A remark on -valued cohomology groups of algebraic group actions

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• 作者：Yongle Jiang ^{yongleji@buffalo.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 22D40 ; 46L10 ; secondary ; 28D05 ; 20J06
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：1 August 2016
• 年：2016
• 卷：271
• 期：3
• 页码：577-592
• 全文大小：383 K

文摘

We prove that for a weakly mixing algebraic action σ:G↷(X,ν)$\sigma :G\curvearrowright (X,\nu )$, the n  th cohomology group Hⁿ(G↷X;T)$H^n(G\curvearrowright X;\mathbb{T})$, after quotienting out the natural subgroup Hⁿ(G,T)$H^n(G,\mathbb{T})$, contains $H^n(G,\hat{X})$ as a natural subgroup for n=1$n=1$. If we further assume the diagonal actions σ²${\sigma }^2$, σ⁴${\sigma }^4$ are T$\mathbb{T}$-cocycle superrigid and $H^2(G,\hat{X})$ is torsion free as an abelian group, then the above also holds true for n=2$n=2$. Applying it for principal algebraic actions when n=1$n=1$, we show that H²(G,ZG)$H^2(G,\mathbb{Z}G)$ 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$n=2$) to answer, negatively, a question of Sorin Popa on the 2nd cohomology group of Bernoulli shift actions of property (T) groups.