Growth and expansion in algebraic groups over finite fields

TL;DR

This paper introduces the study of growth in algebraic groups over finite fields, focusing on groups like SL_2(F_q) and their subgroups, highlighting recent developments and key examples in the area.

Contribution

It provides an accessible overview of growth phenomena in groups of Lie type, synthesizing recent research and course notes for educational purposes.

Findings

Analysis of growth in SL_2(F_q) and subgroups

Connections to algebraic group theory over finite fields

Summary of recent advances in the field

Abstract

This is a brief introduction to the study of growth in groups of Lie type, with $SL_2(\mathbb{F}_q)$ and some of its subgroups as the key examples. They are an edited version of the notes I distributed at the Arizona Winter School in 2016. Those notes were, in turn, based in part on my survey in Bull. Am. Math. Soc. (2015) and in part on the notes for courses I gave on the subject in Cusco (AGRA) and G\"ottingen.

Equations

$$A$$

$$A\cdot A$$

$$A\cdot A\cdot A$$

$$\dots$$

$$A^k$$

$$\text{either}\text{∣A3∣ is much larger than A}\text{or}{A}^3=G,$$

$$\mathcal{A}f(v)=\frac{1}{d}\sum _{w:\text{(v,w) is an edge}}f(w).$$

$$\{e\}=H_0◃H_1◃H_2◃\cdots ◃H_k=G,$$

$$\mathrm{diam}(\Gamma (G,A))\ll (\log |G|{)}^C,$$

$$|A^3|\ge |A{|}^{1+\delta }$$

$$A^3=G,$$

$$\mathrm{diam}(G,A)=e^{O\left((\log n{)}^4(\log \log n)\right)}=e^{O_{ϵ}\left((\log \log |G|{)}^{4+ϵ}\right)}$$

$$|A^3|=\left|\{\sigma ,{\sigma }^{-1},e\}\cdot H\cdot \{\sigma ,{\sigma }^{-1},e\}\cup H{\sigma }^{\pm 1}H\right|\le 9m!+2(m+1)!\le (2m+11)|A|.$$

$$\{e\}=H_0◃H_1◃\cdots ◃H_k=G$$

$$|A^3|\ge C|A|$$

$$N\subset (A\cup A^{-1}\cup \{e\}{)}^k,\left|(A\cup A^{-1}\cup \{e\}{)}^k\cap G_0\right|\ge C^{-O_n(1)}|A|,$$

$$\left\{\left(\begin{array}{cc}r& x\\ 0& 1\end{array}\right):r\in K^{\ast },x\in K\right\}.$$

$$1_A(x)=\{\begin{array}{ll}1& \text{if x∈A,}\\ 0& \text{otherwise.}\end{array}$$

$$|A{C}^{-1}||B|\le |A{B}^{-1}||B{C}^{-1}|.$$

$$\frac{|(A\cup A^{-1}\cup \{e\}{)}^3|}{|A|}\le {\left(3\frac{|A^3|}{|A|}\right)}^3$$

$$\frac{|A^k|}{|A|}\le {\left(\frac{|A^3|}{|A|}\right)}^{k-2}.$$

$$\frac{|(A\cup A^{-1}\cup \{e\}{)}^k|}{|A|}\le 3^{k-2}{\left(\frac{|A^3|}{|A|}\right)}^{3(k-2)}$$

$$|(A^{-1}A)\cap \mathrm{Stab}(x)|\ge \frac{|A|}{|Ax|}.$$

$$|BA|\ge |A\cap \mathrm{Stab}(x)||Bx|.$$

$$|H\cap \mathrm{Stab}(x)|=\frac{|H|}{|Hx|}.$$

$$g\mapsto (h\mapsto g\cdot h)$$

$$g\mapsto (h\mapsto h\cdot g^{-1}),$$

$$g\mapsto (h\mapsto gh{g}^{-1}).$$

$$C(h)=C_G(h)=\{g\in G:gh=hg\};$$

$$\mathrm{Cl}(h)=\{gh{g}^{-1}:g\in G\}.$$

$$|A^2\cap C(g)|\ge \frac{|A|}{|A^{l+2}\cap \mathrm{Cl}(g)|}.$$

$$|A^2\cap H|\ge \frac{|A|}{r},$$

Full text

1. 1 Introduction

2. 1.1 Basic questions and concepts: diameter, growth, expansion

3. 1.2 A brief overview of results on growth and diameter

4. 1.2.1 Simple groups: what to expect?

5. 1.2.2 Simple groups of Lie type (and bounded rank)

6. 1.2.3 The simple group $\operatorname{Alt}(n)$

7. 1.2.4 Solvable and nilpotent groups

8. 1.2.5 Groups over $\mathbb{R}$ or $\mathbb{C}$

9. 1.3 Notation

10. 2 Elementary tools

11. 2.1 Additive combinatorics

12. 2.2 The orbit-stabilizer theorem for sets

13. 3 Growth in a solvable group

14. 3.1 Remarks on abelian groups

15. 3.2 The affine group

16. 3.2.1 Growth in the affine group

17. 3.2.2 Brief remarks on a generalization and an application

18. 3.2.3 Diameter bounds in a remaining case

19. 4 Intersections with varieties

20. 4.1 Preliminaries from algebraic geometry and algebraic groups

21. 4.1.1 Basic definitions.

22. 4.1.2 Degrees. Bézout’s theorem.

23. 4.1.3 Morphisms.

24. 4.1.4 Tangent spaces and derivatives

25. 4.1.5 Linear algebraic groups

26. 4.1.6 Lie algebras

27. 4.1.7 Finite groups of Lie type

28. 4.2 Escape from subvarieties

29. 4.3 Dimensional estimates

30. 5 Growth and diameter in $\operatorname{SL}_{2}(K)$

31. 5.1 Growth in $\operatorname{SL}_{2}(K)$, $K$ arbitrary

32. 5.2 The case of large subsets

33. 6 Further perspectives and open problems

34. 6.1 Expansion, random walks and the affine sieve

35. 6.2 Algorithmic and probabilistic questions

36. 6.3 Final remarks

1. Introduction

This text is meant to serve as a brief introduction to the study of growth in groups of Lie type, with $\operatorname{SL}_{2}(\mathbb{F}_{q})$ and some of its subgroups as the key examples. They are an edited version of the notes I distributed at the Arizona Winter School in 2016. Those notes were, in turn, based in part on the survey [Hel15] and in part on the notes for courses I gave on the subject in Cusco [Hel] and Göttingen.

Given the format of the Arizona Winter School, the emphasis here is on reaching the frontiers of current research as soon as possible, and not so much on giving a comprehensive overview of the field. For that the reader is referred to [Hel15] and its bibliography, or to [Kow13] and [Tao15]. At the same time – again motivated by the school’s demands – we will take a brief look at several applications at the end.

It will be necessary to be minimally conversant with some of the basic classical vocabulary of algebraic geometry (as in the first chapter of Mumford’s Red Book [Mum99]), and with some notions on algebraic groups (such as $\operatorname{SL}_{2}$) and Lie algebras (such as $\mathfrak{sl}_{2}$). A very brief compendium of what will be needed can be found in §4.1. It is often helpful (and only rarely misleading) to be willing to believe that matters work out in much the same way over finite field as they do over the reals.

The purpose of these notes is expository, not historical, though I have tried to give key references. The origins of several ideas are traced in greater detail in [Hel15]. In §1.2, we will give a summary of the results we later prove and also of results and open questions of the same kind. We will go over some important related questions and applications later, in §6.

Acknowledgements. I was supported by ERC Consolidator grant 648329 (codename GRANT) and by funds from my Humboldt professorship. Many thanks are due to a helpful and spirited anonymous referee. Thanks are due as well to Lifan Guan, for providing a useful reference and catching several typos, and to the audiences both at the Arizona Winter School and at the Hausdorff Institute (HIM), for real-time feedback.

1.1. Basic questions and concepts: diameter, growth, expansion

Let $A$ be a finite subset of a group $G$. Consider the sets

[TABLE]

Write $|S|$ for the size of a finite set $S$, meaning simply the number of elements of $S$. A question arises naturally: how does $|A^{k}|$ grow as $k$ grows?

This kind of question has been studied from the perspective of additive combinatorics (for $G$ abelian) and geometric group theory ($G$ infinite, $k\to\infty$). There are also some crucial related concepts coming from other fields: diameters and expanders, to start with.

Diameters. Let $A$ be a set of generators of $G$. When $G$ is infinite, a central question is how $|A^{k}|$ behaves as a function of $k$ as $k\to\infty$. When $G$ is finite, that question does not make much sense, as $|A^{k}|$ obviously stays constant as soon as $A^{k}=G$. Instead, let us ask ourselves what is the least value of $k$ such that $A^{k}=G$. This value of $k$ is called the diameter. It is finite because, for $A$ generating $G$, $A^{j}\neq G$ implies $|A^{j+1}|>|A^{j}|$. (Why is this last statement true?)

The term diameter comes from geometry. What we have is not just an analogy – we can actually put our basic terms in a geometrical framework, as geometric group theory does. A Cayley graph $\Gamma(G,A)$ is the graph having $V=G$ as its set of vertices and $E=\{(g,ag):g\in G,a\in A\}$ as its set of edges. Define the length of a path in the graph as the number of edges in it, and the distance $d(v,w)$ between two vertices $v$, $w$ in the graph as the length of the shortest path between them. The diameter of a graph is the maximum of the distance $d(v,w)$ over all vertices $v$, $w$. It is easy to see that the diameter of $G$ with respect to $A$, as we defined it above, equals the diameter of the graph $\Gamma(G,A)$.

Product theorems. A central question of additive combinatorics is as follows: for finite subsets $A$ of an abelian group $(G,+)$, when exactly is it that $A+A$ is much larger than $A$? In non-abelian groups $(G,\cdot)$, the right form of the question turns out to be: given a set of generators $A$ of $G$, when is $A^{3}$ much larger than $A$? (We will see later why it is better to ask about $A^{3}=A\cdot A\cdot A$ rather than $A^{2}=A\cdot A$ here.)

It is clear that, if we show that, for any generating set $A$ of $G$,

[TABLE]

then $A^{k}$ grows rapidly until roughly the point where $A^{k}=G$: simply apply (1.1) to $A$, $A^{3}$, $A^{9}$, etc., in place of $A$. In particular, (1.1) yields an upper bound on the diameter of $G$ with respect to $A$. We call a result of the form (1.1) a product theorem.

Expansion. We say that a graph is an vertex expander with parameter $\delta>0$ (or $\delta$-vertex expander) if, for every subset $S$ of the set of vertices $V$ satisfying (say) $|S|\leq|V|/2$, the number of vertices $v\in V$ not in $S$ such that at least one edge connects $v$ to some element of $S$ is at least $\delta|S|$. (We may think of $S$ as being a set of infected individuals; then we are saying that the number of the newly infected will always be at least $\delta|S|$, unless the disease has reached a near-saturation point.)

Two closely connected notions are that of edge expansion and spectral expansion. First, some basic terms. A graph is regular if, for any vertex $v$, the number of vertices $w$ such that $(v,w)$ is an edge equals a constant $d$, and the number of vertices $w$ such that $(w,v)$ is an edge also equals a constant (which must also be $d$, by a simple counting argument). We call $d$ the degree or valency of the graph. A Cayley graph $\Gamma(G,A)$ is always regular of degree $d=|A|$.

A regular graph $\Gamma=(V,E)$ of degree $d$ is a $\delta$-edge expander if, for every $S\subset V$ satisfying $|S|\leq|V|/2$, the number of edges having one vertex in $S$ and one outside $S$ is at least $\delta d|S|$. It is clear that, if $\Gamma$ is a $\delta$-vertex expander, then it is a $(\delta/d)$-edge expander, and, if it is a $\delta$-edge expander, then it is a $\delta$-vertex expander.

We say that a graph $\Gamma$ is symmetric to mean that $(v,w)$ is an edge if and only if $(w,v)$ is an edge. If $\Gamma$ is a Cayley graph $\Gamma(G,A)$, then $\Gamma$ is symmetric provided that $A^{-1}=\{g^{-1}:g\in A\}$ equals $A$. We will generally assume that $A^{-1}=A$ without much loss of generality. (Replace $A$ by $A\cup A^{-1}$ otherwise.)

Given a regular graph $\Gamma$ with a set of vertices $V$, the adjacency operator $\mathscr{A}$ is the linear operator taking any given function $f:V\to\mathbb{C}$ to the function $\mathscr{A}f:V\to\mathbb{C}$ defined by

[TABLE]

Assume that the graph $\Gamma$ is symmetric. Then $\mathscr{A}$ is a symmetric operator, and thus has full real spectrum. Its largest eigenvalue is $1$; it corresponds to constant eigenfunctions. If every eigenvalue $\lambda$ of $\mathscr{A}$ corresponding to non-constant eigenfunctions satisfies $\lambda\leq 1-\delta$ for some $\delta>0$, we say that $\Gamma$ is a $\delta$-spectral expander, or a $\delta$-expander for short.

If a regular, symmetric graph is a $\delta$-spectral expander, then it is a $(\delta/2)$-edge expander, and, if it is a $\delta$-edge expander, then it is a $(\delta^{2}/2)$-spectral expander. This fact is non-trivial; it is called the Cheeger-Alon-Milman inequality [AM85], by analogy with the Cheeger inequality on manifolds [Che70].

The notion of spectral expansion is natural, not just because of the analogy with surfaces and their Laplacians, but, among other reasons, because of random walks: a drunken mathematician left to wander in a spectral expander $\Gamma$ will be anywhere with about the same probability after only a short while. To put matters more formally – as we shall see in §6.1, spectral expansion implies small mixing time.

Since the diameter of a graph is bounded by its ($\ell_{\infty}$-)mixing time, it follows immediately that spectral expansion implies small diameter. We can also prove this implication going through edge and vertex expansion: if a graph is a $\delta$-vertex expander, it is very easy to see that its diameter is $\ll(\log|G|)/\delta$; apply, then, the Cheeger-Alon-Milman inequality.

1.2. A brief overview of results on growth and diameter

Let us first review some basic terms from group theory. A group $G$ is simple if it has no normal subgroups other than itself and the identity. A subnormal series of a group $G$ is a sequence of subgroups

[TABLE]

i.e., $H_{i}$ is normal in $H_{i+1}$ for every $0\leq i<k$. A decomposition series is a subnormal series in which every quotient $H_{i+1}/H_{i}$ is simple. It is clear that every finite group has a decomposition series.

In some limited sense, questions on growth behave well under taking quotients, and thus reduce to the case of simple groups, at least if our decomposition series of bounded length. (To be precise: for how product theorems behave under taking quotients, see exercises 2.8 and 2.9). For the behavior of diameters under quotients, look up Schreier generators.) It thus makes sense to focus on simple groups.

1.2.1. Simple groups: what to expect?

Some special cases of the following conjecture are arguably older “folklore”.

Conjecture 1**.**

$($Babai, [BS92, Conj. 1.7]$)$ Let $G$ be finite, simple and non-abelian. Let $A$ be any set of generators of $G$. Then

[TABLE]

where $C$ and the implied constant are absolute constants.

(See §1.3 for definitions of asymptotic notation.)

What about finite, simple, abelian groups $G$? They are the groups $G=\mathbb{Z}/p\mathbb{Z}$. In that case, diameters can be very large: for instance, $\operatorname{diam}\Gamma(\mathbb{Z}/p\mathbb{Z},\{1\})=p-1$. In general, when $G$ is abelian, the question of which subsets $A\subset\mathbb{Z}/p\mathbb{Z}$ satisfy $|A+A|>K|A|$ for given $K$ is classical, and difficult; for $K$ a constant, it is answered by a suitable generalization of Freiman’s theorem [GR07]. (Freiman had done the case $G=\mathbb{Z}$; see [Fre73], or the exposition [Bil99].) The strongest result on the abelian case to date is that of Sanders ([San12]; based in part on [CS10]).

The Classification of Finite Simple Groups111Famed in mathematical lore as the theorem whose proof would be of the size of a large encyclopedia, were it all in one place. tells us that all finite, simple, non-abelian groups $G$ fall into three classes:

1. (a)

simple groups of Lie type, that is, matrix groups over finite fields (such as $\operatorname{PSL}_{n}(\mathbb{F}_{q})$ or $\operatorname{PSp}_{2n}(\mathbb{F}_{q})$), including some generalizations (twisted groups); 2. (b)

alternating groups $\operatorname{Alt}(n)$. The simple group $\operatorname{Alt}(n)$ is the unique subgroup of index $2$ of the group $\operatorname{Sym}(n)$ of all permutations of $n$ elements; 3. (c)

a finite list of exceptions, including, for example, the “monster group”.

