Super-approximation II

I will explain the Bourgain-Gamburd method of proving spectral gap. The spirit of this method is behind all the recent results on this subject. Very roughly it says in order to prove spectral gap it is enough to show that a “generic” subset with positive “box dimension” generates a “large” subgroup in a controlled number of steps (bounded generation). The vague terms will be explained, and we will see their connections with understanding the structure of a “generic” approximate subgroup with positive “box dimension”.