超逼近，I：p-adic半单情形
Super-approximation, I: p-adic semisimple case
摘要：
设$k$是数域，$Ω$是$\mathbb{GL}_{n_0}(k)$的有限对称子集，而$Γ=\langle Ω\rangle$。设\[ C(Γ):=\{\mathfrak{p}\in V_f(k)|\hspace{1mm} Γ\text{is a bounded subgroup of} \mathbb{GL}_{n_0}(k_{\mathfrak{p}})\}, \]和$Γ_{\mathfrak{p}}$是$\mathbb{GL}_{n_0}(k_{\mathfrak{p}})$中$Γ$的闭包。在假定$Γ$的Zerkiki闭包是半单的条件下，证明了左平移作用量族$\{Γ\curvearrowright Γ_{\mathfrak{p}}\}_{\mathfrak{p}\in C(Γ)}$具有{ \em一致谱隙}.作为推论，我们得到左平移作用量$Γ\curvearrowright G$具有{ \em局部谱间隙}，如果$Γ$是半单$p$-adic解析群$G$的可数稠密子群，且Ad $(Γ)$由代数元在Lie $(G)$的某个$\mathbb{Q}_p$ -基中的矩阵组成.这可以被看作是\cite [定理A]{BISG}的一个（更强的）$p$ -adic版本，它使我们能够应用于Banach-Ruziewicz问题和轨道等价刚性。
英文摘要：
Let $k$ be a number field, $Ω$ be a finite symmetric subset of $\mathbb{GL}_{n_0}(k)$, and $Γ=\langle Ω\rangle$. Let \[ C(Γ):=\{\mathfrak{p}\in V_f(k)|\hspace{1mm} Γ\text{is a bounded subgroup of} \mathbb{GL}_{n_0}(k_{\mathfrak{p}})\}, \] and $Γ_{\mathfrak{p}}$ be the closure of $Γ$ in $\mathbb{GL}_{n_0}(k_{\mathfrak{p}})$. Assuming that the Zariski-closure of $Γ$ is semisimple, we prove that the family of left translation actions $\{Γ\curvearrowright Γ_{\mathfrak{p}}\}_{\mathfrak{p}\in C(Γ)}$ has {\em uniform spectral gap}. As a corollary we get that the left translation action $Γ\curvearrowright G$ has {\em local spectral gap} if $Γ$ is a countable dense subgroup of a semisimple $p$-adic analytic group $G$ and Ad$(Γ)$ consists of matrices with algebraic entries in some $\mathbb{Q}_p$-basis of Lie$(G)$. This can be viewed as a (stronger) $p$-adic version of \cite[Theorem A]{BISG}, which enables us to give applications to the Banach-Ruziewicz problem and orbit equivalence rigidity.
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。
