A continuum of non-measure equivalent groups

2025/12/04 by Adrian Ioana, Ioana, Adrian, Robin Tucker-Drob +1
Mathematics · #Advanced Topology and Set Theory #Advanced Operator Algebra Research #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2512.04531

Abstract

We construct a continuum sized family \G_x\_x∈\0,1\^{\mathbb N} of pairwise non-measure equivalent countable groups which have property (T) (hence are finitely generated), have zero ℓ²-Betti numbers of all orders, and are torsion-free. We also prove that the equivalence relation ≃^fg_ME of measure equivalence between finitely generated groups is non-smooth, resolving a question of S. Thomas. Our proof moreover shows that ≃^fg_ME sits above every countable Borel equivalence relation in the Borel reducibility hierarchy.

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• Higher property T and below-rank phenomena of lattices

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