Growth in groups: ideas and perspectives

TL;DR

This survey reviews recent methods in proving growth in non-commutative groups, highlighting techniques from additive combinatorics and group theory with examples like linear algebraic and permutation groups.

Contribution

It provides a comprehensive overview of recent advances and ideas in growth results for non-commutative groups, emphasizing methodological insights.

Findings

Techniques from additive combinatorics and group theory are central to recent growth proofs.

Linear algebraic groups, such as SL_2(Z/pZ), serve as key examples.

The survey clarifies the underlying ideas behind these methods.

Abstract

This is a survey of methods developed in the last few years to prove results on growth in non-commutative groups. These techniques have their roots in both additive combinatorics and group theory, as well as other fields. We discuss linear algebraic groups, with SL_2(Z/pZ) as the basic example, as well as permutation groups. The emphasis lies on the ideas behind the methods.