AI 中文总结
该文研究高阶单李群的离散子群，证明非格子群的单射半径增长率下界，给出Nevo-Stück-Zimmer定理的简化证明，得到格的测度刻画。
AI 中文摘要
Fraczyk和Gelander在文献[FG]中证明，对于任何高阶单李群$G$及每个非格离散子群$\Gamma\leq G$，齐性空间$G/\Gamma$中点的单射半径是无界的，解决了Margulis的一个猜想。本工作中，我们得到了$G/\Gamma$中随半径增长的球上最大单射半径增长率的显式下界。更明确地说，我们证明对任意$R>0$，可在$G/\Gamma$中嵌入一个半径为$c\log^{(4)}R$的球，球心为某点$[g]\in G/\Gamma$，其中$g$取自$G_R$，$c=c(G,\Gamma)$为常数。特别地，我们证明对一般离散子群$\Gamma$，若$G/\Gamma$中单射半径增长慢于$\log^{(4)}$，则$\Gamma$必为格。此外，我们给出了Nevo-Stück-Zimmer定理的一个新的更简短简单的证明，该定理指出每个具有性质$(T)$的高阶单群的作用要么是本质自由的，要么是本质传递的。本文的结果是利用配套论文中测度的几乎结构及额外几何考量得到的。作为证明的一步，我们给出了格的如下刻画：离散子群$\Gamma\leq G$是格当且仅当$G/\Gamma$上存在一个在$G$下足够几乎不变的概率测度。更精确地说，假设$\Gamma\leq G$是离散子群，且存在$G/\Gamma$上的概率测度$\nu$使得对某个$\eps_0(\Gamma)>0$有$W_1^{b}(g\nu,\nu)\leq \eps_0$，则$\Gamma$是格。
英文摘要
Fraczyk and Gelander proved in \cite{FG} that for any simple Lie group $G$ of high rank and for every non-lattice discrete subgroup $Γ\leq G$, the injectivity radius of points in $G/Γ$ is unbounded, resolving a conjecture of Margulis. In this work we obtain an explicit lower bound on the growth rate of the maximal injectivity radius of points taken from growing balls in $G/Γ$. More explicitly, we prove that for any $R>0$, one can embed a ball of radius $c\log^{(4)}R$ in $G/Γ$ centered at some point $[g]\in G/Γ$ where $g$ is taken from $G_R$ and for some constant $c=c(G,Γ)$. In particular, we show that for a general discrete subgroup $Γ$, if the injectivity radius growth in $G/Γ$ is slower than $\log^{(4)}$, $Γ$ must be a lattice. Additionally, we give a new, shorter and simpler proof of the Nevo-Stuck-Zimmer Theorem, saying that every action of a high rank simple group with property $(T)$ is either essentially free or essentially transitive. The results in this paper are obtained using the almost structure of measures from the accompanying paper, together with additional geometric considerations. As a step in the proof, we develop the following characterization for lattices. A discrete subgroup $Γ\leq G$ is a lattice if and only if there is a probability measure on $G/Γ$ which is sufficiently almost invariant under $G$. More precisely, suppose $Γ\leq G$ is a discrete subgroup for which there exists a probability measure $ν$ on $G/Γ$ for which $W_1^{b}(gν,ν)\leq \eps_0$ for some $\eps_0(Γ)>0$, then $Γ$ is a lattice.
