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 $n\ne m$.

arXiv commentCommenta are welcome! 25 pages

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• The bounded Borel class and complex representations of 3-manifold groups
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• Maximal representations of complex hyperbolic lattices in SU(m,n)
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Cited by in corpus (4)

• Integrable tautness of isometries of complex hyperbolic spaces
• Multiplicative constants and maximal measurable cocycles in bounded cohomology
• Parametrized Euler class and semicohomology theory
• Boundaries and equivariant maps for ergodic groupoids