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

### Alireza Salehi Golsefidy

University of California at San Diego, La Jolla, USA

## Abstract

Let $\Omega$ be a finite symmetric subset of GL$_n(\mathbb Z[1/q_0])$, and $\Gamma:=\langle \Omega \rangle$, 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.

## Cite this article

Alireza Salehi Golsefidy, Super-approximation, II: the $p$-adic case and the case of bounded powers of square-free integers. J. Eur. Math. Soc. 21 (2019), no. 7, pp. 2163–2232

DOI 10.4171/JEMS/883