A continuum of expanders

A regular equivalence between two graphs $Γ,Γ'$ is a pair of uniformly proper Lipschitz maps $VΓ\to VΓ'$ and $VΓ'\to VΓ$. 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.

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