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 $\mathrm{CAT}(0)$-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 $\hat{\mathbf{X}}$ 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 Lécureux. When $\Gamma <\mathrm{PU}(n,1)$ is a torsion-free lattice and the $\mathrm{CAT}(0)$-space is $\mathcal{X}(p,\infty )$, we show that a maximal cocycle $\sigma :\Gamma \times \Omega \to \mathrm{PU}(p,\infty )$ with a suitable boundary map is finitely reducible. As a consequence, we prove an infinite-dimensional rigidity phenomenon for maximal cocycles in $\mathrm{PU}(1,\infty )$.