Growth in solvable subgroups of GL_r(Z/pZ)

Let $K=Z/pZ$ and let A be a subset of ᵣ(K) such that $$ is solvable. We reduce the study of the growth of $A$ under the group operation to the nilpotent setting. Specifically we prove that either $A$ grows rapidly (meaning $|A· A· A| |A|1+δ$), or else there are groups $U_R$ and $S$, with $S/U_R$ nilpotent such that $A_k∩ S$ is large and $U_R⊆ A_k$, where $k$ is a bounded integer and $A_k = \{x_1 x_2...b x_k : x_i ∈ A ∪ A⁻¹ ∪ {1}}$. The implied constants depend only on the rank $r$ of $_r(K)$. When combined with recent work by Pyber and Szabó, the main result of this paper implies that it is possible to draw the same conclusions without supposing that $$ is solvable.