On the trivializability of rank-one cocycles with an invariant field of projective measures

TL;DR

This paper proves that certain rank-one cocycles with an invariant field of measures are essentially trivial, extending the result to complex hyperbolic cases, under specific compatibility conditions.

Contribution

It establishes the trivializability of rank-one cocycles with invariant measure fields under compatibility assumptions, generalizing previous results to complex hyperbolic settings.

Findings

Cocycles with invariant measure fields are trivializable under compatibility.

Results extend to complex hyperbolic cases.

Provides conditions for triviality of rank-one cocycles.

Abstract

Let $G$ be $\text{SO}^\circ(n,1)$ for $n \geq 3$ and consider a lattice $\Gamma < G$. Given a standard Borel probability $\Gamma$-space $(\Omega,\mu)$, consider a measurable cocycle $\sigma:\Gamma \times \Omega \rightarrow \mathbf{H}(\kappa)$, where $\mathbf{H}$ is a connected algebraic $\kappa$-group over a local field $\kappa$. Under the assumption of compatibility between $G$ and the pair $(\mathbf{H},\kappa)$, we show that if $\sigma$ admits an equivariant field of probability measures on a suitable projective space, then $\sigma$ is trivializable. An analogous result holds in the complex hyperbolic case.