Introduction to approximate groups, IV
Today, I will enter deeper into the proof of the structure theorem.
Theorem 1 (Breuillard-Green-Tao) Let . Let be a finite group such that . Then there exists a virtually nilpotent subgroup of and such that . Furthermore, has a normal nilpotent finite index subgroup with number of generators and nilpotency class .
This is still a weak version. In the strong version, where is contained in the product of bounded many copies of .
It is enough to prove the theorem when is a -approximate subgroup, i.e. , and , . Indeed, we have seen in lecture 1 that, with some combinatorics, this is equivalent, up to passing to a large subset.
If has finite exponent (i.e. satisfies the law for some integer ), then must be finite. Hence is contained in boundedly many cosets of a finite subgroup of .
0.1. Strategy of proof
Inspired from Gromov, expanded in model theory language by Hrushovski.
This is the model theory of the poor.
Let be a sequence of groups and be finite subsets of which are -approximate subgroups. Fix a nonprincipal ultrafilter on . Form the ultraproduct
Los’ theorem asserts that any first order sentence in the language of groups holds for iff it holds for -almost every . Usually, it is easy to reprove it in simple situations.
Example 1 Let be a finite subset. Let (such subsets are called internal subsets of ). Given , iff a.e..
is agroup, is a subgroup iff is a subgroup a.e., is a -approximate subgroup iff is a -approximate subgroup a.e. However, ultraproducts of finite groups need not be finite, they are by definition pseudofinite.
The structure theorem is equivalent to the following.
Theorem 2 If is a pseudofinite -approximate subgroup of , then there exist internal subgroups such that is normal in , is nilpotent, is pseudofinite, and , with .
Note that this does not give a bound on , i.e. on the number of cosets required.
2. Hrushovski’s Lie model theorem
Theorem 3 Let be a pseudofinite -approximate subgroup. There is a locally compact group and a surjective group homomorphism
- is a compact neighborhood of ,
- For every open set and compact set , there exists and an internal subset such that .
We see that has kernel contained in for some finite . Hence is almost injective.
Also, must be unimodular. Indeed, admits a biinvariant measure, arising as a limit of renormalized counting measures. Its image by is a biinvariant Haar measure on .
The proof will show that has an open subgroup such that is normal and compact and is a nilpotent Lie group.
2.1. Special cases
What if is trivial? Then . By (2), is a subgroup, so a.e. is a finite subgroup, countained is finitely many translates of . This is the structure theorem
What if is compact? The same conclusion holds.
What if has a compact open subgroup ? By (1), is contained in finitely many translates of . Since is internal and contained in , there exists a.e. a finite subgroup such that is contained in bounded many copies of .
Clearly, if , approximation by finite subgroups does not hold, so does not contain any compact open subgroup.
Corollary 4 (Hrushovski) Given integers , , if is a group of exponent and a -approximate subgroup of ., then there exists a finite subgroup such that and is countained in a bounded number of translates of .
It suffices to prove this for pseudofinite groups of finite exponent. has finite exponent. This implies that is totally disconnected, hence it has a compact open subgroup (Dantzig’s theorem).
3. Proof of the Lie model theorem
No Lie group today, they will arise next time. The point is to find a (possibly non-Hausdorff) locally compact topology on . Then one divides by the closure of 1.
One would like to say that is close to the identity if is small, e.g.
The following combinatorial lemma helps.
Lemma 5 (Hrushovski, Sanders, Croot-Sisak) Given integers , let be a finite -approximate subgroup. Then there exists a subset ,
- , for all .
Indeed, applying the Lemma to in , find sets . Furthermore, inductively, one can assume that
Let be their ultraproduct. These sets form a basis for a topology. Their intersection is a normal subgroup. One shows that mod intersection is compact. This a bit like the bounded covering property characterization of approximate subgroups.
The proofs by Sanders or Croot-Sisak are elementary. Croot-Sisak consider sets
In model theory, this topology come out naturally. It is called the logic topology, compactness follows from general principles. See Ben Green’s Bourbaki seminar.