Boundaries, Weyl groups, and Superrigidity

Abstract

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.