{"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"}