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 2 A generalized arithmetic progression (GAP) in an abelian group is where
is 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 1 If is a generating system and , then the corresponding GAP contains ball and is contained in .
Example 2 In Heisenberg group with the standard pair of generators, is roughly given in coordinates by . It satisfies and .
So is an instance of approximate subgroup.
Definition 3 A nilprogression of step and rank is a GAP of rank generated by ‘s that generate an -step nilpotent subgroup.
Proposition 4 There exists and such that any -step nilprogression of rank with side length is a -approximate group.
1.2. Strong structure theorem
Theorem 5 For 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 controlled on both sides by a nilprogression.
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 .
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 8 Given 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.
3.1. 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, .
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.