Growth in solvable subgroups of GL_r(Z/pZ)

Nick Gill, Harald Andres Helfgott

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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 languageEnglish
Pages (from-to)157
Number of pages52
JournalMathematische Annalen
Volume360
Issue number1-2
DOIs
Publication statusPublished - 26 Mar 2014

Keywords

  • math.GR
  • math.CO
  • 20G40, 11B30

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