{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2009:V3XFJTFP3LMF6VQ6NMNJIVAOVS","short_pith_number":"pith:V3XFJTFP","schema_version":"1.0","canonical_sha256":"aeee54ccafdad85f561e6b1a94540eaca33442c960579b08debb5c3307db358b","source":{"kind":"arxiv","id":"0901.1977","version":5},"attestation_state":"computed","paper":{"title":"Free Groups in Quaternion Algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.GR","authors_text":"A. C. Souza Filho, S. O. Juriaans","submitted_at":"2009-01-14T13:26:34Z","abstract_excerpt":"In \\cite{jpsf} we constructed pairs of units $u,v$ in $\\Z$-orders of a quaternion algebra over $\\Q (\\sqrt{-d})$, $d \\equiv 7 \\pmod 8$ positive and square free, such that $< u^ n,v^n>$ is free for some $n\\in \\mathbb{N}$. Here we extend this result to any imaginary quadratic extension of $\\ \\mathbb{Q}$, thus including matrix algebras. More precisely, we show that $< u^n,v^n> $ is a free group for all $n\\geq 1$ and $d>2$ and for $d=2$ and all $n\\geq 2$. The units we use arise from Pell's and Gauss' equations. A criterion for a pair of homeomorphisms to generate a free semigroup is also establishe"},"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":"0901.1977","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2009-01-14T13:26:34Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"9b61b6ddcb1919d0cf8b6a3da1bd750cdd97f6f99286011dd874d22df2456845","abstract_canon_sha256":"09302880335d1a46a863c93eefecdb12cb80d3fb4bb4adbb7d8b0cab884c615a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:39:59.671810Z","signature_b64":"fuG5xSScVWYCAxCYt+v80DxLzQvu9CfQtUV3/Ob7FrljGaHLB1l2NZjj728a7LWIUv2tF9FdNsEJvZ39fZM/AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aeee54ccafdad85f561e6b1a94540eaca33442c960579b08debb5c3307db358b","last_reissued_at":"2026-05-18T04:39:59.671209Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:39:59.671209Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Free Groups in Quaternion Algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.GR","authors_text":"A. C. Souza Filho, S. O. Juriaans","submitted_at":"2009-01-14T13:26:34Z","abstract_excerpt":"In \\cite{jpsf} we constructed pairs of units $u,v$ in $\\Z$-orders of a quaternion algebra over $\\Q (\\sqrt{-d})$, $d \\equiv 7 \\pmod 8$ positive and square free, such that $< u^ n,v^n>$ is free for some $n\\in \\mathbb{N}$. Here we extend this result to any imaginary quadratic extension of $\\ \\mathbb{Q}$, thus including matrix algebras. More precisely, we show that $< u^n,v^n> $ is a free group for all $n\\geq 1$ and $d>2$ and for $d=2$ and all $n\\geq 2$. The units we use arise from Pell's and Gauss' equations. A criterion for a pair of homeomorphisms to generate a free semigroup is also establishe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0901.1977","kind":"arxiv","version":5},"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":"0901.1977","created_at":"2026-05-18T04:39:59.671282+00:00"},{"alias_kind":"arxiv_version","alias_value":"0901.1977v5","created_at":"2026-05-18T04:39:59.671282+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0901.1977","created_at":"2026-05-18T04:39:59.671282+00:00"},{"alias_kind":"pith_short_12","alias_value":"V3XFJTFP3LMF","created_at":"2026-05-18T12:26:02.257875+00:00"},{"alias_kind":"pith_short_16","alias_value":"V3XFJTFP3LMF6VQ6","created_at":"2026-05-18T12:26:02.257875+00:00"},{"alias_kind":"pith_short_8","alias_value":"V3XFJTFP","created_at":"2026-05-18T12:26:02.257875+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/V3XFJTFP3LMF6VQ6NMNJIVAOVS","json":"https://pith.science/pith/V3XFJTFP3LMF6VQ6NMNJIVAOVS.json","graph_json":"https://pith.science/api/pith-number/V3XFJTFP3LMF6VQ6NMNJIVAOVS/graph.json","events_json":"https://pith.science/api/pith-number/V3XFJTFP3LMF6VQ6NMNJIVAOVS/events.json","paper":"https://pith.science/paper/V3XFJTFP"},"agent_actions":{"view_html":"https://pith.science/pith/V3XFJTFP3LMF6VQ6NMNJIVAOVS","download_json":"https://pith.science/pith/V3XFJTFP3LMF6VQ6NMNJIVAOVS.json","view_paper":"https://pith.science/paper/V3XFJTFP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0901.1977&json=true","fetch_graph":"https://pith.science/api/pith-number/V3XFJTFP3LMF6VQ6NMNJIVAOVS/graph.json","fetch_events":"https://pith.science/api/pith-number/V3XFJTFP3LMF6VQ6NMNJIVAOVS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/V3XFJTFP3LMF6VQ6NMNJIVAOVS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/V3XFJTFP3LMF6VQ6NMNJIVAOVS/action/storage_attestation","attest_author":"https://pith.science/pith/V3XFJTFP3LMF6VQ6NMNJIVAOVS/action/author_attestation","sign_citation":"https://pith.science/pith/V3XFJTFP3LMF6VQ6NMNJIVAOVS/action/citation_signature","submit_replication":"https://pith.science/pith/V3XFJTFP3LMF6VQ6NMNJIVAOVS/action/replication_record"}},"created_at":"2026-05-18T04:39:59.671282+00:00","updated_at":"2026-05-18T04:39:59.671282+00:00"}