  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.