We can put (c) out of our minds, since it has a finite number of elements, and we are aiming for asymptotic statements.

1.2.2. Simple groups of Lie type (and bounded rank)

Our main goal in these notes will be to prove the following theorem.

Theorem 1.1**.**

Let $G=\operatorname{SL}_{2}(K)$ or $G=\operatorname{PSL}_{2}(K)$, $K$ a field. Let $A\subset G$ be a set of generators of $G$. Then either

[TABLE]

or

[TABLE]

where $\delta>0$ is an absolute constant.

Here $\operatorname{PSL}_{2}(K)=\operatorname{SL}_{2}(K)/\{I,-I\}$, where $\operatorname{SL}_{2}(K)$ is, of course, the group of $2$-by-$2$ matrices with entries in a field $K$ and determinant $1$. The group $\operatorname{PSL}_{2}(K)$ is simple for $K=\mathbb{F}_{q}$ finite. It is a group of Lie type; indeed, it will be our white mouse, in that it is convenient to work with, but sufficiently complex to be a good example of a large class.

Theorem 1.1 was first proved in [Hel08] for $K=\mathbb{F}_{p}$, with $A^{k}=G$ ($k$ a constant) instead of $A^{3}=G$. It then underwent a series of generalizations ([BG08a], [Din11], [Hel11], [GH11], [BGT11] and [PS16], among others). By now, we know it for every simple group of Lie type of bounded rank ([BGT11], [PS16]). The “bounded rank” condition means simply that the constant $\delta$ in the inequality $|A^{3}|\geq|A|^{1+\delta}$ depends on the rank of the group. (The rank of $\operatorname{SL}_{n}$ is $n-1$, that of $\operatorname{SO}_{n}$ is $\lfloor n/2\rfloor$, etc.) In fact, there are examples (due to Pyber) that show that $\delta$ has to depend on the rank.

We will give a proof of Thm. 1.1 that descends from, but is not the same as, the proof in [Hel08]; it has strong influences from [Hel11], [BGT11] and [PS16]. In particular, the proof we shall give generalizes readily to $\operatorname{SL}_{n}$ and other higher-rank groups; many of our intermediate results will be stated for $\operatorname{SL}_{n}$, and the ideas carry over to other group families.

Exercise 1.2**.**

Let $K$ be a finite field. Let $G=\operatorname{PSL}_{2}(K)$ or $G=\operatorname{SL}_{2}(K)$. Let $S\subset G$ generate $G$. Using Thm. 1.1, prove that the diameter of $\Gamma(G,S)$ is $\ll(\log|G|)^{C}$, where $C$ and the implied constant are absolute. Indeed, $C=O(1/\delta)$, where $\delta$ is the absolute constant in (1.4). Hint: apply Thm. 1.1 repeatedly, with $S$ equal to $A$, $A^{3}$, $A^{9}$,…

In other words, Babai’s conjecture holds for $G=\operatorname{PSL}_{2}(\mathbb{F}_{q})$. The bound $\operatorname{diam}\Gamma(G,A)\ll(\log|G|)^{C}$ also holds for all other simple groups of Lie type, only then $C$ depends on the rank, since $\delta$ does.

Before [Hel08], $\Gamma(G,A)$ was known to be an expander for some particular sets of generators $A$ of $G=\operatorname{SL}_{2}(\mathbb{F}_{q})$. In those cases, then, the diameter bound $\operatorname{diam}\Gamma(G,A)\ll\log|G|$ was also known. The main element of the proof came from modular forms (Selberg’s spectral gap [Sel65]).

Impatient readers may now jump to the body of the text and leave the rest of the introduction for later. They should certainly read §6.1, on applications of Theorem 1.1 to expander graphs.

1.2.3. The simple group $\operatorname{Alt}(n)$

For $G=\operatorname{Alt}(n)$, we have a statement that is somewhat weaker than Babai’s conjecture.

Theorem 1.3**.**

$($Helfgott-Seress, [HS14]$)$ Let $G=\operatorname{Sym}(n)$ or $G=\operatorname{Alt}(n)$. Let $A\subset G$ be a set of generators of $G$. Then

[TABLE]

for $\epsilon>0$ arbitrary.

In fact, the bound $\operatorname{diam}(G,A)=\exp(O((\log n)^{4}(\log\log n)))$ holds for all transitive groups $G<\operatorname{Sym}(n)$, and can be deduced from Thm. 1.3. We could state this result as follows: let us be given a permutation puzzle with $n$ pieces that has a solution and satisfies transitivity (that is, any piece can be sent to any other one by some succession of moves). Then there is always a short solution, starting from any reachable position. Incidentally, non-transitive puzzles, such as Rubik’s cube, can be reduced to transitive ones at some cost, by means of Schreier generators.

We cannot have a product theorem just like Thm. 1.1 in $\operatorname{Alt}(n)$ or $\operatorname{Sym}(n)$.

Counterexample 1** (Pyber, Spiga).**

Let $H$ be the subgroup of $\operatorname{Sym}(n)$ consisting of all permutations of $\{1,\dotsc,m\}$. Let $\sigma$ be the cycle taking $i$ to $i+1$ ($i\leq n-1$) and $n$ to $1$. Let $A=H\cup\{\sigma,\sigma^{-1}\}$. Then

[TABLE]

The factor $(2m+11)$ compared to $|A|$ for $A$ large; if we set, say, $m\sim n/2$, then $(2m+11)\ll|A|^{3/n}$.

It might be that one of $|A^{O(n^{C})}|\geq|A|^{1+\delta}$ or $A^{O(n^{C})}=G$ always holds. Even having one of $\left|A^{O(n^{C})}\right|\geq|A|^{1+\delta/\log n}$ or $A^{O(n^{C})}=G$ would be a definite improvement over Thm. 1.3. The exponents $4$ in (1.6) would become $3$, and, at any rate, as we shall later see, product theorems have consequences other than diameter bounds.

It would be natural to hope that some ideas in 1.3, or its later version [Hel19], or future strengthenings thereof, will be useful in addressing Babai’s conjecture over groups of Lie type of unbounded rank. It is not just that the known counterexamples to strong product theorems over $\operatorname{Sym}(n)$ and $\operatorname{SL}_{n}$ are related. There are ways to define the “field with one element” $\mathbb{F}_{\operatorname{un}}$, and objects over it; then one generally obtains that $\operatorname{Sym}(n)\sim\operatorname{SL}_{n}(\mathbb{F}_{\operatorname{un}})$. See, e.g., [Lor18].

1.2.4. Solvable and nilpotent groups

A group $G$ is solvable if it has a subnormal series

[TABLE]

all of whose quotients $H_{i+1}/H_{i}$ are abelian. As we said before, questions on growth behave well under quotients, but such a reduction does not help us as much as we would like, since the best results available for the abelian case are considerably less strong than $|A\cdot A\cdot A|\geq|A|^{1+\delta}$.

A solvable group is nilpotent if it has a subnormal series (1.7) with $G_{i+1}/G_{i}$ contained in the center of $G/G_{i}$ for every $0\leq i<k$. Nilpotent groups can often be seen as “almost abelian”, and our context is no exception. One should not hope to get stronger results on growth in nilpotent groups than for abelian groups – and, on the positive side, one can study nilpotent groups with Freiman’s and Ruzsa’s tools, supplemented by a Lie-algebra framework ([Toi14]; see also [FKP10] and Tao [Tao10]).

What one can aim for is to show that, given a set $A$ in a solvable group, either $A$ grows rapidly, or we are really in a nilpotent case. We can make such a statement precise as follows.

Conjecture 2**.**

Let $A\subset\operatorname{GL}_{n}(K)$, $K$ a field. Assume that the group $\langle A\rangle$ generated by $A$ is solvable. Then, for any $C\geq 1$, either

[TABLE]

or there are subgroups $N\triangleleft G_{0}\triangleleft\langle A\rangle$ such that $G_{0}/N$ is nilpotent and

[TABLE]

where $k$ depends only on $n$.

We can, of course, set $C=|A|^{\delta}$, so that (1.8) has the familiar form $|A^{3}|\geq|A|^{1+\delta}$.

Gill and Helfgott proved Conjecture 2 for $K=\mathbb{F}_{p}$ [GH14]. The case $K=\mathbb{F}_{q}$ remains open. The case $K=\mathbb{C}$ is relatively straightforward [BG11]; in that case, the group $N$ can be taken to be trivial.

Putting the result for $K=\mathbb{F}_{p}$ together with [PS16], it is simple to show that the same result holds for $A\subset\operatorname{GL}_{n}(\mathbb{F}_{p})$ general, without the assumption that the group $\langle A\rangle$ generated by $A$ be solvable. (What [PS16] does is reduce the general case to the solvable case.) Again, the same conclusion is believed to hold over $\mathbb{F}_{q}$. Breuillard, Green and Tao have proved [BGT12] that, if one is willing to replace $C^{-O_{n}(1)}$ in (1.9) by a factor dependent in an unspecified way on $C$ (but still independent of $|A|$), one does not even need to assume that $A$ is contained in $\operatorname{GL}_{n}(K)$; they start from a completely general, abstract group. They kindly gave the name Helfgott-Lindenstrauss conjecture to the statement they proved, though I would personally give that name to Conj. 2.

We shall study what is arguably the simplest interesting solvable case, namely, the affine group

[TABLE]

over a field $K$. As we shall see, the question of growth in it is essentially equivalent to the sum-product theorem over a field. Indeed, our treatment (§3.2) will show how to take one of the ideas of proofs of the sum-product theorem over finite fields (as in [BKT04] or [BGK06]) and reinterpret it in the context of groups (“pivoting”). A version of the same idea (really just a form of induction) will appear again in our treatment of $\operatorname{SL}_{2}(K)$.

1.2.5. Groups over $\mathbb{R}$ or $\mathbb{C}$

The proof we shall give of Theorem 1.1 also works for $K$ infinite. Even the first proof worked for $K=\mathbb{R}$, indeed more easily than over $\mathbb{Z}/p\mathbb{Z}$. Actually the statement of Theorem 1.1 turns out to have already been known over $\mathbb{R}$: the proof of [EK01, Thm. 2] suffices to establish it.

Some results in combinatorics – such as the sum-product theorem, which underlay the first proof [Hel08] of Thm. 1.1, or Beck’s theorem [Bec83], on which [EK01] relies – are both stronger and easier to prove over the reals than over finite fields. In fact, some results are known only over $\mathbb{R}$, or were known only over $\mathbb{R}$ for many years. The reason is that, over $\mathbb{R}$, the topology of the real plane can be used in the solution of geometrical problems. A line divides the real plane into two halves; such a statement does not hold or even make sense over $\mathbb{Z}/p\mathbb{Z}$.

As it turns out, for many applications, we need to know not just a statement such as Theorem 1.1 for a linear group over the reals, but a stronger version thereof. To be precise: one needs to show that the maximal number $n_{\delta}(A)$ of points in $A$ separated by $\delta$ in the real or complex metric grows: $n_{\delta}(A^{3})\geq n_{\delta}(A)^{1+\delta}$.

Fortunately, as Bourgain and Gamburd first made clear [BG08a], existing proofs of Theorem 1.1 and its generalizations can be modified to yield such stronger variants. They worked with the proof in [Hel08], but the same should hold of later proofs. The applications they found consisted in or involved expander graphs. We will discuss results on expander graphs in §6.1.

1.3. Notation

By $f(n)\ll g(n)$, $g(n)\gg f(n)$ and $f(n)=O(g(n))$ we mean the same thing, namely, that there are $N>0$, $C>0$ such that $|f(n)|\leq C\cdot g(n)$ for all $n\geq N$. We write $\ll_{a}$, $\gg_{a}$, $O_{a}$ if $N$ and $C$ depend on $a$ (say).

As usual, $f(n)=o(g(n))$ means that $|f(n)|/g(n)$ tends to [math] as $n\to\infty$. We write $O^{*}(x)$ to mean any quantity at most $x$ in absolute value. Thus, if $f(n)=O^{*}(g(n))$, then $f(n)=O(g(n))$ (with $N=1$ and $C=1$).

Given a subset $A\subset X$, we let $1_{A}:X\to\mathbb{C}$ be the characteristic function of $A$:

[TABLE]

2. Elementary tools

2.1. Additive combinatorics

Some of additive combinatorics can be described as the study of sets that grow slowly. In abelian groups, results are often stated so as to classify sets $A$ such that $|A^{2}|$ is not much larger than $|A|$; in non-abelian groups, works starting with [Hel08] classify sets $A$ such that $|A^{3}|$ is not much larger than $|A|$. Why?

In an abelian group, if $|A^{2}|<K|A|$, then $|A^{k}|<K^{O(k)}|A|$ – i.e., if a set does not grow after one multiplication with itself, it will not grow under several. This is a result of Plünnecke [Plü70] and Ruzsa [Ruz89]. (Petridis [Pet12] recently gave a purely additive-combinatorial proof.)

In a non-abelian group $G$, there can be sets $A$ breaking this rule.

Exercise 2.1**.**

Let $G$ be a group. Let $H<G$, $g\in G\setminus H$ and $A=H\cup\{g\}$. Then $|A^{2}|<3|A|$, but $A^{3}\supset HgH$, and $HgH$ may be much larger than $A$. Give an example with $G=\operatorname{SL}_{2}(\mathbb{F}_{p})$. Hint: let $H$ is the subgroup of $G$ consisting of the elements $g\in G$ leaving the basis vector $e_{1}=(1,0)$ fixed.

However, Ruzsa’s ideas do carry over to the non-abelian case, as was pointed out in [Hel08] and [Tao08]. We must assume that $|A^{3}|$ is small, not just $|A^{2}|$, and then it does follow that $|A^{k}|$ is small. The formal statement is Exercise 2.3, below. To prove it, we need the following lemma.

Lemma 2.2** (Ruzsa triangle inequality).**

Let $A$, $B$ and $C$ be finite subsets of a group $G$. Then

[TABLE]

Commutativity is not needed. In fact, what is being used is in some sense more basic than a group structure; as shown in [GHR15], the same argument works naturally in any abstract projective plane endowed with the little Desargues axiom.

Proof.

We will construct an injection $\iota:AC^{-1}\times B\hookrightarrow AB^{-1}\times BC^{-1}$. For every $d\in AC^{-1}$, choose $(f_{1}(d),f_{2}(d))=(a,c)\in A\times C$ such that $d=ac^{-1}$. Define $\iota(d,b)=(f_{1}(d)b^{-1},b(f_{2}(d))^{-1})$. We can recover $d=f_{1}(d)(f_{2}(d))^{-1}$ from $\iota(d,b)$; hence we can recover $(f_{1},f_{2})(d)=(a,c)$, and thus $b$ as well. Therefore, $\iota$ is an injection. ∎

Exercise 2.3**.**

Let $G$ be a group. Prove that

[TABLE]

for every finite subset $A$ of $G$. Show as well that, if $A=A^{-1}$ (i.e., if $g^{-1}\in A$ for every $g\in A$), then

[TABLE]

for every $k\geq 3$. Conclude that

[TABLE]

for every $A\subset G$ and every $k\geq 3$.

Inequalities (2.2)–(2.4) go back to Ruzsa (or Ruzsa-Turjányi [RT85]), at least for $G$ abelian.

This means that, from now on, we can generally focus on studying when $|A^{3}|$ is or isn’t much larger than $|A|$. Thanks to (2.2), we can also assume in many contexts that $e\in A$ and $A=A^{-1}$ without loss of generality.

2.2. The orbit-stabilizer theorem for sets

A theme recurs in work on growth in groups: results on subgroups can often be generalized to subsets. This is especially the case if the proofs are quantitative, constructive, or, as we shall later see, probabilistic.

The orbit-stabilizer theorem for sets is a good example, both because of its simplicity (it should really be called a lemma) and because it underlies a surprising number of other results on growth. It also helps to put forward a case for seeing group actions, rather than groups themselves, as the main object of study.

We recall that an action $G\curvearrowright X$ is a homomorphism from a group $G$ to the group of automorphisms of a set $X$. (The automorphisms of a set $X$ are just the bijections from $X$ to $X$; we will see actions on objects with richer structures later.) For $A\subset G$ and $x\in X$, the orbit $Ax$ is the set $Ax=\{g\cdot x:g\in A\}$. The stabilizer $\operatorname{Stab}(x)\subset G$ is given by $\operatorname{Stab}(x)=\{g\in G:g\cdot x=x\}$.

