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Approximate Subgroups of Linear Groups

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Abstract

We establish various results on the structure of approximate subgroups in linear groups such as SL n (k) that were previously announced by the authors. For example, generalising a result of Helfgott (who handled the cases n = 2 and 3), we show that any approximate subgroup of \({{\rm SL}_{n}({\mathbb {F}}_{q})}\) which generates the group must be either very small or else nearly all of \({{\rm SL}_{n}({\mathbb {F}}_{q})}\). The argument generalises to other absolutely almost simple connected (and non-commutative) algebraic groups G over an arbitrary field k and yields a classification of approximate subgroups of G(k). In a subsequent paper, we will give applications of this result to the expansion properties of Cayley graphs.

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Authors and Affiliations

  1. Laboratoire de Mathématiques, Université Paris Sud 11, Bâtiment 425, 91405, Orsay, France

    Emmanuel Breuillard

  2. Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WA, England

    Ben Green

  3. Department of Mathematics, UCLA, 405 Hilgard Ave., Los Angeles, CA, 90095, USA

    Terence Tao

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  1. Emmanuel Breuillard
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  2. Ben Green
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  3. Terence Tao
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Correspondence to Ben Green.

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Breuillard, E., Green, B. & Tao, T. Approximate Subgroups of Linear Groups. Geom. Funct. Anal. 21, 774–819 (2011). https://doi.org/10.1007/s00039-011-0122-y

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