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 to either ${\mathbb{H}}^2$ 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 $\rho :\Gamma \to G$ with a measurable $\rho$-equivariant map $\psi :Y\to {\mathcal{X}}_G$. Connell-Farb described a way to construct a map $F:Y\to {\mathcal{X}}_G$ 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 $\sigma :\Gamma \times \Omega \to G$ be a measurable cocycle. We construct a measurable map $F:Y\times \Omega \to {\mathcal{X}}_G$ 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.

19 pages; Added Lemma 3.1 + new references + corrected typo; to appear on Pacific J. Math

References in corpus (4)

• Non-ergodic actions, cocycles and superrigidity
• Natural maps for measurable cocycles of compact hyperbolic manifolds
• A Matsumoto-Mostow result for Zimmer's cocycles of hyperbolic lattices
• Integrable tautness of isometries of complex hyperbolic spaces