论文标题
在Furstenberg边界上的一些显式Cocycles of双曲空间的异构体和$ \ Mathrm {SL}（3，\ Mathbb {K}）$
Some explicit cocycles on the Furstenberg boundary for products of isometries of hyperbolic spaces and $\mathrm{SL}(3,\mathbb{K})$
论文作者
论文摘要
尼古拉斯·莫诺德（Nicolas Monod）表明，评估图$ h^*_ m（g \ curveArrowright g/p）\ longrightArrow h^*_ m（g）$之间可测量的共同的共同的共同的共同体学的共同体共同的共同体$ g $ g $ g $ g $在其fursstenberg边界$ g/p $ g $ g $ a $ g $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ be n中的表现上最大分裂的圆环$ a <g $的共同体中不变。在最近的一篇论文中，作者完善了monod的结果，并特别表明，在$ g/p $上的非替代共体的共同体学，即位于交替图的内核的人，通常不是很琐碎，并在评估的内核中含义。在本文中，我们在低度上明确描述了$ g/p $上的这种非替代和交替的共生时，当$ g $是真实双曲线空间的异构体的产物，或者$ g = \ mathrm {slrm {Slrm {sl}（3，\ mathbb {k}）$，wher结果，我们推断出比较映射$ h^*_ {m，b}（g）\ rightArrow h^*_ m（g）$从可测量的有界共同体学中是$ 3 $的，这对于非底线的非底线产物是新的。
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 the maximal split torus $A<G$. In a recent paper the authors refine Monod's result and show in particular that the cohomology of non-alternating cocycles on $G/P$, namely those lying in the kernel of the alternation map, 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 $3$, which is new for nontrivial products of isometries of hyperbolic spaces.
