Super-approximation, II: the $p$-adic case and the case of bounded powers of square-free integers

Abstract

Let $\Omega$ be a finite symmetric subset of GL${}_n(\mathbb{Z}[1/q_0])$, and $\Gamma :=⟨\Omega ⟩$, and let ${\pi }_m$ be the group homomorphism induced by the quotient map $\mathbb{Z}[1/q_0]\to \mathbb{Z}[1/q_0]/m\mathbb{Z}[1/q_0]$. Then the family {Cay $({\pi }_m(\Gamma ),{\pi }_m(\Omega )){\}}_m$ of Cayley graphs 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 and largeness of certain $\ell$-adic Galois representations, are also discussed.