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In this paper, we show that, for any r >= 1, there exists a non-elementary Kleinian group whose Jorgensen number is equal to r. This answers a question posed by Oichi and Sato. We also present our computer generated picture which estimates Jorgensen numbers from above in the diagonal slice of Schottky space."},"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":"1703.07732","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2017-03-22T16:26:27Z","cross_cats_sorted":[],"title_canon_sha256":"4909682b32c6aae6a8254c374fc5ef085d86fc8854e8cfdf91ac689c3b094dca","abstract_canon_sha256":"11c4aebeba139a32dbd9eb8ef10d211debd296dfdfcd9cda7d3de579c7c729a7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:52:44.544155Z","signature_b64":"qX8Na3Ztu2P5zmXgRZVvw0eemYOmr4Z/etoXQHb8+cyC+qU3IE08VNET5Tnn/1AFIC+RzJnyzgBQwkprs1zCBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"77066bbf0ad6315fcc646cbe75b28abb7b424539c4db635e6b0afbba7d2225b6","last_reissued_at":"2026-05-17T23:52:44.543644Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:52:44.543644Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The realization problem for J{\\o}rgensen numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Ryosuke Yamazaki, Yasushi Yamashita","submitted_at":"2017-03-22T16:26:27Z","abstract_excerpt":"Let G be a two generator subgroup of PSL(2,C). The Jorgensen number J(G) of G is defined by\n  J(G)=inf{ |tr^2 A-4|+|tr[A,B]-2| ; G=<A,B>}.\n  If G is a non-elementary Kleinian group, then J(G) >= 1. This inequality is called Jorgensen's inequality. In this paper, we show that, for any r >= 1, there exists a non-elementary Kleinian group whose Jorgensen number is equal to r. This answers a question posed by Oichi and Sato. We also present our computer generated picture which estimates Jorgensen numbers from above in the diagonal slice of Schottky space."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.07732","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":"1703.07732","created_at":"2026-05-17T23:52:44.543734+00:00"},{"alias_kind":"arxiv_version","alias_value":"1703.07732v1","created_at":"2026-05-17T23:52:44.543734+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.07732","created_at":"2026-05-17T23:52:44.543734+00:00"},{"alias_kind":"pith_short_12","alias_value":"O4DGXPYK2YYV","created_at":"2026-05-18T12:31:34.259226+00:00"},{"alias_kind":"pith_short_16","alias_value":"O4DGXPYK2YYV7TDE","created_at":"2026-05-18T12:31:34.259226+00:00"},{"alias_kind":"pith_short_8","alias_value":"O4DGXPYK","created_at":"2026-05-18T12:31:34.259226+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/O4DGXPYK2YYV7TDENS7HLMUKXN","json":"https://pith.science/pith/O4DGXPYK2YYV7TDENS7HLMUKXN.json","graph_json":"https://pith.science/api/pith-number/O4DGXPYK2YYV7TDENS7HLMUKXN/graph.json","events_json":"https://pith.science/api/pith-number/O4DGXPYK2YYV7TDENS7HLMUKXN/events.json","paper":"https://pith.science/paper/O4DGXPYK"},"agent_actions":{"view_html":"https://pith.science/pith/O4DGXPYK2YYV7TDENS7HLMUKXN","download_json":"https://pith.science/pith/O4DGXPYK2YYV7TDENS7HLMUKXN.json","view_paper":"https://pith.science/paper/O4DGXPYK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1703.07732&json=true","fetch_graph":"https://pith.science/api/pith-number/O4DGXPYK2YYV7TDENS7HLMUKXN/graph.json","fetch_events":"https://pith.science/api/pith-number/O4DGXPYK2YYV7TDENS7HLMUKXN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/O4DGXPYK2YYV7TDENS7HLMUKXN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/O4DGXPYK2YYV7TDENS7HLMUKXN/action/storage_attestation","attest_author":"https://pith.science/pith/O4DGXPYK2YYV7TDENS7HLMUKXN/action/author_attestation","sign_citation":"https://pith.science/pith/O4DGXPYK2YYV7TDENS7HLMUKXN/action/citation_signature","submit_replication":"https://pith.science/pith/O4DGXPYK2YYV7TDENS7HLMUKXN/action/replication_record"}},"created_at":"2026-05-17T23:52:44.543734+00:00","updated_at":"2026-05-17T23:52:44.543734+00:00"}