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.

We present Breuillard, Green and Tao’s theorem that a finite approximate subgroup of a complex linear group of bounded degree is contained in a union of a few cosets of a nilpotent subgroup of bounded step. We state two substantial ingredients without proof. The first is a result of Mal’cev and Platinov that a virtually soluble complex linear group of bounded degree has a soluble subgroup of bounded index. The second is Breuillard’s uniform Tits alternative, which states that if a finitely generated complex linear group of bounded degree is not virtually soluble then there exist two free generators of a free subgroup that can be expressed as products of boundedly many generators. The third main ingredient is a result, due independently to Sanders and to Croot and Sisask, that if A is an arbitrary approximate group then there is a relatively large neighbourhood of the identity S with the property that a large power of S is contained in a small power of A; we prove this in full.