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This is an addendum to last quarter’s course notes on Hilbert’s fifth problem, which I am in the process of reviewing in order to transcribe them into a book (as was done similarly for several other sets of lecture notes on this blog). When reviewing the zeroth set of notes in particular, I found that I had made a claim (Proposition 11 from those notes) which asserted, roughly speaking, that any sufficiently large nilprogression was an approximate group, and promised to prove it later in the course when we had developed the ability to calculate efficiently in nilpotent groups. As it turned out, I managed finish the course without the need to develop these calculations, and so the proposition remained unproven. In order to rectify this, I will use this post to lay out some of the basic algebra of nilpotent groups, and use it to prove the above proposition, which turns out to be a bit tricky. (In my paper with Breuillard and Green, we avoid the need for this proposition by restricting attention to a special type of nilprogression, which we call a nilprogression in {C}-normal form, for which the computations are simpler.)

There are several ways to think about nilpotent groups; for instance one can use the model example of the Heisenberg group

\displaystyle  H(R) :=\begin{pmatrix} 1 & R & R \\ 0 & 1 & R\\ 0 & 0 & 1 \end{pmatrix}

over an arbitrary ring {R} (which need not be commutative), or more generally any matrix group consisting of unipotent upper triangular matrices, and view a general nilpotent group as being an abstract generalisation of such concrete groups. (In the case of nilpotent Lie groups, at least, this is quite an accurate intuition, thanks to Engel’s theorem.) Or, one can adopt a Lie-theoretic viewpoint and try to think of nilpotent groups as somehow arising from nilpotent Lie algebras; this intuition is rigorous when working with nilpotent Lie groups (at least when the characteristic is large, in order to avoid issues coming from the denominators in the Baker-Campbell-Hausdorff formula), but also retains some conceptual value in the non-Lie setting. In particular, nilpotent groups (particularly finitely generated ones) can be viewed in some sense as “nilpotent Lie groups over {{\bf Z}}“, even though Lie theory does not quite work perfectly when the underlying scalars merely form an integral domain instead of a field.

Another point of view, which arises naturally both in analysis and in algebraic geometry, is to view nilpotent groups as modeling “infinitesimal” perturbations of the identity, where the infinitesimals have a certain finite order. For instance, given a (not necessarily commutative) ring {R} without identity (representing all the “small” elements of some larger ring or algebra), we can form the powers {R^j} for {j=1,2,\ldots}, defined as the ring generated by {j}-fold products {r_1 \ldots r_j} of elements {r_1,\ldots,r_j} in {R}; this is an ideal of {R} which represents the elements which are “{j^{th}} order” in some sense. If one then formally adjoins an identity {1} onto the ring {R}, then for any {s \geq 1}, the multiplicative group {G := 1+R \hbox{ mod } R^{s+1}} is a nilpotent group of step at most {s}. For instance, if {R} is the ring of strictly upper {s \times s} matrices (over some base ring), then {R^{s+1}} vanishes and {G} becomes the group of unipotent upper triangular matrices over the same ring, thus recovering the previous matrix-based example. In analysis applications, {R} might be a ring of operators which are somehow of “order” {O(\epsilon)} or {O(\hbar)} for some small parameter {\epsilon} or {\hbar}, and one wishes to perform Taylor expansions up to order {O(\epsilon^s)} or {O(\hbar^s)}, thus discarding (i.e. quotienting out) all errors in {R^{s+1}}.

From a dynamical or group-theoretic perspective, one can also view nilpotent groups as towers of central extensions of a trivial group. Finitely generated nilpotent groups can also be profitably viewed as a special type of polycylic group; this is the perspective taken in this previous blog post. Last, but not least, one can view nilpotent groups from a combinatorial group theory perspective, as being words from some set of generators of various “degrees” subject to some commutation relations, with commutators of two low-degree generators being expressed in terms of higher degree objects, and all commutators of a sufficiently high degree vanishing. In particular, generators of a given degree can be moved freely around a word, as long as one is willing to generate commutator errors of higher degree.

With this last perspective, in particular, one can start computing in nilpotent groups by adopting the philosophy that the lowest order terms should be attended to first, without much initial concern for the higher order errors generated in the process of organising the lower order terms. Only after the lower order terms are in place should attention then turn to higher order terms, working successively up the hierarchy of degrees until all terms are dealt with. This turns out to be a relatively straightforward philosophy to implement in many cases (particularly if one is not interested in explicit expressions and constants, being content instead with qualitative expansions of controlled complexity), but the arguments are necessarily recursive in nature and as such can become a bit messy, and require a fair amount of notation to express precisely. So, unfortunately, the arguments here will be somewhat cumbersome and notation-heavy, even if the underlying methods of proof are relatively simple.

