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

Abstract

Let G be SO • (n, 1) for n ≥ 3 and consider a lattice Γ < G. Given a standard Borel probability Γ-space (Ω, µ), consider a measurable cocycle σ : Γ × Ω → H(κ), where H is a connected algebraic κ-group over a local field κ. Under the assumption of compatibility between G and the pair (H, κ), we show that if σ admits an equivariant field of probability measures on a suitable projective space, then σ is trivializable.An analogous result holds in the complex hyperbolic case.