Groups that produce expander graphs

TL;DR

This paper surveys group properties necessary for finite groups to produce expander graphs and proves that certain classes like amenable and solvable groups do not generate expanders, highlighting limitations in detecting expansion properties.

Contribution

It generalizes previous results by showing amenable and solvable groups cannot produce expander Schreier graphs, with simplified proofs and insights into group action properties.

Findings

Amenable groups do not produce expander Schreier graphs

Solvable groups of bounded derived length do not produce expanders

Expansion properties are not detectable via abelian sections or stabilizer representations

Abstract

We survey the known group properties that a sequence of finite groups or group actions needs to satisfy to admit subsets of bounded cardinality producing expander Cayley or Schreier graphs. We prove that an infinite amenable group and solvable groups of bounded derived length do not produce expander Schreier graphs, generalizing with easier proofs results of Lubotzky and Weiss for Cayley graphs. In particular, the poor expansion properties of a group action cannot in general be detected by looking at the abelian sections or at the representations above the stabilizer of a point.