Growth in solvable subgroups of GLr(Z/pZ)

Gill, Nick and Helfgott, Harald Andrés (2014). Growth in solvable subgroups of GLr(Z/pZ). Mathematische Annalen, 360(1-2) pp. 157–208.

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

Abstract

Let K = Z/pZ and let $A$ be a subset of GLr(K) such that ‹A› is solvable. We reduce the study of the growth of A under the group operation to the nilpotent setting. Fix a positive number C ≥ 1; we prove that either A grows (meaning|A3|≥ C|A|), or else there are groups UR and S, with UR\ ⊴ S ⊴ ‹A›, such that S/UR is nilpotent, AkS is large and U RAk, where k depends only on the rank r of GLr(K). Here Ak = {x1 x2 ... xk : xɩAA-1 ∪{1}\}, and the implied constants depend only on the rank r of GLr(K). 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 ‹A› is solvable.

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