Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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F. Sarti - A. Savini

Superrigidity of maximal measurable cocycles of complex hyperbolic lattices

created by sarti on 29 Jun 2021
modified on 24 May 2023


Published Paper

Inserted: 29 jun 2021
Last Updated: 24 may 2023

Journal: Math. Z.
Pages: 23
Year: 2022

ArXiv: 2002.03628 PDF


Let $\Gamma$ be a torsion-free lattice of $\text{PU}(p,1)$ with $p \geq 2$ and let $(X,\mu_X)$ be an ergodic standard Borel probability $\Gamma$-space. We prove that any maximal Zariski dense measurable cocycle $\sigma: \Gamma \times X \longrightarrow \text{SU}(m,n)$ is cohomologous to a cocycle associated to a representation of $\text{PU}(p,1)$ into $\text{SU}(m,n)$, with $1 < m \leq n$. 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 $n\neq m$.

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