The statement we are about to give is as in [HS14, §3.1].

Lemma 2.4** (Orbit-stabilizer theorem for sets).**

Let $G$ be a group acting on a set $X$. Let $x\in X$, and let $A\subseteq G$ be non-empty. Then

[TABLE]

Moreover, for every $B\subseteq G$,

[TABLE]

The usual orbit-stabilizer theorem – usually taught as part of a first course in group theory – states that, for $H$ a subgroup of $G$,

[TABLE]

This the special case $A=B=H$ of the Lemma we (or rather you) are about to prove.

Exercise 2.5**.**

Prove Lemma 2.4. Suggestion: for (2.5), use the pigeonhole principle.

If we try to apply Lemma 2.4 to the (left) action of the group $G$ on itself by left multiplication

[TABLE]

or to the (left) action by right multiplication

[TABLE]

we do not get anything interesting: the stabilizer of any element is trivial. The same is of course true of the right actions $g\mapsto(h\mapsto g^{-1}h)$ and $g\mapsto(h\mapsto h\dot{g})$. However, we also have the action by conjugation

[TABLE]

The stabilizer of a point $h\in G$ is its centralizer

[TABLE]

the orbit of a point $h\in G$ under the action of the group $G$ is the conjugacy class

[TABLE]

Thus, we obtain the following result, which will show itself to be crucial later. Its importance resides in making upper bounds on intersections of $A$ (or rather $A^{l+2}$) with $\operatorname{Cl}(g)$ imply lower bounds on intersections of $A^{2}$ with $C(g)$. In other words, the plan is to show that there are not too many elements of $A^{l+2}$ of a special form, and then Lemma 2.6 will imply that there are many elements of $A^{2}$ of another special form. Having many elements of a special form will be very useful.

Lemma 2.6**.**

Let $A\subset G$ be a non-empty set with $A=A^{-1}$. Then, for every $g\in A^{l}$, $l\geq 1$,

[TABLE]

Proof.

Let $G\curvearrowright G$ be the action of $G$ on itself by conjugation. Apply (2.5) with $x=g$; the orbit of $g$ under conjugation by $A$ is contained in $A^{l+2}\cap\operatorname{Cl}(g)$. ∎

It is instructive to see some other consequences of Lemma 2.4.

Exercise 2.7**.**

Let $G$ be a group and $H$ a subgroup thereof. Let $A\subset G$ be a set with $A=A^{-1}$. Then

[TABLE]

where $r$ is the number of cosets of $H$ intersecting $A$.

Hint: Consider the action $G\curvearrowright X=G/H$ by left multiplication, that is, $g\mapsto(aH\mapsto gaH)$. Then apply (2.5).

The following exercise tells us that, if we show that the intersection of $A$ with a subgroup $H$ grows rapidly, then we know that $A$ itself grows rapidly.

Exercise 2.8**.**

Let $G$ be a group and $H$ a subgroup thereof. Let $A\subset G$ be a non-empty set with $A=A^{-1}$. Prove that, for any $k>0$,

[TABLE]

Hint: Consider the action $G\curvearrowright G/H$ again, and apply both (2.6) and (2.5).

Exercise 2.9**.**

Let $G$ be a group and $H$ a subgroup thereof. Write $\pi_{G/H}:G\to G/H$ for the quotient map. Let $A\subseteq G$ be a non-empty set with $A=A^{-1}$. Then, for any $k>0$,

[TABLE]

3. Growth in a solvable group

3.1. Remarks on abelian groups

Let $G$ be an abelian group and $A$ be a finite subset of $G$. This is the classical setup for what nowadays is called additive combinatorics – a field that may be said to have started to split off from additive number theory with Roth [Rot53] and Freiman [Fre73].

In general, for $G$ abelian, $A\subset G$ may be such that $|A+A|$ is barely larger than $|A|$, and that is the case even if we assume that $A$ generates $G$. For instance, take $A$ to be a segment of an arithmetic progression: $A=\{2,5,8,\dotsc,3m-1\}$. Then $|A|=m$ and $|A+A|=2m-1<2|A|$.

Freiman’s theorem [Fre73] (generalized first to abelian groups of bounded torsion [Ruz99] and then to arbitrary abelian groups [GR07]) tells us that, in a very general sense, this is the only kind of set that grows slowly. We have to start by giving a generalization of what we just called a segment of an arithmetic progression.

Definition 3.1**.**

Let $G$ be a group. A centered convex progression of dimension $d$ is a set $P\subset G$ such that there exist

1. (a)

a convex subset $Q\subset\mathbb{R}^{d}$ that is also symmetric ($Q=-Q$), 2. (b)

a homomorphism $\phi:\mathbb{Z}^{d}\to G$,

for which $\phi(\mathbb{Z}^{d}\cap Q)=P$. We say $P$ is proper if $\phi|_{\mathbb{Z}^{d}\cap Q}$ is injective.

Proposition 3.1** (Freiman; Ruzsa-Green).**

Let $G$ be an abelian group. Let $A\subset G$ be finite. Assume that $|A+A|\leq K|A|$ for some $K$. Then $A$ is contained in at most $c_{K,1}$ copies of $P+H$ for some proper, centered convex progression $P$ of dimension $\leq c_{K,2}$ and some finite subgroup $H<G$ such that $|P+H|\ll e^{c_{K,2}}|A|$. Here $c_{K,1},c_{K,2}>0$ depend only on $K$.

The best known bounds are essentially those of Sanders [San12], as improved by Konyagin (see [San13]): $c_{K,1},c_{K_{2}}\ll(\log K)^{3+o(1)}$.

This is a broad field into which we will not venture further. Notice just that, in spite of more than forty years of progress, we do not yet have what is conjectured to be the optimal result, namely, the above with $f(K),g(K)\ll\log K$ (the “polynomial Freiman-Ruzsa conjecture”). Thus the state of our knowledge here is in some sense less satisfactory than in the case of simple groups, as will later become clear.

The situation for nilpotent groups is much like the situation for abelian groups: there is a generalization of the Freiman-Ruzsa theorem to the nilpotent case, due to Tointon [Toi14] (see also Tessera-Tointon [TT16]), based on groundwork laid by Fisher-Katz-Peng [FKP10] and Tao [Tao10].

Brief excursus. There is of course also the matter of the role of nilpotent groups in the study of growth in a different if related sense, within geometric group theory: for $A$ a subset of an infinite group $G$, how does $|A^{k}|$ behave as $k\to\infty$? It is easy to see that, if $G$ is nilpotent, then $|A^{k}|$ grows polynomially on $k$. Gromov’s theorem [Gro81], a deep and celebrated result, states the converse: if $|A^{k}|$ is bounded by a polynomial on $k$, then $\langle A\rangle$ has a nilpotent subgroup of finite index. There are several clearly distinct proofs of Gromov’s theorem by now; of them, the one closest to the study of “growth” in the sense of the present paper is clearly [Hru12]. See [BGT12] for further work in that direction.

3.2. The affine group

3.2.1. Growth in the affine group

We defined the affine group $G$ over a field $K$ in (1.10). (If we were to insist on using language in exactly the same way as later, we would say that the affine group is an algebraic group $G$ (a variety with morphisms defining the group operations) and that (1.10) describes the group $G(K)$ consisting of its rational points. For the sake of simplicity, we avoid this sort of distinction here. We will go over most of these terms once the time to use them has come.)

Consider the following subgroups of $G$:

[TABLE]

These are simple examples of a solvable group $G$, of a maximal unipotent subgroup $U$ and of a maximal torus $T$. In general, in $\operatorname{SL}_{n}$, a maximal torus is just the group of matrices that are diagonal with respect to some fixed basis of $\overline{K}^{n}$, or, what is the same, the centralizer of any element that has $n$ distinct eigenvalues. Here, in our group $G$, the centralizer $C(g)$ of any element $g$ of $G$ not in $U$ is a maximal torus.

When we are looking at what elements of the group $G$ do to each other by the group operation, we are actually looking at two actions: that of $U$ on itself (by the group operation) and that of $T$ on $U$ (by conjugation; $U$ is a normal subgroup of $G$). They turn out to correspond to addition and multiplication in $K$, respectively:

[TABLE]

Thus, we see that growth in $U$ under the actions of $U$ and $T$ is tightly linked to growth in $K$ under addition and multiplication. This can be seen as motivation for studying growth in the affine group $G$. Perhaps we need no such motivation: we are studying growth in general, through a series of examples, and the affine group is arguably the simplest interesting example of a solvable group.

At the same time, the study of growth in a field under addition and multiplication was historically important in the passage from the study of problems in commutative groups (additive combinatorics) to the study of problems in noncommutative groups by related tools. (Growth in noncommutative groups had of course been studied before, but from very different perspectives, e.g., that of geometric group theory.) Some of the ideas we are about to see in the context of groups come ultimately from [BKT04] and [GK07], which are about finite fields, not about groups.

Of course, the way we choose to develop matters emphasizes what the approach to the affine group has in common with the approach to other, not necessarily solvable groups. The idea of pivoting will appear again when we study $\operatorname{SL}_{2}$.

Lemma 3.2**.**

Let $G$ be the affine group over $\mathbb{F}_{p}$. Let $U$ be the maximal unipotent subgroup of $G$, and $\pi:G\to G/U$ the quotient map.

Let $A\subset G$, $A=A^{-1}$. Assume $A\not\subset U$; let $x$ be an element of $A$ not in $U$. Then

[TABLE]

for $T=C(x)$.

Recall $U$ is given by (3.1). Since $x\not\in U$, its centralizer $T=C(x)$ is a maximal torus.

Proof.

By (2.7), $A_{u}:=A^{2}\cap U$ has at least $|A|/|\pi(A)|$ elements. Consider the action of $G$ on itself by conjugation. Then, by Lemma 2.4, $|A^{2}\cap T|\geq|A|/|A(x)|$. (Here $A(x)$ is the orbit of $x$ under the action of $A$ by conjugation, and $\operatorname{Stab}(x)=C(g)=T$ is the stabilizer of $g$ under conjugation.) We set $A_{t}:=A^{2}\cap T$. Clearly, $|A(x)|=|A(x)x^{-1}|$. Since the derived group of $G$ is $U$ (meaning, in particular, that $axa^{-1}x^{-1}\in U$ for any $a$ and $x$), we see that $A(x)x^{-1}\subset A^{4}\cap U$, and so $|A(x)|\leq|A^{4}\cap U|$. At the same time, by (2.6) applied to the action $G\curvearrowright G/U$ by left multiplication, $|A^{5}|=|A^{4}A|\geq|A^{4}\cap U|\cdot|\pi(A)|$. Hence

[TABLE]

∎

The proof of the following proposition will proceed essentially by induction. This may be a little unexpected, since we are in a group $G$, not in, say, $\mathbb{Z}$, which has a natural ordering. However, as the proof will make clear, one can do induction on a group with a finite set of generators, even in the absence of an ordering.

Proposition 3.3**.**

Let $G$ be the affine group over $\mathbb{F}_{p}$, $U$ the maximal unipotent subgroup of $G$, and $T$ a maximal torus. Let $A_{u}\subset U$, $A_{t}\subset T$. Assume $A_{u}=A_{u}^{-1}$, $e\in A_{t},A_{u}$ and $A_{u}\neq\{e\}$. Then

[TABLE]

To be clear: here

[TABLE]

where $t(u)=tut^{-1}$, since $T$ acts on $U$ by conjugation.

Proof.

Call $a\in U$ a pivot if the function $\phi_{a}:A_{u}\times A_{t}\to U$ given by

[TABLE]

is injective.

Case (a): There is a pivot $a$ in $A_{u}$. Then $|\phi_{a}(A_{u},A_{t})|=|A_{u}||A_{t}|$, and so

[TABLE]

This is the motivation for the name “pivot”: the element $a$ is the pivot on which we build an injection $\phi_{a}$, giving us the growth we want.

Case (b): There are no pivots in $U$. As we are about to see, this case can arise only if either $A_{u}$ or $A_{t}$ is large with respect to $p$. Say that $(u_{1},t_{1}),(u_{2},t_{2})\in A_{u}\times A_{t}$ collide for $a\in U$ if $\phi_{a}(u_{1},t_{1})=\phi_{a}(u_{2},t_{2})$. Saying that there are no pivots in $U$ is the same as saying that, for every $a\in U$, there are at least two distinct $(u_{1},t_{1}),(u_{2},t_{2})\in A_{u}\times A_{t}$ that collide for $a$. Now, two distinct $(u_{1},t_{1})$, $(u_{2},t_{2})$ can collide for at most one $a\in U\setminus\{e\}$. (As one can easily see, such an $a$ corresponds to a solution to a non-trivial linear equation, which can have at most one solution.) Hence, if there are no pivots, $|A_{u}|^{2}|A_{t}|^{2}\geq|U\setminus\{e\}|=p-1$, i.e., $|A_{u}|\cdot|A_{t}|$ is large ($\geq\sqrt{p-1}$). This fact already hints that this case will not be hard.

Let $\kappa_{a}$ denote the number of collisions for a given $a\in U$:

[TABLE]

As we were saying, two distinct $(u_{1},t_{1})$, $(u_{2},t_{2})$ collide for at most one $a\in U\setminus\{e\}$. Hence the total number of collisions $\sum_{a\in U\setminus\{e\}}\kappa_{a}$ is $\leq|A_{u}|^{2}|A_{t}|^{2}$, and so there is an $a\in U\setminus\{e\}$ such that

[TABLE]

Now,

[TABLE]

where the inequality is just Cauchy-Schwarz. Thus, $|\phi_{a}(A_{u},A_{t})|\geq|A_{u}|^{2}|A_{t}|^{2}/\kappa_{a}$, and so

[TABLE]

We are not quite done, since $a$ may not be in $A$. Since $a$ is not a pivot (as there are none), there exist distinct $(u_{1},t_{1})$, $(u_{2},t_{2})$ such that $\phi_{a}(u_{1},t_{1})=\phi_{a}(u_{2},t_{2})$. Then $t_{1}\neq t_{2}$ (why?), and so the map $\psi_{t_{1},t_{2}}:U\to U$ given by $u\mapsto t_{1}(u)(t_{2}(u))^{-1}$ is injective. The idea is that the very non-injectivity of $\phi_{a}$ gives an implicit definition of it, much like a line that passes through two distinct points is defined by them.

What follows may be thought of as the “unfolding” step, in that we wish to remove an element $a$ from an expression, and we do so by applying to the expression a map that will send $a$ to something known. We will be using the commutativity of $T$ here.

For any $u\in U$, $t\in T$, since $T$ is abelian,

[TABLE]

where $\psi_{t_{1},t_{2}}(a)=u_{1}^{-1}u_{2}$ holds because $\phi_{a}(u_{1},t_{1})=\phi_{a}(u_{2},t_{2})$. Note that $a$ has disappeared from the last expression in (3.4). We obtain

[TABLE]

Since $\psi_{t_{1},t_{2}}$ is injective, we conclude that

[TABLE]

that is to say, at most a single element of $U$ is missing from $(A_{t}(A_{u}))^{4}$. Since $A_{u}$ contains at least one element besides $e$, we obtain immediately that

[TABLE]

There is an idea here that we are about to see again: any element $a$ that is not a pivot can, by this very fact, be given in terms of some $u_{1},u_{2}\in A_{u}$, $t_{1},t_{2}\in A_{t}$, and so an expression involving $a$ can often be transformed into one involving only elements of $A_{u}$ and $A_{t}$.

Case (c): There are pivots and non-pivots in $U$. Here comes what we can think of as the inductive step. Since $A_{u}\neq\{e\}$, $A_{u}$ generates $U$. Thus, there is a non-pivot $a\in U$ and a $g\in A_{u}$ such that $ga$ is a pivot. Then $\phi_{ag}:A_{u}\times A_{t}\to U$ is injective. Much as in (3.4), we unfold:

[TABLE]

where $(u_{1},t_{1})$, $(u_{2},t_{2})$ are distinct pairs such that $\phi_{a}(u_{1},t_{1})=\phi_{a}(u_{2},t_{2})$. Just as before, $\psi_{t_{1},t_{2}}$ is injective. Hence

[TABLE]

and we are done.

