8 - Residually Nilpotent Approximate Groups

Summary

We prove Tointon’s theorem that a finite approximate subgroup of a residually nilpotent group is contained in a union of a few cosets of a finite-by-nilpotent group in which the nilpotent quotient is of bounded step. We first prove it in the special case in which G is nilpotent of unbounded step, and finish the chapter by showing how to extend this to the general residually nilpotent case. As part of the proof we show that if a nilpotent group G is a central extension of a finite approximate group A then the commutator subgroup of G is contained in a bounded power of A. We also show that if A is an approximate subgroup of a nilpotent group then a large piece of A can be written as a bounded series of some bounded extensions and some central extensions.