A continuum of expanders

Abstract

A regular equivalence between two graphs $\Gamma ,{\Gamma }^{\prime }$ is a pair of uniformly proper Lipschitz maps $V\Gamma \to V{\Gamma }^{\prime }$ and $V{\Gamma }^{\prime }\to V\Gamma$. 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.

arXiv comment11 pages, accepted for publication in Fundamenta Mathematicae