Super-approximation, II: the p-adic and bounded power of square-free integers cases

Let $\Omega$ be a finite symmetric subset of GL$_n(\mathbb{Z}[1/q_0])$, and $\Gamma:=\langle \Omega \rangle$. Then the family of Cayley graphs $\{{\rm Cay}(\pi_m(\Gamma),\pi_m(\Omega))\}_m$ 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 $\Gamma$ is perfect. Some of the immediate applications, e.g. orbit equivalence rigidity, {\em largeness} of certain $\ell$-adic Galois representations, are also discussed.