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Growth in SL2 over finite fields

Dinai, Oren

In: Journal of Group Theory, 2011, vol. 14, no. 2, p. 273-297

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    Title
    • Growth in SL2 over finite fields
    Author
    • Dinai, Oren. Department of Mathematics, Pool Gruppe 1, ETH Zürich, Rämistr. 101, 8092 Zürich, Switzerland
    Document Type
    • Postprint
    Language
    • English
    Published in
    • Journal of Group Theory, 2011, vol. 14, no. 2, p. 273-297. Walter de Gruyter GmbH & Co. KG
    Other Version
    Classification
    • Health
    OAI-PMH Identifier
    • oai:doc.rero.ch:294623
    Summary
    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}