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

TL;DR

This paper demonstrates that the continuous cohomology of the orientation-preserving isometry group of hyperbolic space can be realized on the boundary, establishing injectivity of the comparison map in degree 3 and proving stability results.

Contribution

It introduces a boundary model for continuous cohomology of hyperbolic isometry groups, showing injectivity of the comparison map and establishing stability results.

Findings

Cohomology can be realized on the boundary of hyperbolic space.

The comparison map from bounded to continuous cohomology is injective in degree 3.

Stability results for the cohomology of hyperbolic isometry groups.

Abstract

We prove that the continuous cohomology of $\text{Isom}^+(\mathbb{H}^n)$ can be measurably realized on the boundary of hyperbolic space. This implies in particular that for $\text{Isom}^+(\mathbb{H}^n)$ 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 $\text{Isom}(\mathbb{H}^n)$ and $\text{Isom}(\mathbb{H}_{\mathbb{C}}^n)$.