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.

Bibliographical: preprint (38 pages).

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[Papers] [Alex Furman]