Superrigidity of maximal measurable cocycles of complex hyperbolic lattices

Let $\mathrm{\Gamma }$ be a torsion-free lattice of $\text{PU}(p,1)$ with $p\ge 2$ and let $(X,{\mu }_X)$ be an ergodic standard Borel probability $\mathrm{\Gamma }$-space. We prove that any maximal Zariski dense measurable cocycle $\sigma: \Gamma \times X \longrightarrow $\text{SU}$(m,n)$ is cohomologous to a cocycle associated to a representation of $\text{PU}(p,1)$ into $\text{SU}(m,n)$, with $1<m\le n$. 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, it cannot exist a maximal measurable cocycle with the above properties when $n\mathrm{\ne }m$.