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

TL;DR

This paper investigates the structure of $T$-valued cohomology groups for algebraic group actions, revealing new subgroup relations and torsion properties, especially under superrigidity assumptions and property (T).

Contribution

It establishes new subgroup inclusions in cohomology groups for algebraic actions and applies superrigidity to determine torsion properties and answer open questions.

Findings

$H^n(G,T)$ contains $H^n(G,Xhat)$ as a subgroup for $n=1$.

Under superrigidity and torsion-free conditions, the results extend to $n=2$.

$H^2(G,ZG)$ is torsion free for groups with property (T).

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.