#### Vol. 312, No. 2, 2021

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

### Alessio Savini

Vol. 312 (2021), No. 2, 505–525
##### Abstract

Let $G$ be a semisimple Lie group of noncompact type and let ${\mathsc{𝒳}}_{G}$ be the Riemannian symmetric space associated to it. Suppose ${\mathsc{𝒳}}_{G}$ 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 $\psi :Y\to {\mathsc{𝒳}}_{G}$. Connell and Farb described a way to construct a map $F:Y\to {\mathsc{𝒳}}_{G}$ 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 $\sigma :\Gamma ×\Omega \to G$ be measurable cocycle. We construct a measurable map $F:Y×\Omega \to {\mathsc{𝒳}}_{G}$ 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