[Paper Review] Freiman's theorem for solvable groups

Terence Tao|arXiv (Cornell University)|Jun 18, 2009

Limits and Structures in Graph Theory19 references31 citations

TL;DR

This paper extends Freiman's theorem to solvable groups of bounded derived length by introducing 'coset nilprogressions' as the controlling structure for sets of small doubling. It shows such sets are efficiently controlled by sets with polynomial growth, leading to a strengthened version of the Milnor-Wolf theorem where only one large ball needs to be of polynomial size.

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.

Motivation & Objective

• To generalize Freiman's theorem from abelian to solvable groups of bounded derived length.
• To identify a suitable replacement for coset progressions in non-abelian settings, introducing the concept of coset nilprogressions.
• To establish that sets of small doubling in such groups are controlled by sets with polynomial growth, lying within virtually nilpotent subgroups.
• To strengthen the Milnor-Wolf theorem by showing that solvable groups of polynomial growth are virtually nilpotent if only one large ball has polynomial size.

Proposed method

• Introduce the notion of a coset nilprogression as a generalization of coset progressions for solvable groups.
• Use structural results from the theory of nilpotent and solvable groups to control the growth of product sets.
• Apply tools from additive combinatorics, particularly those related to doubling constants and energy estimates.
• Leverage the bounded derived length condition to inductively build the controlling coset nilprogression.
• Employ the idea of 'efficient containment' to show that small doubling implies polynomial growth in the controlling set.
• Use the structure of solvable groups to reduce the problem to nilpotent subgroups via a finite-index subgroup construction.

Experimental results

Research questions

• RQ1 Can Freiman-type theorems be extended from abelian to solvable groups of bounded derived length?
• RQ2 What is the appropriate generalization of coset progressions for non-abelian solvable groups?
• RQ3 Under what conditions does small doubling in a solvable group imply that the set is controlled by a set of polynomial growth?
• RQ4 Can the Milnor-Wolf theorem be strengthened by requiring only one large ball to have polynomial size?
• RQ5 Is every set of small doubling in a solvable group of bounded derived length contained in a virtually nilpotent subgroup?

Key findings

• Sets of small doubling in solvable groups of bounded derived length are efficiently controlled by a coset nilprogression.
• The controlling set has polynomial growth, and its iterated products grow polynomially.
• Such sets are contained in a virtually nilpotent subgroup, extending the Green-Ruzsa framework to non-abelian settings.
• The Milnor-Wolf theorem is strengthened: a solvable group is virtually nilpotent if only one large ball has polynomial size.
• The result provides a structural characterization of small doubling in solvable groups analogous to Freiman’s theorem in abelian groups.