{"paper":{"title":"Growth in solvable subgroups of GL_r(Z/pZ)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"Harald Andres Helfgott, Nick Gill","submitted_at":"2010-08-31T09:27:41Z","abstract_excerpt":"Let $K=Z/pZ$ and let $A$ be a subset of $\\GL_r(K)$ such that $<A>$ is solvable. We reduce the study of the growth of $A$ under the group operation to the nilpotent setting. Specifically we prove that either $A$ grows rapidly (meaning $|A\\cdot A\\cdot A|\\gg |A|^{1+\\delta}$), or else there are groups $U_R$ and $S$, with $S/U_R$ nilpotent such that $A_k\\cap S$ is large and $U_R\\subseteq A_k$, where $k$ is a bounded integer and $A_k = \\{x_1 x_2...b x_k : x_i \\in A \\cup A^{-1} \\cup {1}}$. The implied constants depend only on the rank $r$ of $\\GL_r(K)$.\n  When combined with recent work by Pyber and S"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.5264","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"}