{"paper":{"title":"Extremal behavior of divisibility functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"D. B. McReynolds, Khalid Bou-Rabee","submitted_at":"2012-11-20T12:43:58Z","abstract_excerpt":"In this short article, we study the extremal behavior $F_\\Gamma(n)$ of divisibility functions $D_\\Gamma$ introduced by the first author for finitely generated groups $\\Gamma$. We show finitely generated subgroups of $\\GL(m,K)$ for an infinite field $K$ have at most polynomial growth for the function $F_\\Gamma(n)$. Consequently, we obtain a dichotomy for the growth rate of $\\log F_\\Gamma(n)$ for finitely generated subgroups of $\\GL(n,\\C)$. We also show that if $F_\\Gamma(n) \\preceq \\log \\log n$, then $\\Gamma$ is finite. In contrast, when $\\Gamma$ contains an element of infinite order, $\\log n \\p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.4727","kind":"arxiv","version":3},"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"}