Let $$\Gamma $$ Γ be a torsion-free lattice of $$PU (p,1)$$ P U ( p , 1 ) with $$p \ge 2$$ p ≥ 2 and let $$(X,\mu _X)$$ (… Click to show full abstract
Let $$\Gamma $$Γ be a torsion-free lattice of $$PU (p,1)$$PU(p,1) with $$p \ge 2$$p≥2 and let $$(X,\mu _X)$$(X,μX) 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⟶SU(m,n) is cohomologous to a cocycle associated to a representation of $$PU (p,1)$$PU(p,1) into $$SU (m,n)$$SU(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.
