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We show that, if $\\mathcal{G}$ is a finite index subgroup of $\\mathcal{M} (N)$ and $\\varphi: \\mathcal{G} \\to \\mathcal{M}  (N)$ is an injective homomorphism, then there exists $f_0 \\in \\mathcal{M}  (N)$ such that $\\varphi (g) = f_0 g f_0^{-1}$ for all $g \\in \\mathcal{G}$. We deduce that the abstract commensurator of $\\mathcal{M}  (N)$ coincides with $\\mathcal{M}  (N)$."},"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":"1708.00218","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2017-08-01T09:28:28Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"b1f1f1c1a6c9d4577f37d7850466b024800a57f10a2c854c4b0b50105b0e5cbc","abstract_canon_sha256":"8d79b68a065945da59192fba0376d0b67ef7c22b773dd7f6b78177577369ff13"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:39:05.783826Z","signature_b64":"L4JN0pq5gmkddRVmftG1Rc2lcroeyQhbaOXZGgXOMEYGW4/xTxvkdjqJ/H+X9Ol+rBqax/Vc0OdKyKnWCrCwBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ee3c83ade83c3bea5be2c9508e3332710770e4fb2051350e5b66731816692594","last_reissued_at":"2026-05-18T00:39:05.783244Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:39:05.783244Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Injective homomorphisms of mapping class groups of non-orientable surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.GT","authors_text":"Elmas Irmak, Luis Paris (IMB)","submitted_at":"2017-08-01T09:28:28Z","abstract_excerpt":"Let $N$ be a compact, connected, non-orientable surface of genus $\\rho$ with $n$ boundary components, with $\\rho \\ge 5$ and $n \\ge 0$, and let $\\mathcal{M} (N)$ be the mapping class group of $N$. We show that, if $\\mathcal{G}$ is a finite index subgroup of $\\mathcal{M} (N)$ and $\\varphi: \\mathcal{G} \\to \\mathcal{M}  (N)$ is an injective homomorphism, then there exists $f_0 \\in \\mathcal{M}  (N)$ such that $\\varphi (g) = f_0 g f_0^{-1}$ for all $g \\in \\mathcal{G}$. We deduce that the abstract commensurator of $\\mathcal{M}  (N)$ coincides with $\\mathcal{M}  (N)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.00218","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1708.00218","created_at":"2026-05-18T00:39:05.783331+00:00"},{"alias_kind":"arxiv_version","alias_value":"1708.00218v1","created_at":"2026-05-18T00:39:05.783331+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.00218","created_at":"2026-05-18T00:39:05.783331+00:00"},{"alias_kind":"pith_short_12","alias_value":"5Y6IHLPIHQ56","created_at":"2026-05-18T12:31:03.183658+00:00"},{"alias_kind":"pith_short_16","alias_value":"5Y6IHLPIHQ56UW7C","created_at":"2026-05-18T12:31:03.183658+00:00"},{"alias_kind":"pith_short_8","alias_value":"5Y6IHLPI","created_at":"2026-05-18T12:31:03.183658+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5Y6IHLPIHQ56UW7CZFII4MZSOE","json":"https://pith.science/pith/5Y6IHLPIHQ56UW7CZFII4MZSOE.json","graph_json":"https://pith.science/api/pith-number/5Y6IHLPIHQ56UW7CZFII4MZSOE/graph.json","events_json":"https://pith.science/api/pith-number/5Y6IHLPIHQ56UW7CZFII4MZSOE/events.json","paper":"https://pith.science/paper/5Y6IHLPI"},"agent_actions":{"view_html":"https://pith.science/pith/5Y6IHLPIHQ56UW7CZFII4MZSOE","download_json":"https://pith.science/pith/5Y6IHLPIHQ56UW7CZFII4MZSOE.json","view_paper":"https://pith.science/paper/5Y6IHLPI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1708.00218&json=true","fetch_graph":"https://pith.science/api/pith-number/5Y6IHLPIHQ56UW7CZFII4MZSOE/graph.json","fetch_events":"https://pith.science/api/pith-number/5Y6IHLPIHQ56UW7CZFII4MZSOE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5Y6IHLPIHQ56UW7CZFII4MZSOE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5Y6IHLPIHQ56UW7CZFII4MZSOE/action/storage_attestation","attest_author":"https://pith.science/pith/5Y6IHLPIHQ56UW7CZFII4MZSOE/action/author_attestation","sign_citation":"https://pith.science/pith/5Y6IHLPIHQ56UW7CZFII4MZSOE/action/citation_signature","submit_replication":"https://pith.science/pith/5Y6IHLPIHQ56UW7CZFII4MZSOE/action/replication_record"}},"created_at":"2026-05-18T00:39:05.783331+00:00","updated_at":"2026-05-18T00:39:05.783331+00:00"}