The idea to recall here is that, if $S$ is a subset of an orbit $\mathscr{O}=\langle A\rangle x$ such that $S\neq\emptyset$ and $S\neq\mathscr{O}$, then there is an $s\in S$ and a $g\in A$ such that $gs\not\in S$. It is in this fashion that we can use induction even in the absence of a natural ordering of $\langle A\rangle$. ∎

We are using the fact that $G$ is the affine group over $\mathbb{F}_{p}$ (and not over some other field) only at the beginning of case (c), when we say that, for $A_{u}\subset U$, $A_{u}\neq\{e\}$ implies $\langle A_{u}\rangle=U$.

Proposition 3.4**.**

Let $G$ be the affine group over $\mathbb{F}_{p}$. Let $U$ be the maximal unipotent subgroup of $G$, and $\pi:G\to G/U$ the quotient map.

Let $A\subset G$, $A=A^{-1}$, $e\in A$. Assume $A$ is not contained in any maximal torus. Then either

[TABLE]

or

[TABLE]

The exponents $72$, $73$ in (3.6) are not optimal. For instance, one can obtain $52$, $53$ by looking closer at the proof of Prop. 3.3.

Proof.

We can assume $A\not\subset U$, as otherwise what we are trying to prove is trivial. Let $g$ be an element of $A$ not in $U$; its centralizer $C(g)$ is a maximal torus $T$. By assumption, there is an element $h$ of $A$ not in $T$. Then $hgh^{-1}g^{-1}\neq e$. At the same time, $hgh^{-1}g^{-1}$ does lie in $A^{4}\cap U$, and so $A^{4}\cap U$ is not $\{e\}$.

Let $A_{u}=A^{4}\cap U$, $A_{t}=A^{2}\cap T$; their size is bounded from below by (3.2). Applying Prop. 3.3, we obtain

[TABLE]

By (2.6), $|A^{73}|\geq|A^{72}\cap U|\cdot|\pi(A)|$. Clearly, if $|A|/|A^{5}|<1/\sqrt{|\pi(A)|}$, then $|A^{57}|\geq|A^{5}|>\sqrt{|\pi(A)|}\cdot|A|$. If $|A|/|A^{5}|\geq 1/\sqrt{|\pi(A)|}$, then either $|A^{72}\cap U|\geq|A|/\sqrt{|\pi(A)|}$ and so $|A^{73}|\geq\sqrt{|\pi(A)|}\cdot|A|$, or $|A^{72}\cap U|=p$ and so $U\subset A^{72}$. ∎

For $A\subset U$, getting a better-than-trivial lower bound on $|A^{k}|$, $k$ a constant, amounts to Freiman’s theorem in $\mathbb{F}_{p}$, and getting a growth factor of the form $|\pi(A)|^{\delta}$, $\delta>0$, would involve proving a version of Freiman’s theorem of polynomial strength. As we discussed before, that is a difficult open problem.

3.2.2. Brief remarks on a generalization and an application

We can see Prop. 3.4 as a very simple result of the “classification of approximate subgroups” kind. If a set $A$ (with $A=A^{-1}$, $e\in A$) in the affine group over $\mathbb{F}_{p}$ grows slowly ($|A^{k}|\leq|A|^{1+\delta}$, $k=73$, $\delta$ small) then either (i) $A$ is contained in a maximal torus, (ii) $A$ is contained in a few cosets of the maximal unipotent subgroup $U$ (that is, $|\pi(A)|\leq|A|^{2\delta}$), or (iii) $A^{k}$ contains a subgroup (namely, $U$) such that $\langle A\rangle/H$ is nilpotent (here, in fact, abelian).

Exercise 3.5**.**

Give examples of subsets $A$ of the affine group over $\mathbb{F}_{p}$ that fail to grow for each of the reasons above: a set contained in a maximal torus, a set almost contained in $U$, and a set containing $U$.

The following more general statement has been proved for $K=\mathbb{F}_{p}$ [GH11]. (It remains open for general finite $K$.) Let $A\subset G=\operatorname{GL}_{n}(K)$ ($A=A^{-1}$, $e\in A$) be such that $\langle A\rangle$ is solvable. Then, for any $\delta>0$, if $|A^{3}|<|A|^{1+\delta}$, there are a subgroup $S\triangleleft\langle A\rangle$ and a unipotent subgroup $U\triangleleft S$ such that (a) $S/U$ is nilpotent, (b) $U\subset A^{k}$, where $k=O_{n}(1)$, (c) $A$ is contained in $|A|^{O_{n}(\delta)}$ cosets of $U$.

Exercise 3.6**.**

Verify that each of the cases (i)-(iii) enumerated above in the case of the affine group satisfies this description, i.e., there are $S$ and $U$ such that (a)–(c) are fulfilled.

What is also interesting is that the results we have proved on growth in the affine linear group can be interpreted as a sum-product theorem.

Exercise 3.7**.**

Let $X\subset\mathbb{F}_{p}$, $Y\subset\mathbb{F}_{p}^{*}$ be given with $X=-X$, $0\in X$, $1\in Y$. Using Prop. 3.3, show that

[TABLE]

This is almost exactly [GK07], Corollary 3.5], say.

Using (3.8), or any estimate like it, one can prove the following.

Theorem 3.8** **(Sum-product theorem [BKT04],

[BGK06]; see also [EM03]).

For any $A\subset\mathbb{F}_{p}^{*}$ with $|A|\leq p^{1-\epsilon}$, $\epsilon>0$, we have

[TABLE]

where $\delta>0$ depends only on $\epsilon$.

In fact, the proof we have given of Prop. 3.3 takes its ideas from proofs of the sum-product theorem. In particular, the idea of pivoting is already present in them. We will later see how to apply it in a broader context.

3.2.3. Diameter bounds in a remaining case

We have proved that growth occurs in $\operatorname{SL}_{2}$ under some weak conditions. This leaves open the question of what happens with $A^{k}$, $k$ unbounded, for $A$ not obeying those conditions. In particular: what happens when $A$, while not contained in the maximal unipotent group $U$, is contained in the union of few cosets of $U$?

One thing that is certainly relevant here is that, in general, there is no vertex expansion in the affine group, and thus no expansion. Indeed, the purpose of this subsection is to give a glimpse of the issue of diameter bounds in situations in which neither expansion nor rapid growth hold.

Let us state the lack of vertex expansion in elementary terms.

Proposition 3.9**.**

For any $\lambda_{1},\dotsc,\lambda_{k}\in\mathbb{Z}$, and any $\epsilon>0$ , there is a constant $C$ depending on $\epsilon$ such that, for every prime $p>C$, there is a set $S\subset\mathbb{F}_{p}$, $0<|S|\leq p/2$, such that

[TABLE]

Exercise 3.10**.**

Prove Proposition 3.9. Hints: prove this for $k=1$ first; you can assume $\lambda=\lambda_{1}$ is $\geq 2$. Here is a plan. We want to show that $|S\cup(S+1)\cup\lambda S|\leq(1+\epsilon)|S|$. For $|S\cup(S+1)|$ to be $\leq(1+\epsilon/2)|S|$, it is enough that $S$ be a union of intervals of length $>2/\epsilon$. (By an interval we mean the image of an interval $[a,b]\cap\mathbb{Z}$ under the map $\mathbb{Z}\to\mathbb{Z}/p\mathbb{Z}\sim\mathbb{F}_{p}$.) We also want $|S\cup\lambda S|\leq(1+\epsilon)|S|$; this will be the case if $S$ is the union of disjoint sets of the form $V$, $\lambda^{-1}V$, …, $\lambda^{-r}V$, $r\geq\epsilon/2$. Now, in $\mathbb{F}_{p}$, if $I$ is an interval of length $\ell$, then $\lambda^{-1}I$ is the union of $\lambda$ intervals (why? of what length?). Choose $V$ so that $V,\lambda^{-1}V,\dotsc,\lambda^{-r}V$ are disjoint. Let $S$ be the union of these sets; verify that it fulfills (3.9).

The following exercise shows that Prop. 3.9 is closely connected to the fact that a certain group is amenable.

Exercise 3.11**.**

Let $\lambda\geq 2$ be an integer. Define the Baumslag-Solitar group $\operatorname{BS}(1,\lambda)$ by

[TABLE]

1. (a)

A group $G$ with generators $a_{1},\dotsc,a_{\ell}$ is called amenable if, for every $\epsilon>0$, there is a finite $S\subset G$ such that

[TABLE]

Show that $\operatorname{BS}(1,\lambda)$ is amenable. Hint: to construct $F$, take your inspiration from Exercise 3.10. 2. (b)

Express the subgroup of the affine group over $\mathbb{F}_{p}$ generated by the set

[TABLE]

as a quotient of $\operatorname{BS}(1,\lambda)$, i.e., as the image of a homomorphism $\pi_{p}$ defined on $\operatorname{BS}(1,\lambda)$. 3. (c)

Displace or otherwise modify your sets $F$ so that, for each of them, $\pi_{p}|_{F}$ is injective for $p$ larger than a constant. Conclude that $S=\pi_{p}(F)$ satisfies (3.9), thus giving a (slightly) different proof of exercise 3.10.

Amenability is not good news when we are trying to prove that a diameter is small, in that it closes a standard path towards showing that it is logarithmic in the size of the group. However, it does not imply that the diameter is not small.

Let us first be clear about what we can prove or rather about what we cannot hope to prove. We should not aim at a bound on the diameter of the affine group $G$ with respect to an arbitrary set of generators $A$: it is easy to choose $A$ so that the diameter of $\Gamma(G,A)$ is very large.

Exercise 3.12**.**

Let $A_{\lambda}$ be as in (3.10) for $\lambda$ a generator of $\mathbb{F}_{p}^{*}$. Let $A=A_{\lambda}\cup A_{\lambda}^{-1}$. Then $A$ generates the affine group $G$ over $\mathbb{F}_{p}$. Show that $\operatorname{diam}\Gamma(G,A)=(p-1)/2$.

Rather, we should aim for a bound on the diameter of the Schreier graph of the action of the affine group $G$ by conjugation on its maximal unipotent subgroup $U$. In general, the Schreier graph of an action $G\curvearrowright X$ of a group $G$ on a set $X$ with respect to a set of generators $A$ of $G$ is the graph having $X$ as its set of vertices and $\{(x,ax):x\in X,a\in A\}$ as its set of edges. In our case ($X=U$, $A=A_{\lambda}\cup A_{\lambda}^{-1}$, $\lambda\in\mathbb{F}_{p}^{*}$), the Schreier graph is isomorphic to the graph $\Gamma_{p,\lambda}$ with vertex set $\mathbb{F}_{p}$ and edge set

[TABLE]

We are not avoiding the problem posited by the fact that the Baumslag-Solitar group $\operatorname{BS}(1,\lambda)$ is amenable, since what amenability impedes is precisely a natural approach to prove logarithmic diameter bounds on $\Gamma_{p,\lambda}$. If Proposition 3.9 were not true, then the diameter of $\Gamma_{p,\lambda}$ would be $O(\log p)$. (Why?)

If $\lambda$ is the projection of a fixed integer $\lambda_{0}$, then it is possible, and easy, to give a logarithmic diameter bound nevertheless.

Exercise 3.13**.**

Let $\lambda_{0}\geq 2$ be an integer. Let $\lambda=\lambda_{0}\operatorname{\,mod}p$, which lies in $\mathbb{F}_{p}^{*}$ for $p>\lambda_{0}$. Show that the diameter of the graph $\Gamma_{p,\lambda}$ is $O(\lambda_{0}\log p)$. Hint: lift elements of $\mathbb{F}_{p}$ to $\mathbb{Z}\cap[0,p-1]$, and write them out in base $\lambda_{0}$.

It turns out to be possible to give a polylogarithmic bound for general $\lambda\in\mathbb{F}_{p}^{*}$:

[TABLE]

where the implied constants are independent of $p$ and $\lambda$. Here we need not assume that $\lambda$ generates $\mathbb{F}_{p}^{*}$, but we do assume that the order of $\lambda$ is $\gg\log p$. (Indeed, if the order of $\lambda$ is very small, viz., $o((\log p)/\log\log p)$, then (3.11) cannot hold; why?)

The proof of (3.11) was the outcome of a series of discussions among B. Bukh, A. Harper, E. Lindenstrauss and the author. It is essentially an exercise in Fourier analysis using bounds on exponential sums due to Konyagin [Kon92].

Exercise 3.14**.**

Let $p$ be a prime, $\lambda\in\mathbb{F}_{p}^{*}$. Assume $\lambda$ has order $\geq\log p$. Write $e(t)=e^{2\pi it}$ and $e_{p}(t)=e^{2\pi it/p}$. Konyagin [Kon92, Lemma 6] showed that, for any $\epsilon>0$, there is a $c_{\epsilon}>0$ such that, for any $p\geq c_{\epsilon}$ prime and $\alpha,\lambda\in(\mathbb{Z}/p\mathbb{Z})^{*}$ with $\lambda$ of order $\geq c_{\epsilon}(\log p)/(\log\log p)^{1-\epsilon}$ in the group $(\mathbb{Z}/p\mathbb{Z})^{*}$,

[TABLE]

where $J=\lfloor c_{\epsilon}\log p(\log\log p)^{4}\rfloor$ and $\{x\}$ is the element of $(-1/2,1/2]$ such that $x-\{x\}$ is an integer.

1. (a)

Show that (3.12) implies that $S(\alpha)=\sum_{j=0}^{J}e_{p}(\alpha\lambda^{j})$ satisfies $|S(\alpha)|\leq J+1-1/(\log p)^{3^{\epsilon/4}/2}$ for every $\alpha\in(\mathbb{Z}/p\mathbb{Z})^{*}$. 2. (b)

Deduce that every element of $\mathbb{Z}/p\mathbb{Z}$ can be written as a sum $\sum_{i=1}^{K}\lambda^{j_{i}}$, where $0\leq j_{i}\leq J$ and $K$ is bounded by

[TABLE]

To do so, show first that for any sequence $r_{0},\dotsc,r_{j}\in\mathbb{Z}/p\mathbb{Z}$, the number of ways of expressing $x\in\mathbb{Z}/p\mathbb{Z}$ as a sum of $K$ elements (not necessarily distinct) of a subset $A\subset\mathbb{Z}/p\mathbb{Z}$ equals

[TABLE]

where $S_{A}(\alpha)=\sum_{a\in A}e(\alpha a)$. This approach is the circle method over $\mathbb{Z}/p\mathbb{Z}$. 3. (c)

Conclude that the graph $\Gamma_{p,\lambda}$ with vertex set $\mathbb{F}_{p}$ and edge set

[TABLE]

has diameter $\ll_{\epsilon}(\log p)^{5/2+\epsilon}$.

4. Intersections with varieties

Let $G$ a linear algebraic group defined over a field $K$. Let $A$ be a finite set of generators of the set of points of $G$ over $K$.

We will first show that, unless all the points of $G$ over $K$ lie in $V$, there are (plenty of) elements of $A^{k}$, $k$ bounded, that do not lie on $V$ (escape from subvarieties). Here the constant $k$ depends only on some invariants of $V$ (its number of components, their degree and their dimension), not on $K$ or on other properties of $V$.

Our main aim will then be to show that, if $A$ grows slowly, then $A$ is truly a beautiful object, very regular from many points of view. Of course, this is a strategy for showing in the following section that $A$ does not exist (or is almost all of $G$).

“Very regular” here means “behaving well with respect to the algebraic geometry of the ambient group $G$”. To be precise: the intersection of a slowly growing set $A$ with any variety $V$ will be bounded by not much more than $|A|^{\dim(V)/\dim(G)}$ (Theorem 4.4; the dimensional estimate).

Here is an intuitive image. Thinking for a moment in three dimensions (that is, $\dim(G)=3$), one might say that this estimate means that $A$ is very regular in the sense of being a roughly spherical blob, as its intersection with any line, or any curve of bounded degree, is bounded by $O(|A|^{1/3})$, and its intersection with any plane, or any surface of bounded degree, is bounded by $O(|A|^{2/3})$.

Finally, we will see that for some kinds of varieties $V$ – namely, centralizers – we can give a lower bound on the intersection of $A$ with $V$, roughly of the same order as the upper bound above. This fact will be a crucial tool in §5.

