Let
$G$
be a semisimple Lie group of noncompact type and let
${\mathcal{\mathcal{X}}}_{G}$
be the Riemannian symmetric space associated to it. Suppose
${\mathcal{\mathcal{X}}}_{G}$ has
dimension
$n$
and does not contain any factor isometric to either
${\mathbb{H}}^{2}$ or
$SL\left(3,\mathbb{R}\right)\u2215SO\left(3\right)$. Given a closed
$n$dimensional complete
Riemannian manifold
$N$,
let
$\Gamma ={\pi}_{1}\left(N\right)$ be its fundamental
group and
$Y$
its universal cover. Consider a representation
$\rho :\Gamma \to G$ with a measurable
$\rho $equivariant
map
$\psi :Y\to {\mathcal{\mathcal{X}}}_{G}$.
Connell and Farb described a way to construct a map
$F:Y\to {\mathcal{\mathcal{X}}}_{G}$ which is smooth,
$\rho $equivariant
and with uniformly bounded Jacobian.
We extend the construction of Connell and Farb to the context of measurable cocycles. More precisely,
if
$\left(\Omega ,{\mu}_{\Omega}\right)$ is a standard Borel
probability
$\Gamma $space,
let
$\sigma :\Gamma \times \Omega \to G$
be measurable cocycle. We construct a measurable map
$F:Y\times \Omega \to {\mathcal{\mathcal{X}}}_{G}$ which is
$\sigma $equivariant,
whose slices are smooth and they have uniformly bounded Jacobian. For such
equivariant maps we define also the notion of volume and we prove a sort of mapping
degree theorem in this particular context.
