PO(p,q)中Anosov表示的拓扑障碍
Topological obstruction for Anosov representations into $\operatorname{PO}(p, q)$
AI总结：
该研究针对可将Anosov表示嵌入到PO(p,q)或G₂'的双曲群，建立其虚上同调维数的拓扑限制，证明相关齐性空间不存在紧商空间。
摘要：
我们对可将双曲群以Anosov表示嵌入到PO(p,q)（p>q≥2）或例外李群G₂'中的群的虚上同调维数建立新限制。主要结果表明，若Pₖ是PO(p,q)中稳定化ℝ^{p,q}内各向同性k平面的抛物子群，则可将{P₁,P_q}-Anosov表示嵌入到PO(p,q)的群，其虚上同调维数严格小于p。由此推出，PO(p,q)的H^{p,q-1}凸余紧子群无法在H^{p,q-1}上真不连续且余紧作用。主定理还意味着，可将P_α-Anosov表示嵌入到例外李群G₂'的群，其虚上同调维数严格小于4，其中α是G₂'的短限制根。作为应用，我们证明齐性空间G₂'/SU(2,1)不存在任何紧商空间。
英文摘要：
We establish new restrictions on the virtual cohomological dimension of word hyperbolic groups admitting Anosov representations into $\operatorname{PO}(p, q)$ ($p > q \geq 2$) or the exceptional Lie group $\operatorname{G}_2'$. Our main result states that, if $P_k$ is the parabolic subgroup of $\operatorname{PO}(p, q)$ stabilizing an isotropic $k$-plane in $\mathbb{R}^{p, q}$, a group admitting a $\{P_1,P_q\}$-Anosov representation into $\operatorname{PO}(p, q)$ has virtual cohomological dimension strictly less than $p$. As a result, a $\mathbf{H}^{p, q-1}$-convex cocompact subgroup of $\operatorname{PO}(p, q)$ cannot act properly discontinuously and cocompactly on $\mathbf{H}^{p, q-1}$. The main theorem also implies that a group admitting a $P_α$-Anosov representation into the exceptional Lie group $\operatorname{G}_2'$ has virtual cohomological dimension strictly less than $4$, where $α$ is a short restricted root of $\operatorname{G}_2'$. As an application, we prove that the homogeneous space $\operatorname{G}_2'/ \operatorname{SU}(2, 1)$ does not admit any compact quotient.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
