The boundary model for the continuous cohomology of \(\mathrm{Isom}^+ (\mathbb H^n)\) (Q2418866)

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