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 $\\mathcal{X}_G$ be the Riemannian symmetric space associated to it. Suppose $\\mathcal{X}_G$ has dimension $n$ and it has no factor isometric either to $\\mathbb{R},\\mathbb{H}^2$ or $\\text{SL}(3,\\mathbb{R})/\\text{SO}(3)$. Given a closed $n$-dimensional Riemannian manifold $N$, let $\\Gamma=\\pi_1(N)$ be its fundamental group and $Y$ its universal cover. Consider a representation $\\rho:\\Gamma \\rightarrow G$ with a measurable $\\rho$-equivariant map $\\psi:Y \\rightarrow \\mathcal{X}_G$. Connell-Farb described a way to construct a map $F:Y\\rightarrow \\mathcal{X}_G$ which is smooth, $\\rho$-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 $(\\Omega,\\mu_\\Omega)$ is a standard Borel probability $\\Gamma$-space, let $\\sigma:\\Gamma \\times \\Omega \\rightarrow G$ be measurable cocycle which admits a measurable $\\sigma$-equivariant map $\\psi:Y \\times \\Omega \\rightarrow \\mathcal{X}_G$. We construct a measurable map $F: Y \\times \\Omega \\rightarrow \\mathcal{X}_G$ 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.