{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:VATD5NY7BHAMTRFSG3RS3NFN4E","short_pith_number":"pith:VATD5NY7","schema_version":"1.0","canonical_sha256":"a8263eb71f09c0c9c4b236e32db4ade1212c059a6ad456a05b0e3743f0ab65c4","source":{"kind":"arxiv","id":"2512.04531","version":2},"attestation_state":"computed","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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2512.04531","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2025-12-04T07:27:34Z","cross_cats_sorted":["math.DS","math.OA"],"title_canon_sha256":"ca9237cc76222d6677e0f862329656b6745f6d5665d4597eb0cbdc2aee772860","abstract_canon_sha256":"297a4664951adc9f5163dfe4719519dee21e4cd2e29b0509b1240142f0be95cc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-14T00:18:21.062148Z","signature_b64":"NF4lGllAFyh3CqDCGOB3rLWisXfFfDBCNdFJxBQ27v1AxbttFa2YrKWbuZxuooF3wHDp5LxnAY4xXd2ej61ECA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a8263eb71f09c0c9c4b236e32db4ade1212c059a6ad456a05b0e3743f0ab65c4","last_reissued_at":"2026-07-14T00:18:21.061272Z","signature_status":"signed_v1","first_computed_at":"2026-07-14T00:18:21.061272Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2512.04531","created_at":"2026-07-14T00:18:21.061684+00:00"},{"alias_kind":"arxiv_version","alias_value":"2512.04531v2","created_at":"2026-07-14T00:18:21.061684+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2512.04531","created_at":"2026-07-14T00:18:21.061684+00:00"},{"alias_kind":"pith_short_12","alias_value":"VATD5NY7BHAM","created_at":"2026-07-14T00:18:21.061684+00:00"},{"alias_kind":"pith_short_16","alias_value":"VATD5NY7BHAMTRFS","created_at":"2026-07-14T00:18:21.061684+00:00"},{"alias_kind":"pith_short_8","alias_value":"VATD5NY7","created_at":"2026-07-14T00:18:21.061684+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2601.00074","citing_title":"Kazhdan groups of dimension $16$ with prescribed second $\\ell^2$-Betti number","ref_index":9,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VATD5NY7BHAMTRFSG3RS3NFN4E","json":"https://pith.science/pith/VATD5NY7BHAMTRFSG3RS3NFN4E.json","graph_json":"https://pith.science/api/pith-number/VATD5NY7BHAMTRFSG3RS3NFN4E/graph.json","events_json":"https://pith.science/api/pith-number/VATD5NY7BHAMTRFSG3RS3NFN4E/events.json","paper":"https://pith.science/paper/VATD5NY7"},"agent_actions":{"view_html":"https://pith.science/pith/VATD5NY7BHAMTRFSG3RS3NFN4E","download_json":"https://pith.science/pith/VATD5NY7BHAMTRFSG3RS3NFN4E.json","view_paper":"https://pith.science/paper/VATD5NY7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2512.04531&json=true","fetch_graph":"https://pith.science/api/pith-number/VATD5NY7BHAMTRFSG3RS3NFN4E/graph.json","fetch_events":"https://pith.science/api/pith-number/VATD5NY7BHAMTRFSG3RS3NFN4E/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VATD5NY7BHAMTRFSG3RS3NFN4E/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VATD5NY7BHAMTRFSG3RS3NFN4E/action/storage_attestation","attest_author":"https://pith.science/pith/VATD5NY7BHAMTRFSG3RS3NFN4E/action/author_attestation","sign_citation":"https://pith.science/pith/VATD5NY7BHAMTRFSG3RS3NFN4E/action/citation_signature","submit_replication":"https://pith.science/pith/VATD5NY7BHAMTRFSG3RS3NFN4E/action/replication_record"}},"created_at":"2026-07-14T00:18:21.061684+00:00","updated_at":"2026-07-14T00:18:21.061684+00:00"}