超逼近，II：无平方整数的p-adic和有界幂
Super-approximation, II: the p-adic and bounded power of square-free integers cases
摘要：
设$Ω$是GL $_n(\mathbb{Z}[1/q_0])$的有限对称子集，且$Γ:=\langle Ω\rangle$ .则Cayley图族$\{{\rm Cay}(π_m(Γ),π_m(Ω))\}_m$是一个扩张子族，因为$m$的值域是关于与$q_0$互质的无平方整数的固定幂和素数的固定幂的当且仅当$Γ$的Zerkiki-闭包的连通分支是完美的.一些直接的应用，例如轨道等价刚性，{ \em largeness}的某些$\ell$ -adic伽罗瓦表示，也进行了讨论。
英文摘要：
Let $Ω$ be a finite symmetric subset of GL$_n(\mathbb{Z}[1/q_0])$, and $Γ:=\langle Ω\rangle$. Then the family of Cayley graphs $\{{\rm Cay}(π_m(Γ),π_m(Ω))\}_m$ is a family of expanders as $m$ ranges over fixed powers of square-free integers and powers of primes that are coprime to $q_0$ if and only if the connected component of the Zariski-closure of $Γ$ is perfect. Some of the immediate applications, e.g. orbit equivalence rigidity, {\em largeness} of certain $\ell$-adic Galois representations, are also discussed.
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