## Notes of Emmanuel Breuillard’s sixth Cambridge lecture 26-04-2017

Introduction to approximate groups, VI

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

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 ${|A\cap gH|\geq \frac{1}{C(K)}|A|}$.

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 ${G}$ is ${\pi(B)}$ where

$\displaystyle \begin{array}{rcl} B=\prod_{i=1}^d [-N_i,N_i]\subset {\mathbb Z}^d \end{array}$

is a large box and ${\pi:{\mathbb Z}^d\rightarrow G}$ is a group homomorphism.

In an arbitrary group, fix elements ${g_1,\ldots,g_d\in G}$, numbers ${N_i\in{\mathbb N}}$ and consider the set ${P}$ of products of ${g_i^{\pm 1}}$ where the total number of occurrences of ${g_i}$ or ${g_i^{-1}}$ is at most ${N_i}$. This is the GAP of rank ${d}$ generated by the ${g_i}$‘s with side length ${N_i}$.

Example 1 If ${\{g_i^{\pm 1}\}}$ is a generating system and ${N_1=\cdots=N_d=N}$, then the corresponding GAP ${P}$ contains ball ${B(N)}$ and is contained in ${B(dN)}$.

Example 2 In Heisenberg group ${G}$ with the standard pair of generators, ${P=P(N_1,N_2)}$ is roughly given in coordinates by ${\{|x|\leq N_1,\,|y|\leq N_2,\,|z|\leq N_1 N_2\}}$. It satisfies ${|P|\sim (N_1 N_2)^2}$ and ${|PP|\sim|P|}$.

So ${P}$ is an instance of approximate subgroup.

1.1. Nilprogressions

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

Proposition 4 There exists ${K=K(d,s)}$ and ${C=C(d,s)}$ such that any ${s}$-step nilprogression of rank ${d}$ with side length ${\geq C}$ is a ${K}$-approximate group.

1.2. Strong structure theorem

Theorem 5 For all ${K\geq 1}$ there exists ${C(K)\geq 1}$ such that if ${A\subset G}$ is a finite subset of a group ${G}$ that satisfies ${|AA|\leq K|A|}$, then ${A\subset XH}$ where

• ${|X|\leq C}$,
• ${H}$ is virtually nilpotent.

Furthermore, there is an exact sequence

$\displaystyle \begin{array}{rcl} 1\rightarrow N\rightarrow H\rightarrow L\rightarrow 1 \end{array}$

where ${N}$ is finite, ${L}$ is nilpotent, and there exists a nilprogression ${P\subset L}$ such that

• ${\pi^{-1}(P)\subset A^{-2}A^{2}}$,
• ${A \subset X\pi^{-1}(P)}$, ${|X|\leq C}$.

In other words, every approximate group is controlled on both sides by a nilprogression.

2. Proof

A sequence of ${K}$-approximate groups ${A_n\subset G_n}$ gives rise to a pseudo-finite ${K}$-approximate group ${\mathbb{A}\subset\mathbb{G}}$ in an ultraproduct.

Hrushovski’s model theorem states that there exists a locally compact group ${G}$ and an epimorphism ${\pi:\langle \mathbb{A}\rangle\rightarrow G}$ such that

• ${\pi(\mathbb{A}^2)}$ is a compact neighborhood of ${1\in G}$,
• 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 ${\pi^{-1}(K)\subset\mathbb{X}\subset\pi^{-1}(U)}$.

