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

Let G be \(SO ^\circ (n,1)\) $S{O}^{\circ }(n,1)$ for \(n \geqslant 3\) $n⩾3$ and consider a lattice \(\Gamma < G\) $\mathrm{\Gamma }<G$ . Given a standard Borel probability \(\Gamma \) $\mathrm{\Gamma }$ -space \((\Omega ,\mu )\) $(\mathrm{\Omega },\mu )$ , consider a measurable cocycle \(\sigma :\Gamma \hspace{1.111pt}{\times }\hspace{1.111pt}\Omega \rightarrow {\textbf{H}}(\kappa )\) $\sigma :\mathrm{\Gamma }\times \mathrm{\Omega }\to \mathbf{H}(\kappa )$ , where \({\textbf{H}}\) $\mathbf{H}$ is a connected algebraic \(\kappa \) $\kappa$ -group over a local field \(\kappa \) $\kappa$ . Under the assumption of compatibility between G and the pair \(({\textbf{H}},\kappa )\) $(\mathbf{H},\kappa )$ , we show that if \(\sigma \) $\sigma$ admits an equivariant field of probability measures on a suitable projective space, then \(\sigma \) $\sigma$ is trivializable. An analogous result holds in the complex hyperbolic case.