Superrigidity of maximal measurable cocycles of complex hyperbolic lattices

Let Γ be a torsion-free lattice of PU(p, 1) with p≥ 2 and let (X, μX) be an ergodic standard Borel probability Γ -space. We prove that any maximal Zariski dense measurable cocycle σ: Γ × X⟶ SU(m, n) is cohomologous to a cocycle associated to a representation of PU(p, 1) into SU(m, n) , with 1 ≤ m≤ n. The proof follows the line of Zimmer’ Superrigidity Theorem and requires the existence of a boundary map, that we prove in a much more general setting. As a consequence of our result, there cannot exist maximal measurable cocycles with the above properties when 1 < m< n.