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}.