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 Γ and a boundary action Γ B we associate a certain generalized Weyl group WΓ,B and show that any representation with a Zariski dense unbounded image in a simple algebraic group, ρ : Γ → H, defines a special homomorphism WΓ,B → WeylH. This general fact allows the deduction of the aforementioned superrigidity results.