English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

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 furtherassume the diagonal actions $\sigma^2, \sigma^4$ are $\mathbb{T}$-cocyclesuperrigid and $H^2(G, \widehat{X})$ is torsion free as an abelian group, thenthe above also holds true for $n=2$. Applying it for principal algebraicactions when $n=1$, we show that $H^2(G,\mathbb{Z}G)$ is torsion free as anabelian group when $G$ has property (T) as a direct corollary of Sorin Popa'scocycle superrigidity theorem; we also use it (when $n=2$) to answer,negatively, a question of Sorin Popa on the 2nd cohomology group of Bernoullishift actions of property (T) groups.

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