On the trivializability of rank-one cocycles with an invariant field of projective measures

Abstract

Let $G$ be ${\text{SO}}^{\circ }(n,1)$ for $n\ge 3$ and consider a lattice $\Gamma <G$. Given a standard Borel probability $\Gamma$-space $(\Omega ,\mu )$, consider a measurable cocycle $\sigma :\Gamma \times \Omega \to \mathbf{H}(\kappa )$, where $\mathbf{H}$ is a connected algebraic $\kappa$-group over a local field $\kappa$. Under the assumption of compatibility between $G$ and the pair $(\mathbf{H},\kappa )$, we show that if $\sigma$ admits an equivariant field of probability measures on a suitable projective space, then $\sigma$ is trivializable. An analogous result holds in the complex hyperbolic case.

arXiv comment24 pages, I improved the exposition following the referee report, to appear on Europ. J. Math