[Papers] [Alex Furman]

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^1) as the target group. We are motivated by Ghys' theorem stating that any representation of an irreducible lattice 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->PSL(2,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 of n>1 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.

Authors: U. Bader, A. Furman, Ali Shaker.
Bibliographical: preprint (38 pages).