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Let $G\\ast_{\\phi}$ be the ascending HNN extension of $G$ with respect to $\\phi$; that is, $G\\ast_{\\phi}$ is given by the presentation $$G\\ast_{\\phi}= < G, t \\ |\\ t^{-1}gt = \\phi(g)\\ \\{for all}\\ g\\in G >.$$ Furthermore, let $\\hat{G\\ast_{\\phi}}$ be the profinite completion of $G\\ast_{\\phi}$. We prove that, for any finite discrete $\\hat{G\\ast_{\\phi}}$-module $A$, the map $H^*(\\hat{G\\ast_{\\phi}}, A)\\to H^*(G\\ast_{\\phi},A)$ induced by the canonical map $G\\ast_{\\phi}\\to \\hat{G\\ast_{\\phi}}$ 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":"1009.2645","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-09-14T13:05:24Z","cross_cats_sorted":["math.KT"],"title_canon_sha256":"a3d1b4a4bb4244be28d1799ef2123f7368cd410405e45b1798b8f8572ba9f2ab","abstract_canon_sha256":"70f7f470b57d6b0b2a1e826fae7d8ef04652fb39260f5239560900f07c12c301"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:37:10.040950Z","signature_b64":"Q0uvluTlyWoBbhdZZQn4zg6Zv8TV/LEkl5aKtkGFslaPwnBFLtRsG9p6XkkFNlnPMohd525lBSuFmUPOSJa+Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"800d8387e4835f2ac05e179e8448bbc83f765c5f04577cb547bfb8d4dad22794","last_reissued_at":"2026-05-18T04:37:10.040254Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:37:10.040254Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ascending HNN extensions of polycyclic groups have the same cohomology as their profinite completions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.KT"],"primary_cat":"math.GR","authors_text":"Karl Lorensen","submitted_at":"2010-09-14T13:05:24Z","abstract_excerpt":"Assume $G$ is a polycyclic group and $\\phi:G\\to G$ an endomorphism. Let $G\\ast_{\\phi}$ be the ascending HNN extension of $G$ with respect to $\\phi$; that is, $G\\ast_{\\phi}$ is given by the presentation $$G\\ast_{\\phi}= < G, t \\ |\\ t^{-1}gt = \\phi(g)\\ \\{for all}\\ g\\in G >.$$ Furthermore, let $\\hat{G\\ast_{\\phi}}$ be the profinite completion of $G\\ast_{\\phi}$. We prove that, for any finite discrete $\\hat{G\\ast_{\\phi}}$-module $A$, the map $H^*(\\hat{G\\ast_{\\phi}}, A)\\to H^*(G\\ast_{\\phi},A)$ induced by the canonical map $G\\ast_{\\phi}\\to \\hat{G\\ast_{\\phi}}$ is an isomorphism."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.2645","kind":"arxiv","version":5},"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":"1009.2645","created_at":"2026-05-18T04:37:10.040362+00:00"},{"alias_kind":"arxiv_version","alias_value":"1009.2645v5","created_at":"2026-05-18T04:37:10.040362+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.2645","created_at":"2026-05-18T04:37:10.040362+00:00"},{"alias_kind":"pith_short_12","alias_value":"QAGYHB7EQNPS","created_at":"2026-05-18T12:26:12.377268+00:00"},{"alias_kind":"pith_short_16","alias_value":"QAGYHB7EQNPSVQC6","created_at":"2026-05-18T12:26:12.377268+00:00"},{"alias_kind":"pith_short_8","alias_value":"QAGYHB7E","created_at":"2026-05-18T12:26:12.377268+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/QAGYHB7EQNPSVQC6C6PIISF3ZA","json":"https://pith.science/pith/QAGYHB7EQNPSVQC6C6PIISF3ZA.json","graph_json":"https://pith.science/api/pith-number/QAGYHB7EQNPSVQC6C6PIISF3ZA/graph.json","events_json":"https://pith.science/api/pith-number/QAGYHB7EQNPSVQC6C6PIISF3ZA/events.json","paper":"https://pith.science/paper/QAGYHB7E"},"agent_actions":{"view_html":"https://pith.science/pith/QAGYHB7EQNPSVQC6C6PIISF3ZA","download_json":"https://pith.science/pith/QAGYHB7EQNPSVQC6C6PIISF3ZA.json","view_paper":"https://pith.science/paper/QAGYHB7E","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1009.2645&json=true","fetch_graph":"https://pith.science/api/pith-number/QAGYHB7EQNPSVQC6C6PIISF3ZA/graph.json","fetch_events":"https://pith.science/api/pith-number/QAGYHB7EQNPSVQC6C6PIISF3ZA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QAGYHB7EQNPSVQC6C6PIISF3ZA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QAGYHB7EQNPSVQC6C6PIISF3ZA/action/storage_attestation","attest_author":"https://pith.science/pith/QAGYHB7EQNPSVQC6C6PIISF3ZA/action/author_attestation","sign_citation":"https://pith.science/pith/QAGYHB7EQNPSVQC6C6PIISF3ZA/action/citation_signature","submit_replication":"https://pith.science/pith/QAGYHB7EQNPSVQC6C6PIISF3ZA/action/replication_record"}},"created_at":"2026-05-18T04:37:10.040362+00:00","updated_at":"2026-05-18T04:37:10.040362+00:00"}