EQUIVARIANT MAPS FOR MEASURABLE COCYCLES WITH VALUES INTO HIGHER RANK LIE GROUPS

Let G be a semisimple Lie group of noncompact type and let XG be the Rie-mannian symmetric space associated to it. Suppose XG has dimension n and does not contain any factor isometric to either (Formula presented) or SL(3, R)/SO(3). Given a closed n-dimensional complete Riemannian manifold N, let Γ = π1(N) be its fundamental group and Y its universal cover. Consider a representation ρ: Γ G with a measurable ρ-equivariant map ψ: Y XG. Connell and Farb described a way to construct a map F: Y XG which is smooth, ρ -equivariant and with uniformly bounded Jacobian. We extend the construction of Connell and Farb to the context of measurable cocycles. More precisely, if (Ω, µ.Ω) is a standard Borel probability Γ-space, let σ: Γ x Ω G be measurable cocycle. We construct a measurable map F: Y x Ω XG which is σ-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.