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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 (Digital Object Identifier) Link: https://doi.org/10.1007/s00208-014-1008-8
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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.

Item Type: Journal Item
Copyright Holders: 2014 Springer-Verlag Berlin Heidelberg
ISSN: 1432-1807
Academic Unit/School: Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
Faculty of Science, Technology, Engineering and Mathematics (STEM)
Item ID: 39345
Depositing User: Nick Gill
Date Deposited: 29 Jan 2014 12:32
Last Modified: 08 Dec 2018 11:38
URI: http://oro.open.ac.uk/id/eprint/39345
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