{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2006:TRX6SEPXWZIVQQC2JBDLAQHPKW","short_pith_number":"pith:TRX6SEPX","schema_version":"1.0","canonical_sha256":"9c6fe911f7b65158405a4846b040ef55bd3a14bb231ee58948b0ea232b330d2b","source":{"kind":"arxiv","id":"math/0607671","version":2},"attestation_state":"computed","paper":{"title":"Deficiency and abelianized deficiency of some virtually free groups","license":"","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Martin R. Bridson, Michael Tweedale","submitted_at":"2006-07-26T18:03:53Z","abstract_excerpt":"Let $Q_m$ be the HNN extension of $\\Z/m \\times \\Z/m$ where the stable letter conjugates the first factor to the second. We explore small presentations of the groups $\\Gamma_{m,n}=Q_m \\ast Q_n$. We show that for certain choices of (m,n), for example (2,3), the group $\\Gamma_{m,n}$ has a relation gap unless it admits a presentation with at most 3 defining relations, and we establish restrictions on the possible form of such a presentation. We then associate to each (m,n) a 3-complex with 16 cells. This 3-complex is a counterexample to the D(2) conjecture if $\\Gamma_{m,n}$ has a relation gap."},"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/0607671","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.GR","submitted_at":"2006-07-26T18:03:53Z","cross_cats_sorted":[],"title_canon_sha256":"72233cbe1426c96e766b7f004bdec2200e90ea86cc4937a3900152cb20e7f703","abstract_canon_sha256":"7b7dfec4d8dd059c622327140e74e4397a290ed67d3cb75020ffa70b3f0a9d59"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:47:21.401750Z","signature_b64":"azeSSz6fmUldXEV1c8DZoUDLxUwbDTkuRfD2D21Nj3UtH+U5ZxngW38tEPFC1MiQL7gZ/N20RmR65S10tceWBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9c6fe911f7b65158405a4846b040ef55bd3a14bb231ee58948b0ea232b330d2b","last_reissued_at":"2026-05-17T23:47:21.401370Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:47:21.401370Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Deficiency and abelianized deficiency of some virtually free groups","license":"","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Martin R. Bridson, Michael Tweedale","submitted_at":"2006-07-26T18:03:53Z","abstract_excerpt":"Let $Q_m$ be the HNN extension of $\\Z/m \\times \\Z/m$ where the stable letter conjugates the first factor to the second. We explore small presentations of the groups $\\Gamma_{m,n}=Q_m \\ast Q_n$. We show that for certain choices of (m,n), for example (2,3), the group $\\Gamma_{m,n}$ has a relation gap unless it admits a presentation with at most 3 defining relations, and we establish restrictions on the possible form of such a presentation. We then associate to each (m,n) a 3-complex with 16 cells. This 3-complex is a counterexample to the D(2) conjecture if $\\Gamma_{m,n}$ has a relation gap."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0607671","kind":"arxiv","version":2},"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/0607671","created_at":"2026-05-17T23:47:21.401433+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0607671v2","created_at":"2026-05-17T23:47:21.401433+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0607671","created_at":"2026-05-17T23:47:21.401433+00:00"},{"alias_kind":"pith_short_12","alias_value":"TRX6SEPXWZIV","created_at":"2026-05-18T12:25:54.717736+00:00"},{"alias_kind":"pith_short_16","alias_value":"TRX6SEPXWZIVQQC2","created_at":"2026-05-18T12:25:54.717736+00:00"},{"alias_kind":"pith_short_8","alias_value":"TRX6SEPX","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/TRX6SEPXWZIVQQC2JBDLAQHPKW","json":"https://pith.science/pith/TRX6SEPXWZIVQQC2JBDLAQHPKW.json","graph_json":"https://pith.science/api/pith-number/TRX6SEPXWZIVQQC2JBDLAQHPKW/graph.json","events_json":"https://pith.science/api/pith-number/TRX6SEPXWZIVQQC2JBDLAQHPKW/events.json","paper":"https://pith.science/paper/TRX6SEPX"},"agent_actions":{"view_html":"https://pith.science/pith/TRX6SEPXWZIVQQC2JBDLAQHPKW","download_json":"https://pith.science/pith/TRX6SEPXWZIVQQC2JBDLAQHPKW.json","view_paper":"https://pith.science/paper/TRX6SEPX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0607671&json=true","fetch_graph":"https://pith.science/api/pith-number/TRX6SEPXWZIVQQC2JBDLAQHPKW/graph.json","fetch_events":"https://pith.science/api/pith-number/TRX6SEPXWZIVQQC2JBDLAQHPKW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TRX6SEPXWZIVQQC2JBDLAQHPKW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TRX6SEPXWZIVQQC2JBDLAQHPKW/action/storage_attestation","attest_author":"https://pith.science/pith/TRX6SEPXWZIVQQC2JBDLAQHPKW/action/author_attestation","sign_citation":"https://pith.science/pith/TRX6SEPXWZIVQQC2JBDLAQHPKW/action/citation_signature","submit_replication":"https://pith.science/pith/TRX6SEPXWZIVQQC2JBDLAQHPKW/action/replication_record"}},"created_at":"2026-05-17T23:47:21.401433+00:00","updated_at":"2026-05-17T23:47:21.401433+00:00"}