Approximate groups and doubling metrics

TL;DR

This paper extends Freiman's theorem to certain non-abelian groups by using a stronger growth condition, resulting in structural descriptions involving translation-invariant pseudo-metrics.

Contribution

It introduces a non-abelian version of Freiman's theorem for groups with polynomial growth, replacing the small doubling condition with a relative growth hypothesis.

Findings

Provides a structural description of sets with polynomial growth in non-abelian groups.

Identifies balls in translation-invariant pseudo-metrics as the key structures.

Complements existing approaches by other researchers in non-abelian additive combinatorics.

Abstract

We develop a version of Freiman's theorem for a class of non-abelian groups, which includes finite nilpotent, supersolvable and solvable A-groups. To do this we have to replace the small doubling hypothesis with a stronger relative polynomial growth hypothesis akin to that in Gromov's theorem (although with an effective range), and the structures we find are balls in (left and right) translation invariant pseudo-metrics with certain well behaved growth estimates. Our work complements three other recent approaches to developing non-abelian versions of Freiman's theorem by Breuillard and Green, Fischer, Katz and Peng, and Tao.