Boundary maps and reducibility for cocycles into the isometries of CAT(0)-spaces

Abstract: 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 $\mathrm{PU}(1,\infty)$.