Sieve in expansion

TL;DR

This survey discusses recent advances in sieve methods and expansion properties of Cayley graphs of finite linear groups, highlighting work by Bourgain, Gamburd, and Sarnak on sieve in orbits.

Contribution

It provides a comprehensive overview of recent developments in sieve techniques and expansion in finite linear groups, emphasizing new methods and results.

Findings

Overview of sieve in orbits methodology

Summary of expansion properties of Cayley graphs

Discussion of recent breakthroughs by Bourgain, Gamburd, and Sarnak

Abstract

This is a survey report for the Bourbaki Seminar (Exp. no. 1028, November 2010) concerning sieve and expanders, in particular the recent works of Bourgain, Gamburd and Sarnak introducing "sieve in orbits", and the related developments concerning expansion properties of Cayley graphs of finite linear groups.