{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2008:7PQWA6R3CSVVOGQQT5AZPP2RHF","short_pith_number":"pith:7PQWA6R3","schema_version":"1.0","canonical_sha256":"fbe1607a3b14ab571a109f4197bf513943362f7e539fbd65875d5c928d6345cd","source":{"kind":"arxiv","id":"0812.1575","version":1},"attestation_state":"computed","paper":{"title":"Reversible biholomorphic germs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.DS","authors_text":"Anthony G. O'Farrell, Patrick Ahern","submitted_at":"2008-12-08T21:32:32Z","abstract_excerpt":"Let $G$ be a group. We say that an element $f\\in G$ is {\\em reversible in} $G$ if it is conjugate to its inverse, i.e. there exists $g\\in G$ such that $g^{-1}fg=f^{-1}$. We denote the set of reversible elements by $R(G)$. For $f\\in G$, we denote by $R_f(G)$ the set (possibly empty) of {\\em reversers} of $f$, i.e. the set of $g\\in G$ such that $g^{-1}fg=f^{-1}$. We characterise the elements of $R(G)$ and describe each $R_f(G)$, where $G$ is the the group of biholomorphic germs in one complex variable. That is, we determine all solutions to the equation $ f\\circ g\\circ f = g$, in which $f$ and $"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"0812.1575","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2008-12-08T21:32:32Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"cfc526327fc8012fea351e1e615b710e572b83e0d91c6d851837c5404c1f6d4b","abstract_canon_sha256":"b14f5ebc8b7e104144056c2f5881575deba09a4f3dc79a0d40f38dae7f285541"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:59:45.734534Z","signature_b64":"UCepwlnYU9Q2tA9fvvuoZ9QM6HQ9agmEg+sMwIoLnRAuw+2dydag3XPAlh8sg3MxHoWbwMN+n9bwOGnEO3XxCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fbe1607a3b14ab571a109f4197bf513943362f7e539fbd65875d5c928d6345cd","last_reissued_at":"2026-05-18T02:59:45.733693Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:59:45.733693Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Reversible biholomorphic germs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.DS","authors_text":"Anthony G. O'Farrell, Patrick Ahern","submitted_at":"2008-12-08T21:32:32Z","abstract_excerpt":"Let $G$ be a group. We say that an element $f\\in G$ is {\\em reversible in} $G$ if it is conjugate to its inverse, i.e. there exists $g\\in G$ such that $g^{-1}fg=f^{-1}$. We denote the set of reversible elements by $R(G)$. For $f\\in G$, we denote by $R_f(G)$ the set (possibly empty) of {\\em reversers} of $f$, i.e. the set of $g\\in G$ such that $g^{-1}fg=f^{-1}$. We characterise the elements of $R(G)$ and describe each $R_f(G)$, where $G$ is the the group of biholomorphic germs in one complex variable. That is, we determine all solutions to the equation $ f\\circ g\\circ f = g$, in which $f$ and $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0812.1575","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"0812.1575","created_at":"2026-05-18T02:59:45.733840+00:00"},{"alias_kind":"arxiv_version","alias_value":"0812.1575v1","created_at":"2026-05-18T02:59:45.733840+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0812.1575","created_at":"2026-05-18T02:59:45.733840+00:00"},{"alias_kind":"pith_short_12","alias_value":"7PQWA6R3CSVV","created_at":"2026-05-18T12:25:56.245647+00:00"},{"alias_kind":"pith_short_16","alias_value":"7PQWA6R3CSVVOGQQ","created_at":"2026-05-18T12:25:56.245647+00:00"},{"alias_kind":"pith_short_8","alias_value":"7PQWA6R3","created_at":"2026-05-18T12:25:56.245647+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7PQWA6R3CSVVOGQQT5AZPP2RHF","json":"https://pith.science/pith/7PQWA6R3CSVVOGQQT5AZPP2RHF.json","graph_json":"https://pith.science/api/pith-number/7PQWA6R3CSVVOGQQT5AZPP2RHF/graph.json","events_json":"https://pith.science/api/pith-number/7PQWA6R3CSVVOGQQT5AZPP2RHF/events.json","paper":"https://pith.science/paper/7PQWA6R3"},"agent_actions":{"view_html":"https://pith.science/pith/7PQWA6R3CSVVOGQQT5AZPP2RHF","download_json":"https://pith.science/pith/7PQWA6R3CSVVOGQQT5AZPP2RHF.json","view_paper":"https://pith.science/paper/7PQWA6R3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0812.1575&json=true","fetch_graph":"https://pith.science/api/pith-number/7PQWA6R3CSVVOGQQT5AZPP2RHF/graph.json","fetch_events":"https://pith.science/api/pith-number/7PQWA6R3CSVVOGQQT5AZPP2RHF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7PQWA6R3CSVVOGQQT5AZPP2RHF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7PQWA6R3CSVVOGQQT5AZPP2RHF/action/storage_attestation","attest_author":"https://pith.science/pith/7PQWA6R3CSVVOGQQT5AZPP2RHF/action/author_attestation","sign_citation":"https://pith.science/pith/7PQWA6R3CSVVOGQQT5AZPP2RHF/action/citation_signature","submit_replication":"https://pith.science/pith/7PQWA6R3CSVVOGQQT5AZPP2RHF/action/replication_record"}},"created_at":"2026-05-18T02:59:45.733840+00:00","updated_at":"2026-05-18T02:59:45.733840+00:00"}