Growth in solvable subgroups of GL_r(Z/pZ)

2010/08/31 by Nick Gill, H. A. Helfgott, Gill, Nick +2
Mathematics · #11B30 #20G40 #Advanced Algebra and Geometry #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Limits and Structures in Graph Theory #math.CO #math.GR #msc:11B30 #msc:20G40

paper · pdf · doi:10.48550/arxiv.1008.5264

46 pages. This version includes revisions recommended by an anonymous referee including, in particular, the statement of a new theorem, Theorem 3

openalex publication_date 2010/08/31 · arxiv created 2013/09/10 · arxiv updated 2013/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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⋅ A⋅ A|≫ |A|^1+δ), or else there are groups U_R and S, with S/U_R nilpotent such that A_k∩ S is large and U_R⊆ A_k, where k is a bounded integer and A_k = \x₁ x₂...b x_k : x_i ∈ A ∪ A^-1 ∪ 1. The implied constants depend only on the rank r of \GL_r(K). When combined with recent work by Pyber and Szabó, the main result of this paper implies that it is possible to draw the same conclusions without supposing that <A> is solvable.

Citations

• Growth in finite simple groups of Lie type of bounded rank

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