NATURAL MAPS FOR MEASURABLE COCYCLES OF COMPACT HYPERBOLIC MANIFOLDS

Let G(n) be equal to either PO(n,1), PU(n,1) or PSp(n,1) and let Γ ≤ G(n) be a uniform lattice. Denote by Hn K 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 Γ-space (X,μX), we assume that a measurable cocycle σ : Γ×X→G(m) admits an essentially unique boundary map ø : δHnK×X→ δHmKwhose slices øx : Hn K→HmKare atomless for almost every x ∈ X. Then there exists a σ-equivariant measurable map F : HnK×X→ HmKwhose slices Fx : HnK→ HmKare differentiable for almost every x ∈ X and such that Jaca Fx ≤ 1 for every a ∈ HnKand almost every x ∈X. This allows us to define the natural volume NV(σ) of the cocycle σ. This number satisfies the inequality NV(σ) ≤ Vol(Γ\HnK). Additionally, the equality holds if and only if σ is cohomologous to the cocycle induced by the standard lattice embedding i : Γ→G(n) ≤ G(m), modulo possibly a compact subgroup of G(m) when m > n. Given a continuous map f : M→N between compact hyperbolic manifolds, we also obtain an adaptation of the mapping degree theorem to this context.