{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2006:LIEPDDRDUFG47GRAVXYYY4IER4","short_pith_number":"pith:LIEPDDRD","schema_version":"1.0","canonical_sha256":"5a08f18e23a14dcf9a20adf18c71048f2f648d8db6d672c4db008db641ef6452","source":{"kind":"arxiv","id":"math/0603682","version":1},"attestation_state":"computed","paper":{"title":"The Tits alternative for generalized triangle groups of type (3,4,2)","license":"","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Gerald Williams, James Howie","submitted_at":"2006-03-29T12:47:32Z","abstract_excerpt":"A generalized triangle group is a group that can be presented in the form $G = < x,y | x^p=y^q=w(x,y)^r=1>$, where $p,q,r\\geq 2$ and $w(x,y)$ is a cyclically reduced word of length at least 2 in the free product $\\Z_p*\\Z_q=< x,y | x^p=y^q=1>$. Rosenberger has conjectured that every generalized triangle group $G$ satisfies the Tits alternative. It is known that the conjecture holds except possibly when the triple $(p,q,r)$ is one of $(2,3,2), (2,4,2),(2,5,2),(3,3,2),(3,4,2)$, or $(3,5,2)$. In this paper we show that the Tits alternative holds in the case $(p,q,r)=(3,4,2)$."},"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":"math/0603682","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.GR","submitted_at":"2006-03-29T12:47:32Z","cross_cats_sorted":[],"title_canon_sha256":"948510449eb2781ea3060f696cd83d895496e35cd494a2db53132b8e557942be","abstract_canon_sha256":"7a3b6f386723d262de70215f4af36d0fa5929e5e7e041cde93c73b8c9f7f262b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:33:27.513042Z","signature_b64":"mEDSY2R1QdXxSiO76R2WJSPLUwQ6hrbBCG+7Y5IgnTdW6fKXTvcJkOLQ1lucfYpOxaXL5Lt490I6evMog8GNDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5a08f18e23a14dcf9a20adf18c71048f2f648d8db6d672c4db008db641ef6452","last_reissued_at":"2026-05-18T04:33:27.512568Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:33:27.512568Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Tits alternative for generalized triangle groups of type (3,4,2)","license":"","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Gerald Williams, James Howie","submitted_at":"2006-03-29T12:47:32Z","abstract_excerpt":"A generalized triangle group is a group that can be presented in the form $G = < x,y | x^p=y^q=w(x,y)^r=1>$, where $p,q,r\\geq 2$ and $w(x,y)$ is a cyclically reduced word of length at least 2 in the free product $\\Z_p*\\Z_q=< x,y | x^p=y^q=1>$. Rosenberger has conjectured that every generalized triangle group $G$ satisfies the Tits alternative. It is known that the conjecture holds except possibly when the triple $(p,q,r)$ is one of $(2,3,2), (2,4,2),(2,5,2),(3,3,2),(3,4,2)$, or $(3,5,2)$. In this paper we show that the Tits alternative holds in the case $(p,q,r)=(3,4,2)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0603682","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":"math/0603682","created_at":"2026-05-18T04:33:27.512642+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0603682v1","created_at":"2026-05-18T04:33:27.512642+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0603682","created_at":"2026-05-18T04:33:27.512642+00:00"},{"alias_kind":"pith_short_12","alias_value":"LIEPDDRDUFG4","created_at":"2026-05-18T12:25:54.717736+00:00"},{"alias_kind":"pith_short_16","alias_value":"LIEPDDRDUFG47GRA","created_at":"2026-05-18T12:25:54.717736+00:00"},{"alias_kind":"pith_short_8","alias_value":"LIEPDDRD","created_at":"2026-05-18T12:25:54.717736+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/LIEPDDRDUFG47GRAVXYYY4IER4","json":"https://pith.science/pith/LIEPDDRDUFG47GRAVXYYY4IER4.json","graph_json":"https://pith.science/api/pith-number/LIEPDDRDUFG47GRAVXYYY4IER4/graph.json","events_json":"https://pith.science/api/pith-number/LIEPDDRDUFG47GRAVXYYY4IER4/events.json","paper":"https://pith.science/paper/LIEPDDRD"},"agent_actions":{"view_html":"https://pith.science/pith/LIEPDDRDUFG47GRAVXYYY4IER4","download_json":"https://pith.science/pith/LIEPDDRDUFG47GRAVXYYY4IER4.json","view_paper":"https://pith.science/paper/LIEPDDRD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0603682&json=true","fetch_graph":"https://pith.science/api/pith-number/LIEPDDRDUFG47GRAVXYYY4IER4/graph.json","fetch_events":"https://pith.science/api/pith-number/LIEPDDRDUFG47GRAVXYYY4IER4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LIEPDDRDUFG47GRAVXYYY4IER4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LIEPDDRDUFG47GRAVXYYY4IER4/action/storage_attestation","attest_author":"https://pith.science/pith/LIEPDDRDUFG47GRAVXYYY4IER4/action/author_attestation","sign_citation":"https://pith.science/pith/LIEPDDRDUFG47GRAVXYYY4IER4/action/citation_signature","submit_replication":"https://pith.science/pith/LIEPDDRDUFG47GRAVXYYY4IER4/action/replication_record"}},"created_at":"2026-05-18T04:33:27.512642+00:00","updated_at":"2026-05-18T04:33:27.512642+00:00"}