Let
$\operatorname {\mathrm {{\rm G}}}(n)$
be equal to either
$\operatorname {\mathrm {{\rm PO}}}(n,1),\operatorname {\mathrm {{\rm PU}}}(n,1)$
or
$\operatorname {\mathrm {\textrm {PSp}}}(n,1)$
and let
$\Gamma \leq \operatorname {\mathrm {{\rm G}}}(n)$
be a uniform lattice. Denote by
$\operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}}$
the hyperbolic space associated to
$\operatorname {\mathrm {{\rm G}}}(n)$
, where
$\operatorname {\mathrm {{\rm K}}}$
is a division algebra over the reals of dimension d. Assume
$d(n-1) \geq 2$
.
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
$\sigma :\Gamma \times X \rightarrow \operatorname {\mathrm {{\rm G}}}(m)$
admits an essentially unique boundary map
$\phi :\partial _\infty \operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}} \times X \rightarrow \partial _\infty \operatorname {\mathrm {\mathbb {H}^m_{{\rm K}}}}$
whose slices
$\phi _x:\operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}} \rightarrow \operatorname {\mathrm {\mathbb {H}^m_{{\rm K}}}}$
are atomless for almost every
$x \in X$
. Then there exists a
$\sigma $
-equivariant measurable map
$F: \operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}} \times X \rightarrow \operatorname {\mathrm {\mathbb {H}^m_{{\rm K}}}}$
whose slices
$F_x:\operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}} \rightarrow \operatorname {\mathrm {\mathbb {H}^m_{{\rm K}}}}$
are differentiable for almost every
$x \in X$
and such that
$\operatorname {\mathrm {\textrm {Jac}}}_a F_x \leq 1$
for every
$a \in \operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}}$
and almost every
$x \in X$
. This allows us to define the natural volume
$\operatorname {\mathrm {\textrm {NV}}}(\sigma )$
of the cocycle
$\sigma $
. This number satisfies the inequality
$\operatorname {\mathrm {\textrm {NV}}}(\sigma ) \leq \operatorname {\mathrm {\textrm {Vol}}}(\Gamma \backslash \operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}})$
. Additionally, the equality holds if and only if
$\sigma $
is cohomologous to the cocycle induced by the standard lattice embedding
$i:\Gamma \rightarrow \operatorname {\mathrm {{\rm G}}}(n) \leq \operatorname {\mathrm {{\rm G}}}(m)$
, modulo possibly a compact subgroup of
$\operatorname {\mathrm {{\rm G}}}(m)$
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.