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

Abstract

Nicolas Monod showed that the evaluation map $H_m^{\ast }(G\curvearrowright G/P)⟶H_m^{\ast }(G)$ 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 $H_{m,b}^{\ast }(G)\to H_m^{\ast }(G)$ 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.

38 pages, 3 Figures