Superrigidity for homomorphisms into isometry groups of CAT(?1) spaces

Abstract

CAT(−1) spaces are generalizations of manifolds with negative curvature. In this paper, we prove three types of rigidity results related to CAT(−1) spaces, namely the rigidity of the isometric actions on CAT(−1) spaces under the commensurability subgroups, the higher rank lattices and certain ergodic cocycles. The main idea for our approach relies on a study of the boundary theory we established for the general CAT(−1) spaces.