Rigidity Theorems for Higher Rank Lattice Actions

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.