Some explicit cocycles on the Furstenberg boundary for products of isometries of hyperbolic spaces and $\mathrm{SL}(3,\mathbb{K})$

TL;DR

This paper explicitly describes certain cocycles on the Furstenberg boundary for products of hyperbolic space isometries and SL(3,K), revealing new insights into their cohomological properties and bounded cohomology injectivity.

Contribution

It provides explicit descriptions of non-alternating and alternating cocycles in low degrees for specific groups, refining previous results and proving bounded cohomology injectivity in new cases.

Findings

Non-alternating cocycles are non-trivial and in the kernel of the evaluation map.

The comparison map from bounded to measurable cohomology is injective in degree 3 for certain groups.

Alternative proof of injectivity for SL(3,K) when K is real or complex.

Abstract

Nicolas Monod showed that the evaluation map $H^*_m(G\curvearrowright G/P)\longrightarrow H^*_m(G)$ between the measurable cohomology of the action of a connected semisimple Lie group $G$ on its Furstenberg boundary $G/P$ and the measurable cohomology of $G$ is surjective with a kernel that can be entirely described in terms of invariants in the cohomology of a maximal split torus $A<G$. In a recent paper we refine Monod's result and show in particular that the cohomology of non-alternating cocycles on $G/P$ is in general not trivial and lies in the kernel of the evaluation. In this paper we describe explicitly such non-alternating and alternating cocycles on $G/P$ in low degrees when $G$ is either a product of isometries of real hyperbolic spaces or $G=\mathrm{SL}(3,\mathbb{K})$, where $\mathbb{K}$ is either the real or the complex field. As a consequence, we deduce that the comparison map $H^*_{m,b}(G)\rightarrow H^*_m(G)$ from the measurable bounded cohomology is injective in degree $3$ for nontrivial products of isometries of hyperbolic spaces. We get also another proof of the injectivity for $G=\mathrm{SL}(3,\mathbb{K})$, when $\mathbb{K}$ is either the real field or the complex one.