Continuous cochains on Furstenberg boundaries and injectivity of the comparison map
Michelle Bucher, Alessio Savini
TL;DR
The paper advances boundary realizations of (bounded) cohomology for connected semisimple groups by showing that cohomology classes can be represented by cocycles continuous on generic boundary tuples, and establishes an isomorphism between continuous bounded alternating boundary cohomology and its measurable counterpart. A continuous (and a bounded) bicomplex built from $G$-actions on $G/P$ and generic tuples, together with a $G$-equivariant barycenter map, yields vanishing of higher columns in the spectral sequence and identifies the boundary realization with the standard cohomology. These methods enable injectivity results for the comparison map in low degrees: in particular, injectivity in degree $3$ for $G=\mathrm{Isom}^\circ(\mathbb{H}^n_{\mathbb{C}})$ and in degree $4$ for $G=\mathrm{Isom}^\circ(\mathbb{H}^n_{\mathbb{R}})$ with $n\ge 2$, extending known low-degree injectivity results. The results sharpen our understanding of rigidity phenomena in low-degree cohomology and provide elementary proofs in rank-one real/complex hyperbolic settings, complementing prior PDE-based approaches.
Abstract
Monod proved that any continuous cohomology of a semisimple Lie group $G$ can be represented by a measurable cocycle on the associated Furstenberg boundary, which we upgraded to an alternating cocycle. In the current paper we improve that result by showing that we can actually take a representing cocycle which is continuous on an explicit subset of generic tuples. We give an analogous result in the case of bounded cohomology. Finally, we exploit this characterization to prove the injectivity of the comparison map in degree $3$ for $\mathrm{Isom}^\circ(\mathbb{H}_{\mathbb{C}}^n)$, when $n \geq 2$, and in degree $4$ for $\mathrm{Isom}^\circ(\mathbb{H}^n_{\mathbb{R}})$, when $n \geq 2$.