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Here, $u_{r_{m,i}}$ is the single relator of the upper presentation of the $2$-bridge link group of slope $r_{m,i}$, where $r_{m,0}=[m+1,m,m]$ and $r_{m,i}=[m+1,m-1,(i-1)\\langle m \\rangle,m+1,m]$ in continued fraction expansion for every integer $i \\ge 1$."},"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":"1609.04288","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-09-14T14:23:10Z","cross_cats_sorted":[],"title_canon_sha256":"b65780dcdf658681e679698a83e36b353cfa77d914ea748f36a046872f4a7b43","abstract_canon_sha256":"2ade3658964285cd9c0a54458fd0583359485023d3e616085a50ac159fd374b9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:38.374865Z","signature_b64":"3ZXv9N3e7AW8UFxxb5kXU1BN80bKCsagRJREvDP7d07Qm2LftGRPudmj6mcdS7fkkE6D2Ul/4MDeaDdm35xvCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6103d71edb5853db585f7f192bf29cefae86dfe6737a266422ae39fdf024e006","last_reissued_at":"2026-05-18T01:04:38.374149Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:38.374149Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A family of two generator non-Hopfian groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Donghi Lee, Makoto Sakuma","submitted_at":"2016-09-14T14:23:10Z","abstract_excerpt":"We construct $2$-generator non-Hopfian groups $G_m, m=3, 4, 5, \\dots$, where each $G_m$ has a specific presentation $G_m=\\langle a, b \\, | \\, u_{r_{m,0}}=u_{r_{m,1}}=u_{r_{m,2}}= \\cdots =1 \\rangle$ which satisfies small cancellation conditions $C(4)$ and $T(4)$. Here, $u_{r_{m,i}}$ is the single relator of the upper presentation of the $2$-bridge link group of slope $r_{m,i}$, where $r_{m,0}=[m+1,m,m]$ and $r_{m,i}=[m+1,m-1,(i-1)\\langle m \\rangle,m+1,m]$ in continued fraction expansion for every integer $i \\ge 1$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.04288","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":"1609.04288","created_at":"2026-05-18T01:04:38.374284+00:00"},{"alias_kind":"arxiv_version","alias_value":"1609.04288v1","created_at":"2026-05-18T01:04:38.374284+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.04288","created_at":"2026-05-18T01:04:38.374284+00:00"},{"alias_kind":"pith_short_12","alias_value":"MEB5OHW3LBJ5","created_at":"2026-05-18T12:30:32.724797+00:00"},{"alias_kind":"pith_short_16","alias_value":"MEB5OHW3LBJ5WWC7","created_at":"2026-05-18T12:30:32.724797+00:00"},{"alias_kind":"pith_short_8","alias_value":"MEB5OHW3","created_at":"2026-05-18T12:30:32.724797+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/MEB5OHW3LBJ5WWC7P4MSX4U456","json":"https://pith.science/pith/MEB5OHW3LBJ5WWC7P4MSX4U456.json","graph_json":"https://pith.science/api/pith-number/MEB5OHW3LBJ5WWC7P4MSX4U456/graph.json","events_json":"https://pith.science/api/pith-number/MEB5OHW3LBJ5WWC7P4MSX4U456/events.json","paper":"https://pith.science/paper/MEB5OHW3"},"agent_actions":{"view_html":"https://pith.science/pith/MEB5OHW3LBJ5WWC7P4MSX4U456","download_json":"https://pith.science/pith/MEB5OHW3LBJ5WWC7P4MSX4U456.json","view_paper":"https://pith.science/paper/MEB5OHW3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1609.04288&json=true","fetch_graph":"https://pith.science/api/pith-number/MEB5OHW3LBJ5WWC7P4MSX4U456/graph.json","fetch_events":"https://pith.science/api/pith-number/MEB5OHW3LBJ5WWC7P4MSX4U456/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MEB5OHW3LBJ5WWC7P4MSX4U456/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MEB5OHW3LBJ5WWC7P4MSX4U456/action/storage_attestation","attest_author":"https://pith.science/pith/MEB5OHW3LBJ5WWC7P4MSX4U456/action/author_attestation","sign_citation":"https://pith.science/pith/MEB5OHW3LBJ5WWC7P4MSX4U456/action/citation_signature","submit_replication":"https://pith.science/pith/MEB5OHW3LBJ5WWC7P4MSX4U456/action/replication_record"}},"created_at":"2026-05-18T01:04:38.374284+00:00","updated_at":"2026-05-18T01:04:38.374284+00:00"}