LAUSR

Let $$\Gamma $$ Γ be a torsion-free lattice of $$PU (p,1)$$ P U ( p , 1 ) with $$p \ge 2$$ p ≥ 2 and let $$(X,\mu _X)$$ ( X , μ X ) be an ergodic standard Borel probability $$\Gamma $$ Γ -space. We prove that any maximal Zariski dense measurable cocycle $$\sigma : \Gamma \times X \longrightarrow SU (m,n)$$ σ : Γ × X ⟶ S U ( m , n ) is cohomologous to a cocycle associated to a representation of $$PU (p,1)$$ P U ( p , 1 ) into $$SU (m,n)$$ S U ( m , n ) , with $$1 \le m \le n$$ 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 $$1< m < n$$ 1 < m < n .