Linear approximate groups

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].

Locations

• Electronic Research Announcements - View - PDF
• arXiv (Cornell University) - View - PDF
• arXiv (Cornell University) - View - PDF
• arXiv (Cornell University) - PDF
• Electronic Research Announcements - View - PDF
• arXiv (Cornell University) - View - PDF
• arXiv (Cornell University) - View - PDF
• arXiv (Cornell University) - PDF
• Electronic Research Announcements - View - PDF
• arXiv (Cornell University) - View - PDF
• arXiv (Cornell University) - View - PDF
• arXiv (Cornell University) - PDF