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

Alessio Savini

Vol. 312 (2021), No. 2, 505–525

DOI: 10.2140/pjm.2021.312.505

Abstract

Let $G$ be a semisimple Lie group of noncompact type and let ${\mathcal{𝒳}}_G$ be the Riemannian symmetric space associated to it. Suppose ${\mathcal{𝒳}}_G$ has dimension $n$ and does not contain any factor isometric to either ${ℍ}^2$ or $SL\left(3,ℝ\right)∕SO\left(3\right)$. Given a closed $n$-dimensional complete Riemannian manifold $N$, let $\Gamma ={\pi }_1\left(N\right)$ 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 and Farb described a way to construct a map $F:Y\to {\mathcal{𝒳}}_G$ 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 $\left(\Omega ,{\mu }_{\Omega }\right)$ is a standard Borel probability $\Gamma$-space, let $\sigma :\Gamma \times \Omega \to G$ be 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.

Keywords

uniform lattice, Zimmer cocycle, Patterson–Sullivan measure, natural map, Jacobian, mapping degree

Mathematical Subject Classification

Primary: 22E40, 57M50

Milestones

Received: 10 February 2020

Revised: 28 April 2021

Accepted: 30 April 2021

Published: 31 August 2021