Natural maps for measurable cocycles of compact hyperbolic manifolds

Alessio Savini

Published 2019-09-17Version 1

Let $\text{G}(n)$ be equal either to $\text{PO}(n,1),\text{PU}(n,1)$ or $\text{PSp}(n,1)$ and let $\Gamma \leq \text{G}(n)$ be a uniform lattice. Denote by $\mathbb{H}^n_K$ the hyperbolic space associated to $\text{G}(n)$, where $K$ is a division algebra over the reals of dimension $d=\dim_{\mathbb{R}} K$. Assume $d(n-1) \geq 2$. In this paper we define the notion of natural map in the setting of Zimmer's cocycles theory. More precisely, let $(X,\mu_X)$ be a standard Borel probability $\Gamma$-space without atoms. Assume that a measurable cocycle $\sigma:\Gamma \times X \rightarrow \text{G}(m)$ admits an essentially unique boundary map $\phi:\partial_\infty \mathbb{H}^n_K \times X \rightarrow \partial_\infty \mathbb{H}^m_K$ whose slices $\phi_x:\mathbb{H}^n_K \rightarrow \mathbb{H}^m_K$ are essentially injective for almost every $x \in X$. Then there exists a $\sigma$-equivariant measurable map $F: \mathbb{H}^n_K \times X \rightarrow \mathbb{H}^m_K$ whose slices $F_x:\mathbb{H}^n_K \rightarrow \mathbb{H}^m_K$ are differentiable for almost every $x \in X$ and such that $\text{Jac}_a F_x \leq 1$ for every $a \in \mathbb{H}^n_K$ and almost every $x \in X$. The previous properties allow us to define the natural volume $\text{NV}(\sigma)$ of the cocycle $\sigma$. This number is constant along the $\text{G}(m)$-cohomology class of $\sigma$ and it satisfies the Milnor-Wood type inequality $\text{NV}(\sigma) \leq \text{Vol}(\Gamma \backslash \mathbb{H}^n_K)$. Additionally the equality holds if and only if $\sigma$ is cohomologous to the cocycle induced by the standard lattice embedding $i:\Gamma \rightarrow \text{G}(n) \leq \text{G}(m)$. Given a continuous map $f:M \rightarrow N$ between compact hyperbolic manifolds, we also obtain an adaptation of the mapping degree theorem to this context and a characterization of maps homotopic to local isometries in terms of maximal cocycles.