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Gill, Nick and Helfgott, Harald Andrés
(2014).
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, Ak ∩S is large and U R ⊆ Ak, where k depends only on the rank r of GLr(K). Here Ak = {x1 x2 ... xk : xɩ ∈ A ∪A-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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About
- Item ORO ID
- 39345
- Item Type
- Journal Item
- ISSN
- 1432-1807
- Academic Unit or School
-
Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
Faculty of Science, Technology, Engineering and Mathematics (STEM) - Copyright Holders
- © 2014 Springer-Verlag Berlin Heidelberg
- Depositing User
- Nick Gill