A continuum of expanders

TL;DR

This paper demonstrates the existence of a continuum of regular equivalence classes among expander graphs and certain finitely generated groups, using separation profiles to distinguish them.

Contribution

It introduces a method to classify expander graphs and groups via regular equivalence, revealing a vast diversity of such structures.

Findings

There are uncountably many regular equivalence classes of expanders.

A continuum of finitely generated groups contain expanders isometrically.

Separation profiles effectively distinguish these classes.

Abstract

A regular equivalence between two graphs $\Gamma,\Gamma'$ is a pair of uniformly proper Lipschitz maps $V\Gamma\to V\Gamma'$ and $V\Gamma'\to V\Gamma$. Using separation profiles we prove that there are $2^{\aleph_0}$ regular equivalence classes of expander graphs, and of finitely generated groups with a representative which isometrically contains expanders.