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 ${\simeq }_{\mathrm{ME}}^{\mathrm{fg}}$ of measure equivalence between finitely generated groups is non-smooth, resolving a question of S. Thomas. Our proof moreover shows that ${\simeq }_{\mathrm{ME}}^{\mathrm{fg}}$ sits above every countable Borel equivalence relation in the Borel reducibility hierarchy.

Added a new result showing that the ME-relation on the space of finitely generated groups is not smooth