Reduction of cocycles with hyperbolic targets

We show that any cocycle from an ergodic, finite measure preserving action of a higher-rank group to a closed subgroup of the isometry group of a proper, geodesic hyperbolic, 'at most exponential' metric space is necessarily cohomologous to a cocycle with values in a compact subgroup. Philosophically, this says that higher-rank dynamics is incompatible with hyperbolic dynamics; since hyperbolicity is, in some sense, generic among finitely presented groups, this places strong restrictions on the possible dynamics of higher-rank groups. We derive some applications of the main result. When the target group of the cocycle has no small subgroups, we show that the main result holds for a wider class of domain groups.