Approximate Subgroups of Linear Groups

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 which generates the group must be either very small or else nearly all of . 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.