A remark on $\mathbb{T}$-valued cohomology groups of algebraic group actions

Abstract

We prove that for a weakly mixing algebraic action $\sigma: G\curvearrowright(X,\nu)$, the $n$-cohomology group $H^n(G\curvearrowright X; \mathbb{T})$, after quotienting out the natural subgroup $H^n(G,\mathbb{T})$, contains $H^n(G,\widehat{X})$ as a natural subgroup for $n=1$. If we further assume the diagonal actions $\sigma^2, \sigma^4$ are $\mathbb{T}$-cocycle superrigid and $H^2(G, \widehat{X})$ is torsion free as an abelian group, then the above also holds true for $n=2$. Applying it for principal algebraic actions when $n=1$, we show that $H^2(G,\mathbb{Z}G)$ 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 $n=2$) to answer, negatively, a question of Sorin Popa on the 2nd cohomology group of Bernoulli shift actions of property (T) groups.