{"paper":{"title":"Heegaard splittings and 1-relator groups","license":"","headline":"","cross_cats":["math.GR"],"primary_cat":"math.GT","authors_text":"Joseph D. Masters","submitted_at":"2006-08-25T16:28:29Z","abstract_excerpt":"We show that if $M$ is a fibered, orientable 3-manifold, and if $\\pi_1 M$ has 1-relator presentation, then the presentation is induced by a Heegaard splitting of $M$.\n  A corollary is that, for these manifolds, the rank of $\\pi_1 M$ is equal to the \"restricted\" Heegaard genus of $M$.\n  We also explore the analogy between 1-relator groups and Haken 3-manifolds, showing that every 1-relator group possesses a \"1-relator hierarchy\"."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0608635","kind":"arxiv","version":4},"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"}