{"paper":{"title":"Generic elements of a Zariski-dense subgroup form an open subset","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.GR","authors_text":"Andrei S. Rapinchuk, Gopal Prasad","submitted_at":"2017-07-09T00:45:49Z","abstract_excerpt":"Let G be a semi-simple algebraic group over a finitely generated field K of characteristic zero, and let \\Gamma < G(K) be a finitely generated Zariski-dense subgroup. In this note we prove that the set of K-generic elements of \\Gamma (whose existence was established earlier in [9]) is open in the profinite topology of \\Gamma. We then extend this result to the fields of positive characteristic, and also prove the existence of generic elements in this case."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.02508","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"}