Freiman's theorem for solvable groups

TL;DR

This paper extends Freiman's theorem to solvable groups of bounded derived length, showing small doubling sets are controlled by polynomial growth sets within virtually nilpotent groups, and strengthens the Milnor-Wolf theorem.

Contribution

It generalizes Freiman's theorem to solvable groups using coset nilprogressions and improves the Milnor-Wolf theorem for polynomial growth groups.

Findings

Small doubling sets are controlled by polynomial growth sets

Sets are contained inside virtually nilpotent groups

Strengthens the Milnor-Wolf theorem for solvable groups

Abstract

Freiman's theorem asserts, roughly speaking, if that a finite set in a torsion-free abelian group has small doubling, then it can be efficiently contained in (or controlled by) a generalised arithmetic progression. This was generalised by Green and Ruzsa to arbitrary abelian groups, where the controlling object is now a coset progression. We extend these results further to solvable groups of bounded derived length, in which the coset progressions are replaced by the more complicated notion of a "coset nilprogression". As one consequence of this result, any subset of such a solvable group of small doubling is is controlled by a set whose iterated products grow polynomially, and which are contained inside a virtually nilpotent group. As another application we establish a strengthening of the Milnor-Wolf theorem that all solvable groups of polynomial growth are virtually nilpotent, in which only one large ball needs to be of polynomial size. This result complements recent work of Breulliard-Green, Fisher-Katz-Peng, and Sanders.