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In this paper we give a finite presentation for the commutator subgroup $TW_{m+2}'$, and prove that $TW_{m+2}'$ has rank $2m-1$. We derive that $TW_{m+2}'$ is free if and only if $m \\leq 3$. From this it follows that $TW_{m+2}$ is word-hyperbolic and does not contain a surface group if and only if $m \\leq 3$. It also follows that the automorphism group of $TW_{m+2}$ is finitely presented for $m \\leq 3$."},"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":"1804.05375","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2018-04-15T16:02:42Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"3398bc7b32014d509b5a3820b8893a5113064e831842b20af0d8893cadb7e484","abstract_canon_sha256":"7755569fdab3695b4daf174e56265970fa5caab7d3eeabce3f83e9ed3d730642"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:16:49.938796Z","signature_b64":"KOlFl0lNQU7vUDvf5p/BdDY4YHR9KXdyKpqqPumk/0+WkBNp6/g/uxvS0+bFpcumexWE8YCYzfKUJLxnBGSfBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"46f6824987278add00d0b559bf073771bdd0c7c41c8610a675047bf76219019d","last_reissued_at":"2026-05-18T00:16:49.938076Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:16:49.938076Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Commutator Subgroups of Twin Groups and Grothendieck's Cartographical Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.GT","authors_text":"Krishnendu Gongopadhyay, Soumya Dey","submitted_at":"2018-04-15T16:02:42Z","abstract_excerpt":"Let $TW_n$ be the twin group on $n$ arcs, $n \\geq 2$. 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