{"paper":{"title":"Model theoretic connected components of finitely generated nilpotent groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.GR"],"primary_cat":"math.LO","authors_text":"Cong Chen, Jakub Gismatullin, Nathan Bowler","submitted_at":"2011-12-14T18:02:03Z","abstract_excerpt":"We prove that for a finitely generated infinite nilpotent group G with a first order structure (G,*,...), the connected component G*0 of a sufficiently saturated extension G* of G exists and equals $\\bigcap_{n\\in\\N} {g^n : g\\in G^*}$. We construct a first order expansion of Z by a predicate (Z,+,P) such that the type-connected component Z*00_{\\emptyset} is strictly smaller than Z*0. We generalize this to finitely generated virtually solvable groups. As a corollary of our construction we obtain an optimality result for the van der Waerden theorem."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.3292","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"}