A continuum of non-measure equivalent groups

Abstract

We construct a continuum sized family $\{G_x\}_{x\in\{0,1\}^{\mathbb N}}$ of pairwise non-measure equivalent countable groups which have property (T) (hence are finitely generated), have zero $\ell^2$-Betti numbers of all orders, and are torsion-free. We also prove that the equivalence relation $\simeq^{\mathrm{fg}}_{\mathrm{ME}}$ of measure equivalence between finitely generated groups is non-smooth, resolving a question of S. Thomas. Our proof moreover shows that $\simeq^{\mathrm{fg}}_{\mathrm{ME}}$ sits above every countable Borel equivalence relation in the Borel reducibility hierarchy.