Linear approximate groups

Emmanuel Breuillard; Ben Green; Terence Tao

doi:10.3934/era.2010.17.57

Article Contents

2010, Volume 17: 57-67. Doi: 10.3934/era.2010.17.57 open access

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Linear approximate groups

1.
Laboratoire de Mathématiques, Bâtiment 425, Université Paris Sud 11, 91405 Orsay
2.
Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA
3.
Department of Mathematics, UCLA, 405 Hilgard Ave, Los Angeles, CA 90095

Received: December 31, 2009

Published: January 2010

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Abstract

This is an informal announcement of results to be described and proved in detail in [3]. We give various results on the structure of approximate subgroups in linear groups such as ${\rm{S}}{{\rm{L}}_n}(k)$. For example, generalizing a result of Helfgott (who handled the cases $n = 2$ and $3$), we show that any approximate subgroup of ${\rm{S}}{{\rm{L}}_n}({\mathbb{F}_q})$ which generates the group must be either very small or else nearly all of ${\rm{S}}{{\rm{L}}_n}({\mathbb{F}_q})$. The argument is valid for all Chevalley groups $G(\mathbb{F}_q)$. Extending work of Bourgain-Gamburd we also announce some applications to expanders, which will be proven in detail in [4].

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Mathematics Subject Classification: 20G40, 20N99.

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