Super-approximation I

Abstract: Let Γ be a finitely generated subgroup of a compact group G. 1. I will recall what it means to say the action of Γ on G by left translation has spectral gap, and mention some examples and applications, e.g. Banach-Ruziewicz problem, explicit construction of expanders. 2. I will give the precise formulation of super-approximation conjecture and present the best known results. 3. I will mention some new applications, e.g. orbit equivalence rigidity, variation of Galois representations.