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Moreover every element of the commutator subgroup of a free group rationally bounds an injective map of a surface group.\n  The proof of these facts"},"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":"0802.1352","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2008-02-10T21:04:53Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"1db23eb5ef2df8a8dab0b68afa7517a5ffc3f801764d1cbffd4b08fe40476151","abstract_canon_sha256":"c7f041d24f45988fa89ca924fb71289843db61862b8c6b246527824b145e8d90"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:16:01.180389Z","signature_b64":"tym4IK8tdQGAt8TyqWaCMFWgkjy1AiotOIlIR6L3924OAwoCjEXYxe3318q18EuUBuT0s71xXtrq67r6NyITCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0c212b9c93e0c89167e2da68db5d4ea79b947b90d8babb37b2dad09ed521d341","last_reissued_at":"2026-05-18T02:16:01.179953Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:16:01.179953Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Stable commutator length is rational in free groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.GR","authors_text":"Danny Calegari","submitted_at":"2008-02-10T21:04:53Z","abstract_excerpt":"For any group, there is a natural (pseudo-)norm on the vector space B1 of real (group) 1-boundaries, called the stable commutator length norm. 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