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The \\emph{${\\mathscr C}$-congruence completion $\\check{\\Gamma}(S)^{\\mathscr C}$ of $\\Gamma(S)$} is the profinite completion induced by the embedding $\\Gamma(S)\\hookrightarrow{\\operatorname{Out}}(\\hat{\\pi}_1(S)^{\\mathscr C})$. In this paper, we begin a systematic study of such completions for different ${\\mathscr C}$. We show that"},"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":"1804.06322","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-04-17T15:33:53Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"7032749c03d9889fcec87d5dc84a202dab13c6ffcd0d8f06b9f76f6e2bf5c983","abstract_canon_sha256":"11bba6cada39a68717b0bc60be81f2876a939c3b0a3f22f771bb31149f20a721"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:18:21.007106Z","signature_b64":"STgsayoyRxGPSGDWRLxHogG0XzzUi/kQis3zmU/d3OqwqYTDEoaOqFrHVOqY20D4oh4vP1gp1lWw+xbtB6tKAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"28159f9725848759d4596156126f4021955b8729d37dedb0c64728abd06802a9","last_reissued_at":"2026-05-18T00:18:21.006410Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:18:21.006410Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Congruence topologies on the mapping class group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.GR","authors_text":"Marco Boggi","submitted_at":"2018-04-17T15:33:53Z","abstract_excerpt":"Let $\\Gamma(S)$ be the pure mapping class group of a connected orientable surface $S$ of negative Euler characteristic. For ${\\mathscr C}$ a class of finite groups, let $\\hat{\\pi}_1(S)^{\\mathscr C}$ be the pro-${\\mathscr C}$ completion of the fundamental group of $S$. The \\emph{${\\mathscr C}$-congruence completion $\\check{\\Gamma}(S)^{\\mathscr C}$ of $\\Gamma(S)$} is the profinite completion induced by the embedding $\\Gamma(S)\\hookrightarrow{\\operatorname{Out}}(\\hat{\\pi}_1(S)^{\\mathscr C})$. In this paper, we begin a systematic study of such completions for different ${\\mathscr C}$. 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