Abstract
By using tools from additive combinatorics, invariant theory and bounds on the size of the minimal generating sets of PSL2(𝔽 q ), we prove the following growth property. There exists ɛ > 0 such that the following holds for any finite field 𝔽 q . Let G be the group SL2(𝔽 q ), or PSL2(𝔽 q ), 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(𝔽 p ): p prime}.
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