双曲空间等距群与 $\mathrm{SL}(3,\mathbb{K})$ 乘积的 Furstenberg 边界上若干显式上闭链
Some explicit cocycles on the Furstenberg boundary for products of isometries of hyperbolic spaces and $\mathrm{SL}(3,\mathbb{K})$
AI总结：
本文在低次数情形显式构造了双曲空间等距群乘积及 $\mathrm{SL}(3,\mathbb{K})$ 的 Furstenberg 边界上的非交错与交错上闭链，并由此证明相应可测有界上同调比较映射在次数 3 处的单射性。
摘要：
Nicolas Monod 证明了连通半单李群 $G$ 作用在其 Furstenberg 边界 $G/P$ 上的可测上同调与 $G$ 的可测上同调之间的赋值映射 $H^*_m(G\curvearrowright G/P)\longrightarrow H^*_m(G)$ 是满射，且其核可以完全用极大分裂环面 $A<G$ 的上同调中的不变量来描述。在最近的一篇论文中，我们细化了 Monod 的结果，并特别证明了 $G/P$ 上非交错上闭链的上同调一般并非平凡，且位于赋值映射的核中。本文在低次数情形下显式描述了这样的非交错与交错上闭链，其中 $G$ 要么是实双曲空间等距群的乘积，要么是 $G=\mathrm{SL}(3,\mathbb{K})$，而 $\mathbb{K}$ 为实数域或复数域。作为推论，我们推出对于双曲空间等距群的非平凡乘积，从可测有界上同调出发的比较映射 $H^*_{m,b}(G)\rightarrow H^*_m(G)$ 在次数 $3$ 处是单射。当 $\mathbb{K}$ 为实数域或复数域时，我们也得到了 $G=\mathrm{SL}(3,\mathbb{K})$ 情形单射性的另一个证明。
英文摘要：
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.
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