{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:6DHYB2D4J2D5I2NOVCJWDECRVI","short_pith_number":"pith:6DHYB2D4","canonical_record":{"source":{"id":"1105.4908","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-05-24T23:01:08Z","cross_cats_sorted":[],"title_canon_sha256":"e0cc4d143fffa64ef9b2af8a2f0abd630969f4ae0cb34d63ebd60727fc458e8c","abstract_canon_sha256":"60b1e85cdc7474b07929366da38bb77d402619fe5ec5b170ce03c1fd6f6a5af0"},"schema_version":"1.0"},"canonical_sha256":"f0cf80e87c4e87d469aea893619051aa29801eccffd11f71cb2198b00a350da0","source":{"kind":"arxiv","id":"1105.4908","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1105.4908","created_at":"2026-05-18T02:48:26Z"},{"alias_kind":"arxiv_version","alias_value":"1105.4908v3","created_at":"2026-05-18T02:48:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.4908","created_at":"2026-05-18T02:48:26Z"},{"alias_kind":"pith_short_12","alias_value":"6DHYB2D4J2D5","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"6DHYB2D4J2D5I2NO","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"6DHYB2D4","created_at":"2026-05-18T12:26:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:6DHYB2D4J2D5I2NOVCJWDECRVI","target":"record","payload":{"canonical_record":{"source":{"id":"1105.4908","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-05-24T23:01:08Z","cross_cats_sorted":[],"title_canon_sha256":"e0cc4d143fffa64ef9b2af8a2f0abd630969f4ae0cb34d63ebd60727fc458e8c","abstract_canon_sha256":"60b1e85cdc7474b07929366da38bb77d402619fe5ec5b170ce03c1fd6f6a5af0"},"schema_version":"1.0"},"canonical_sha256":"f0cf80e87c4e87d469aea893619051aa29801eccffd11f71cb2198b00a350da0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:48:26.287785Z","signature_b64":"+hz50e63mkQcnsVAS+hkeDg6QxcrTdfZwfriYSO3ssNk/vsawWpWUSdUEHUR5n4ok/IkbpuAGk0OGABT4gxZAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f0cf80e87c4e87d469aea893619051aa29801eccffd11f71cb2198b00a350da0","last_reissued_at":"2026-05-18T02:48:26.287305Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:48:26.287305Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1105.4908","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:48:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"obiS3q1PmxLroEVTx5NswD+lIsK7R3r+n4rz3sU4fgzYcOE/KUSDikuBXiMH1ud16/JlZCG1qiv1EPO99p4kCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T09:17:00.821413Z"},"content_sha256":"86b3cce9813d35de7f703197472dc002156af8e848487a9d2249fdba6e42dcca","schema_version":"1.0","event_id":"sha256:86b3cce9813d35de7f703197472dc002156af8e848487a9d2249fdba6e42dcca"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:6DHYB2D4J2D5I2NOVCJWDECRVI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Girth Alternative for Subgroups of PL_o(I)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Azer Akhmedov","submitted_at":"2011-05-24T23:01:08Z","abstract_excerpt":"We prove the Girth Alternative for finitely generated subgroups of PL_o(I). We also prove that a finitely generated subgroup of Homeo(I) which is sufficiently rich with hyperbolic-like elements has infinite girth."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.4908","kind":"arxiv","version":3},"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":"2026-05-18T02:48:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/D2g4e9Sc0bC//HeLLKZyWajXm1w5KfTcGi7zX5iaq3xtgY4U7IgQh2jYHxKcS32aJdRdrOWgX5usAg1BDInDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T09:17:00.821759Z"},"content_sha256":"d5e221f33dc0283c359b2ee3a23a6c8935b148b26ca4f8a488715c517f25ae25","schema_version":"1.0","event_id":"sha256:d5e221f33dc0283c359b2ee3a23a6c8935b148b26ca4f8a488715c517f25ae25"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6DHYB2D4J2D5I2NOVCJWDECRVI/bundle.json","state_url":"https://pith