Notes of Emmanuel Breuillard’s fourth Cambridge lecture 11-04-2017

Introduction to approximate groups, IV

Today, I will enter deeper into the proof of the structure theorem.

Theorem 1 (Breuillard-Green-Tao) Let {k\geq 1}. Let {A\subset G} be a finite group such that {|AA|\leq K|A|}. Then there exists a virtually nilpotent subgroup {H} of {G} and {g\in G} such that {|A\cap gH|\geq \frac{1}{C(K)}|A|}. Furthermore, {H} has a normal nilpotent finite index subgroup {N} with number of generators and nilpotency class {\leq 6\log K}.

This is still a weak version. In the strong version, {H=FN} where {F} is contained in the product of bounded many copies of {A}.

It is enough to prove the theorem when {A} is a {K}-approximate subgroup, i.e. {1\in A}, {A=A^{-1}} and {AA\subset XA}, {|X|\leq K}. Indeed, we have seen in lecture 1 that, with some combinatorics, this is equivalent, up to passing to a large subset.

If {G} has finite exponent (i.e. satisfies the law {x^e=1} for some integer {e}), then {H} must be finite. Hence {A} is contained in boundedly many cosets of a finite subgroup of {G}.

0.1. Strategy of proof

Inspired from Gromov, expanded in model theory language by Hrushovski.

1. Ultraproducts

This is the model theory of the poor.

Let {G_n} be a sequence of groups and {A_n\subset G_n} be finite subsets of {G_n} which are {K}-approximate subgroups. Fix a nonprincipal ultrafilter {\mathcal{U}} on {{\mathbb N}}. Form the ultraproduct

\displaystyle  \begin{array}{rcl}  \mathbb{G}=\prod_{\mathcal{U}}G_n \end{array}

Los’ theorem asserts that any first order sentence {\phi} in the language of groups holds for {\mathbb{G}} iff it holds for {\mathcal{U}}-almost every {G_n}. Usually, it is easy to reprove it in simple situations.

Example 1 Let {X_n\subset G_n} be a finite subset. Let {\mathbb{X}=\prod_{\mathcal{U}}X_n} (such subsets are called internal subsets of {\mathbb{G}}). Given {k\in{\mathbb N}}, {|\mathbb{X}|=k} iff {|X_n|} a.e..

{\mathbb{G}} is agroup, {\mathbb{X}} is a subgroup iff {X_n} is a subgroup a.e., {\mathbb{X}} is a {K}-approximate subgroup iff {X_n} is a {K}-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 {\mathbb{A}} is a pseudofinite {K}-approximate subgroup of {\mathbb{G}}, then there exist internal subgroups {\mathbb{M}<\mathbb{H}<\mathbb{G}} such that {\mathbb{M}} is normal in {\mathbb{H}}, {\mathbb{M}} is nilpotent, {\mathbb{H}_{|\mathbb{M}}} is pseudofinite, and {\mathbb{A}\subset\mathbb{H}X}, with {|X|<\infty}.

Note that this does not give a bound on {|X|}, i.e. on the number of cosets required.

2. Hrushovski’s Lie model theorem

Theorem 3 Let {\mathbb{A}\subset\mathbb{G}} be a pseudofinite {K}-approximate subgroup. There is a locally compact group {G} and a surjective group homomorphism

\displaystyle  \begin{array}{rcl}  \pi:\langle \mathbb{A}\rangle \rightarrow G \end{array}

such that

  1. {\pi(\mathbb{A}^2)} is a compact neighborhood of {1\in G},
  2. For every open set {U\subset G} and compact set {K\subset U}, there exists {k\in{\mathbb N}} and an internal subset {\mathbb{X}\subset\mathbb{A}^k} such that {\pi^{-1}(K)\subset\mathbb{X}\subset\pi^{-1}(U)}.

We see that {\pi} has kernel contained in {\mathbb{A}^{k_0}} for some finite {k_0}. Hence {\pi} is almost injective.

Also, {G} must be unimodular. Indeed, {\langle \mathbb{A}\rangle} admits a biinvariant measure, arising as a limit of renormalized counting measures. Its image by {\pi} is a biinvariant Haar measure on {G}.

The proof will show that {G} has an open subgroup {G'=KL} such that {K} is normal and compact and {L} is a nilpotent Lie group.

2.1. Special cases

What if {G} is trivial? Then {\langle \mathbb{A}\rangle\subset \mathbb{A}^{k_0}}. By (2), {\mathbb{A}^{k_0}} is a subgroup, so a.e. {A_n^{k_0}} is a finite subgroup, countained is finitely many translates of {A_n}. This is the structure theorem

What if {G} is compact? The same conclusion holds.

What if {G} has a compact open subgroup {H}? By (1), {\pi(\mathbb{A}^2)} is contained in finitely many translates of {H}. Since {\pi^{-1}(H)} is internal and contained in {\langle \mathbb{A}\rangle\subset \mathbb{A}^{k_0}}, there exists a.e. a finite subgroup {K_n<G_n} such that {A_n} is contained in bounded many copies of {K_n}.

Clearly, if {A_n=[-n,n]\subset {\mathbb Z}}, approximation by finite subgroups does not hold, so {G} does not contain any compact open subgroup.

Corollary 4 (Hrushovski) Given integers {K}, {e}, if {G} is a group of exponent {e} and {A} a {K}-approximate subgroup of {G}., then there exists a finite subgroup {H<G} such that {|H|\leq C|A|} and {A} is countained in a bounded number of translates of {H}.

It suffices to prove this for pseudofinite groups of finite exponent. {G} has finite exponent. This implies that {G} 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 {\langle\mathbb{A}\rangle}. Then one divides by the closure of 1.

One would like to say that {g} is close to the identity if {g\mathbb{A}\delta \mathbb{A}} is small, e.g.

\displaystyle  \begin{array}{rcl}  |g\mathbb{A}\Delta \mathbb{A}|<\epsilon|A|. \end{array}

The following combinatorial lemma helps.

Lemma 5 (Hrushovski, Sanders, Croot-Sisak) Given integers {k,K}, let {A\subset G} be a finite {k}-approximate subgroup. Then there exists a subset {S\subset A^4},

  1. {aS^ka^{-1}\subset A^4}, for all {a\in A}.
  2. {|S|\geq \frac{1}{c(k,K)}|A|}.

Indeed, applying the Lemma to {A_n} in {G_n}, find sets {S_{n,k}}. Furthermore, inductively, one can assume that

\displaystyle  \begin{array}{rcl}  (S_{n,k+1})^2\subset S_{n,k}. \end{array}

Let {\mathbb{S}_k} be their ultraproduct. These sets form a basis for a topology. Their intersection is a normal subgroup. One shows that { \mathbb{A}} 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

\displaystyle  \begin{array}{rcl}  S_\epsilon=\{g\in G\,;\,\|1_A\star 1_A-g1_A\star 1_A\|_2<\epsilon\|1_A\star 1_A\|_2\}. \end{array}

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.


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