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Approximate subgroups of linear groups. (English) Zbl 1229.20045

The authors establish various results on the structure of approximate subgroups in linear groups such as \(\text{SL}_n(k)\) that were previously announced by them. For example, generalizing a result of Helfgott (who handled the cases \(n=2\) and 3), they show that any approximate subgroup of \(\text{SL}_n(\mathbb F_q)\) which generates the group must be either very small or else nearly all of \(\text{SL}_n(\mathbb F_q)\). The argument generalizes to other absolutely almost simple connected (and non-commutative) algebraic groups \(\mathbf G\) over an arbitrary field \(k\) and yields a classification of approximate subgroups of \(\mathbf G(k)\). In a subsequent paper, they promise to give applications of this result to the expansion properties of Cayley graphs.

MSC:

20G40 Linear algebraic groups over finite fields
11B75 Other combinatorial number theory
20F05 Generators, relations, and presentations of groups
20E07 Subgroup theorems; subgroup growth
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