Growth in solvable subgroups of GLᵣ(Z/pZ)

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Let K=Z/pZ and let A be a subset of 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 Uᵣ and S, with S/Uᵣ nilpotent such that Aₖ∩ S is large and Uᵣ Aₖ, where k is a bounded integer and Aₖ = {x₁ x₂...b xₖ: xᵢ ∈ A ∪ A⁻¹ ∪ 1. The implied constants depend only on the rank r of 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.