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.

arXiv comment27 pages, to appear on J. Inst. Math. Jussieu

References in corpus (11)

• Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds
• Non-ergodic actions, cocycles and superrigidity
• Fundamental classes not representable by products
• Fundamental groups of aspherical manifolds and maps of non-zero degree
• Volume rigidity at ideal points of the character variety of hyperbolic 3-manifolds
• Rigidity at infinity for lattices in rank-one Lie groups
• A Matsumoto-Mostow result for Zimmer's cocycles of hyperbolic lattices
• Algebraic hull of maximal measurable cocycles of surface groups into Hermitian Lie groups
• Integrable tautness of isometries of complex hyperbolic spaces
• Ideal simplicial volume of manifolds with boundary
• Multiplicative constants and maximal measurable cocycles in bounded cohomology

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