Natural maps for measurable cocycles of compact hyperbolic manifolds

Let $text{G}(n)$ be equal either to $text{PO}(n,1),text{PU}(n,1)$ or $text{PSp}(n,1)$ and let $Gamma leq text{G}(n)$ be a uniform lattice. Denote by $mathbb{H}^n_K$ the hyperbolic space associated to $text{G}(n)$, where $K$ is a division algebra over the reals of dimension $d=dim_{mathbb{R}} K$. Assume $d(n-1) geq 2$. In this paper we generalize natural maps to measurable cocycles. Given a standard Borel probability $Gamma$-space $(X,mu_X)$, we assume that a measurable cocycle $sigma:Gamma times X rightarrow text{G}(m)$ admits an essentially unique boundary map $phi:partial_infty mathbb{H}^n_K times X rightarrow partial_infty mathbb{H}^m_K$ whose slices $phi_x:mathbb{H}^n_K rightarrow mathbb{H}^m_K$ are atomless for almost every $x in X$. Then, there exists a $sigma$-equivariant measurable map $F: mathbb{H}^n_K times X rightarrow mathbb{H}^m_K$ whose slices $F_x:mathbb{H}^n_K rightarrow mathbb{H}^m_K$ are differentiable for almost every $x in X$ and such that $text{Jac}_a F_x leq 1$ for every $a in mathbb{H}^n_K$ and almost every $x in X$. The previous properties allow us to define the natural volume $text{NV}(sigma)$ of the cocycle $sigma$. This number satisfies the inequality $text{NV}(sigma) leq text{Vol}(Gamma backslash mathbb{H}^n_K)$. Additionally, the equality holds if and only if $sigma$ is cohomologous to the cocycle induced by the standard lattice embedding $i:Gamma rightarrow text{G}(n) leq text{G}(m)$, modulo possibly a compact subgroup of $text{G}(m)$ when $m>n$. Given a continuous map $f:M rightarrow N$ between compact hyperbolic manifolds, we also obtain an adaptation of the mapping degree theorem to this context.