{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:OY5IRPSVI44M26RT7MMVQ3XDZQ","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":"539b8aae5ed4a60e6631da891005c60ec286dfd28e82f4c91d790f8409178818","cross_cats_sorted":["math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-11-22T06:29:20Z","title_canon_sha256":"ced697ac5df8877c098a2adb664a504a89ca6b0d151e75efe627cbc22202be12"},"schema_version":"1.0","source":{"id":"1011.4739","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.4739","created_at":"2026-05-18T04:33:06Z"},{"alias_kind":"arxiv_version","alias_value":"1011.4739v2","created_at":"2026-05-18T04:33:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.4739","created_at":"2026-05-18T04:33:06Z"},{"alias_kind":"pith_short_12","alias_value":"OY5IRPSVI44M","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_16","alias_value":"OY5IRPSVI44M26RT","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_8","alias_value":"OY5IRPSV","created_at":"2026-05-18T12:26:12Z"}],"graph_snapshots":[{"event_id":"sha256:0f3bec89fb70b5e11ccdb8ea16ed32cc778af8b699d8372226fa6c013ed56fa7","target":"graph","created_at":"2026-05-18T04:33:06Z","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 show that the first $L^2$-betti number of a finitely generated residually finite group can be estimated from below by using ordinary first betti numbers of finite index normal subgroups. As an application we construct a finitely generated infinite residually finite torsion group with positive first $L^2$-betti number.","authors_text":"D. Osin, W. L\\\"uck","cross_cats":["math.OA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-11-22T06:29:20Z","title":"Approximating the first $L^2$-betti number of residually finite groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.4739","kind":"arxiv","version":2},"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":"sha256:7459845aca3797c2b5ca83dc86c4d063a159b91ff9467e31a7d4f91dc7fd1b8a","target":"record","created_at":"2026-05-18T04:33:06Z","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":"539b8aae5ed4a60e6631da891005c60ec286dfd28e82f4c91d790f8409178818","cross_cats_sorted":["math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-11-22T06:29:20Z","title_canon_sha256":"ced697ac5df8877c098a2adb664a504a89ca6b0d151e75efe627cbc22202be12"},"schema_version":"1.0","source":{"id":"1011.4739","kind":"arxiv","version":2}},"canonical_sha256":"763a88be554738cd7a33fb19586ee3cc2d02ef9b2b2eff70b44de05740e6dc3f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"763a88be554738cd7a33fb19586ee3cc2d02ef9b2b2eff70b44de05740e6dc3f","first_computed_at":"2026-05-18T04:33:06.747962Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:33:06.747962Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7QMAeuNBvlI98/Vo5N3FM7Zrj6Wjv5QN7kNBgW7Kz+0E5dVQRD0sfJEUQophfZEXAdPVpwvjo0368E3aVHajDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:33:06.748659Z","signed_message":"canonical_sha256_bytes"},"source_id":"1011.4739","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7459845aca3797c2b5ca83dc86c4d063a159b91ff9467e31a7d4f91dc7fd1b8a","sha256:0f3bec89fb70b5e11ccdb8ea16ed32cc778af8b699d8372226fa6c013ed56fa7"],"state_sha256":"7250d3c62b2924937c958d592196bd3aa58ecf729358f7fe01d75f477643d3e5"}