Equivariant maps for measurable cocycles of higher rank Lie groups

Let G be a semisimple Lie group of noncompact type and let X-G be the Riemannian symmetric space associated to it. Suppose chi(G) has dimension n and does not contain any factor isometric to either H-2 or SL(3, R)/SO(3). Given a closed n-dimensional complete Riemannian manifold N, let Gamma = pi(1)(N) be its fundamental group and Y its universal cover. Consider a representation rho : Gamma -> G with a measurable p-equivariant map psi : Y -> X-G. Connell and Farb described a way to construct a map F : Y -> 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 (Omega, mu(Omega)) is a standard Borel probability Gamma-space, let sigma : Gamma x Omega -> G be measurable cocycle. We construct a measurable map F : Y x Omega -> X-G which is -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.