Natural maps for measurable cocycles of compact hyperbolic manifolds

Abstract

Let $\text{G}(n)$ be equal either to $\text{PO}(n,1),\text{PU}(n,1)$ or $\text{PSp}(n,1)$ and let $\Gamma \le \text{G}(n)$ be a uniform lattice. Denote by ${\mathbb{H}}_K^n$ 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 $d(n-1)\ge 2$. In this paper we generalize natural maps to measurable cocycles. Given a standard Borel probability $\Gamma$-space $(X,{\mu }_X)$, we assume that a measurable cocycle $\sigma :\Gamma \times X\to \text{G}(m)$ admits an essentially unique boundary map $\varphi :{\partial }_{\infty }{\mathbb{H}}_K^n\times X\to {\partial }_{\infty }{\mathbb{H}}_K^m$ whose slices ${\varphi }_x:{\mathbb{H}}_K^n\to {\mathbb{H}}_K^m$ are atomless for almost every $x\in X$. Then, there exists a $\sigma$-equivariant measurable map $F:{\mathbb{H}}_K^n\times X\to {\mathbb{H}}_K^m$ whose slices $F_x:{\mathbb{H}}_K^n\to {\mathbb{H}}_K^m$ are differentiable for almost every $x\in X$ and such that ${\text{Jac}}_a{F}_x\le 1$ for every $a\in {\mathbb{H}}_K^n$ 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 satisfies the inequality $\text{NV}(\sigma )\le \text{Vol}(\Gamma \backslash {\mathbb{H}}_K^n)$. Additionally, the equality holds if and only if $\sigma$ is cohomologous to the cocycle induced by the standard lattice embedding $i:\Gamma \to \text{G}(n)\le \text{G}(m)$, modulo possibly a compact subgroup of $\text{G}(m)$ when $m>n$. Given a continuous map $f:M\to N$ between compact hyperbolic manifolds, we also obtain an adaptation of the mapping degree theorem to this context.

27 pages, to appear on J. Inst. Math. Jussieu

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• Non-ergodic actions, cocycles and superrigidity
• Fundamental classes not representable by products
• A Matsumoto-Mostow result for Zimmer's cocycles of hyperbolic lattices
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Cited by in corpus (3)

• Integrable tautness of isometries of complex hyperbolic spaces
• Multiplicative constants and maximal measurable cocycles in bounded cohomology
• Equivariant maps for measurable cocycles with values into higher rank Lie groups