Superrigidity of maximal measurable cocycles of complex hyperbolic lattices

Abstract

Let $\Gamma$ 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 $\Gamma$-space. We prove that any maximal Zariski dense measurable cocycle $\sigma :\Gamma \times X⟶\text{SU}(m,n)$ 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 $n\ne m$.

Commenta are welcome! 25 pages

References in corpus (3)

• Boundaries, rigidity of representations, and Lyapunov exponents
• Harmonic maps and representations of non-uniform lattices of PU(m,1)
• Maximal representations of complex hyperbolic lattices in SU(m,n)

Cited by in corpus (3)

• Integrable tautness of isometries of complex hyperbolic spaces
• Parametrized Euler class and semicohomology theory
• Boundaries and equivariant maps for ergodic groupoids