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Abstract
By using tools from additive combinatorics, invariant theory and bounds on the size of the minimal generating sets of PSL2(𝔽q), we prove the following growth property. There exists ɛ > 0 such that the following holds for any finite field 𝔽q. Let G be the group SL2(𝔽q), or PSL2(𝔽q), and let A be a generating set of G. Then
|A · A · A| ⩾ min {|A|1 +ɛ, |G|}.
Our work extends the work of Helfgott [Helfgott, Ann. of Math. 167: 601–623, 2008] who proved similar results for the family {SL2(𝔽p): p prime}.
Received: 2010-02-07
Revised: 2010-05-08
Published Online: 2010-10-13
Published in Print: 2011-April
© de Gruyter 2011
