{"paper":{"title":"Commutator Subgroups of Virtual and Welded Braid Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.GT","authors_text":"Krishnendu Gongopadhyay, Mikhail V. Neshchadim, Valeriy G. Bardakov","submitted_at":"2018-02-05T13:39:09Z","abstract_excerpt":"Let $VB_n$, resp. $WB_n$ denote the virtual, resp. welded, braid group on $n$ strands. We study their commutator subgroups $VB_n' = [VB_n, VB_n]$ and, $WB_n' = [WB_n, WB_n]$ respectively. We obtain a set of generators and defining relations for these commutator subgroups. In particular, we prove that $VB_n'$ is finitely generated if and only if $n \\geq 4$, and $WB_n'$ is finitely generated for $n \\geq 3$. Also we prove that $VB_3'/VB_3'' =\\mathbb{Z}_3 \\oplus \\mathbb{Z}_3 \\oplus\\mathbb{Z}_3 \\oplus \\mathbb{Z}^{\\infty}$, $VB_4' / VB_4'' = \\mathbb{Z}_3 \\oplus \\mathbb{Z}_3 \\oplus \\mathbb{Z}_3$, $WB"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.01383","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"}