{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:GZQPEOSEC2OXZCPHQVDFKMLZJ6","short_pith_number":"pith:GZQPEOSE","canonical_record":{"source":{"id":"1312.0129","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-11-30T17:29:54Z","cross_cats_sorted":[],"title_canon_sha256":"bf631711a1a4b16c736f82952724f60185f4b48f6856c6c9860baddf1709a182","abstract_canon_sha256":"b2e4cd30c08aff17ed94fb1e2833730078431dd7ec26b4e45de1f31f0e912f71"},"schema_version":"1.0"},"canonical_sha256":"3660f23a44169d7c89e785465531794fb96ce8b2167942a62c1d1a3b3d6802bc","source":{"kind":"arxiv","id":"1312.0129","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.0129","created_at":"2026-05-18T02:46:31Z"},{"alias_kind":"arxiv_version","alias_value":"1312.0129v2","created_at":"2026-05-18T02:46:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.0129","created_at":"2026-05-18T02:46:31Z"},{"alias_kind":"pith_short_12","alias_value":"GZQPEOSEC2OX","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"GZQPEOSEC2OXZCPH","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"GZQPEOSE","created_at":"2026-05-18T12:27:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:GZQPEOSEC2OXZCPHQVDFKMLZJ6","target":"record","payload":{"canonical_record":{"source":{"id":"1312.0129","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-11-30T17:29:54Z","cross_cats_sorted":[],"title_canon_sha256":"bf631711a1a4b16c736f82952724f60185f4b48f6856c6c9860baddf1709a182","abstract_canon_sha256":"b2e4cd30c08aff17ed94fb1e2833730078431dd7ec26b4e45de1f31f0e912f71"},"schema_version":"1.0"},"canonical_sha256":"3660f23a44169d7c89e785465531794fb96ce8b2167942a62c1d1a3b3d6802bc","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:46:31.116010Z","signature_b64":"G7eY1GZhSKEfdkdEueYkB9ofieHVUFLcnXF2tgCBccCjVvuLz9H5hCcmmK40zmLcw5U3fUC41ZxtPZHiDARkBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3660f23a44169d7c89e785465531794fb96ce8b2167942a62c1d1a3b3d6802bc","last_reissued_at":"2026-05-18T02:46:31.115606Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:46:31.115606Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1312.0129","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:46:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Kt9M1kNT4uUrMymibAQRFMrJJchxDlcl/vjf2Q1FkmYeMYW5556ztHCseKtVDtzCiiIJt2/d2Y83vIwjW06vAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T10:21:12.053507Z"},"content_sha256":"87a3bb8df29f0f5678cbdf9518a01ac90a5665e7bba7f011fe6ee9390387ebf1","schema_version":"1.0","event_id":"sha256:87a3bb8df29f0f5678cbdf9518a01ac90a5665e7bba7f011fe6ee9390387ebf1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:GZQPEOSEC2OXZCPHQVDFKMLZJ6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Subnormal subgroups in free groups, their growth and cogrowth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Alexander Olshanskii","submitted_at":"2013-11-30T17:29:54Z","abstract_excerpt":"In this paper, the author (1) compares subnormal closures of finite sets in free groups; (2) proves that the exponential growth rate (e.g.r.), i.e., the limit of the n-th roots of g(n), where g(n) is the growth function of a subgroup H with respect to a finite free basis of F, exists for any subgroup H of the free group F; (3) gives sharp estimates from below for the e.g.r. of subnormal subgroups in free groups; and (4) finds cogrowth for the subnormal closures of free generators in F."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.0129","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:46:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WiWt7O23j3D2RV7pDPo8PzjeCwWe5ElKm4/rTo5EokJ5qS4b2+J+uoXHg1txgWvGWXurAZBUOEJ4lvh9eHNaDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T10:21:12.053857Z"},"content_sha256":"34fe33b5307dfa6099f5d166a1cc81ac858bb4fb11306487bee7eaf1badc007c","schema_version":"1.0","event_id":"sha256:34fe33b5307dfa6099f5d166a1cc81ac858bb4fb11306487bee7eaf1badc007c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GZQPEOSEC2OXZCPHQVDFKMLZJ6/bundle.json","state_url":"https://pith.science/pith/GZQPEOSEC2OXZCPHQVDFKMLZJ6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GZQPEOSEC2OXZCPHQVDFKMLZJ6/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-26T10:21:12Z","links":{"resolver":"https://pith.science/pith/GZQPEOSEC2OXZCPHQVDFKMLZJ6","bundle":"https://pith.science/pith/GZQPEOSEC2OXZCPHQVDFKMLZJ6/bundle.json","state":"https://pith.science/pith/GZQPEOSEC2OXZCPHQVDFKMLZJ6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GZQPEOSEC2OXZCPHQVDFKMLZJ6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:GZQPEOSEC2OXZCPHQVDFKMLZJ6","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":"b2e4cd30c08aff17ed94fb1e2833730078431dd7ec26b4e45de1f31f0e912f71","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-11-30T17:29:54Z","title_canon_sha256":"bf631711a1a4b16c736f82952724f60185f4b48f6856c6c9860baddf1709a182"},"schema_version":"1.0","source":{"id":"1312.0129","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.0129","created_at":"2026-05-18T02:46:31Z"},{"alias_kind":"arxiv_version","alias_value":"1312.0129v2","created_at":"2026-05-18T02:46:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.0129","created_at":"2026-05-18T02:46:31Z"},{"alias_kind":"pith_short_12","alias_value":"GZQPEOSEC2OX","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"GZQPEOSEC2OXZCPH","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"GZQPEOSE","created_at":"2026-05-18T12:27:46Z"}],"graph_snapshots":[{"event_id":"sha256:34fe33b5307dfa6099f5d166a1cc81ac858bb4fb11306487bee7eaf1badc007c","target":"graph","created_at":"2026-05-18T02:46:31Z","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":"In this paper, the author (1) compares subnormal closures of finite sets in free groups; (2) proves that the exponential growth rate (e.g.r.), i.e., the limit of the n-th roots of g(n), where g(n) is the growth function of a subgroup H with respect to a finite free basis of F, exists for any subgroup H of the free group F; (3) gives sharp estimates from below for the e.g.r. of subnormal subgroups in free groups; and (4) finds cogrowth for the subnormal closures of free generators in F.","authors_text":"Alexander Olshanskii","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-11-30T17:29:54Z","title":"Subnormal subgroups in free groups, their growth and cogrowth"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.0129","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":"sha256:87a3bb8df29f0f5678cbdf9518a01ac90a5665e7bba7f011fe6ee9390387ebf1","target":"record","created_at":"2026-05-18T02:46:31Z","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":"b2e4cd30c08aff17ed94fb1e2833730078431dd7ec26b4e45de1f31f0e912f71","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-11-30T17:29:54Z","title_canon_sha256":"bf631711a1a4b16c736f82952724f60185f4b48f6856c6c9860baddf1709a182"},"schema_version":"1.0","source":{"id":"1312.0129","kind":"arxiv","version":2}},"canonical_sha256":"3660f23a44169d7c89e785465531794fb96ce8b2167942a62c1d1a3b3d6802bc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3660f23a44169d7c89e785465531794fb96ce8b2167942a62c1d1a3b3d6802bc","first_computed_at":"2026-05-18T02:46:31.115606Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:46:31.115606Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"G7eY1GZhSKEfdkdEueYkB9ofieHVUFLcnXF2tgCBccCjVvuLz9H5hCcmmK40zmLcw5U3fUC41ZxtPZHiDARkBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:46:31.116010Z","signed_message":"canonical_sha256_bytes"},"source_id":"1312.0129","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:87a3bb8df29f0f5678cbdf9518a01ac90a5665e7bba7f011fe6ee9390387ebf1","sha256:34fe33b5307dfa6099f5d166a1cc81ac858bb4fb11306487bee7eaf1badc007c"],"state_sha256":"5d7ed98d4e9d58a310d339d7b2ce9e8d92e171c38709de3532324452ffa65c4d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uvyjZmdQkgsYk9aCVIzxyLKvFNEC+dKMlvub0w1pN3ntg20xdblpk5PmPMtkpTN+dmz8axPr+blseLfxiwhACw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T10:21:12.056161Z","bundle_sha256":"4f3387b4b59b289b2151acc1bca8fa61ea86413b770df30d2427a326ce04deb2"}}