Growth in solvable subgroups of GL_r(Z/pZ)

Gill, Nick and Helfgott, Harald Andrés (2014). Growth in solvable subgroups of GL_r(Z/pZ). Mathematische Annalen, 360(1-2) pp. 157–208.

DOI: https://doi.org/10.1007/s00208-014-1008-8

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. Fix a positive number C ≥ 1; we prove that either A grows (meaning|A₃|≥ C|A|), or else there are groups U_R and S, with U_R\ ⊴ S ⊴ ‹A›, such that S/U_R is nilpotent, A_k ∩S is large and U _R ⊆ A_k, where k depends only on the rank r of GL_r(K). Here A_k = {x₁ x₂ ... x_k : x_ɩ ∈ A ∪A^-1 ∪{1}\}, and 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 ‹A› is solvable.

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