Growth in solvable subgroups of \(\mathrm{GL}_r(\mathbb Z/p\mathbb Z)\). (Q464139)

Let \(K=\mathbb Z/p\mathbb Z\) and let \(A\) be a subset of \(\mathrm{GL}_r(K)\) such that \(\langle A\rangle\) is solvable. The authors reduce the study of the growth of \(A\) under the group operation to the nilpotent setting. Fix a positive number \(C\geq 1\); they prove that either \(A\) grows (meaning \(|A_3|\geq C|A|\)), or else there are groups \(U_R\) and \(S\), with \(U_R\trianglelefteq S\trianglelefteq\langle A\rangle\), such that \(S/U_R\) is nilpotent, \(A_k\cap S\) is large and \(U_R\subseteq A_k\), where \(k\) depends only on the rank \(r\) of \(\mathrm{GL}_r(K)\). Here \(A_k=\{x_1x_2\cdots x_k:x_i\in A\cup A^{-1}\cup\{1\}\}\). 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 \(\langle A\rangle\) is solvable.