Equivariant maps for measurable cocycles with values into higher rank Lie groups

Let $G$ a semisimple Lie group of non-compact type and let ${\mathcal{X}}_G$ be the Riemannian symmetric space associated to it. Suppose ${\mathcal{X}}_G$ has dimension $n$ and it has no factor isometric to either ${\mathbb{H}}^2$ or $\text{SL}(3,\mathbb{R})\mathrm{/}\text{SO}(3)$. Given a closed $n$-dimensional Riemannian manifold $N$, let $\mathrm{\Gamma }={\pi }_1(N)$ be its fundamental group and $Y$ its universal cover. Consider a representation $\rho :\mathrm{\Gamma }\to G$ with a measurable $\rho$-equivariant map $\psi :Y\to {\mathcal{X}}_G$. Connell-Farb described a way to construct a map $F:Y\to {\mathcal{X}}_G$ which is smooth, $\rho$-equivariant and with uniformly bounded Jacobian. In this paper we extend the construction of Connell-Farb to the context of measurable cocycles. More precisely, if $(\mathrm{\Omega },{\mu }_{\mathrm{\Omega }})$ is a standard Borel probability $\mathrm{\Gamma }$-space, let $\sigma:\Gamma \times \Omega \rightarrow G$beameasurablecocycle\mathrm{.}Weconstructameasurablemap$F: Y \times \Omega \rightarrow $\mathcal{X}$_G$whichis$\sigma$-equivariant, whose slices are smooth and they have uniformly bounded Jacobian. For such equivariant maps we define also the notion of volume and we prove a sort of mapping degree theorem in this particular context.