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

Let Γ be a discrete countable group acting isometrically on a measurable field X of CAT(0)-spaces of finite telescopic dimension over some ergodic standard Borel probability Γ-space (Ω, μ). If X does not admit any invariant Euclidean subfield, we prove that the measurable field X(Equation presented) extended to a Γ-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 Lécureux. When Γ < PU(n, 1) is a torsion-free lattice and the CAT(0)-space is X(p, ∞), we show that a maximal cocycle σ: Γ × Ω → PU(p, ∞) with a suitable boundary map is finitely reducible. As a consequence, we prove an infinite-dimensional rigidity phenomenon for maximal cocycles in PU(1, ∞).