{"paper":{"title":"A continuum of non-measure equivalent groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.OA"],"primary_cat":"math.GR","authors_text":"Adrian Ioana, Robin Tucker-Drob","submitted_at":"2025-12-04T07:27:34Z","abstract_excerpt":"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.\n  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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2512.04531","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2512.04531/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}