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Let $\\Pi_{g,n}$ be the fundamental group of $S_{g,n}$, with a given base point, and $\\hat{\\Pi}_{g,n}$ its profinite completion. There is then a natural faithful representation $\\Gamma_{g,n}\\hookrightarrow Out(\\hat{\\Pi}_{g,n})$. The procongruence completion $\\check{\\Gamma}_{g,n}$ of the Teichm\\\"uller group is define"},"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":"0910.4305","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2009-10-22T12:39:04Z","cross_cats_sorted":["math.GR","math.NT"],"title_canon_sha256":"5d8a619c2bf2ec4e64e002ad881ef6478906cf28cb4f9ebd92ff4a33dcf828b5","abstract_canon_sha256":"87d9c349a8eeb91d26c9b0800c40122017c10ea662cef7060722c332765f36f3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:36:06.761097Z","signature_b64":"TtZZ+4STpLfX7MLQS3SLrqHZqhv3oycAani4UGDwSWRTbzCkyG7Uj3VcUTV72hxtzYCTIomOqD868HaVgFtmAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c2936532238f6149b1bfff342c3571e20ddba5f06d9ebffaf05238a43d28ed47","last_reissued_at":"2026-05-18T03:36:06.760459Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:36:06.760459Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the procongruence completion of the Teichm\\\"uller modular group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.NT"],"primary_cat":"math.AG","authors_text":"Marco Boggi","submitted_at":"2009-10-22T12:39:04Z","abstract_excerpt":"For $2g-2+n>0$, the Teichm\\\"uller modular group $\\Gamma_{g,n}$ of a compact Riemann surface of genus $g$ with $n$ points removed $S_{g,n}$ is the group of homotopy classes of diffeomorphisms of $S_{g,n}$ which preserve the orientation of $S_{g,n}$ and a given order of its punctures. Let $\\Pi_{g,n}$ be the fundamental group of $S_{g,n}$, with a given base point, and $\\hat{\\Pi}_{g,n}$ its profinite completion. There is then a natural faithful representation $\\Gamma_{g,n}\\hookrightarrow Out(\\hat{\\Pi}_{g,n})$. 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