4.1. Preliminaries from algebraic geometry and algebraic groups

We will have the choice of working sometimes over linear algebraic groups and sometimes over Lie algebras (as in [Hel15], following [Hel11]) or solely over linear algebraic groups (as in [Tao15], which follows [BGT11]). We will follow the first path. Naturally, we will need some preliminaries on varieties, their behavior under mappings, the derivatives of such mappings, and so forth. It will all be a quick review for some readers. When it comes to basic algebraic geometry, we will cite mainly [Mum99] and [Har77], as they are standard sources for English speakers. In the case of either source, we will limit ourselves to the first chapter, that is, to classical foundations. Our definitions for terms related to algebraic groups come mostly from [Spr98] and [Bor91]; basic facts on finite groups of Lie type come from [MT11, ch. 21 and 24].

.

4.1.1. Basic definitions.

We will need some basic terms from algebraic geometry. Let $K$ be a field; denote by $\overline{K}$ an algebraic closure of $K$. For us, a variety $V$ will simply be an affine or a projective variety – that is, the algebraic set consisting of the solutions in $\mathbb{A}^{n}$ to a system of polynomial equations, or the solutions in $\mathbb{P}^{n}$ to a system of homogeneous polynomial equations. We say $V$ is defined over $K$ if $V$ can be described by polynomial equations with coefficients in $K$. Given a field $L$ containing $K$, we write $V(L)$ for the set of solutions with coordinates in $L$. When we simply say “points on $V$”, we mean elements of $V(\overline{K})$.

Abstract algebraic varieties (as in, say, [Mum99, Def. I.6.2]) will not really be needed, although they do give a very natural way to handle a variety that parametrizes a family of varieties, among many other things. For instance, we will tacitly refer to the variety of all $d$-dimensional planes in projective space, and, while that variety (a Grassmanian) can indeed be defined as an algebraic set in projective space, that is a non-obvious though standard fact.

The Zariski topology on $\mathbb{A}^{n}$ or $\mathbb{P}^{n}$ is the topology whose open sets are the complements of varieties (affine ones if we work in $\mathbb{A}^{n}$, projective ones if we work in $\mathbb{P}^{n}$). It induces a topology, also called Zariski topology, on any variety $V$; its open sets are the complements $V\setminus W$ of subvarieties $W$ of $V$. (A subvariety of $V$ is a variety contained in $V$.) The Zariski closure $\overline{S}$ of a subset $S$ of $V$ is its closure in the Zariski topology.

A variety $V$ is irreducible if it is not the union of two varieties $V_{1},V_{2}\neq\emptyset,V$. (Note that many authors call an algebraic set a variety only if it is irreducible.) Every variety $V$ can be written as a finite union of irreducible varieties $V_{i}$, with $V_{i}\not\subset V_{j}$ for $i\neq j$; they are called the irreducible components (or simply the components) of $V$.

When we say “property $P$ holds for a generic point in the variety $V$”, we simply means that there is a dense open subset $U\subset V$ such that property $P$ holds for every point on $U$. It is easy to see that a non-empty open subset of an irreducible variety is always dense.

The dimension $\dim V$ of an irreducible variety $V$ is the largest $d$ such that there exists a chain of irreducible varieties

[TABLE]

The union of several irreducible varieties of dimension $d$ is called a pure-dimensional variety of dimension $d$. If $W$ is a pure-dimensional proper subvariety of an irreducible variety $V$, then $\dim W<\dim V$ [Mum99, Cor. I.7.1]. (A subvariety $W\subset V$ is proper if $W\neq V$.)

The direct product $V\times W$ of irreducible varieties $V$, $W$ is an irreducible variety of dimension is $\dim V+\dim W$ ([Har77, Exer. I.3.15 and I.2.14] or [Mum99, Prop. I.6.1, Thm. I.6.3 and Prop. I.7.5]).

4.1.2. Degrees. Bézout’s theorem.

The degree of a pure-dimensional variety $V$ in $\mathbb{A}^{n}$ or $\mathbb{P}^{n}$ of dimension $d$ is its number of points of intersection with a generic plane of dimension $n-d$. (See? We just referred tacitly to…)

Bézout’s theorem, in its classical formulation, states that, for any two distinct irreducible curves $C_{1}$, $C_{2}$ in $\mathbb{A}^{2}$, the number of points of intersection $(C_{1}\cap C_{2})(\overline{K})$ is at most $d_{1}d_{2}$. (In fact, for $C_{1}$ and $C_{2}$ generic, the number of points of intersection is exactly $d_{1}d_{2}$; the same is true for all distinct $C_{1}$, $C_{2}$ if we count points of intersection with multiplicity.)

In general, if $V_{1}$ and $V_{2}$ are irreducible varieties, and we write $V_{1}\cap V_{2}$ as a union of irreducible varieties $W_{1},W_{2},\dotsc,W_{k}$ with $W_{i}\not\subset W_{j}$ for $i\neq j$, a generalization of Bézout’s theorem tells us that

[TABLE]

See, for instance, [DS98, p.251], where Fulton and MacPherson are mentioned in connection to this and even more general statements.

Inequality (4.1) implies immediately that, if a variety $V$ is defined by at most $m$ equations of degree at most $d$, then the number and degrees of the irreducible components of $V$ are bounded in terms of $m$ and $d$ alone.

4.1.3. Morphisms.

A morphism from a variety $V_{1}\subset\mathbb{A}^{m}$ to a variety $V_{2}\subset\mathbb{A}^{n}$ is simply a map $f:V_{1}\to V_{2}$ of the form

[TABLE]

where $P_{1},\dots,P_{n}$ are polynomials. It is clear that the preimage $f^{-1}(W)$ of a subvariety $W\subset V_{2}$ is a subvariety of $V_{1}$.

What is not at all evident a priori is that, for $W\subset V_{1}$ a subvariety, the image $\phi(W)$ is a constructible set, meaning a finite union of terms of the form $W\setminus W^{\prime}$, where $W$ and $W^{\prime}\subset W$ are varieties. (For instance, if $V\subset\mathbb{A}^{2}$ is the variety given by $x_{1}x_{2}=1$ (a hyperbola), then its image under the morphism $\phi(x_{1},x_{2})=x_{1}$ is the constructible set $\mathbb{A}^{1}\setminus\{0\}$.) This result is due to Chevalley [Mum99, Cor. I.8.2].222As R. Vakil says of the closely related statement that the image of a projective variety under a morphism is a projective variety: “a great deal of classical algebra and geometry is contained in this theorem as special cases.” In model-theoretical terms, we are talking of quantifier elimination.

Let $V$ be irreducible and let $f:V\to\mathbb{A}^{n}$ be a morphism. It is easy to see that the Zariski closure $\overline{f(V)}$ must be irreducible, and that $\dim\overline{f(V)}\leq\dim V$. Let $d=\dim V-\overline{f(V)}$. Then there is a Zariski open subset $U\subset\overline{f(V)}$ such that, for every $x\in U$, the preimage $f^{-1}(\{x\})$ is a pure-dimensional variety of dimension $d$ [Mum99, Thm. I.8.3].

It is easy to see (by Bézout (4.1)) that the degree of $f^{-1}(\{x\})$ is bounded in terms of $\deg(V)$, $n$ and the degrees of the polynomials $P_{1},\dotsc,P_{n}$ defining $f$. If $\dim V=\overline{f(V)}$, $f^{-1}(\{x\})$ is [math]-dimensional, and so its number of points is bounded by its degree, by the definition of degree.

4.1.4. Tangent spaces and derivatives

Let $V\subset\mathbb{A}^{n}$ be a variety of dimension $d$ defined by equations $P_{i}(x_{1},\dotsc,x_{n})=0$, $1\leq i\leq k$. The tangent space $T_{x}V$ of $V$ at $x$ is the kernel of the linear map from $\mathbb{A}^{n}$ to $\mathbb{A}^{k}$ given by the matrix $\mathscr{P}|_{x}=\left(\partial P_{i}/\partial x_{j}\right)_{1\leq i\leq k,1\leq j\leq n}$. (These are formal partial derivatives.) A point $x$ on $V$ is non-singular if $\dim T_{x}V=\dim V$, and singular otherwise. The set of singular points is a proper subvariety of $V$ [Har77, Thm. I.5.3].

Let $V\subset\mathbb{A}^{n}$, $W\subset\mathbb{A}^{m}$ be varieties and let $f:V\to W$ be a morphism. At any point $\vec{x}$ on $V$, the linear map given by the matrix $J|_{x}=\left(\frac{\partial f_{i}}{\partial x_{j}}\right)_{1\leq i\leq m,1\leq j\leq n}$ restricts to a linear map $Df|_{x}:T_{x}V\to T_{x}W$ (as follows from the chain rule). For any $r\geq 0$, the set of non-singular points on $V$ such that the rank of $Df|_{x}$ is at least $r$ is Zariski-open in $V$. This fact is easy to see for $V=\mathbb{A}^{n}$: the rank is then $<r$ if and only if every $r$-by-$r$ minor of $J|_{x}$ is [math], a condition that defines a subvariety. For $V$ general, define a new matrix by putting the matrix $\mathscr{P}|_{x}$ on top of the matrix $J|_{x}$, and note that the new matrix will have rank at least $n-\dim(V)+r$ if and only if $Df|_{x}$ has rank at least $r$; thus we can proceed as for $V=\mathbb{A}^{n}$.

Exercise 4.1**.**

Let $V$, $W$ be varieties, $V$ irreducible, $f:V\to W$ a morphism, and $x$ a non-singular point on $V$. Prove that, if the rank of $Df|_{x}$ is at least $r$, then the dimension of $\overline{f(V)}$ is at least $r$.

4.1.5. Linear algebraic groups

A linear algebraic group over a field $K$ is a subvariety $G$ of $\operatorname{GL}_{n}$, defined over $K$, that is closed under multiplication and inversion.333Alternatively, we could define a linear algebraic group $G$ to be an affine variety with two morphisms $\cdot:G\times G\mapsto G$ and ${}^{-1}:G\to G$ satisfying the usual rules, and then prove that $G$ is isomorphic to a subvariety of $\operatorname{GL}_{n}$ with the multiplication and inversion morphisms it inherits from $\operatorname{GL}_{n}$ [Bor91, Prop. 1.10]. We thus have morphisms $\cdot:G\times G\mapsto G$ and ${}^{-1}:G\to G$. An algebraic or closed subgroup of $G$ is a subvariety $H$ of $G$ that is also closed under multiplication and inversion.

We will assume that the field of definition $K$ is perfect, meaning that every finite extension of $k$ is separable; this assumption will save us from possible trouble. Finite fields, fields of characteristic [math] and algebraically closed fields are always perfect fields.

A linear algebraic group $G$ is semisimple if it has no connected, non-trivial and solvable normal algebraic subgroups, even defined over $\overline{K}$. (“Connected” means “connected in the Zariski topology; an algebraic group is connected if and only if it is irreducible [Spr98, Prop. 2.2.1]. For algebraic groups, being solvable is defined analogously as for groups [Bor91, §2.4].) We say $G$ is simple (over $K$) if it is semisimple, connected and has no connected, proper and non-trivial normal algebraic subgroups defined over $K$.444Some sources (e.g., [Bor91, §22.8]) give the name almost-simple to what we call simple.

Let $G$ be an arbitrary linear algebraic group over a field $K$. An element $g\in G(\overline{K})$. is semisimple if it is diagonalizable over $\overline{K}$. Note that, by [Bor91, §4.3, Prop.] and the first definition in [Bor91, §4.5], the semisimplicity of $g$ is invariant under isomorphisms of $G$, i.e., it does not actually depend on the embedding of $G$ into $\operatorname{GL}_{n}$.

A torus $T<\operatorname{GL}_{n}$ is an algebraic group isomorphic to $\operatorname{GL}_{1}^{r}$ over $\overline{K}$ for some $r\geq 1$. A torus defined over $K$ is always diagonalizable over $\overline{K}$ [Bor91, §8.5, Prop.]; that is, there exists $g\in\operatorname{GL}_{n}(\overline{K})$ such that $gTg^{-1}$ is a subgroup of the group of diagonal matrices in $\operatorname{GL}_{n}$. A maximal torus of a connected linear algebraic group $G$ is a torus $T<G$ with $r$ maximal. We call $r$ the rank of $G$. If $G$ is connected, then every semisimple $g\in G(\overline{K})$ lies in a maximal torus [Spr98, Thm. 6.4.5(ii)].

The centralizer $C(g)$ of a semisimple point $g$ in $G$ has dimension at least $r=\operatorname{rank}(G)$; if $\dim C(g)=\operatorname{rank}(G)$, we say $g$ is regular. When $G$ is semisimple, a semisimple element $g\in G(\overline{K})$ is regular if and only if the connected component $C(g)^{\circ}$ of $C(g)$ containing the identity is a maximal torus ([Bor91, §12.2, Prop., and §13.17, Cor. 2(c)]). A regular semisimple element $g\in G(\overline{K})$ lies in exactly one maximal torus [Bor91, §12.2, Prop.]. For $G$ semisimple, regular semisimple elements form a non-empty open subset of $G$ [Ste65, §2.14].

4.1.6. Lie algebras

A Lie algebra is a vector space $\mathfrak{g}$ over a field $K$ together with a bilinear map $[\cdot,\cdot]:\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}$ satisfying the identities

[TABLE]

An ideal of a Lie algebra is a subspace $\mathfrak{v}\subsetneq\mathfrak{g}$ such that $[g,\mathfrak{v}]\subset\mathfrak{v}$. We say a Lie algebra is simple if it has no ideals other than $(0)$.

A linear algebraic group $G$ acts on its tangent space $\mathfrak{g}=T_{e}G$ at the origin by conjugation: for $g\in G$, we define the linear map $\operatorname{Ad}_{g}:\mathfrak{g}\to\mathfrak{g}$ to be the derivative of $y\mapsto gyg^{-1}$. The derivative of $\operatorname{Ad}_{g}$ with respect to $g$ can be written as a bilinear map $\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}$, which we call $[\cdot,\cdot]$; it is fairly straightforward to check that it satisfies the identities in (4.2), and thus makes $\mathfrak{g}$ into a Lie algebra.

It is easy to see that, if a subspace $\mathfrak{v}$ of the Lie algebra $\mathfrak{g}$ of a linear algebraic group $G$ is invariant under $\operatorname{Ad}_{g}$ for every $g\in G$, then $\mathfrak{v}$ is an ideal. Thus, if $G$ is not simple, then $\mathfrak{g}$ is not simple.

It would be convenient if $G$ simple implied $\mathfrak{g}$ simple, but that is not quite true555To the contrary of what was carelessly stated in the proof of Prop. 5.3 in the survey [Hel15].. However, there are only a few exceptions, all in small characteristic. To summarize: for $G=\operatorname{SL}_{n}$, the Lie algebra $\mathfrak{g}=\mathfrak{sl}_{n}$ is simple provided that the characteristic $p$ of the field $K$ does not divide $n$. (If $p|n$, then $\mathfrak{sl}_{n}$ has non-trivial center, namely, the multiples of the diagonal matrix $I$.) For almost simple Lie groups $G$ such that $\mathfrak{g}$ is not isomorphic to $\mathfrak{sl}_{n}$, we have that $\mathfrak{g}$ is simple provided that $\operatorname{char}(K)>3$ [Hog82, Table 1]. (The assumption in [Hog82] that the ground field is algebraically closed is harmless, as, if $\mathfrak{g}$ is simple over $\overline{K}$, it follows trivially that $\mathfrak{g}$ is simple over $K$: a decomposition over $K$ would also be valid over $\overline{K}$.) In fact, $\operatorname{char}(K)>2$ is enough for all Lie algebras of type other than $A_{n}$ (corresponding to $\operatorname{SL}_{n}$), $E_{6}$ and $G_{2}$, by the same table.

In spite of this small-characteristic phenomenon, we will nevertheless descend from the algebraic groups to Lie algebra at an important step (proof of Lemma 4.6), as then matters arguably become particularly clear and straightforward.

4.1.7. Finite groups of Lie type

The general definition of a finite group of Lie type is that it is the group $G^{F}$ of points on a semisimple algebraic group $G$ defined over a finite field $\mathbb{F}_{q}$ that are left fixed by a Steinberg endomorphism $F:G\to G$. A Steinberg endomorphism is an endomorphism $F:G\to G$ such that, for some $m\geq 1$, $F^{m}$ is the Frobenius map with respect to $\mathbb{F}_{q}$. The Frobenius map with respect to $\mathbb{F}_{q}$ is the map sending every element $g\in G(\overline{\mathbb{F}_{q}})$ with entries $g_{i,j}$ to the element with entries $g_{i,j}^{q}$. It fixes precisely the elements of $G(\mathbb{F}_{q})$.

