In this talk we introduce the Apollonian group, sometimes coined the “quintessential”
thin group, which is the underlying symmetry group of Apollonian circle packings. We
review some of the exciting results that have been proven with respect to number-theoretic
aspects of these packings, and indicate where super-approximation plays a role.
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
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”.