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Journal of Group Theory

Editor-in-Chief: Wilson, John S.

Managing Editor: Howie, James / Kramer, Linus / Parker, Christopher W.

Editorial Board Member: Abért, Miklós / Borovik, Alexandre V. / Boston, Nigel / Bridson, Martin R. / Caprace, Pierre-Emmanuel / Giovanni, Francesco / Guralnick, Robert / Jaikin Zapirain, Andrei / Kessar, Radha / Khukhro, Evgenii I. / Kochloukova, Dessislava H. / Malle, Gunter / Olshanskii, Alexander / Remy, Bertrand / Robinson, Derek J.S. / Willis, George

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Growth in SL2 over finite fields

1Department of Mathematics, Pool Gruppe 1, ETH Zürich, Rämistr. 101, 8092 Zürich, Switzerland

Citation Information: Journal of Group Theory. Volume 14, Issue 2, Pages 273–297, ISSN (Online) 1435-4446, ISSN (Print) 1433-5883, DOI: 10.1515/jgt.2010.056, October 2010

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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}.

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