# 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 $Ω$ be a finite symmetric subset of GL$_{n}(Z[1/q_{0}])$, and $Γ:=⟨Ω⟩$, and let $π_{m}$ be the group homomorphism induced by the quotient map $Z[1/q_{0}]→Z[1/q_{0}]/mZ[1/q_{0}]$. Then the family {Cay $(π_{m}(Γ),π_{m}(Ω))}_{m}$ 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 $Γ$ is perfect. Some of the immediate applications, e.g. orbit equivalence rigidity and largeness of certain $ℓ$-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