ar X iv : m at h / 06 05 27 6 v 4 [ m at h . D S ] 1 6 Ju n 20 06 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 lattice Γ 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 → PSL2(R). 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 = G1×· · ·Gn of n ≥ 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 Ã2 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. Dedicated to Ali (A.F. and U.B.)