{"paper":{"title":"Gradients of sequences of subgroups in a direct product","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Mark Shusterman, Nikolay Nikolov, Zvi Shemtov","submitted_at":"2016-09-28T13:14:34Z","abstract_excerpt":"For a sequence $\\{U_n\\}_{n = 1}^\\infty$ of finite index subgroups of a direct product $G = A \\times B$ of finitely generated groups, we show that $$\\lim_{n \\to \\infty} \\frac{\\min\\{|X| : \\langle X \\rangle = U_n\\}}{[G : U_n]} = 0$$ once $[A : A \\cap U_n], [B : B \\cap U_n] \\to \\infty$ as $n \\to \\infty$. Our proof relies on the classification of finite simple groups. For $A,B$ that are finitely presented we show that $$ \\lim_{n \\to \\infty} \\frac{\\log |\\mathrm{Torsion}(U_n^{\\mathrm{ab}})|}{[G : U_n]} = 0. $$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.08900","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"}