非紧型对称空间的分裂秩与有界上同调
On splitting rank of non-compact type symmetric spaces and bounded cohomology
摘要：
设$X=G/K$是高秩非紧型对称空间，其中$G$是$X$的等距群的连通分支.定义$X$的分裂秩为全测地子流形$Y\subset X$分裂等距$\mathbb R$ -因子的最大维数，记为$\text{srk}(X)$.我们明确地计算每个不可约对称空间的分裂秩。对于任意的对称空间（不一定是不可约的），证明了比较映射$η:H^{*}_{c,b}(G,\mathbb{R})\rightarrow H^{*}_c(G,\mathbb{R})$在$*\geq \text{srk}(X)+2$的度上是满射的，只要$X$没有小的直因子.
英文摘要：
Let $X=G/K$ be a higher rank symmetric space of non-compact type, where $G$ is the connected component of the isometry group of $X$. We define the splitting rank of $X$, denoted by $\text{srk}(X)$, to be the maximal dimension of a totally geodesic submanifold $Y\subset X$ which splits off an isometric $\mathbb R$-factor. We compute explicitly the splitting rank for each irreducible symmetric space. For an arbitrary (not necessarily irreducible) symmetric space, we show that the comparison map $η:H^{*}_{c,b}(G,\mathbb{R})\rightarrow H^{*}_c(G,\mathbb{R})$ is surjective in degrees $*\geq \text{srk}(X)+2$, provided $X$ has no small direct factors.
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