Let Γ be a weakly irreducible lattice in a higher rank semisimple algebraic Lie group G without property (T) such as \(\mathrm {SL}_{2}({\mathbb{Z}}[\sqrt{2}])\) in \(\mathrm {SL}_{2}({\mathbb{R}})\times \mathrm {SL}_{2}({\mathbb{R}})\).

In this paper, for such Γ, we prove a cocycle superrigidity, Dynamical cocycle superrigidity, for Γ-actions under L² integrability and irreducibility assumptions. It gives a similar result to Zimmer’s cocycle superrigidity, so we can apply it instead of Zimmer’s cocycle superrigidity in many situations. For instance, in this paper, we obtain global rigidity of Anosov Γ-actions on nilmanifolds under the irreducibility assumption on a fully supported invariant measure.

令 Γ 为无属性 (T) 的高阶半单代数李群 G 中的弱不可约格，例如 \(\mathrm {SL}_{2}({\mathbb{Z}}[\sqrt{2}]) \) 中 \(\mathrm {SL}_{2}({\mathbb{R}})\times \mathrm {SL}_{2}({\mathbb{R}})\)。

在本文中，对于这样的 Г，我们证明了 L ² 可积性和不可约性假设下的 Г 作用的余循环超刚性，动态余循环超刚性。它给出了与 Zimmer 的余循环超刚度类似的结果，因此我们可以在许多情况下应用它来代替 Zimmer 的余循环超刚度。例如，在本文中，我们在完全支持的不变测度的不可约性假设下获得了尼尔流形上 Anosov Γ 作用的全局刚性。