9 - Soluble Approximate Subgroups of GLn(C)

Summary

We prove Breuillard and Green’s theorem that a finite approximate subgroup of a soluble complex linear group G of bounded degree is contained in a union of a few cosets of a nilpotent group of bounded step. We first treat the special case in which G is an upper-triangular group. An important ingredient is Solymosi’s sum-product theorem over the complex numbers, which we state and prove. We introduce some basic representation theory and use it to prove that a soluble complex linear group of bounded degree has a subgroup of bounded index that is conjugate to an upper-triangular group; this is a special case of a result of Mal’cev. We then use this to extend from the upper-triangular case to the general soluble case.