Lectures on approximate groups and Hilbert's 5th problem

TL;DR

This paper reviews the theory of approximate groups, the Gleason-Yamabe theorem, and their applications to finite group diameter bounds and graph scaling limits, connecting group structure with geometric and combinatorial properties.

Contribution

It synthesizes key ideas from approximate groups and Hilbert's 5th problem, providing insights and applications in group theory and graph theory.

Findings

Uniform diameter bounds for finite groups

Scaling limits of vertex transitive graphs

Connections between approximate groups and locally compact groups

Abstract

This paper gathers four lectures, based on a mini-course at IMA in 2014, whose aim was to discuss the structure of approximate subgroups of an arbitrary group, following the works of Hrushovski and of Green, Tao and the author. Along the way we discuss the proof of the Gleason-Yamabe theorem on Hilbert's 5th problem about the structure of locally compact groups and explain its relevance to approximate groups. We also present several applications, in particular to uniform diameter bounds for finite groups and to the determination of scaling limits of vertex transitive graphs with large diameter.