{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:XBJZ6WRXXUHEHHG7T6KNGMSUVO","short_pith_number":"pith:XBJZ6WRX","canonical_record":{"source":{"id":"1004.5308","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-04-29T14:41:17Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"d6e3aac5fc23660a53a7f97841233d3c99ed452546dbb0788bf6589cd2f4179c","abstract_canon_sha256":"58ae3d07bc11dee2a4f4f7527758b0fe366c7aeeaf990d816059ec43b1fbd564"},"schema_version":"1.0"},"canonical_sha256":"b8539f5a37bd0e439cdf9f94d33254ab8e3ec08e48216fc9b6ce5b2c9bb661f3","source":{"kind":"arxiv","id":"1004.5308","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.5308","created_at":"2026-05-18T02:24:15Z"},{"alias_kind":"arxiv_version","alias_value":"1004.5308v2","created_at":"2026-05-18T02:24:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.5308","created_at":"2026-05-18T02:24:15Z"},{"alias_kind":"pith_short_12","alias_value":"XBJZ6WRXXUHE","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"XBJZ6WRXXUHEHHG7","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"XBJZ6WRX","created_at":"2026-05-18T12:26:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:XBJZ6WRXXUHEHHG7T6KNGMSUVO","target":"record","payload":{"canonical_record":{"source":{"id":"1004.5308","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-04-29T14:41:17Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"d6e3aac5fc23660a53a7f97841233d3c99ed452546dbb0788bf6589cd2f4179c","abstract_canon_sha256":"58ae3d07bc11dee2a4f4f7527758b0fe366c7aeeaf990d816059ec43b1fbd564"},"schema_version":"1.0"},"canonical_sha256":"b8539f5a37bd0e439cdf9f94d33254ab8e3ec08e48216fc9b6ce5b2c9bb661f3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:24:15.026613Z","signature_b64":"ETChH6royzEv/iMuK5Ml/AXvD2OYQAJ0CT2wY+KL1+wDFXhqUcMpMLCIVaZY5yuBYNoi31PDfocYy8qL4brXBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b8539f5a37bd0e439cdf9f94d33254ab8e3ec08e48216fc9b6ce5b2c9bb661f3","last_reissued_at":"2026-05-18T02:24:15.025951Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:24:15.025951Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1004.5308","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":"2026-05-18T02:24:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8snndZE7UXWGDBO2wncq1Efrf4q2xUn1pyge3lQassWXzk8unmsanqrzRB6Nz/wm5IBSeiWWSlS9DbczMZEyCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T15:59:08.798096Z"},"content_sha256":"e8dfcfde4ee3fb81050a581946377364643063b9a8ee43738a623df4063a8538","schema_version":"1.0","event_id":"sha256:e8dfcfde4ee3fb81050a581946377364643063b9a8ee43738a623df4063a8538"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:XBJZ6WRXXUHEHHG7T6KNGMSUVO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Periodic elements in Garside groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.GT","authors_text":"Eon-Kyung Lee, Sang-Jin Lee","submitted_at":"2010-04-29T14:41:17Z","abstract_excerpt":"Let $G$ be a Garside group with Garside element $\\Delta$, and let $\\Delta^m$ be the minimal positive central power of $\\Delta$. An element $g\\in G$ is said to be 'periodic' if some power of it is a power of $\\Delta$. In this paper, we study periodic elements in Garside groups and their conjugacy classes.\n  We show that the periodicity of an element does not depend on the choice of a particular Garside structure if and only if the center of $G$ is cyclic; if $g^k=\\Delta^{ka}$ for some nonzero integer $k$, then $g$ is conjugate to $\\Delta^a$; every finite subgroup of the quotient group $G/<\\Delt"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.5308","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":"2026-05-18T02:24:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zXWtO7UE08Gu2ZMKCBFQ3oAhi4bjGMebK3U7ErEewlkX0MfIJFQblbBWPvZ293q5QRmGe1cnAG6wc5iCpHKfCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T15:59:08.798451Z"},"content_sha256":"994cd3f06e62e36ee8a2f464f0ca4698d84d74fbba8ecba5c822798c7caf7c06","schema_version":"1.0","event_id":"sha256:994cd3f06e62e36ee8a2f464f0ca4698d84d74fbba8ecba5c822798c7caf7c06"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XBJZ6WRXXUHEHHG7T6KNGMSUVO/bundle.json","state_url":"