** Introduction to approximate groups, VI **

Recall the weak version of the structure theorem for approximate groups.

Theorem 1 (Breuillard-Green-Tao)Let . Let be a finite group such that . Then there exists a virtually nilpotent subgroup of and such that .

Before we state the strong version, let us define generalized arithmetic progressions.

**1. Generalized arithmetic progressions **

Definition 2A generalized arithmetic progression (GAP) in an abelian group is whereis a large box and is a group homomorphism.

In an arbitrary group, fix elements , numbers and consider the set of products of where the total number of occurrences of or is at most . This is the GAP of rank generated by the ‘s with side length .

Example 1If is a generating system and , then the corresponding GAP contains ball and is contained in .

Example 2In Heisenberg group with the standard pair of generators, is roughly given in coordinates by . It satisfies and .

So is an instance of approximate subgroup.

** 1.1. Nilprogressions **

Definition 3A nilprogression of step and rank is a GAP of rank generated by ‘s that generate an -step nilpotent subgroup.

Proposition 4There exists and such that any -step nilprogression of rank with side length is a -approximate group.

** 1.2. Strong structure theorem **

Theorem 5For all there exists such that if is a finite subset of a group that satisfies , then where

- ,
- is virtually nilpotent.
Furthermore, there is an exact sequence

where is finite, is nilpotent, and there exists a nilprogression such that

- ,
- , .

In other words, every approximate group is controlledon both sidesby a nilprogression.

**2. Proof **

A sequence of -approximate groups gives rise to a pseudo-finite -approximate group in an ultraproduct.

Hrushovski’s model theorem states that there exists a locally compact group and an epimorphism such that

- is a compact neighborhood of ,
- For every open set and compact set , there exists and an internal subset such that .

** 2.1. Aside **

Let me answer to a question of Cornelia Drutu: can be a compact Lie group in Hrushovski’s theorem? Answer is positive. For instance, produce an epimorphism . However, in this case, is abelian. Indeed, no simple compact Lie group can be a homomorphic image of a product of finite groups. This is a recent result of Nikolov-Schreider-Thom, relying on a theorem of Nikolov-Segal that states that given , there exists such that for every finite group and generating set ,

In particular, for every word , for every finite group and every elements ,

This cannot happen in a compact Lie group. Indeed, such a group admit sa conjugation-invariant neighborhood of .

** 2.2. Continuation of the proof **

The difficult case is when has no compact open subgroup. The Gleason-Yamabe theorem applied to provides an approximation of by Lie groups. Hence we can assume that is a Lie group. Then the proof is modelled on the proof of the Gleason-Yamabe theorem: reduction to NSS, and proof that NSS groups are Lie groups.

Proposition 6 (Reduction to NSS)Let be a pseudo-finite -approximate group. Then there exists such that is a subgroup whose normalizer contains a subgroup which is a substantial part of and contains no proper subgroups at all.

This follows from

Proposition 7 (Reduction to NSS)Let be a pseudo-finite -approximate group which is NSS. Then there exists which is a pseudo-nilprogression and with bounded.

Indeed, given a -approximate group , define the escape norm with respect to and prove and approximate Gleason-Yamabe Lemma.

Lemma 8Given a -approximate group , there exists a -approximate subgroup such that

From the Lemma, it follows that is a subgroup normalized by . Thus is NSS, this proves Proposition 1.

If is finite, there is an element in of minimal escape norm. This is centralized by the small elements of . If is pseudo-finite, one produces a non-trivial 1-parameter subgroup of which is central in . One can mod out by this 1-parmeter subgroup. By induction on the dimension of , a nilpotent subgroup and a nilprogression is obtained.

There is a technical point. Modding out by the entire 1-parmeter subgroup may produce torsion, hence a group which is not NSS any more. One needs to mod out by the germ only, hence a discussion of local groups.

**3. Application **

** 3.1. A theorem by Benjamini-Finucane-Tessera **

Given integers , consider the class of graphs of finite vertex transitive graphs such that

Theorem 9 (Benjamini-Finucane-Tessera)Every sequence of graphs in has a Gromov-Hausdorff converging subsequence to a flat Finsler -torus, .

** 3.2. Proof **

Let be the automorphism group of the -th graph, the subset of elements that move the base point at most distance 1 away. Then generates . The assumption provides a polynomial growth bound. Therefore, at some scale, doubling occurs. The strong form of the structure theorem provides doubling at every larger scale. Indeed, if is a nilprogression, so is with the same rank and step, hence Proposition 4 applies.

Thus Gromov’s compactness criterion applies: some subsequence Gromov-Haisdorff converges to a locally connected homogeneous metric space . Montgomery-Zippin (in fact, Peter-Weyl) implies is a homogeneous space of a Lie group. Finite group -acts on . Every such -homomorphism to a Lie group is close to a true homomorphism (it is a early result by Alan Turing). Since is approximable by finite groups, it is abelian, hence is a torus.