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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 SL_n(F_q) which generates the
group must be either very small or else nearly all of SL_n(F_q). The argument
generalises to other absolutely almost simple connected (and non-commutative)
algebraic groups G over a finite field k. In a subsequent paper, we will give
applications of this result to the expansion properties of Cayley graphs.Comment: 48 page
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