关于在边界上遍历作用的离散群的若干方面
On some aspects of discrete groups acting ergodically on the boundary
浏览论文内容
中文总结 AI 辅助
该研究针对实半单李群的慢增长离散子群，证明其在Furstenberg边界上的完全耗散作用，构造了SO(n,2)的遍历作用离散子群反例，还提出了关联格形变与光滑群作用形变的新方法。
中文摘要 AI 辅助
我们证明：若G为实半单李群，Γ<G为具有慢增长性的离散子群（其增长指示函数小于ρ），则Γ在G的Furstenberg边界上完全耗散地作用。此外，对于G=SO(n,2)且n≥3，我们构造了在G的Furstenberg边界上遍历作用的无限共体积离散子群，为Margulis关于G=SO(n,2)（n>2）的猜想提供了反例。这些例子源于H=SO(n,1)中的格Γ及其在G中的形变。更重要的是，我们描述了一种研究此类格形变的新方法，将其与SO(n-1,1)在Γ\backslash H上的光滑右平移作用的形变关联起来。这种对应使我们能应用DeWitt和Dolgopyat关于光滑群作用的最新结果，给出SO(n-1,1)在Γ\backslash H上的右平移作用不具备C⁰-局部刚性的例子。
英文摘要
We show that if $G$ is a real semisimple Lie group and $Γ<G$ is a discrete subgroup with slow growth, in the sense that its growth indicator function is smaller than $ρ$, then $Γ$ acts totally dissipatively on the Furstenberg boundary of $G$. Moreover, for $G= SO(n,2)$ and $n\geq 3$, we construct infinite-covolume discrete subgroups that act ergodically on the Furstenberg boundary of $G$, providing a counterexample to a conjecture of Margulis for $G = SO(n,2), n > 2$. The examples arise from lattices $Γ<H= SO(n,1)$ and their deformations in $G$. Perhaps more importantly, we describe a new approach to studying deformations of such lattices by relating them to deformations of the smooth right-translation action of $SO(n-1,1)$ on $Γ\backslash H$. This correspondence allows us to apply recent results of DeWitt and Dolgopyat on smooth group actions and to give examples where the right-translation action of $SO(n-1,1)$ on $Γ\backslash H$ can fail to be $C^0$-locally rigid.
