The structure of approximate groups

Abstract

Let K 1 be a parameter. A K-approximate group is a finite set A in a (local) group which contains the identity, is symmetric, and such that A · A is covered by K left translates of A.The main result of this paper is a qualitative description of approximate groups as being essentially finite-by-nilpotent, answering a conjecture of H. Helfgott and E. Lindenstrauss. This may be viewed as a generalisation of the Freiman-Ruzsa theorem on sets of small doubling in the integers to arbitrary groups.We begin by establishing a correspondence principle between approximate groups and locally compact (local) groups that allows us to recover many results recently established in a fundamental paper of Hrushovski. In particular we establish that approximate groups can be approximately modeled by Lie groups.To prove our main theorem we apply some additional arguments essentially due to Gleason. These arose in the solution of Hilbert's fifth problem in the 1950s.Applications of our main theorem include a finitary refinement of Gromov's theorem, as well as a generalized Margulis lemma conjectured by Gromov and a result on the virtual nilpotence of the fundamental group of Ricci almost nonnegatively curved manifolds.Résumé. Soit K 1 un paramètre. On appelle groupe K-approximatif toute partie finie A d'un groupe (ou d'un groupe local) qui est symétrique, contient l'identité et est telle que A · A peutêtre recouvert par au plus K translatésà gauche de A.Le résultat principal de cet article montre que tout groupe approximatif est, grossièrement, finipar-nilpotent, ce qui répond par l'affirmativeà une conjecture de H.Helfgott et de E.Lindenstrauss.On peut interpréter ce théorème comme une généralisationà un groupe quelconque du théorème de Freiman-Ruzsa sur la structure des parties finiesà petit doublement du groupe additif des entiers relatifs.Nous commençons parétablir un principe de correspondence entre les groupes approximatifs et les groupes (ou groupes locaux) localement compacts, puis nous en déduisons de nombreuses conséquences issues d'un important article récent de Hrushovski. En particulier, nous montrons que tout groupe approximatif peut-être représenté par un groupe de Lie.Pour démontrer notre théorème principal, nous appliquons des arguments, en substance dusà Gleason pour la plupart, qui ont vus le jour dans le contexte de la solution du cinquième problème de Hilbert dans les années 50.En guise d'application du théorème principal, nous montrons une version fine du théorème de Gromov, ainsi qu'un lemme de Margulis généralisé conjecturé par Gromov, et un résultat sur la presque nilpotence des groupes fondamentaux des variétésà courbure de Ricci presque positive.

Coarse Geometry and Randomness

“…The proof relies on a recent quantitative version of Gromov's theorem on groups with polynomial growth obtained by Breuillard, Green and Tao [BGT12] and a scaling limit theorem for nilpotent groups by Pansu. A quantitative version of this theorem can be useful in establishing the resistance conjecture from [BK05] and the polynomial case of the conjectures in [ABS04]. In [BGT12] a strong isoperimetric inequality for finite vertex transitive is established.…”

A nilpotent Freiman dimension lemma

“…In [5], we established a structure theorem for sets of small doubling and for approximate subgroups of arbitrary groups. Approximate subgroups are symmetric finite subsets A of an ambient group whose product set AA can be covered by a bounded number of translates of the set.…”

“…One of the main theorems proved in [5] asserts, roughly speaking, that approximate subgroups can be covered by a bounded number of cosets of a certain finite-by-nilpotent subgroup with bounded complexity. The bounds on the number of cosets and on the complexity (rank and step) of the nilpotent subgroup depend only on the doubling parameter K.…”

“…In [5], this analogy was worked out making use of ultrafilters in order to build a certain limit approximate group to which we then applied an adequately modified version of the tools developed in the 1950s for the solution of Hilbert's fifth problem on the structure of locally compact groups.…”

“…The use of ultrafilters makes the results of [5] ineffective in a key way. The aim of the present note is to give, in a fairly special case, a simple argument independent of [5] but which achieves the same goal with explicit bounds.…”