.science/pith/6DHYB2D4J2D5I2NOVCJWDECRVI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6DHYB2D4J2D5I2NOVCJWDECRVI/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":"2026-06-22T09:17:00Z","links":{"resolver":"https://pith.science/pith/6DHYB2D4J2D5I2NOVCJWDECRVI","bundle":"https://pith.science/pith/6DHYB2D4J2D5I2NOVCJWDECRVI/bundle.json","state":"https://pith.science/pith/6DHYB2D4J2D5I2NOVCJWDECRVI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6DHYB2D4J2D5I2NOVCJWDECRVI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:6DHYB2D4J2D5I2NOVCJWDECRVI","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":"60b1e85cdc7474b07929366da38bb77d402619fe5ec5b170ce03c1fd6f6a5af0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-05-24T23:01:08Z","title_canon_sha256":"e0cc4d143fffa64ef9b2af8a2f0abd630969f4ae0cb34d63ebd60727fc458e8c"},"schema_version":"1.0","source":{"id":"1105.4908","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1105.4908","created_at":"2026-05-18T02:48:26Z"},{"alias_kind":"arxiv_version","alias_value":"1105.4908v3","created_at":"2026-05-18T02:48:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.4908","created_at":"2026-05-18T02:48:26Z"},{"alias_kind":"pith_short_12","alias_value":"6DHYB2D4J2D5","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"6DHYB2D4J2D5I2NO","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"6DHYB2D4","created_at":"2026-05-18T12:26:22Z"}],"graph_snapshots":[{"event_id":"sha256:d5e221f33dc0283c359b2ee3a23a6c8935b148b26ca4f8a488715c517f25ae25","target":"graph","created_at":"2026-05-18T02:48:26Z","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 prove the Girth Alternative for finitely generated subgroups of PL_o(I). We also prove that a finitely generated subgroup of Homeo(I) which is sufficiently rich with hyperbolic-like elements has infinite girth.","authors_text":"Azer Akhmedov","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-05-24T23:01:08Z","title":"Girth Alternative for Subgroups of PL_o(I)"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.4908","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":"sha256:86b3cce9813d35de7f703197472dc002156af8e848487a9d2249fdba6e42dcca","target":"record","created_at":"2026-05-18T02:48:26Z","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":"60b1e85cdc7474b07929366da38bb77d402619fe5ec5b170ce03c1fd6f6a5af0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-05-24T23:01:08Z","title_canon_sha256":"e0cc4d143fffa64ef9b2af8a2f0abd630969f4ae0cb34d63ebd60727fc458e8c"},"schema_version":"1.0","source":{"id":"1105.4908","kind":"arxiv","version":3}},"canonical_sha256":"f0cf80e87c4e87d469aea893619051aa29801eccffd11f71cb2198b00a350da0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f0cf80e87c4e87d469aea893619051aa29801eccffd11f71cb2198b00a350da0","first_computed_at":"2026-05-18T02:48:26.287305Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:48:26.287305Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+hz50e63mkQcnsVAS+hkeDg6QxcrTdfZwfriYSO3ssNk/vsawWpWUSdUEHUR5n4ok/IkbpuAGk0OGABT4gxZAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:48:26.287785Z","signed_message":"canonical_sha256_bytes"},"source_id":"1105.4908","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:86b3cce9813d35de7f703197472dc002156af8e848487a9d2249fdba6e42dcca","sha256:d5e221f33dc0283c359b2ee3a23a6c8935b148b26ca4f8a488715c517f25ae25"],"state_sha256":"91505804ffff353cfc34a69049e3744d5ff92532e8eb9cc026c85f77fd753bd9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mhf7/skZH4bcVqu8w0bSsRmwRUvK2egmjRGaclTqmXlkosbldxpbv4L/Es0a2x5Dh/qFgATThodWPGlppMdFCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T09:17:00.823727Z","bundle_sha256":"24707337ca025455450dc48fa6eefb9058f918d62234a42451b5928455a0a483"}}