Boundaries , Weyl Groups , and Superrigidity

This note describes a unified approach to several superrigidity results, old and new, concerning representations of lattices into simple algebraic groups over local fields. For an arbitrary group Γ and a boundary action Γ y B we associate certain generalized Weyl group WΓ,B and show that any representation with a Zariski dense unbounded image in a simple algebraic group, ρ : Γ → H, defines a special homomorphism WΓ,B → WeylH. This general fact allows to deduce the aforementioned superrigidity results. Introduction. This note describes some aspects of a unified approach to a family of ”higher rank superrigidity” results, based on a notion of a generalized Weyl group. While this approach applies equally well to representations of lattices (as in the original work of Margulis [10]), and to measurable cocycles (as in the later work of Zimmer [18]), in this note we shall focus on representations only. Yet, it should be emphasized that our techniques do not involve any cocompactness, or integrability assumptions on lattices, and their generalizations to general measurable cocycles are rather straightforward. Hereafter we consider representations into simple algebraic groups; some other possible target groups are discussed in [1], [2]. Let k be a local field, and H denote the locally compact group of k-points of some connected adjoint k-simple k-algebraic group. Consider representations ρ : Γ→ H with Zariski dense and unbounded image, where Γ is some discrete countable group. We shall outline a unified argument showing that for the following groups G all lattices Γ in G have the property that such a representation ρ : Γ → H can occur only as a restriction of a continuous homomorphism ρ̄ : G→ H; this includes: (a) G = G is the group of `-points of a connected `-simple `-algebraic group of rk`(G) ≥ 2 where ` is a local field (Margulis [10], [12, §VII]), (b) G = G1×G2 for G1, G2 general locally compact groups, where Γ is assumed to be irreducible (cf. [13], [8], [6]), (c) G = Aut(X) where X is an Ã2-building and G has finitely many orbits for its action on the space of chambers of X, Ch(X). New implications of these results include non-linearity of the exotic Ã2-groups (deduced from (c)), and arithmeticity vs. non-linearity dichotomy for irreducible lattices in products of topologically simple groups, as in [15], but with integrability assumptions removed. Cocycle versions of the above results cover more new ground. U.B. and A.F. were supported in part by the BSF grant 2008267. U.B was supported in part by the ISF grant 704/08. A.F. was supported in part by the NSF grants DMS 0905977.