https://pith.science/pith/XBJZ6WRXXUHEHHG7T6KNGMSUVO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XBJZ6WRXXUHEHHG7T6KNGMSUVO/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":"2026-06-23T15:59:08Z","links":{"resolver":"https://pith.science/pith/XBJZ6WRXXUHEHHG7T6KNGMSUVO","bundle":"https://pith.science/pith/XBJZ6WRXXUHEHHG7T6KNGMSUVO/bundle.json","state":"https://pith.science/pith/XBJZ6WRXXUHEHHG7T6KNGMSUVO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XBJZ6WRXXUHEHHG7T6KNGMSUVO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:XBJZ6WRXXUHEHHG7T6KNGMSUVO","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":"58ae3d07bc11dee2a4f4f7527758b0fe366c7aeeaf990d816059ec43b1fbd564","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-04-29T14:41:17Z","title_canon_sha256":"d6e3aac5fc23660a53a7f97841233d3c99ed452546dbb0788bf6589cd2f4179c"},"schema_version":"1.0","source":{"id":"1004.5308","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.5308","created_at":"2026-05-18T02:24:15Z"},{"alias_kind":"arxiv_version","alias_value":"1004.5308v2","created_at":"2026-05-18T02:24:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.5308","created_at":"2026-05-18T02:24:15Z"},{"alias_kind":"pith_short_12","alias_value":"XBJZ6WRXXUHE","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"XBJZ6WRXXUHEHHG7","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"XBJZ6WRX","created_at":"2026-05-18T12:26:17Z"}],"graph_snapshots":[{"event_id":"sha256:994cd3f06e62e36ee8a2f464f0ca4698d84d74fbba8ecba5c822798c7caf7c06","target":"graph","created_at":"2026-05-18T02:24:15Z","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":"Let $G$ be a Garside group with Garside element $\\Delta$, and let $\\Delta^m$ be the minimal positive central power of $\\Delta$. An element $g\\in G$ is said to be 'periodic' if some power of it is a power of $\\Delta$. In this paper, we study periodic elements in Garside groups and their conjugacy classes.\n  We show that the periodicity of an element does not depend on the choice of a particular Garside structure if and only if the center of $G$ is cyclic; if $g^k=\\Delta^{ka}$ for some nonzero integer $k$, then $g$ is conjugate to $\\Delta^a$; every finite subgroup of the quotient group $G/<\\Delt","authors_text":"Eon-Kyung Lee, Sang-Jin Lee","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-04-29T14:41:17Z","title":"Periodic elements in Garside groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.5308","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:e8dfcfde4ee3fb81050a581946377364643063b9a8ee43738a623df4063a8538","target":"record","created_at":"2026-05-18T02:24:15Z","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":"58ae3d07bc11dee2a4f4f7527758b0fe366c7aeeaf990d816059ec43b1fbd564","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-04-29T14:41:17Z","title_canon_sha256":"d6e3aac5fc23660a53a7f97841233d3c99ed452546dbb0788bf6589cd2f4179c"},"schema_version":"1.0","source":{"id":"1004.5308","kind":"arxiv","version":2}},"canonical_sha256":"b8539f5a37bd0e439cdf9f94d33254ab8e3ec08e48216fc9b6ce5b2c9bb661f3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b8539f5a37bd0e439cdf9f94d33254ab8e3ec08e48216fc9b6ce5b2c9bb661f3","first_computed_at":"2026-05-18T02:24:15.025951Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:24:15.025951Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ETChH6royzEv/iMuK5Ml/AXvD2OYQAJ0CT2wY+KL1+wDFXhqUcMpMLCIVaZY5yuBYNoi31PDfocYy8qL4brXBA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:24:15.026613Z","signed_message":"canonical_sha256_bytes"},"source_id":"1004.5308","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e8dfcfde4ee3fb81050a581946377364643063b9a8ee43738a623df4063a8538","sha256:994cd3f06e62e36ee8a2f464f0ca4698d84d74fbba8ecba5c822798c7caf7c06"],"state_sha256":"eacd610ea4363599f6076cdf2f31d134db15a37c35efe93e759a12d4fb5945bf"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fj1hRIzEmJ1wljaMBJWKZ0w2Qmc2rNUBi4gl9iuwp2+kIVEu5OtcQx2/aCe+JvjukBcAIGVpiEpgjB1ILkfeBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T15:59:08.800343Z","bundle_sha256":"bea23b9e4456c70674d4a0a34d78dd1501a7ac2d4eded9f51b8fa7337f6139b4"}}