{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:PKJJPIFQYR67L2PNBW5ONTB2EO","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"231573150ddf245345f38b205b089d7ca56abc530ded5805fd48bbaeb02d513a","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-06-21T07:13:22Z","title_canon_sha256":"b5c804d48dc5b9de092827aaade6ac6fec697a187c407e90215dec6b2f607630"},"schema_version":"1.0","source":{"id":"1206.4793","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.4793","created_at":"2026-05-18T03:34:03Z"},{"alias_kind":"arxiv_version","alias_value":"1206.4793v3","created_at":"2026-05-18T03:34:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.4793","created_at":"2026-05-18T03:34:03Z"},{"alias_kind":"pith_short_12","alias_value":"PKJJPIFQYR67","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"PKJJPIFQYR67L2PN","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"PKJJPIFQ","created_at":"2026-05-18T12:27:18Z"}],"graph_snapshots":[{"event_id":"sha256:e7812f09264ef707acdf7d11ea28a83fb464fc72022494d1c37effda17053c0f","target":"graph","created_at":"2026-05-18T03:34:03Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We study vanishing results for L2-cohomology of countable groups under the presence of subgroups that satisfy some weak normality condition. As a consequence we show that the L2-Betti numbers of SL(n,R) for any infinite integral domain R vanish below degree n-1. We also give a uniform proof for the vanishing of L2-Betti numbers of Thompson's groups F and T.","authors_text":"Alex Furman, Roman Sauer, Uri Bader","cross_cats":["math.AT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-06-21T07:13:22Z","title":"Weak notions of normality and vanishing up to rank in L2-cohomology"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.4793","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:3508bf3bd1abb34e2316ea4fa7341730132a58f246341294af0b10014dc7ff61","target":"record","created_at":"2026-05-18T03:34:03Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"231573150ddf245345f38b205b089d7ca56abc530ded5805fd48bbaeb02d513a","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-06-21T07:13:22Z","title_canon_sha256":"b5c804d48dc5b9de092827aaade6ac6fec697a187c407e90215dec6b2f607630"},"schema_version":"1.0","source":{"id":"1206.4793","kind":"arxiv","version":3}},"canonical_sha256":"7a9297a0b0c47df5e9ed0dbae6cc3a23bebc9ac794726894c8657c733759ad39","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7a9297a0b0c47df5e9ed0dbae6cc3a23bebc9ac794726894c8657c733759ad39","first_computed_at":"2026-05-18T03:34:03.168130Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:34:03.168130Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/QX4mc1v42baB7Lq/DQkQqQgU6nFuHcskG6ahGpbqsfsRDwuAofsoEC/c7YT8Q081UErS3fZ5uovR3jLtgAQCA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:34:03.168906Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.4793","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3508bf3bd1abb34e2316ea4fa7341730132a58f246341294af0b10014dc7ff61","sha256:e7812f09264ef707acdf7d11ea28a83fb464fc72022494d1c37effda17053c0f"],"state_sha256":"36f5263fd9c221a6b20db170208b27513dd4516dfde8e2e2cab5181bf046f29c"}