2.1. Aside

Let me answer to a question of Cornelia Drutu: can ${G}$ be a compact Lie group in Hrushovski’s theorem? Answer is positive. For instance, ${A_n={\mathbb Z}/n{\mathbb Z}}$ produce an epimorphism ${\pi:\langle \mathbb{A}\rangle\rightarrow{\mathbb R}/{\mathbb Z}}$. However, in this case, ${G}$ 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 ${d\in{\mathbb N}}$, there exists ${e=e(d)}$ such that for every finite group ${G}$ and generating set ${S}$,

$\displaystyle \begin{array}{rcl} [G,G]=(\prod_{s\in S} [G,s])^e. \end{array}$

In particular, for every word ${w\in[F_2,F_2]}$, for every finite group and every elements ${g,h\in G}$,

$\displaystyle \begin{array}{rcl} w(g,h)\in \end{array}$

This cannot happen in a compact Lie group. Indeed, such a group admit sa conjugation-invariant neighborhood of ${1}$.

2.2. Continuation of the proof

The difficult case is when ${G}$ has no compact open subgroup. The Gleason-Yamabe theorem applied to ${G}$ provides an approximation of ${G}$ by Lie groups. Hence we can assume that ${G}$ 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 ${\mathbb{A}}$ be a pseudo-finite ${K}$-approximate group. Then there exists ${\mathbb{H}\subset \mathbb{A}^4}$ such that ${\mathbb{H}}$ is a subgroup whose normalizer contains a subgroup ${\mathbb{B}}$ which is a substantial part of ${\mathbb{A}}$ and contains no proper subgroups at all.

This follows from

Proposition 7 (Reduction to NSS) Let ${\mathbb{A}}$ be a pseudo-finite ${K}$-approximate group which is NSS. Then there exists ${\mathbb{P}\subset \mathbb{A}^4}$ which is a pseudo-nilprogression and ${\mathbb{A}\subset X\mathbb{P}}$ with ${|X|}$ bounded.

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

Lemma 8 Given a ${K}$-approximate group ${A}$, there exists a ${C(K)}$-approximate subgroup ${B\subset A^4}$ such that

$\displaystyle \begin{array}{rcl} \|hgh^{-1}\|_B&\leq& C\|g\|_B.\\ \|gh\|_B&\leq& C(\|g\|_B+\|h\|_B).\\ \|[g,h]\|_B&\leq& C\|g\|_B\|h\|_B. \end{array}$

From the Lemma, it follows that ${H=\{h\in B\,;\, \|h\|_B=0\}=\{h\in B\,;\,\langle h \rangle \subset B\}}$ is a subgroup normalized by ${B}$. Thus ${B/H}$ is NSS, this proves Proposition 1.

If ${B}$ is finite, there is an element ${u}$ in ${B}$ of minimal escape norm. This ${u}$ is centralized by the small elements of ${B}$. If ${B}$ is pseudo-finite, one produces a non-trivial 1-parameter subgroup of ${G}$ which is central in ${G}$. One can mod out by this 1-parmeter subgroup. By induction on the dimension of ${G}$, 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 ${c,d}$, consider the class of graphs ${\mathcal{F}_{c,d}}$ of finite vertex transitive graphs such that

$\displaystyle \begin{array}{rcl} \frac{|\textrm{vertices}|}{\textrm{degree}}\leq c\,\mathrm{diameter}^d. \end{array}$

Theorem 9 (Benjamini-Finucane-Tessera) Every sequence of graphs in ${\mathcal{F}_{c,d}}$ has a Gromov-Hausdorff converging subsequence to a flat Finsler ${d'}$-torus, ${d'\leq d}$.

3.2. Proof

Let ${G_n}$ be the automorphism group of the ${n}$-th graph, ${S_n}$ the subset of elements that move the base point at most distance 1 away. Then ${S_n}$ generates ${G_n}$. 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 ${P}$ is a nilprogression, so is ${P^k}$ 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 ${X}$. Montgomery-Zippin (in fact, Peter-Weyl) implies ${X}$ is a homogeneous space of a Lie group. Finite group ${G_n}$ ${\epsilon}$-acts on ${X}$. Every such ${\epsilon}$-homomorphism to a Lie group is close to a true homomorphism (it is a early result by Alan Turing). Since ${Isom(X)^0}$ is approximable by finite groups, it is abelian, hence ${X}$ is a torus.