Growth in solvable subgroups of GL_r(Z/pZ)

TL;DR

This paper investigates the growth behavior of subsets within solvable subgroups of GL_r(Z/pZ), showing that such sets either grow rapidly or are contained in structured, nearly nilpotent groups, extending to non-solvable cases.

Contribution

It reduces the problem of growth in solvable subgroups of GL_r(Z/pZ) to a nilpotent setting and extends results to non-solvable groups using recent advances.

Findings

Either A grows rapidly or is contained in a nearly nilpotent subgroup.

Growth rate is quantified with explicit bounds depending on r.

Results extend to non-solvable groups with recent work.

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 $|A\cdot A\cdot A|\gg |A|^{1+\delta}$), 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 $\GL_r(K)$. When combined with recent work by Pyber and Szab\'o, the main result of this paper implies that it is possible to draw the same conclusions without supposing that $<A>$ is solvable.