Expansion in simple groups

TL;DR

This paper surveys the development of expansion and spectral gap phenomena in finite and compact simple groups, highlighting key contributions and open problems in the field.

Contribution

It provides a comprehensive overview of the history and current challenges in the study of expansion in simple groups, emphasizing the impact of Margulis's seminal work.

Findings

Exploration of the use of Kazhdan's property (T) in constructing expander graphs.

Analysis of the uniqueness of invariant means on compact simple Lie groups.

Identification of open problems in the area of expansion and spectral gaps.

Abstract

Two short seminal papers of Margulis used Kazhdan's property $(T)$ to give, on the one hand, explicit constructions of expander graphs, and to prove, on the other hand, the uniqueness of some invariant means on compact simple Lie groups. These papers opened a rich line of research on expansion and spectral gap phenomena in finite and compact simple groups. In this paper we survey the history of this area and point out a number of problems which are still open.