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

Abstract: 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 $F:Y\rightarrow \mathcal{X}_G$ 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.