Natural maps for measurable cocycles of compact hyperbolic manifolds

Let G(m) be equal to either PO(n,1), PU(n,1) or PSp(n, 1) and let Gamma <= G(n) be a uniform lattice. Denote by H-K(n) the hyperbolic space associated to G(n), where K is a division algebra over the reals of dimension d. Assume d(n - 1) >= 2. In this article we generalise natural maps to measurable cocycles. Given a standard Borel probability Gamma-space (X,mu(X)), we assume that a measurable cocycle sigma : Gamma x X -> G(m) admits an essentially unique boundary map phi : partial derivative H-infinity(K)n x X -> partial derivative infinity H-k(m) whose slices phi x : H-K(n) -> H-k(m) are atomless for almost every x is an element of X. Then there exists a sigma-equivariant measurable map F : H-k(n) x X -> H-K(m) whose slices F-x: H-K(n) -> H-K(m) are differentiable for almost every x is an element of X and such that Jac(a) F-x <= 1 for every alpha is an element of H-K(n) and almost every x is an element of X. This allows us to define the natural volume NV(sigma) of the cocycle sigma. This number satisfies the inequality NV(sigma) <= Vol(Gamma\H-K(n)). Additionally, the equality holds if and only if sigma is cohomologous to the cocycle induced by the standard lattice embedding i : F -> G(n) <= G(m), modulo possibly a compact subgroup of C(m) when m > m. Given a continuous map f : M -> N between compact hyperbolic manifolds, we also obtain an adaptation of the mapping degree theorem to this context.