Cat(−1)-spaces, Divergence Groups and Their Commensurators

A CAT(−1)-space is a metric geodesic space in which every geodesic triangle is thinner than its associated comparison triangle in the hyperbolic plane ([B], [BriHa], [Gr]). The CAT(−1)-property is one among many possible generalizations to singular spaces of the notion of negative curvature. Important examples of CAT(−1)-spaces include Riemannian manifolds of sectional curvature k ≤ −1 and their convex subsets ([B-G-S]), metric trees and piecewise hyperbolic cell complexes ([Mou],[Da],[Hag],[Be 1],[Be 2],[B-Br]). In this paper we establish certain superrigidity results for isometric actions of a group Λ on a CAT(−1)-space in the following two settings: A. The group Λ is a subgroup of a locally compact group G with Γ < Λ < ComGΓ, where Γ < G is a sufficiently large discrete subgroup and ComGΓ = {g ∈ G : g−1Γg and Γ share a subgroup of finite index} is the commensurator of Γ in G. B. The group Λ is an irreducible lattice in G := ∏n α=1Gα(kα), where each Gα is a semisimple algebraic group defined over a local field kα. The issues addressed in this paper are motivated on one hand by earlier work of G.A. Margulis ([Ma]) dealing with the linear representation theory of Λ, where in case A, G is a semisimple group and Γ < G a lattice, and on the other hand by the results of Lubotzky, Mozes and Zimmer ([L-M-Z]) concerning isometric actions of Λ on trees, where Γ < Λ < ComGΓ, G is the group of automorphisms of a regular tree and Γ < G is a lattice. Our approach to establishing superrigidity results is based on ergodic theoretic methods developed by Margulis ([Ma],[Zi 3],[A’C-B]). In this context, the following notion of boundary of a locally compact group Γ will be useful: let B be a standard Borel space on which Γ acts by Borel automorphisms preserving a σ-finite measure class μ.