The most familiar finite groups of Lie type (classical groups and Chevalley groups) are of the form $G(\mathbb{F}_{q})$, $G$ a semisimple algebraic group; they correspond to the case $m=1$. The groups that require $m>1$ are called twisted groups.

We will work out growth in $G(K)$, $G=\operatorname{SL}_{2}$, $K$ finite (or, more generally, perfect) in a way that generalizes easily to other groups of Lie type with $G$ simple. It is possible to include twisted groups, as was shown in [PS16]; however, our notation will be of the form $G(K)$, as is appropriate for $m=1$.

Requiring $G$ to be simple is not quite the same as requiring the group of Lie type $G^{F}=G(K)$ to be simple. The simple groups coming from groups of Lie type are of the form $G^{F}/Z(G^{F})$, $G$ simple.666Two comments for the sake of precision are in order. (a) There is one group in the classification of finite simple groups that is almost but not quite of the type $G^{F}/Z(G^{F})$: the Tits group [MT11, p. 213]. As we said before, we need not care about individual groups in the classification, since we aim at asymptotic statements. (b) By a result of Tits [MT11, Thm. 24.17], given $G$ simple and simply connected [MT11, Def. 9.14], the group $G^{F}/Z(G^{F})$ will be simple, provided we are not in a finite list of exceptions. Notably, $\operatorname{SO}_{n}$ is not simply connected; one uses a simply-connected finite cover of $\operatorname{SO}_{n}$ in its stead. The center $Z(G^{F})$ is described in [MT11, Table 24.2]. It is very easy to pass from statements on growth in $G^{F}$ to statements on growth in $G^{F}/Z(G^{F})$, as we will see in the case for $G=\operatorname{SL}_{2}$, where $Z(G^{F})=\{I,-I\}$.

4.2. Escape from subvarieties

We are working with a finite subset $A$ of a group $G$. At some points in the argument, we will need to make sure that we can find an element $g\in A^{k}$ ($k$ small) that is not special: for example, we want to be able to use a $g$ that is not unipotent, that does not have a given $\vec{v}$ as an eigenvector, that is regular semisimple, etc.

It is possible to give a completely general argument of this form. Let us first set the framework. Let $G$ be a group acting by linear transformations on $n$-dimensional space $\mathbb{A}^{n}$ over a field $K$. In other words, we are given a homomorphism $\phi:G\to\operatorname{GL}_{n}(K)$ from $G$ to the group of invertible matrices $\operatorname{GL}_{n}(K)$. Let $W$ be a proper subvariety of $\mathbb{A}^{n}$. We may think of points on $W$ as being special, and points outside $W$ as being generic. We start with a point $x$ of $\mathbb{A}^{n}$, and a subset $A$ of $G$. The following proposition ensures us that, if, starting from $x$ and acting on it repeatedly by $A$, we can eventually escape from $W$, then we can escape from it in a bounded number of steps, and in many ways.

The proof777The statement of the proposition is as in [Hel11], based closely on [EMO05], but the idea is probably older. proceeds by induction on the dimension, with the degree kept under control. What is crucial for us is that the dimension is an integer, and thus can be used as a counter for induction. (Alternatively, we could say that the kind of induction we are about to undertake works because the ring $K[x_{1},\dotsc,x_{n}]$ is Noetherian.)

Proposition 4.2**.**

Let us be given

• •

$G$* a group acting linearly on affine space $\mathbb{A}^{n}$ over a field $K$,*

• •

$W\subsetneq\mathbb{A}^{n}$, a subvariety,

• •

$A$* a set of generators of $G$ with $A=A^{-1}$, $e\in A$,*

• •

$x\in\mathbb{A}^{n}$* such that the orbit $G\cdot x$ of $x$ is not contained in $W$.*

Then there are constants $k$, $c$ depending only the number, dimension and degree of the irreducible components of $W$ such that there are at least $\max(1,c|A|)$ elements $g\in A^{k}$ for which $gx\notin W(K)$.

Proof for a special case.

Let us first do the special case of $W$ an irreducible linear subvariety. We will proceed by induction on the dimension of $W$. If $\dim(W)=0$, then $W$ consists of a single point, and the statement is clear: since $G\cdot x\not\subset\{x\}$ and $A$ generates $G$, there exists a $g\in A$ such that $gx\neq x$; if there are fewer than $|A|/2$ such elements of $A$, we let $g_{0}$ be one of them, and note that any product $g^{-1}g_{0}$ with $gx=x$ satisfies $g^{-1}g_{0}x\neq x$; there are $>|A|/2$ such products.

Assume, then, that $\dim(W)>0$, and that the statement has been proven for all $W^{\prime}$ with $\dim(W^{\prime})<\dim(W)$. If $gW=W$ for all $g\in A$, then either (a) $gx$ does not lie on $W$ for any $g\in A$, proving the statement, or (b) $gx$ lies on $W$ for every $g\in G=\langle A\rangle$, contradicting the assumption. Assume that $gW\neq W$ for some $g\in A$; then $W^{\prime}=gW\cap W$ is an irreducible linear variety with $\dim(W^{\prime})<\dim(W)$. Thus, by the inductive hypothesis, there are at least $\max(1,c^{\prime}|A|)$ elements $g^{\prime}\in A^{k^{\prime}}$ ($c^{\prime}$, $k^{\prime}$ depending only on $\dim(W^{\prime})$) such that $g^{\prime}x$ does not lie on $W^{\prime}=gW\cap W$. Hence, for each such $g^{\prime}$, either $g^{-1}g^{\prime}x$ or $g^{\prime}x$ does not lie on $W$. We have thus proven the statement with $c=c^{\prime}/2$, $k=k^{\prime}+1$. ∎

Exercise 4.3**.**

Generalize the proof so that it works without the assumptions that $W$ be linear or irreducible. Sketch: work first towards removing the assumption of irreducibility. Let $W$ be the union of $r$ components, not necessarily all of the same dimension. The intersection $W^{\prime}=gW\cap W$ may also have several components, but no more than $r^{2}$; this is what we meant by “keeping the degree under control”. Now pay attention to $d$, the maximum of the dimensions of the components of a variety, and $m$, the number of components of maximal dimension. Show that either (1) $d$ is lower for $W^{\prime}=gW\cap W$ than for $W$, or (2) $d$ is the same in both cases, but $m$ is lower for $W^{\prime}$ than for $W$, or (3) $x$ does not lie in any component of $W$ of dimension $d$, and thus we may work instead with $W$ with those components removed. Use this fact to carry out the inductive process.

Now note that you never really used the fact that $W$ is linear. Instead of keeping track of the number of components $r$, keep track of the sum of their degrees. Control that using the generalized form (4.1) of Bézout’s theorem.

4.3. Dimensional estimates

By a dimensional estimate we mean a lower or upper bound on an intersection of the form $A^{k}\cap V$, where $A\subset G(K)$, $V$ is a subvariety of $G$ and $G/K$ is an algebraic group. As you will notice, the bounds that we obtain will be meaningful when $A$ grows relatively slowly. However, no assumption on $A$ is made, other than that it generate $G(K)$.

Of course, Proposition 4.2 may already be seen as a dimensional estimate of sorts, in that it tells us that $\gg|A|$ elements of $A^{k}$, $k$ bounded, lie outside $W$. We are now aiming at much stronger bounds; Proposition 4.2 will be a useful tool along the way.

We aim for the estimates whose most general form is as follows.

Theorem 4.4**.**

Let $G<\operatorname{GL}_{n}$ be a simple linear algebraic group over a finite field $K$. Let $A\subset G(K)$ be a finite set of generators of $G(K)$. Assume $A=A^{-1}$, $e\in A$. Let $V$ be a pure-dimensional subvariety of $G$. Then

[TABLE]

where $k$ and the implied constant depend only on $n$ and on $\deg(V)$.

Estimates of this form can be traced in part to [LP11] ($A$ a subgroup, $V$ general) and in part to [Hel08] y [Hel11] ($A$ an arbitrary set, but $V$ special). We now have Theorem 4.4, thanks to [BGT11] and [PS16]. In fact, [PS16] gives a more general statement, in that twisted groups of Lie type are covered. Actually, one can state Theorem 4.4 in an even more general form, in that the assumption that $K$ is finite can be dropped, and the condition that $A$ generate $G(K)$ can be replaced by a condition that $\langle A\rangle$ be “Zariski-dense enough”, meaning not contained in a union of $\leq C$ varieties of degree $\leq C$, where $C$ depends only on $n$ and $\deg(V)$.

We will show how to prove the estimate (4.3) in the case we actually need, but in a way that can be generalized to arbitrary $V$ and arbitrary simple $G$. We will give a detailed outline of how to obtain the generalization.

Actually, as a first step towards the general strategy, let us study a particular $V$ that we will not use in the end; it was crucial in earlier versions of the proof, and, more importantly, it makes several of the key ideas clear quickly. The proof is basically the same as in [Hel08, §4]. In particular, it will not look as if we used any algebraic geometry; however, the concrete procedure we follow here will then lead us naturally to a general procedure that will ask for the language and the basic tools of algebraic geometry.

Lemma 4.5**.**

Let $G=\operatorname{SL}_{2}$, $K$ a field. Let $A\subset G(K)$ be a finite set of generators of $G(K)$. Assume $A=A^{-1}$, $e\in A$. Let $T$ be a maximal torus of $G$. Then

[TABLE]

where $k$ and the implied constant are absolute.

Proof.

We can assume without loss of generality that $|K|$ and $|A|$ are greater than a constant, as otherwise the statement is trivial. We can also write the elements of $T$ as diagonal matrices, by conjugation by an element of $\operatorname{SL}_{2}(\overline{K})$.

Let

[TABLE]

be any element of $\operatorname{SL}_{2}(\overline{K})$ with $abcd\neq 0$. Consider the map $\phi:T(K)\times T(K)\times T(K)\to G(K)$ given by

[TABLE]

We would like to show that this map is in some sense almost injective. (What for? If the map were injective, and we had $g\in A^{\ell}$, $\ell$ bounded by a constant, we would have

[TABLE]

which would imply immediately the result we are trying to prove. Here we are simply using the fact that the image $\phi(D)$ of an injection $\phi$ has the same number of elements as the domain $D$.)

Multiplying matrices, we see that, for

[TABLE]

$\phi((x,y,z))$ equals

[TABLE]

Let $s\in\overline{K}$ be such that $s^{-1}-s\neq 0$ and $sad-s^{-1}bc\neq 0$. A brief calculation shows then that $\phi^{-1}(\{\phi((x,y,z))\})$ has at most $16$ elements: we have

[TABLE]

and, since $abcd\neq 0$, at most $4$ values of $s$ can give the same value $-(s-s^{-1})^{2}abcd$ (the product of the top right and bottom left entries of ((4.6)); for each such value of $s$, the product and the quotient of the upper left and upper right entries of (4.6) determine $r^{2}$ and $t^{2}$, respectively, and obviously there are at most $2$ values of $r$ and $2$ values of $t$ for $r^{2}$, $t^{2}$ given.

Now, there are at most $4$ values of $s$ such that $s^{-1}-s=0$ or $sad-s^{-1}bc=0$. Hence,

[TABLE]

and, at the same time, $\phi(A\cap T(K),A\cap T(K),A\cap T(K))\subset AA^{\ell}AA^{-\ell}A=A^{3+2\ell}$, as we said before. If $|A\cap T(K)|$ is less than $8$ (or any other constant), conclusion (4.4) is trivial. Therefore,

[TABLE]

i.e., (4.4) holds.

It only remains to verify that there exists an element (4.5) of $A^{\ell}$ with $abcd\neq 0$. Now, $abcd=0$ defines a subvariety $W$ of $\mathbb{A}^{4}$, where $\mathbb{A}^{4}$ is identified with the space of $2$-by-$2$ matrices. Moreover, for $|K|>2$, there are elements of $G(K)$ outside that variety. Hence, the conditions of Prop. 4.2 hold (with $x=e$). Thus, we obtain that there is a $g\in A^{\ell}$ ($\ell$ a constant) such that $g\not\in W(K)$, and that was what we needed. ∎

Let us abstract the essence of what we have just done, so that we can then generalize the result to an arbitrary variety $V$ instead of working just with $T$. For the sake of convenience, we will do the case $\dim V=1$, which is, at any rate, the case we will need. The strategy of the proof of Lemma 4.5 is to construct a morphism $\phi:V\times V\times\dotsb\times V\to G$ ($r$ copies of $V$, where $r=\dim(G)$) of the form

[TABLE]

where $g_{1},g_{2},\dotsc,g_{r-1}\in A^{\ell}$, in such a way that, for $v=(v_{1},\dotsc,v_{r})$ a generic point in $V\times V\times\dotsb\times V$, the preimage $\phi^{-1}(\{\phi(v)\})$ has dimension [math]. Actually, as we have just seen, it is enough to prove that this is true for $(g_{1},g_{2},\dotsc,g_{r-1})$ a generic element of $G^{r-1}$; the escape argument (Prop. 4.2) takes care of the rest.

The following lemma is the same as [Tao15, Prop. 5.5.3], which, in turn, is the same as [LP11, Lemma 4.5]. We will give a proof valid for $\mathfrak{g}$ simple.

Lemma 4.6**.**

Let $G<\operatorname{SL}_{n}$ be a simple algebraic group defined over a field $K$. Let $V,V^{\prime}\subsetneq G$ be irreducible subvarieties with $\dim(V)<\dim(G)$ and $\dim(V^{\prime})>0$. Then, for every $g\in G(\overline{K})$ outside a subvariety $W\subsetneq G$ depending on $V$ and $V^{\prime}$, the variety $\overline{VgV^{\prime}}$ has dimension $>\dim(V)$.

Moreover, the number and degrees of the irreducible components of $W$ are bounded by a constant that depends only on $n$ and $\deg(V)$ and $\deg(V^{\prime})$.

In fact, the proof we will now see bounds the number and degrees of the components of $W$ in terms of $n$ alone.

Proof for $\mathfrak{g}$ simple.

We can assume without loss of generality – replacing $V$ and $V^{\prime}$ by varieties $Vh$ and $h^{\prime}V$, $h,h^{\prime}\in G(\overline{K})$, if necessary – that $V$ and $V^{\prime}$ go through the origin, and that the origin is a non-singular point for $V$ and $V^{\prime}$. We may also assume without loss of generality that $K$ is algebraically closed.

Let $\mathfrak{v}$ and $\mathfrak{v}^{\prime}$ be the tangent spaces to $V$ and $V^{\prime}$ at the origin. The tangent space to $\overline{VgV^{\prime}}g^{-1}$ at the identity is $\mathfrak{v}+\operatorname{Ad}_{g}\mathfrak{v}^{\prime}$. Thus, for $\overline{V^{\prime}gV}$ to have dimension $>\dim(V)$, it is enough that $\mathfrak{v}+\operatorname{Ad}_{g}\mathfrak{v}^{\prime}$ have dimension $>\dim(\mathfrak{v})=\dim(V)$.

Suppose that this is not the case for any $g$ on $G$. Then the space $\mathfrak{w}$ spanned by all spaces $\operatorname{Ad}_{g}\mathfrak{v}^{\prime}$, for all $g$, is contained in $\mathfrak{v}$. Since $\dim(V)<\dim(G)$, $\mathfrak{v}\subsetneq\mathfrak{g}$. Clearly, $\mathfrak{w}$ is non-empty and invariant under $\operatorname{Ad}_{g}$ for every $g$. Hence it is an ideal. However, we are assuming $\mathfrak{g}$ to be simple. Contradiction.

Thus, $\mathfrak{v}+\operatorname{Ad}_{g}\mathfrak{v}^{\prime}$ has dimension greater than $\dim(\mathfrak{v})$ for some $g$. It is easy to see that the points $g$ where that is not the case are precisely those such that all $(\dim(\mathfrak{v})+1)\times(\dim(\mathfrak{v})+1)$ minors of a matrix – whose entries are polynomial on the entries of $g$ – vanish. We let $W$ be the subvariety of $V$ where those minors all vanish. The claim on the number and degrees of components of $W$ follows by Bézout (4.1). ∎

We can now generalize our proof of Lemma 4.5, and thus prove (4.3) for all varieties of dimension $1$. Before we start, we need a basic counting lemma, left as an exercise.

