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We prove that this is an isomorphism for all $N$ and $G$ if and only if $Q$ is super-perfect and has no proper subgroups of finite index.\n  We prove that there is no algorithm that, given a finitely presented, residually finite group $G$ and a finitely presentable subgroup $P\\subset G$, can determine whether or not $\\hat P\\to\\hat G$ is an isomorphism."},"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":"0907.5010","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2009-07-28T22:05:45Z","cross_cats_sorted":[],"title_canon_sha256":"e4647e02e5ee170ae924591aa6e1266d1730dee52442a8909c3dd5234ac62cfa","abstract_canon_sha256":"04bfe0d90c8b785a7cfb8ecf82057a106ffb229458bf8ab2b4701dd81e8bff04"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:58:02.918155Z","signature_b64":"L42fwE52JdfCWovhZUlto8uI537hCnvlQlRk081XC/VQMzB4C0nEkeMW9HTt8cGZCf9PGZwzIOcaKF+79KehDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c0a5a0168da9a11dcd3a6a382991b1b561e524ca7d7b7b3b8a4def28bff0869b","last_reissued_at":"2026-05-18T02:58:02.917258Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:58:02.917258Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Schur multiplier, profinite completions and decidability","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Martin R Bridson","submitted_at":"2009-07-28T22:05:45Z","abstract_excerpt":"We fix a finitely presented group $Q$ and consider short exact sequences $1\\to N\\to G\\to Q\\to 1$ with $G$ finitely generated. 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