Approximate Subgroups of Linear Groups

作者：Breuillard Emmanuel^*; Green Ben; Tao Terence

来源：Geometric and Functional Analysis, 2011, 21(4): 774-819.

DOI：10.1007/s00039-011-0122-y

• Approximate group
• linear group

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 SL(n)(Fq) which generates the group must be either very small or else nearly all of SL(n)(Fq). 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.

• 出版日期2011-8