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We prove the existence of an epimorphism $\\zeta : \\langle H, K \\rangle \\to F_2$, where $F_2$ is a free group of rank 2, such that the restriction of $\\zeta$ on both $H$ and $K$ is injective and the restriction $\\zeta_0 : H \\cap K \\to \\zeta (H) \\cap \\zeta (K) $ of $\\zeta$ on $H \\cap K $ to $\\zeta (H) \\cap \\zeta (K)$ is surjective. This is obtained as a corollary of a"},"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":"1608.00617","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-08-01T21:02:05Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"4268cdba1d393fd519e80f68d9172762be1b30a13af68bab894146e417e4c08f","abstract_canon_sha256":"e1fdff7565fbf6caa402b6526153a7a7e1eca5f51fff37a799d67007c659a55b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:10:11.372717Z","signature_b64":"a8T6UnsGPSKB1rPoAiFQHu+RwRfQvH2Dc+uuQkW6YpNOsLgo034oVhqJHCRIaQxvDVBLrtdUG/vtGRihY8ouAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7e6774d5b2d9a678e11819510ac39914dbe5a7da126967e9ba375521fb6c6c20","last_reissued_at":"2026-05-18T01:10:11.372299Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:10:11.372299Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On rank of the join of two subgroups in a free group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.GR","authors_text":"Sergei V. Ivanov","submitted_at":"2016-08-01T21:02:05Z","abstract_excerpt":"Let $H, K$ be two finitely generated subgroups of a free group, let $\\langle H, K \\rangle$ denote the subgroup generated by $H, K$, called the join of $H, K$, and let neither of $H$, $K$ have finite index in $\\langle H, K \\rangle$. We prove the existence of an epimorphism $\\zeta : \\langle H, K \\rangle \\to F_2$, where $F_2$ is a free group of rank 2, such that the restriction of $\\zeta$ on both $H$ and $K$ is injective and the restriction $\\zeta_0 : H \\cap K \\to \\zeta (H) \\cap \\zeta (K) $ of $\\zeta$ on $H \\cap K $ to $\\zeta (H) \\cap \\zeta (K)$ is surjective. 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