# Growth in solvable subgroups of GL_r(Z/pZ)

Nick Gill, Harald Andres Helfgott

Research output: Contribution to journalArticlepeer-review

## Abstract

Let $K=Z/pZ$ and let $A$ be a subset of $\GL_r(K)$ 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 $|A\cdot A\cdot A|\gg |A|^{1+\delta}$), 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  is solvable.
Original language English 157 52 Mathematische Annalen 360 1-2 https://doi.org/10.1007/s00208-014-1008-8 Published - 26 Mar 2014

• math.GR
• math.CO
• 20G40, 11B30

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