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Published online by Cambridge University Press: 29 May 2025
We give a description of nongrowing subsets in linear groups over arbitrary fields, which extends the product theorem for simple groups of Lie type. We also give an account of various related aspects of growth in linear groups.
The following theorem was proved independently in 2010 by Breuillard, Green and Tao [Breuillard et al. 2011a] and the authors [Pyber and Szabó 2010].
Theorem 1 (product theorem). Let L be a finite simple group of Lie type of rank r and A a generating set of L. Then either A3 = L or
|A3| > |A|1+ Ɛ,
where Ɛ depends only on r .
For G =PSL(2, p), p prime, this is a famous result of Helfgott [2008]. Other special cases that preceded the general proof include G = PSL(3, p) [Helfgott 2011] and G = PSL(2, q), q a prime power [Dinai 2011; Varjú 2012].
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