{"paper":{"title":"On the Maximal Size of Irredundant Generating Sets in Lie Groups and Algebraic Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A topologically generating set in a connected compact Lie group must be redundant if its size exceeds a polynomial in the group's rank.","cross_cats":[],"primary_cat":"math.GR","authors_text":"Itamar Vigdorovich, Tal Cohen","submitted_at":"2026-03-10T13:15:30Z","abstract_excerpt":"We show the following dichotomy for a connected Lie group $G$: If $G$ is amenable, then any topologically generating set $X\\subset G$ of size larger than a fixed polynomial in the dimension of $G$ must be redundant (i.e., a proper subset of $X$ still generates $G$). If $G$ is non-amenable, then it admits arbitrarily large topologically generating sets that are irredundant, and remain irredundant even after applying Nielsen transformations.\n  The polynomial bound for amenable groups is obtained by reduction to finite simple groups of Lie type via strong approximation. This partially answers two"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We show that a topologically generating set S of a connected compact Lie group G of size larger than a fixed polynomial in the rank of G must be redundant (i.e., some proper subset of S still topologically generates G).","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The quantitative bounds produced by the method are controlled by corresponding bounds for finite simple groups of Lie type; the argument assumes that sufficiently strong polynomial bounds already exist or can be established for those finite groups.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"In connected compact Lie groups the maximal size of an irredundant topologically generating set is bounded by a polynomial in the rank, with analogous statements for amenable Lie groups and reductive algebraic groups.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A topologically generating set in a connected compact Lie group must be redundant if its size exceeds a polynomial in the group's rank.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"32601c2e123281ecdc7f519caed39a0d034aa78546af3db40fbeecd8ed55652a"},"source":{"id":"2603.09640","kind":"arxiv","version":3},"verdict":{"id":"2cfd157d-a888-4086-b779-75fbebd823bd","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T13:14:25.768622Z","strongest_claim":"We show that a topologically generating set S of a connected compact Lie group G of size larger than a fixed polynomial in the rank of G must be redundant (i.e., some proper subset of S still topologically generates G).","one_line_summary":"In connected compact Lie groups the maximal size of an irredundant topologically generating set is bounded by a polynomial in the rank, with analogous statements for amenable Lie groups and reductive algebraic groups.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The quantitative bounds produced by the method are controlled by corresponding bounds for finite simple groups of Lie type; the argument assumes that sufficiently strong polynomial bounds already exist or can be established for those finite groups.","pith_extraction_headline":"A topologically generating set in a connected compact Lie group must be redundant if its size exceeds a polynomial in the group's rank."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2603.09640/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"f76e688a2f051d0477d504f59c01f0f9ca6eb1a202c3aabade42a053d6cb1f2f"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}