An algebraic $G$-action is a homomorphism $\alpha: G\to Aut(X)$ from $G$ to the group of continuous automorphisms of a compact metrizable abelian group $X$. In this talk, I would explain what the cocycle superrigidity problem for algebraic actions is and why we are interested in it. Then we present two results.

(1) Under natural assumptions, we could show that the 1st and 2nd cohomology groups "remember" the "algebraic data" of this action.

(2) If $G$ is a finitely generated non-torsion group, then every continuous cocycle for the full shift $G\curvearrowright A^G$ into any countable group $H$ is trivial iff $G$ has one end.

The 2nd result is based on joint work with Nhan-Phu Chung.