We will talk about rigidity theorems for smooth actions of a higher rank lattice $\Gamma$ on compact manifolds following the philosophy of the Zimmer program. Let $G$ be a semisimple Lie group. Assume that all simple factors of $G$ have a higher rank. If $\Gamma$ is a lattice of such a $G$, then many rigidity phenomena are known due to the presence of "higher rank" and "property (T)". In this case, Zimmer's cocycle superrigidity theorem plays an important role. If higher rank lattice $\Gamma$ does not have the property (T), we can not expect Zimmer's cocycle superrigidity theorem to hold.
In this talk, we will consider higher rank lattices without property (T) such as
$\textrm{SL}_{2}(\mathbb{Z}[\sqrt{17}])$. We will see local and global rigidity theorems for such a lattice action. The main ingredient will be a dynamical superrigidity theorem that is an analog of Zimmer's cocycle superrigidity. Speaker(s): Homin Lee (Indiana University)