of property (T) groups.