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 $text{SL}(3,mathbb{R})/text{SO}(3)$. Given a closed $n$-dimensional Riemannian manifold $N$, let $Gamma=pi_1(N)$ be its fundamental group and $Y$ its universal cover. Consider a representation $rho:Gamma rightarrow G$ with a measurable $rho$-equivariant map $psi:Y rightarrow mathcal{X}_G$. Connell-Farb described a way to construct a map $F:Yrightarrow mathcal{X}_G$ 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 $(Omega,mu_Omega)$ is a standard Borel probability $Gamma$-space, let $sigma:Gamma times Omega rightarrow G$ be a measurable cocycle. We construct a measurable map $F: Y times Omega rightarrow mathcal{X}_G$ which is $sigma$-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.