Approximate Subgroups of Linear Groups
Identifieur interne : 000323 ( Main/Exploration ); précédent : 000322; suivant : 000324Approximate Subgroups of Linear Groups
Auteurs : Emmanuel Breuillard [France] ; Ben Green [Royaume-Uni] ; Terence Tao [États-Unis]Source :
- Geometric and Functional Analysis [ 1016-443X ] ; 2011-08-01.
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Abstract
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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DOI: 10.1007/s00039-011-0122-y
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