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 boundary action $\Gamma$ ↷ $B$ we associate a certain generalized Weyl group $W_{{\Gamma}{B}}$ and show that any representation with a Zariski dense unbounded image in a simple algebraic group, $\rho:\Gamma\to \bf{H}$, defines a special homomorphism $W_{{\Gamma}{B}}\to Weyl_{\bf H}$. This general fact allows the deduction of the aforementioned superrigidity results.