Exercise 4.7**.**

*Let $W\subset\mathbb{A}^{n}$ be a variety defined over $K$ such that every component of $W$ has dimension $\leq d$. Let $S$ be a finite subset of $K$. Then the number of points $(x_{1},\dotsc,x_{n})\in S\times S\times\dotsb\times S$ ($n$ times) lying on $W$ is $\ll|S|^{d}$, where the implied constant depends only on $n$ and on the number and degrees of the components of $W$. *

Proposition 4.8**.**

Let $G\subset\operatorname{SL}_{n}$ be an simple algebraic group over a finite field $K$. Assume that $|G(K)|\geq c|K|^{\dim(G)}$, $c>0$. Let $Z\subset G$ be a variety of dimension $1$. Let $A\subset G(K)$ be a set of generators of $G(K)$ such that $A=A^{-1}$, $e\in A$.

Then

[TABLE]

where $k$ and the implied constant depend only on $n$, $c$, $\deg(G)$ and the number and degrees of the irreducible components of $Z$.

Obviously, $G=\operatorname{SL}_{n}$ is a valid choice, since it is simple and $|\operatorname{SL}_{n}(K)|\gg|K|^{n^{2}-1}=|K|^{\dim(G)}$.

Proof.

We will use Lemma 4.6 repeatedly. When we apply it, we get a subvariety $W\subsetneq G$ such that, for every $g$ outside $W$, some component of $\overline{VgV^{\prime}}$ has dimension $>\dim(V)$ (where $V$ and $V^{\prime}$ are varieties satisfying the conditions of Lemma 4.6). Since $G$ is irreducible, every component of $W$ has dimension less than $\dim(G)$. By Exercise 4.7 (with $S=K$) and the assumption $|G(K)|\geq c|K|^{\dim(G)}$, there is at least one point of $G(K)$ not on $W$, provided that $|K|$ is larger than a constant, as we can indeed assume. Hence, we can use escape from subvarieties (Prop. 4.2) to show that there is a $g\in(A\cup A^{-1}\cup\{e\})^{\ell}$, where $\ell$ depends only on the number and degrees of components of $W$, that is to say – by Lemma 4.6 – only on $n$ and $\deg(G)$.

So: first, we apply Lemma 4.6 with $V=V^{\prime}=Z$; we obtain a variety $V_{2}=\overline{Vg_{1}V^{\prime}}=\overline{Zg_{1}Z}$ with $g\in(A\cup A^{-1}\cup\{e\})^{\ell}$ such that $V_{2}$ has at least one component of dimension $2$. (We might as well assume $V$ is irreducible from now on; then $V_{2}$ is irreducible.) We apply Lemma 4.6 again with $V=V_{2}$, $V^{\prime}=Z$, and obtain a variety $V_{3}=\overline{V_{2}g_{2}Z}=\overline{Zg_{1}Zg_{2}Z}$ of dimension $3$. We go on and on, and get that there are $g_{1},\dotsc,g_{m-1}\in(A\cup A^{-1}\cup\{e\})^{\ell^{\prime}}$, $r=\dim(G)$, such that $\overline{Zg_{1}Zg_{2}\dotsc Zg_{r-1}Z}$ has dimension $r$.

Hence, the variety $W$ of singular points of the map $f$ from $Z^{r}=Z\times Z\times\dotsb\times Z$ ($r$ times) to $G$ given by

[TABLE]

cannot be all of $Z\times\dotsc\times Z$. Thus, since $Z\times\dotsc\times Z$ is irreducible, every component of $W$ is of dimension less than $\dim V$. Again by Exercise 4.7 (with $S=A\cap Z(K)$), at most $O(|A\cap Z(K)|^{r-1})$ points of $(A\cap Z(K))\times\dotsb\times(A\cap Z(K))$ ($r$ times) on $W$. The number of points of $(A\cap Z(K))\times\dotsb\times(A\cap Z(K))$ not on $W$ is at most the degree of $f$ times the number of points on $f(A\cap Z(K),\dotsc,A\cap Z(K))$, which is contained in $A^{k}$ for $k={r+(r-1)\ell^{\prime}}$. Therefore,

[TABLE]

and so we are done. ∎

In general, one can prove (4.3) for $\dim(V)$ arbitrary using very similar arguments, together with an induction on the dimension of the variety $V$ in (4.3). We will demonstrate the basic procedure doing things in detail for $G=\operatorname{SL}_{2}$ and for the kind of variety $V$ for which we really need to prove estimates.

We mean the variety $V_{t}$ defined by

[TABLE]

for $t\neq\pm 2$. Such varieties are of interest to us because, for any regular semisimple $g\in\operatorname{SL}_{2}(K)$ (meaning: any matrix in $\operatorname{SL}_{2}(K)$ having two distinct eigenvalues), the conjugacy class $\operatorname{Cl}(g)$ is contained in $V_{\operatorname{tr}(g)}$.

Proposition 4.9**.**

Let $K$ be a finite field. Let $A\subset\operatorname{SL}_{2}(K)$ be a set of generators of $\operatorname{SL}_{2}(K)$ with $A=A^{-1}$, $e\in A$. Let $V_{t}$ be given by (4.9).

Then, for every $t\in K$ other than $\pm 2$,

[TABLE]

where $k$ and the implied constant are absolute.

Needless to say, $\dim(\operatorname{SL}_{2})=3$ and $\dim(V_{t})=2$, so this is a special case of (4.3).

Proof.

Consider the map $\phi:V_{t}(K)\times V_{t}(K)\to\operatorname{SL}_{2}(K)$ given by

[TABLE]

It is clear that

[TABLE]

Thus, if $\phi$ were injective, we would obtain immediately that $|A\cap V_{t}(K)|^{2}\leq|A^{2}|$. Now, $\phi$ is not injective, not even nearly so. The preimage of $\{h\}$, $h\in\operatorname{SL}_{2}(K)$, is

[TABLE]

We should thus ask ourselves how many elements of $A$ lie on the subvariety $Z_{t,h}$ of $G$ defined by

[TABLE]

For $h\neq\pm e$, $\dim(Z_{t,h})=1$, and the number and degrees of irreducible components of $Z_{t,h}$ are bounded by an absolute constant. Thus, applying Proposition 4.8, we get that, for $h\neq\pm e$,

[TABLE]

where $k^{\prime}$ and the implied constant are absolute.

Now, for every $y_{1}\in V_{t}(K)$, there are at least $|V_{t}(K)|-2$ elements $y_{2}\in V_{t}(K)$ such that $y_{1}y_{2}^{-1}\neq\pm e$. We conclude that

[TABLE]

We can assume that $|A\cap V(K)|\geq 3$, as otherwise the desired conclusion is trivial. We obtain, then, that

[TABLE]

for $k=\max(2,k^{\prime})$, as we wanted. ∎

Now we can finally prove the result we needed.

Corollary 4.10**.**

Let $G=\operatorname{SL}_{2}$, $K$ a finite field. Let $A$ be a set of generators of $G(K)$ with $A=A^{-1}$, $e\in A$. Let $g\in A^{\ell}$ ($\ell\geq 1$) be regular semisimple. Then

[TABLE]

where $k$ and the implied constant are absolute.

In particular, if $|A^{3}|\leq|A|^{1+\delta}$, then

[TABLE]

Proof.

Proposition 4.9 and Lemma 2.6 imply (4.11) immediately, and (4.12) follows readily from (4.11) via (2.4). ∎

Let us now see two problems whose statements we will not use; they are, however, essential if one wishes to work in $\operatorname{SL}_{n}$ for $n$ arbitrary, or in an arbitrary simple algebraic group. The first problem is challenging, but we have already seen and applied the main ideas involved in its solution. In essence, it is a matter of setting up a recursion properly.

Exercise 4.11**.**

Generalize Proposition 4.8 to pure-dimensional varieties $Z$ of arbitrary dimension; that is, prove Theorem 4.4.

The following exercise is easy. In part (b), follow the proof of Corollary 4.10, using Exercise 4.11.

Exercise 4.12**.**

Let $G$ be a simple algebraic group over a finite field $K$. Let $A\subset G(K)$, $A=A^{-1}$, $e\in A$, $\langle A\rangle=G(K)$. Let $g\in A^{\ell}$, $\ell\geq 1$.

1. (a)

Using the material in §4.1.3, show that $\dim G-\dim\overline{\operatorname{Cl}(g)}=\dim C(g)$. 2. (b)

Show that, if $|A|^{3}\leq|A|^{1+\delta}$,

[TABLE]

where the implied constants depend only on $n$.

If $g$ is regular semisimple, then, as we know, $C(g)$ is a maximal torus.

5. Growth and diameter in $\operatorname{SL}_{2}(K)$

5.1. Growth in $\operatorname{SL}_{2}(K)$, $K$ arbitrary

We come to the proof of our main result. Here we will be closer to newer treatments (in particular, [PS16]) than to what was the first proof, given in [Hel08]; these newer versions generalize more easily. We will give the proof only for $\operatorname{SL}_{2}$, and point out the couple of places in the proof where one would has to be especially careful when generalizing matters to $\operatorname{SL}_{n}$, $n>2$, or other linear algebraic groups.

The proof in [Hel08] used the sum-product theorem (Thm. 3.8). We will not use it, but the idea of “pivoting” will reappear. It is also good to note that, just as before, there is an inductive process here, carried out on a group $G$, even though $G$ does not have a natural order ($1,2,3,\dotsc$). All we need for the induction to work is a set of generators $A$ of $G$.

Theorem 5.1** (Helfgott [Hel08]).**

Let $K$ be a finite field. Let $A\subset\operatorname{SL}_{2}(K)$ be a set of generators of $\operatorname{SL}_{2}(K)$ with $A=A^{-1}$, $e\in A$. There either

[TABLE]

where $\delta>0$ is an absolute constant, or

[TABLE]

Actually, [Hel08] proved this result (with $A^{k}$, $k$ a constant, instead of $A^{3}$ in (5.2)) for $K=\mathbb{F}_{p}$; the first generalization to a general finite field $K$ was given by [Din11]. The proof we are about to see works for $K$ general without any extra effort. It works, incidentally, for $K$ infinite as well, dropping the condition $|A|<|\operatorname{SL}_{2}(K)|^{1-\epsilon}$, which becomes trivially true. The case of characteristic [math] is actually easier than the case $K=\mathbb{F}_{p}$; the proof in [Hel08] was already valid for $K=\mathbb{R}$ or $K=\mathbb{C}$, say. However, for applications, the “right” result for $K=\mathbb{R}$ or $K=\mathbb{C}$ is not really Thm. 5.1, but a statement counting how many elements there can be in $A$ and $A\cdot A\cdot A$ that are separated by a given small distance from each other; that was proven in [BG08a], adapting the techniques in [Hel08].

Proof.

We may assume that $|A|$ is larger than an absolute constant, since otherwise the conclusion would be trivial. Let $G=\operatorname{SL}_{2}$.

Suppose that $|A^{3}|<|A|^{1+\delta}$, where $\delta>0$ is a small constant to be determined later. By escape (Prop. 4.2), there is an element $g_{0}\in A^{c}$ that is regular semisimple (that is, $\operatorname{tr}(g_{0})\neq\pm 2$), where $c$ is an absolute constant. (Easy exercise: show we can take $c=2$.) Its centralizer in $G(K)$ is $\mathbf{T}:=C(g)=T(\overline{K})\cap G(K)$ for some maximal torus $T$.

Call $\xi\in G(K)$ a pivot if the map $\phi_{\xi}:A\times\mathbf{T}\to G(K)$ defined by

[TABLE]

is injective as a function from $(\pm e\cdot A)/\{\pm e\}\times\mathbf{T}/\{\pm e\}$ to $G(K)/\{\pm e\}$.

Case (a): There is a pivot $\xi$ in $A$. By Corollary 4.10, there are $\gg|A|^{1/3-O(c\delta)}$ elements of $\mathbf{T}$ in $A^{-1}A$. Hence, by the injectivity of $\phi_{\xi}$,

[TABLE]

At the same time, $\phi_{\xi}(A,A^{2}\cap\mathbf{T})\subset A^{5}$, and thus

[TABLE]

For $|A|$ larger than a constant and $\delta>0$ less than a constant, this inequality gives us a contradiction with $|A^{3}|<|A|^{1+\delta}$ (by Ruzsa (2.3)).

Case (b): There are no pivots $\xi$ in $G(K)$. Then, for every $\xi\in G(K)$, there are $a_{1},a_{2}\in A$, $t_{1},t_{2}\in\mathbf{T}$, $(a_{1},t_{1})\neq(\pm a_{2},\pm t_{2})$ such that $a_{1}\xi t_{1}\xi^{-1}=\pm e\cdot a_{2}\xi t_{2}\xi^{-1}$, and that gives us that

[TABLE]

In other words, for each $\xi\in G(K)$, $A^{2}$ has a non-trivial intersection with the torus $\xi T\xi^{-1}$:

[TABLE]

(Note this means that case (b) never arises for $K$ infinite. Why?)

Choose any $g\in A^{2}\cap\xi\mathbf{T}\xi^{-1}$ with $g\neq\pm e$. Then $g$ is regular semisimple. (This fact is peculiar to $\operatorname{SL}_{2}$, or rather to groups of rank $1$. This is one place in the proof that requires some work when you generalize it to other groups.)

The centralizer $C(g)$ of $g$ equals $\xi\mathbf{T}\xi^{-1}$ (why?). Hence, by Corollary 4.10, we obtain that there are $\geq c^{\prime}|A|^{1/3-O(\delta)}$ elements of $\xi\mathbf{T}\xi^{-1}$ in $A^{2}$, where $c^{\prime}$ and the implied constant are absolute.

At least $(1/2)|G(K)|/|\mathbf{T}|$ maximal tori of $G$ are of the form $\xi T\xi^{-1}$, $\xi\in G(K)$ (check this yourself!). Every semisimple element of $G$ that is not $\pm e$ is regular (again, something peculiar to $\operatorname{SL}_{2}$); thus, every element of $G$ that is not $\pm e$ can lie on at most one maximal torus. Hence

[TABLE]

Therefore, either $|A^{2}|>|A|^{1+2\delta}$ (say) or $|A|\geq|G|^{1-O(\delta)}$. In the first case, we have obtained a contradiction. In the second case, Proposition 5.6 implies that $A^{3}=G$.

Case (c): There are pivots and non-pivots in $G(K)$. Since $\langle A\rangle=G(K)$, this implies that there exists a non-pivot $\xi\in G$ and an $a\in A$ such that $a\xi\in G$ is a pivot. Since $\xi$ is not a pivot, (5.4) holds, and thus there are $|A|^{1/3-O(\delta)}$ elements of $\xi\mathbf{T}\xi^{-1}$ in $A^{k}$.

At the same time, $a\xi$ is a pivot, i.e., the map $\phi_{a\xi}$ defined in (5.3) is injective (considered as an application from $A/\{\pm e\}\times\mathbf{T}/\{\pm e\}$ to $G(K)/\{\pm e\}$). Therefore,

[TABLE]

Since $\phi_{a\xi}(A,\xi^{-1}(A^{2}\cap\xi T\xi^{-1})\xi)\subset A^{5}$, we obtain that

[TABLE]

Thanks again to Ruzsa (2.3), this inequality contradicts $|A^{3}|\leq|A|^{1+\delta}$ for $\delta>0$ smaller than a constant. ∎

The following is a trivial exercise.

Exercise 5.2**.**

Using Theorem 5.1, show that the statement of Thm. 5.1 is also true with $\operatorname{PSL}_{2}$ in place of $\operatorname{SL}_{2}$. This step finishes the proof of Thm. 1.1.

For $\operatorname{SL}_{n}$, $n>2$, or for general algebraic groups, there is, as we have seen, one difficulty in generalizing the above proof: a semisimple element other than $\pm e$ is not necessarily regular. The key to circumventing this difficulty is to use Theorem 4.4 to bound the number of elements on non-maximal subtori of a maximal torus $T$, and, in that way, bound the number of non-semisimple elements of $A^{k}$ on $T$.

Exercise 5.3**.**

Using this observation, modify the proof of Thm. 5.1 so as to work for any simple linear algebraic group $G$.

There remains the question of what the optimal value of $\delta$ in Thm. 5.1 could be. Kowalski [Kow13] proves Thm. 5.1 with $\delta=1/3024$ (under the assumption $A=A^{-1}$). Button and Roney-Dougal prove (under the same assumption) that one cannot do better than $\delta=(\log_{2}7-1)/6\approx 0.3012$ [BRD15].

