Superrigidity, Weyl groups, and actions on the Circle

Abstract

We propose a new approach to superrigidity phenomena and implement it for lattice representations and measurable cocycles with homeomorphisms of the circle as the target group. We are motivated by Ghys' theorem stating that any representation $\ro:\Gamma\to\Homeo_+(S^1)$ of an irreducible lattice $\Gamma$ in a semi-simple real Lie group $G$ of higher rank, either has a finite orbit or, up to a semi-conjugacy, extends to $G$ which acts through an epimorphism $G\to\PSL_{2}(\RR)$. Our approach, based on the study of abstract boundary theory and, specifically, on the notion of a generalized Weyl group, allows: (A) to prove a similar superrigidity result for irreducible lattices in products $G=G_{1}\times... G_{n}$ of $n\ge 2$ general locally compact groups, (B) to give a new (shorter) proof of Ghys' theorem, (C) to establish a commensurator superrigidity for general locally compact groups, (D) to prove first superrigidity theorems for $\tilde{A}_{2}$ groups. This approach generalizes to the setting of measurable circle bundles; in this context we prove cocycle versions of (A), (B) and (D). This is the first part of a broader project of studying superrigidity via generalized Weyl groups.