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We prove that groups of the form $F_2\\rtimes\\mathbb{Z}$ are distinguished from one another by their profinite completions. Thus, regardless of betti number, if $M$ and $N$ are punctured torus bundles over the circle and $M$ is not homeomorphic to $N$, then there is a finite group $G$ such that one of $\\pi_1M$ and $\\pi_1N$ maps onto $G$ and the other does not."},"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":"1610.02410","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-10-07T20:01:35Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"6423fd129f2b9eb95c87e2261def8eb18fcf63f89c8740dfeda148727c522684","abstract_canon_sha256":"5547abfa0ec2db3817800d92016834e34ce6738aa1e2aa804e370dad7e2eb56e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:31.089802Z","signature_b64":"j+QGfM+ICqJiedl9KPtESi2BsAEv8fV/i+PKtEyqihtYw+flelqas3eyG/Ow8mKtfsCPii0hmyEeErL+WwxLAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"76ffc3eb72d18922daea62fa619ea8601853059626a9b316d46105949bf63db6","last_reissued_at":"2026-05-18T00:38:31.089335Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:31.089335Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Profinite rigidity and surface bundles over the circle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.GR","authors_text":"Alan W. Reid, Henry Wilton, Martin R. Bridson","submitted_at":"2016-10-07T20:01:35Z","abstract_excerpt":"If $M$ is a compact 3-manifold whose first betti number is 1, and $N$ is a compact 3-manifold such that $\\pi_1N$ and $\\pi_1M$ have the same finite quotients, then $M$ fibres over the circle if and only if $N$ does. We prove that groups of the form $F_2\\rtimes\\mathbb{Z}$ are distinguished from one another by their profinite completions. Thus, regardless of betti number, if $M$ and $N$ are punctured torus bundles over the circle and $M$ is not homeomorphic to $N$, then there is a finite group $G$ such that one of $\\pi_1M$ and $\\pi_1N$ maps onto $G$ and the other does not."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.02410","kind":"arxiv","version":2},"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":"1610.02410","created_at":"2026-05-18T00:38:31.089398+00:00"},{"alias_kind":"arxiv_version","alias_value":"1610.02410v2","created_at":"2026-05-18T00:38:31.089398+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.02410","created_at":"2026-05-18T00:38:31.089398+00:00"},{"alias_kind":"pith_short_12","alias_value":"O374H23S2GES","created_at":"2026-05-18T12:30:36.002864+00:00"},{"alias_kind":"pith_short_16","alias_value":"O374H23S2GESFWXK","created_at":"2026-05-18T12:30:36.002864+00:00"},{"alias_kind":"pith_short_8","alias_value":"O374H23S","created_at":"2026-05-18T12:30:36.002864+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/O374H23S2GESFWXKML5GDHVIMA","json":"https://pith.science/pith/O374H23S2GESFWXKML5GDHVIMA.json","graph_json":"https://pith.science/api/pith-number/O374H23S2GESFWXKML5GDHVIMA/graph.json","events_json":"https://pith.science/api/pith-number/O374H23S2GESFWXKML5GDHVIMA/events.json","paper":"https://pith.science/paper/O374H23S"},"agent_actions":{"view_html":"https://pith.science/pith/O374H23S2GESFWXKML5GDHVIMA","download_json":"https://pith.science/pith/O374H23S2GESFWXKML5GDHVIMA.json","view_paper":"https://pith.science/paper/O374H23S","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1610.02410&json=true","fetch_graph":"https://pith.science/api/pith-number/O374H23S2GESFWXKML5GDHVIMA/graph.json","fetch_events":"https://pith.science/api/pith-number/O374H23S2GESFWXKML5GDHVIMA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/O374H23S2GESFWXKML5GDHVIMA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/O374H23S2GESFWXKML5GDHVIMA/action/storage_attestation","attest_author":"https://pith.science/pith/O374H23S2GESFWXKML5GDHVIMA/action/author_attestation","sign_citation":"https://pith.science/pith/O374H23S2GESFWXKML5GDHVIMA/action/citation_signature","submit_replication":"https://pith.science/pith/O374H23S2GESFWXKML5GDHVIMA/action/replication_record"}},"created_at":"2026-05-18T00:38:31.089398+00:00","updated_at":"2026-05-18T00:38:31.089398+00:00"}