{"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"}