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 $\mathrm{\Gamma }\le \text{G}(n)$ be a uniform lattice. Denote by ${\mathbb{H}}_K^n$ 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 $d(n-1)\ge 2$. In this paper we generalize natural maps to measurable cocycles. Given a standard Borel probability $\mathrm{\Gamma }$-space $(X,{\mu }_X)$, we assume that a measurable cocycle $\sigma :\mathrm{\Gamma }\times X\to \text{G}(m)$ admits an essentially unique boundary map $\phi:\partial_\infty $\mathbb{H}$^n_K \times X \rightarrow \partial_\infty $\mathbb{H}$^m_K$whoseslices$\phi_x:$\mathbb{H}$^n_K \rightarrow $\mathbb{H}$^m_K$areatomlessforalmostevery$x \in X$. Then, there exists a $\sigma$-equivariant measurable map $F: $\mathbb{H}$^n_K \times X \rightarrow $\mathbb{H}$^m_K$whoseslices$F_x:$\mathbb{H}$^n_K \rightarrow $\mathbb{H}$^m_K$aredifferentiableforalmostevery$x \in X$andsuchthat$$\text{Jac}$_a F_x \leq 1$forevery$a \in $\mathbb{H}$^n_K$andalmostevery$x \in X$. The previous properties allow us to define the natural volume $\text{NV}(\sigma )$ of the cocycle $\sigma$. This number satisfies the inequality $$\text{NV}$(\sigma) \leq $\text{Vol}$(\Gamma \backslash $\mathbb{H}$^n_K)$\mathrm{.}Additionally,theequalityholdsifandonlyif$\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 $f:M\to N$ between compact hyperbolic manifolds, we also obtain an adaptation of the mapping degree theorem to this context.