The boundary model for the continuous cohomology of Isom${}^{+}({\mathbb{H}}^n)$

Abstract

We prove that the continuous cohomology of Isom${}^{+}({\mathbb{H}}^n)$ can be measurably realized on the boundary of hyperbolic space. This implies in particular that for Isom${}^{+}({\mathbb{H}}^n)$ the comparison map from continuous bounded cohomology to continuous cohomology is injective in degree 3. We furthermore prove a stability result for the continuous bounded cohomology of Isom$({\mathbb{H}}^n)$ and Isom$({\mathbb{H}}_{\mathbb{C}}^n)$.