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

Details Growth in solvable subgroups of $GL_r(Z/pZ)$ Growth in solvable subgroups of $GL_r(Z/pZ)$

2010-08-31

arxiv.org/abs/1008.5264v2

Mathematics

Group Theory

42 pages. Some errors are corrected in this version. The main result has been generalized in two directions: first, it is writ

Scientific paper

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. It is our intention to extend the main result of this paper to hold for $\GL_r(\mathbb{F}_q)$, $q$ an arbitrary prime power.

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