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

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Abstract

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. We reduce the study of the growth of \(A\) under the group operation to the nilpotent setting. Fix a positive number \(C\ge 1\); we prove that either \(A\) grows (meaning \(|A_3|\ge C|A|\)), or else there are groups \(U_R\) and \(S\), with \(U_R\unlhd S \unlhd \langle A\rangle \), 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 \(A_k = \{x_1 x_2 \cdots x_k : x_i \in A \cup A^{-1} \cup \{1\}\}\). 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.

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Notes

  1. We thank an anonymous referee for pointing out that our results can be extended in this way, and for sketching the proof.

  2. We caution the reader that there are a number of slightly differing definitions of nilprogressions, and coset nilprogressions, in the literature. In particular the definitions used by Tao in his work on solvable groups (which were the first such definitions to appear in the literature) are slightly different from the definitions we require for Theorem 1.3. For that theorem we use the definitions of [8]; see Sect. 9 for full details.

  3. Our thanks to Simon Goodwin for pointing this out.

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Acknowledgments

Pablo Spiga provided help with group theory results; Martin Kassabov provided significant assistance in understanding solvable algebraic groups. Thanks are also due in this regard to Emmanuel Breuillard, Kevin Buzzard, Simon Goodwin, Alex Gorodnik, Scott Murray, László Pyber and an anonymous referee. In addition Simon Goodwin pointed out an error in the statement of Lemma 3.1 in an earlier version. Part of this work was completed while the first author was visiting the University of Western Australia and the University of Bristol; he would like to thank members of both maths departments for providing excellent working conditions, and for their interest in the work at hand. The second author would like to thank the Ecole Polytechnique Fédérale de Lausanne for hosting him during part of his work on this project. Sect. 8 of this paper is joint work with László Pyber and Endre Szabó; it is a pleasure to thank them for the warm way in which they have shared their considerable insight.

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  1. Escuela de Matemática, Universidad de Costa Rica, 11501 , San Jose, Costa Rica

    Nick Gill

  2. ENS-DMA, 45 rue d’Ulm, 75230 , Paris, France

    Harald Andrés Helfgott

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  1. Nick Gill
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Gill, N., Helfgott, H.A. Growth in solvable subgroups of \({{\mathrm{GL}}}_r({\mathbb {Z}}/p{\mathbb {Z}})\) . Math. Ann. 360, 157–208 (2014). https://doi.org/10.1007/s00208-014-1008-8

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  • DOI: https://doi.org/10.1007/s00208-014-1008-8

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