Boundaries, Weyl groups, and Superrigidity

TL;DR

This paper presents a unified approach to superrigidity results for lattice representations into simple algebraic groups, using generalized Weyl groups associated with boundary actions to establish rigidity phenomena.

Contribution

It introduces a general framework linking boundary actions, Weyl groups, and superrigidity, extending previous results to a broader class of groups and representations.

Findings

Generalized Weyl groups associated with boundary actions.

Homomorphisms from boundary Weyl groups to algebraic Weyl groups.

Unified derivation of superrigidity results.

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.