Let $\Gamma$ be a discrete countable group acting isometrically on a measurable field $\mathbf{ X}$ of CAT(0)-spaces of finite telescopic dimension over some ergodic standard Borel probability $\Gamma$-space $(\Omega,\mu)$. If $\mathbf{ X}$ does not admit any invariant Euclidean subfield, we prove that the measurable field $\widehat{ \mathbf{ X}}$ extended to a $\Gamma$-boundary admits an invariant section. In the case of constant fields this shows the existence of Furstenberg maps for measurable cocycles, extending results by Bader, Duchesne and L\'ecureux. When $\Gamma<\mathrm{ PU}(n,1)$ is a torsion-free lattice and the CAT(0)-space is $\mathcal{ X}(p,\infty)$, we show that a maximal cocycle $\sigma:\Gamma \times \Omega \rightarrow \mathrm{ PU}(p,\infty)$ with a suitable boundary map is finitely reducible. As a consequence, we prove an infinite dimensional rigidity phenomenon for maximal cocycles in $\mathrm{ PU}(1,\infty)$.
Categories: math.GT