Boundaries, Weyl groups, and Superrigidity

This note describes a unified approach to several superrigidity results, old and new, concerning representations of lattices into simple algebraic groups over local fields. For an arbitrary group $\Gamma$ and a $\Gamma$-boundary $B$ we associate certain generalized Weyl group $W_{\Gamma,B}$ and show that any representation with a Zariski dense unbounded image in a simple algebraic group, $\rho:\Gamma\to \mathbf{H}$, defines a special homomorphism $W_{\Gamma,B}\to {\rm Weyl}(\mathbf{H})$. This general fact allows to deduce the aforementioned superrigidity results.Comment: 7 page

Algebraic Representations of Ergodic Actions and Super-Rigidity

We revisit Margulis-Zimmer Super-Rigidity and provide some generalizations. In particular we obtain super-rigidity results for lattices in higher-rank groups or product of groups, targeting at algebraic groups over arbitrary fields with absolute values. We also obtain cocycle super-rigidity results for a wide class of groups with respect to mixing actions. Our approach is based on a systematic study of algebraic representations of ergodic actions.Comment: 25 pages. The file is different from a previously distributed file by the same name. 2nd submission: an inaccuracy in the definition of a morphism of bi-algebraic representations was fixe