Approximate Subgroups of Linear Groups

We establish various results on the structure of approximate subgroups in linear groups such as SLn(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 SLn(Fq) which generates the group must be either very small or else nearly all of SLn(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.