Let K be R, C or a Z/pZ. Let G = SL_2(K). Not long ago, I proved the following theorem: for every subset A of G that is not contained in a proper subgroup, the set A\cdot A\cdot A is much larger than A. A generalisation to groups of higher rank was desired by many, but seemed hard to obtain.
I shall now present a proof somewhat different from the first one. The role of both linearity and the group structure of G should now be clearer. A few ideas towards a generalisation will be discussed, with a focus on the case of SL_3(Z/pZ).
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