To obtain a good value of $\delta$, it seems best to aim for a statement with a conclusion of the form

[TABLE]

instead of (5.1). It may be even better to aim for a result of the form, say,

[TABLE]

where $A_{0}$ is an arbitrary set of generators of $\operatorname{SL}_{2}(K)$. Then, when using our result to prove a diameter bound (as in exercise 1.2), we can set $A_{0}$ to be our initial set of generators $S$, whereas we set $A$ equal to increasing powers of $S$. The resulting constant $C$ in the exponent of the bound $\operatorname{diam}\Gamma(G,S)\ll(\log|G|)^{C}$ should then improve substantially over the value $C=3323$ given in [Kow13].

Of course, we still need to prove Prop. 5.6. Let us do so.

5.2. The case of large subsets

Let us first see how $A$ grows when $A\subset\operatorname{SL}_{2}(\mathbb{F}_{q})$ is large with respect to $G=\operatorname{SL}_{2}(\mathbb{F}_{q})$. In fact, it is not terribly hard to show that, if $|A|\geq|G|^{1-\delta}$, $\delta>0$ a small constant, then $(A\cup A^{-1}\cup\{e\})^{k}=G$, where $k$ is an absolute constant. To proceed as in [Hel08]: we can use (2.7) to pass to the solvable group of upper- or lower-triangular matrices, then go on as in §3.2 to show that the subgroups $U^{\pm}$ of upper- or lower-triangular matrices are contained in $(A\cup A^{-1}\cup\{e\})^{k^{\prime}}$, $k^{\prime}$ a constant; we are then done by $G=U^{-}U^{+}U^{-}U^{+}$.

We will prove a stronger and nicer result: $A^{3}=G$. The proof is due to Nikolov and Pyber [NP11]; it is based on a classical idea, brought to bear to this particular context by Gowers [Gow08]. It will give us the opportunity to revisit the adjacency operator $\mathscr{A}$ and its spectrum.

Recall that a complex representation of a group $G$ is just a homomorphism $\phi:G\to\operatorname{GL}_{d}(\mathbb{C})$; by the dimension of the representation we just mean $d$. A representation $\phi$ is trivial if $\phi(g)=e$ for every $g\in G$.

The following result is due to Frobenius (1896), at least for $q$ prime. It can be proven simply by examining a character table, as in [Sha99]. The same procedure gives analogues of the same result for other groups of Lie type. Alternatively, there is a very nice elementary proof for $q$ prime, to be found, for example, in [Tao15, Lemma 1.3.3].

Proposition 5.4**.**

Let $G=\operatorname{SL}_{2}(\mathbb{F}_{q})$, $q=p^{\alpha}$. Then every non-trivial complex representation of $G$ has dimension $\geq(q-1)/2$.

We recall that the adjacency operator $\mathscr{A}$ on a Cayley graph $\Gamma(G,A)$ is the linear operator that takes a function $f:V\to\mathbb{C}$ to the function $\mathscr{A}f:V\to\mathbb{C}$ given by

[TABLE]

Assume, as usual, that $A=A^{-1}$. Then $\mathscr{A}$ is symmetric and all its eigenvalues are real:

[TABLE]

The largest eigenvalue $\nu_{0}$ corresponds to the eigenspace of constant functions.

Exercise 5.5**.**

Show that no eigenvalue $\nu$ can be larger than $1$. Hint: assume $\nu>1$, and show, using (5.6), that, for $g$ such that $|f(g)|$ is maximal, the equation $\mathscr{A}f(g)=\nu f(g)$ leads to a contradiction.

By an eigenspace of $\mathscr{A}$ we mean, of course, the vector space consisting of functions $f$ such that $\mathscr{A}f=\nu f$ for some fixed eigenvalue $\nu$. It is clear from the definition that every eigenspace of $\mathscr{A}$ is invariant under the action of $G$ by multiplication on the right. Hence, an eigenspace of $\mathscr{A}$ is a complex representation of $G$ – and it can be trivial only if it is the eigenspace of constant functions, i.e., the eigenspace corresponding to $\nu_{0}$. Thus, by Prop. 5.4, all other eigenvalues have multiplicity $\geq(q-1)/2$.

The idea now is to obtain a spectral gap, i.e., a non-trivial upper bound on $\nu_{j}$, $j>0$. It is standard to use the fact that the trace of a power $\mathscr{A}^{r}$ of an adjacency operator $\mathscr{A}$ can be expressed in two ways: as a the number of cycles of length $r$ in the graph $\Gamma(G,A)$ (multiplied by $1/|A|^{r}$), and as the sum of the $r$th powers of the eigenvalues of $\mathscr{A}$. In our case, for $r=2$, this gives us

[TABLE]

for any $j\geq 1$, and hence

[TABLE]

This is a very low upper bound when $|A|$ is large. This means that a few applications of the operator $\mathscr{A}$ are enough to render any function almost uniform, since any component orthogonal to the space of constant functions is multiplied by some $\nu_{j}$, $j\geq 1$, at every step. The following proof puts in practice this observation efficiently.

Proposition 5.6** ([NP11]).**

Let $G=\operatorname{SL}_{2}(\mathbb{F}_{q})$, $q=p^{\alpha}$. Let $A\subset G$, $A=A^{-1}$. Assume $|A|\geq 2|G|^{8/9}$. Then

[TABLE]

Actually, [NP11] proves this result without the assumption $A=A^{-1}$. We need $A=A^{-1}$ for $\mathscr{A}$ to be a symmetric operator, but, thanks to [Gow08], essentially the same argument works in the case $A\neq A^{-1}$.

Proof.

Suppose there is a $g\in G$ such that $g\notin A^{3}$. Then the scalar product

[TABLE]

equals [math], as otherwise there is an $x\in gA$ and an $a\in A$ such that $ax\in A$, and that would imply $g\in A^{-1}AA^{-1}=A^{3}$.

Since $\mathscr{A}$ is symmetric, it has full spectrum, that is, there exists a system of $n=|G|$ orthonormal eigenvectors $v_{0},v_{1},\dotsc$ of $\mathscr{A}$. Here $v_{0}$ is the constant function satisfying $\langle v_{0},v_{0}\rangle=1$, that is, the constant function taking the value $1/\sqrt{|G|}$ everywhere. Then

[TABLE]

Now

[TABLE]

At the same time, by (5.8) and Cauchy-Schwarz,

[TABLE]

Since $|G|=q(q^{2}-1)$, we see that $|A|\geq 2|G|^{8/9}$ implies

[TABLE]

and thus $\langle\mathscr{A}1_{A},1_{gA}\rangle>0$. Contradiction. ∎

6. Further perspectives and open problems

6.1. Expansion, random walks and the affine sieve

Let $G$ be a group, $A\subset G$, $A=A^{-1}$. As we saw in §1.1, the adjacency operator $\mathscr{A}$ has full real spectrum, and we can define what it means for the graph $\Gamma(G,A)$ to be a $\delta$-spectral expander, or simply an $\delta$-expander. An infinite family of graphs $\Gamma(G_{i},A_{i})$ is called an expander family if there is an $\epsilon>0$ such that every $\Gamma(G_{i},A_{i})$ is an $\epsilon$-expander. Of particular interest are expander families with $|A_{i}|$ bounded.

Using Thm. 5.1, Bourgain and Gamburd proved the following result [BG08b].

Theorem 6.1**.**

Let $A_{0}\subset\operatorname{SL}(\mathbb{Z})$. Assume that $A_{0}$ is not contained in any proper algebraic subgroup of $\operatorname{SL}_{2}$. Then

[TABLE]

is an expander family for some constant $C$.

The proof also involves Proposition 5.4 (applied as in [SX91]) as well as a non-commutative version [Tao08] of the Balog-Gowers-Szemerédi theorem from additive combinatorics. There are by now wide-ranging generalizations of Thm. 6.1; see, e.g., [GV12].

A random walk on a graph is what it sounds like: we start at a vertex $v_{0}$, and at every step we move to one of the $d$ neighbors of the vertex we are at – choosing any one of them with probability $1/d$. For convenience we work with a lazy random walk: at every step, we decide to stay where we are with probability $1/2$, and to move to a neighbor with probability $1/2d$. The mixing time is the number of steps it takes for ending point of a lazy random walk to become almost equidistributed (where “almost” is understood in any reasonable metric). In an $\epsilon$-expander graph $\Gamma(G,A)$, the mixing time is $O_{\epsilon}(\log|G|)$, i.e., about as small as it could be: it is easy to see that, for $|A|$ bounded, the mixing time (and even the diameter) has to be $\gg\log|G|$.

Exercise 6.2**.**

Let $G$ be a group, $A\subset G$, $A=A^{-1}$, $\langle A\rangle=G$. Let $\mathscr{A}$ be the adjacency operator on the Cayley graph.

1. (a)

Take a lazy random walk with $k$ steps on the Cayley graph, starting at the identity $e$. Show that the probability of your final position is given by the function $\phi_{k}=((\mathscr{A}+I)/2)^{k}\delta_{e}$, where $\delta_{e}:G\to\mathbb{C}$ is the function taking the value $1$ at $e$ and [math] elsewhere. 2. (b)

Write $\delta_{e}$ as a linear combination $\delta_{e}=\sum_{j}c_{j}v_{j}$, where each $v_{j}$ is an eigenvector of $\mathscr{A}$. What is the coefficient in front of the constant eigenvector $v_{0}$? What is $((\mathscr{A}+I)/2)^{k}\delta_{e}$, as a linear combination of the eigenvectors $v_{j}$? 3. (c)

Assume $\Gamma(G,A)$ is a $\delta$-expander. Show that, for $k\geq(2C/\delta)\log|G|$, $C\geq 1$, the probability distribution $\phi_{k}$ is nearly uniform in both the $\ell^{2}$- and the $\ell^{\infty}$-norms:

[TABLE]

[TABLE]

That is to say, the mixing time with respect to either the $\ell^{2}$- or the $\ell^{\infty}$-norms is $\ll(1/\delta)\log|G|$.

Thus, Thm. 6.1 gives us small mixing times. This fact has made the affine sieve possible [BGS10]. The affine sieve is an analogue of classical sieve methods; they are recast as sieves based on the natural action of $\mathbb{Z}$ on $\mathbb{Z}$, whereas a general affine sieve considers the actions of other groups, such as $\operatorname{SL}_{2}(\mathbb{Z})$.

Expansion had been shown before for some specific $A_{0}$. In particular, when $A_{0}$ generates $\operatorname{SL}_{2}(\mathbb{Z})$ (or a subgroup of finite index before) then the fact that (6.1) is an expander graph can be derived from the Selberg spectral gap [Sel65], i.e., the fact that the Laplacian on the quotient $\operatorname{SL}_{2}(\mathbb{Z})\backslash\mathbb{H}$ of the upper half plane $\mathbb{H}$ has a spectral gap. Nowadays, one can go in the opposite direction: spectral gaps on more general quotients can be proven using Thm. 6.1 [BGS11].

Let us finish this discussion by saying that it is generally held to be plausible that the family of all Cayley graphs of $\operatorname{SL}_{2}(\mathbb{Z}/p\mathbb{Z})$, for all $p$, is an expander family; in other words, there may be an $\epsilon>0$ such that, for every prime $p$ and every generator $A$ of $\operatorname{SL}_{2}(\mathbb{Z}/p\mathbb{Z})$, the graph $\Gamma(\operatorname{SL}_{2}(\mathbb{Z}/p\mathbb{Z}),A)$ is an $\epsilon$-expander. This statement has seemed plausible at least since [LR92], but proving it is an open problem believed to be very hard. It has been shown that there exists a thin family of primes such that the statement is true if those primes are omitted [BG10].

6.2. Algorithmic and probabilistic questions

It is one thing to show that the diameter of a group $G$ is small, that is, to show that every element of $G$ can be written as short word on any set of generators $A$. (By a word on $A$ we mean a product of elements of $A\cup A^{-1}$.) It is quite another to be able to find that word – reasonably quickly, it is understood.

Larsen [Lar03] gave a probabilistic algorithm that expresses an arbitrary $g\in\operatorname{SL}_{2}(\mathbb{Z}/p\mathbb{Z})$ as a word of length $O(\log p\log\log p)$ in the generators

[TABLE]

in time $(\log p)^{O(1)}$. No algorithm is known for arbitrary generators of $\operatorname{SL}_{2}(\mathbb{Z}/p\mathbb{Z})$. Neither do we have an algorithm for finding short words on arbitrary generators of finite simple groups in any other family.

Another question is what happens when $g_{1},g_{2}$ are random elements of a group $G$. For several kinds of groups (linear algebraic, $\operatorname{Alt}(n)$) it is known that, with probability tending to one, $g$ and $h$ generate $G$. What is the diameter of the Cayley graph of $G$ with respect to $\{g,h\}$ likely to be? For $G=\operatorname{SL}_{2}(\mathbb{F}_{p})$, it is known that it is $O(\log|G|)$ with probability tending to one (by [GHS*+*09] taken together with Thm. 5.1). For $\operatorname{Alt}(n)$, it is known to be $O(n^{2}(\log n)^{O(1)})$ with probability tending to one [HSZ15]. Is it actually $O(n(\log n)^{O(1)})$, or even $O(n\log n)$, with probability tending to one?

One can combine algorithmic and probabilistic questions. The proof in [BBS04] (supplemented by [BH05]) yields a probabilistic algorithm that, for a proportion $\to 1$ (as $n\to\infty$) of all pairs of elements $g_{1}$, $g_{2}$ of $\operatorname{Alt}(n)$, expresses any given element $g$ of $\operatorname{Alt}(n)$ as a word of polynomial length on $g_{1}$ and $g_{2}$, and does so in (Las Vegas) polynomial time. (If the algorithm will fail for a given pair $(g_{1},g_{2})$, it states so at an initial stage taking polynomial time.) The procedure in [HSZ15] gives a probabilistic algorithm that finds a word of length $O(n^{2}(\log n)^{O(1)})$ in time $O(n^{2}(\log n)^{O(1)})$ for a proportion $\to 1$ of all pairs $g_{1}$, $g_{2}$ and $g$ arbitrary, as is sketched in [HSZ15, App. B].

No analogous algorithm is known over $\operatorname{SL}_{2}(\mathbb{F}_{q})$, or for any other simple group of Lie type; we do not know how to express an arbitrary element of $\operatorname{SL}_{2}(\mathbb{F}_{q})$ as a word of length $(\log q)^{O(1)}$ on a random pair of generators of $G$ in time $(\log q)^{O(1)}$.

6.3. Final remarks

Let us briefly mention some links with other areas.

Group classification. It is by now clear that it is useful to look at a particular kind of result in group classification: the kind that was developed so as to avoid casework, and to do without the Classification of Finite Simple Groups. (The Classification is now generally accepted, but this was not always the case, and it is still sometimes felt to be better to prove something without it than with it; what we are about to see gives itself some validation to this viewpoint.) While results proven without the Classification are sometimes weaker than others, they are also more robust. Classifying subgroups of a finite group $G$ is the same as classifying subsets $A\subset G$ such that $e\in A$ and $|AA|=|A|$. Some Classification-free classification methods can be adapted to help in classifying subsets $A\subset G$ such that $e\in A$ and $|AAA|\leq|A|^{1+\delta}$ – in other words, precisely what we are studying. It is in this way that [LP11] was useful in [BGT11], and [Bab82], [Pyb93] were useful in [HS14].

Model theory. Model theory is essentially a branch of logic with applications to algebraic structures. Hrushovski and his collaborators [HP95], [HW08], [Hru12] have used model theory to study subgroups of algebraic groups. This was influenced by Larsen-Pink [LP11], and also served to explain it. In turn, [Hru12] influenced later work, especially [BGT12].

Permutation-group algorithms. Much work on permutation groups has been algorithmic in nature. Here a standard reference is [Ser03]. A good example is a problem we mentioned before – that of bounding the diameter of $\operatorname{Sym}(n)$ with respect to a random pair of generators; the approach in [BBS04] combines probabilistic and algorithmic ideas – as does [HSZ15], which builds on [BBS04], and as, for that matter, does [HS14]. The reference [LPW09] treats several of the relevant probabilistic tools.

Geometric group theory. Here much work remains to be done. Geometric group theory, while still a relatively new field, is considerably older than the approach followed in these notes. It is clear that there is a connection, but it has not yet been fully explored. Here it is particularly worth remarking that [Hru12] gave a new proof of Gromov’s theorem by means of the study of sets $A$ that grow slowly in the sense used in these notes.