Superrigidity, Weyl Groups, and Actions on the Circle

We propose a new approach to superrigidity phenomena and implement it for lattice representations and measurable cocycles with Homeo+(S ) as the target group. We are motivated by Ghys’ theorem stating that any representation ̺ : Γ → Homeo+(S ) of an irreducible lattices Γ in semi-simple real Lie groups G of higher rank, either has a finite orbit or, up to a semiconjugacy, extends to G which acts through an epimorphism G → PSL2(R). Our approach, based on the study of abstract boundary theory and specifically on a notion of a generalized Weyl group, allows: (A) to prove a similar superrigidity result for irreducible lattices in products G = G1 × · · ·Gn of n ≥ 2 general locally compact groups, (B) to give a new (shorter) proof of Ghys’ theorem, (C) establish a commensurator superrigidity for general locally compact groups, (D) prove first superrigidity theorems for Ã2 groups. This approach generalizes to the setting of measurable circle bundles, in which context we prove cocycle versions of (A), (B) and (D). This is a first part of a broader project of studying superrigidity via generalized Weyl group. Dedicated to Ali (A.F. and U.B.)