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

  • Alireza Salehi Golsefidy

    University of California at San Diego, La Jolla, USA
Super-approximation, II: the $p$-adic case and the case of bounded powers of square-free integers cover
Download PDF

A subscription is required to access this article.

Abstract

Let Ω\Omega be a finite symmetric subset of GLn(Z[1/q0])_n(\mathbb Z[1/q_0]), and Γ:=Ω\Gamma:=\langle \Omega \rangle, and let πm\pi_m be the group homomorphism induced by the quotient map Z[1/q0]Z[1/q0]/mZ[1/q0]\mathbb Z[1/q_0] \to \mathbb Z[1/q_0] / m \mathbb Z[1/q_0]. Then the family {Cay (πm(Γ),πm(Ω))}m(\pi_m (\Gamma),\pi_m(\Omega))\}_m of Cayley graphs is a family of expanders as mm ranges over fixed powers of square-free integers and powers of primes that are coprime to q0q_0 if and only if the connected component of the Zariski-closure of Γ\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 pp-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