Approximate subgroups and super-strong approximation

TL;DR

This paper reviews recent advances in approximate subgroups and super-strong approximation for thin groups, highlighting key methods, classifications, and applications in spectral gaps, algebraic groups, and group sieves.

Contribution

It introduces the Bourgain-Gamburd method for spectral gaps, classifies approximate subgroups of semisimple algebraic groups over finite fields, and provides a quantitative proof of super-strong approximation.

Findings

Bourgain-Gamburd method for spectral gaps

Classification of approximate subgroups in algebraic groups

Quantitative super-strong approximation proof

Abstract

Surveying some of the recent developments on approximate subgroups and super-strong approximation for thin groups, we describe the Bourgain-Gamburd method for establishing spectral gaps for finite groups and the proof of the classification of approximate subgroups of semisimple algebraic groups over finite fields. We then give a proof of the super-strong approximation for mod $p$ quotients via random matrix products and a quantitative version of strong approximation. Some applications to the group sieve are also presented.