6 - Nilpotent Approximate Groups

Summary

We present Tointon’s proof of Freiman’s theorem in an arbitrary nilpotent group. More specifically, we show that a finite K-approximate subgroup of a nilpotent group of bounded step is contained in a relatively small coset progression of rank bounded by a polynomial in K. We start by treating the torsion-free case, where the details are easiest. As part of our proof of the general case we show that if X is a union of subgroups in an abelian p-group of rank r then the subgroup generated by X has diameter at most r with respect to X. We also show that if H is a subgroup of a nilpotent group G generated by a K-approximate group A, and H is contained in a bounded power of A, then the normal closure of H in G is also contained in a bounded power of A.