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Thus $\\Gamma$ has a presentation of the form $\\Gamma\\cong\\langle f,g: f^4=g^p=w(f,g)=\\cdots=1 \\rangle$ We find that necessarily $p\\in \\{2,3,4,5,6,\\infty\\}$, where $p=\\infty$ denotes that $g$ is a parabolic element, the total degree of the invariant trace field $k\\Gamma=\\mathbb{Q}(\\{\\tr^2(h):h\\in\\Gamma\\})$ is at most $4$, and each orbifold is either a two bridge link of slope $r/s$ "},"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":"2206.14174","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2022-06-28T17:41:00Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"0b930da3e1d1ba6774417cbf3c4db3ba3e86e7ee115a6336a063f8ab5c983858","abstract_canon_sha256":"513ebb93cb82de5bd867ebe9f840f72659d17c024e22811f45b92809ba33c0a3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:35:53.803763Z","signature_b64":"Y3apW59UfA6UlrzfdID/b+sAhWsS59ZMkhcp9wI6bDnxyTFQEhGJNVAnFkv5tzOgg8nf2RPTaNOJX3sNh/aZBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ec1c0dde2735b3dfb6990ed9869b2e659e2631a9f708727917a25760e14fa246","last_reissued_at":"2026-07-05T04:35:53.803365Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:35:53.803365Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The $(4,p)$-arithmetic hyperbolic lattices, $p\\geq 2$, in three dimensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.GT","authors_text":"G.J. Martin, K. Salehi, Y. Yamashita","submitted_at":"2022-06-28T17:41:00Z","abstract_excerpt":"We identify the finitely many arithmetic lattices $\\Gamma$ in the orientation preserving isometry group of hyperbolic $3$-space $\\mathbb{H}^3$ generated by an element of order $4$ and and element of order $p\\geq 2$. Thus $\\Gamma$ has a presentation of the form $\\Gamma\\cong\\langle f,g: f^4=g^p=w(f,g)=\\cdots=1 \\rangle$ We find that necessarily $p\\in \\{2,3,4,5,6,\\infty\\}$, where $p=\\infty$ denotes that $g$ is a parabolic element, the total degree of the invariant trace field $k\\Gamma=\\mathbb{Q}(\\{\\tr^2(h):h\\in\\Gamma\\})$ is at most $4$, and each orbifold is either a two bridge link of slope $r/s$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2206.14174","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2206.14174/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2206.14174","created_at":"2026-07-05T04:35:53.803438+00:00"},{"alias_kind":"arxiv_version","alias_value":"2206.14174v1","created_at":"2026-07-05T04:35:53.803438+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2206.14174","created_at":"2026-07-05T04:35:53.803438+00:00"},{"alias_kind":"pith_short_12","alias_value":"5QOA3XRHGWZ5","created_at":"2026-07-05T04:35:53.803438+00:00"},{"alias_kind":"pith_short_16","alias_value":"5QOA3XRHGWZ57NUZ","created_at":"2026-07-05T04:35:53.803438+00:00"},{"alias_kind":"pith_short_8","alias_value":"5QOA3XRH","created_at":"2026-07-05T04:35:53.803438+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2607.06820","citing_title":"Evaluating SageMath-Augmented LLM Agents for Computational and Experimental Mathematics","ref_index":60,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5QOA3XRHGWZ57NUZB3MYNGZOMW","json":"https://pith.science/pith/5QOA3XRHGWZ57NUZB3MYNGZOMW.json","graph_json":"https://pith.science/api/pith-number/5QOA3XRHGWZ57NUZB3MYNGZOMW/graph.json","events_json":"https://pith.science/api/pith-number/5QOA3XRHGWZ57NUZB3MYNGZOMW/events.json","paper":"https://pith.science/paper/5QOA3XRH"},"agent_actions":{"view_html":"https://pith.science/pith/5QOA3XRHGWZ57NUZB3MYNGZOMW","download_json":"https://pith.science/pith/5QOA3XRHGWZ57NUZB3MYNGZOMW.json","view_paper":"https://pith.science/paper/5QOA3XRH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2206.14174&json=true","fetch_graph":"https://pith.science/api/pith-number/5QOA3XRHGWZ57NUZB3MYNGZOMW/graph.json","fetch_events":"https://pith.science/api/pith-number/5QOA3XRHGWZ57NUZB3MYNGZOMW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5QOA3XRHGWZ57NUZB3MYNGZOMW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5QOA3XRHGWZ57NUZB3MYNGZOMW/action/storage_attestation","attest_author":"https://pith.science/pith/5QOA3XRHGWZ57NUZB3MYNGZOMW/action/author_attestation","sign_citation":"https://pith.science/pith/5QOA3XRHGWZ57NUZB3MYNGZOMW/action/citation_signature","submit_replication":"https://pith.science/pith/5QOA3XRHGWZ57NUZB3MYNGZOMW/action/replication_record"}},"created_at":"2026-07-05T04:35:53.803438+00:00","updated_at":"2026-07-05T04:35:53.803438+00:00"}