{"paper":{"title":"Dependence over subgroups of free groups","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Amnon Rosenmann, Enric Ventura Capell","submitted_at":"2021-07-07T11:30:24Z","abstract_excerpt":"Given a finitely generated subgroup $H$ of a free group $F$, we present an algorithm which computes $g_1,\\ldots,g_m\\in F$, such that the set of elements $g\\in F$, for which there exists a non-trivial $H$-equation having $g$ as a solution, is, precisely, the disjoint union of the double cosets $H\\sqcup Hg_1H\\sqcup \\cdots \\sqcup Hg_mH$. Moreover, we present an algorithm which, given a finitely generated subgroup $H\\leqslant F$ and an element $g\\in F$, computes a finite set of elements of $H * \\langle x \\rangle$ that generate (as a normal subgroup) the ``ideal\" $I_H(g) \\unlhd H * \\langle x \\rangl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2107.03154","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2107.03154/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}