Equivariant maps for measurable cocycles with values into higher rank Lie groups

Abstract

Let G a semisimple Lie group of non-compact type and let X G be the Riemannian symmetric space associated to it. Suppose X G has dimension n and it does not contain any factor isometric either to R, H 2 or SL(3, R)/SO(3). Given a closed n-dimensional 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 → X G. Connell-Farb described a way to construct a map F : Y → X G which is smooth, ρ-equivariant and with uniformly bounded Jacobian.In this paper we extend the construction of Connell-Farb to the context of measurable cocycles. More precisely, if (Ω, µΩ) is a standard Borel probability Γ-space, let σ : Γ × Ω → G be measurable cocycle which admits a measurable σequivariant map ψ : Y × Ω → X G. We construct a measurable map F : Y × Ω → X G 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.