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.


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