Super-approximation, II: the p-adic and bounded power of square-free integers cases

Abstract

Let $\Omega$ be a finite symmetric subset of GL${}_n(\mathbb{Z}[1/q_0])$, and $\Gamma :=⟨\Omega ⟩$. Then the family of Cayley graphs $\{\mathrm{Cay}({\pi }_m(\Gamma ),{\pi }_m(\Omega )){\}}_m$ 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.

Major revision based on recommendations by referee reports. More details are added

References in corpus (1)

• Adelic openness without the Mumford-Tate conjecture

Cited by in corpus (3)

• Sum-product for real Lie groups
• Inducing Super-Approximation
• Sum-product phenomena: p-adic case