单李群的局部谱间隙及其应用
Local spectral gap in simple Lie groups and applications
摘要：
我们引入了一个新的概念{ \it本地频谱间隙}一般，可能是无限的，措施保持行动。当$Γ$是由任意连通单李群$G$的代数元生成的稠密子群时，我们建立了左平移作用量$Γ\curvearrowright G$的局部谱隙.这扩展到非紧凑设置Bourgain和Gamburd \cite{BG06,BG10}，和Benoist和de Saxcé \cite{BdS14}的最新作品。我们提出了几个应用程序的巴拿赫-Ruziewicz问题，轨道等价刚性，连续和单调扩张，有界随机游动$G$。特别地，我们证明了直到一个乘法常数，Haar测度是定义在$G$的所有有界可测子集上的唯一的$Γ$ -不变可加测度.
英文摘要：
We introduce a novel notion of {\it local spectral gap} for general, possibly infinite, measure preserving actions. We establish local spectral gap for the left translation action $Γ\curvearrowright G$, whenever $Γ$ is a dense subgroup generated by algebraic elements of an arbitrary connected simple Lie group $G$. This extends to the non-compact setting recent works of Bourgain and Gamburd \cite{BG06,BG10}, and Benoist and de Saxcé \cite{BdS14}. We present several applications to the Banach-Ruziewicz problem, orbit equivalence rigidity, continuous and monotone expanders, and bounded random walks on $G$. In particular, we prove that, up to a multiplicative constant, the Haar measure is the unique $Γ$-invariant finitely additive measure defined on all bounded measurable subsets of $G$.
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