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

Abstract

Let Ω be a finite symmetric subset of GLn(Z[1/q 0 ]), Γ := Ω , and let πm be the group homomorphism induced by the quotient map Z[1/q 0 ] → Z[1/q 0 ]/mZ[1/q 0 ]. Then the family of Cayley graphs {Cay(πm(Γ), πm(Ω))}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 Γ is perfect. Some of the immediate applications, e.g. orbit equivalence rigidity, largeness of certain ℓ-adic Galois representations, are also discussed. ContentsALIREZA SALEHI GOLSEFIDY 4.3. Bounded generation of perfect groups by commutators. 4.4. Proof of Theorem 1 for bounded powers of square-free integers.5. Super-approximation: the p-adic case. 5.1. Escape from proper subgroups. 5.2. Getting a large ideal by adding/subtracting a congruence subgroup boundedly many times. 5.3. Getting a p-adically large vector in a submodule in boundedly many steps. 5.4. Proof of super-approximation: the p-adic case. 6. Appendix A: quantitative open image for p-adic analytic maps. References