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In this final set of course notes, we discuss how (a generalisation of) the expansion results obtained in the preceding notes can be used for some nnumber-theoretic applications, and in particular to locate almost primes inside orbits of thin groups, following the work of Bourgain, Gamburd, and Sarnak. We will not attempt here to obtain the sharpest or most general results in this direction, but instead focus on the simplest instances of these results which are still illustrative of the ideas involved.

One of the basic general problems in analytic number theory is to locate tuples of primes of a certain form; for instance, the famous (and still unsolved) twin prime conjecture asserts that there are infinitely many pairs {(n_1,n_2)} in the line {\{ (n_1,n_2) \in {\bf Z}^2: n_2-n_1=2\}} in which both entries are prime. In a similar spirit, one of the Landau conjectures (also still unsolved) asserts that there are infinitely many primes in the set {\{ n^2+1: n \in {\bf Z} \}}. The Mersenne prime conjecture (also unsolved) asserts that there are infinitely many primes in the set {\{ 2^n - 1: n \in {\bf Z} \}}, and so forth.

More generally, given some explicit subset {V} in {{\bf R}^d} (or {{\bf C}^d}, if one wishes), such as an algebraic variety, one can ask the question of whether there are infinitely many integer lattice points {(n_1,\ldots,n_d)} in {V \cap {\bf Z}^d} in which all the coefficients {n_1,\ldots,n_d} are simultaneously prime; let us refer to such points as prime points.

At this level of generality, this problem is impossibly difficult. Indeed, even the much simpler problem of deciding whether the set {V \cap {\bf Z}^d} is non-empty (let alone containing prime points) when {V} is a hypersurface {\{ x \in {\bf R}^d: P(x) = 0 \}} cut out by a polynomial {P} is essentially Hilbert’s tenth problem, which is known to be undecidable in general by Matiyasevich’s theorem. So one needs to restrict attention to a more special class of sets {V}, in which the question of finding integer points is not so difficult. One model case is to consider orbits {V = \Gamma b}, where {b \in {\bf Z}^d} is a fixed lattice vector and {\Gamma} is some discrete group that acts on {{\bf Z}^d} somehow (e.g. {\Gamma} might be embedded as a subgroup of the special linear group {SL_d({\bf Z})}, or on the affine group {SL_d({\bf Z}) \ltimes {\bf Z}^d}). In such a situation it is then quite easy to show that {V = V \cap {\bf Z}^d} is large; for instance, {V} will be infinite precisely when the stabiliser of {b} in {\Gamma} has infinite index in {\Gamma}.

Even in this simpler setting, the question of determining whether an orbit {V = \Gamma b} contains infinitely prime points is still extremely difficult; indeed the three examples given above of the twin prime conjecture, Landau conjecture, and Mersenne prime conjecture are essentially of this form (possibly after some slight modification of the underlying ring {{\bf Z}}, see this paper of Bourgain-Gamburd-Sarnak for details), and are all unsolved (and generally considered well out of reach of current technology). Indeed, the list of non-trivial orbits {V = \Gamma b} which are known to contain infinitely many prime points is quite slim; Euclid’s theorem on the infinitude of primes handles the case {V = {\bf Z}}, Dirichlet’s theorem handles infinite arithmetic progressions {V = a{\bf Z} + r}, and a somewhat complicated result of Green, Tao, and Ziegler handles “non-degenerate” affine lattices in {{\bf Z}^d} of rank two or more (such as the lattice of length {d} arithmetic progressions), but there are few other positive results known that are not based on the above cases (though we will note the remarkable theorem of Friedlander and Iwaniec that there are infinitely many primes of the form {a^2+b^4}, and the related result of Heath-Brown that there are infinitely many primes of the form {a^3+2b^3}, as being in a kindred spirit to the above results, though they are not explicitly associated to an orbit of a reasonable action as far as I know).

On the other hand, much more is known if one is willing to replace the primes by the larger set of almost primes – integers with a small number of prime factors (counting multiplicity). Specifically, for any {r \geq 1}, let us call an {r}-almost prime an integer which is the product of at most {r} primes, and possibly by the unit {-1} as well. Many of the above sorts of questions which are open for primes, are known for {r}-almost primes for {r} sufficiently large. For instance, with regards to the twin prime conjecture, it is a result of Chen that there are infinitely many pairs {p,p+2} where {p} is a prime and {p+2} is a {2}-almost prime; in a similar vein, it is a result of Iwaniec that there are infinitely many {2}-almost primes of the form {n^2+1}. On the other hand, it is still open for any fixed {r} whether there are infinitely many Mersenne numbers {2^n-1} which are {r}-almost primes. (For the superficially similar situation with the numbers {2^n+1}, it is in fact believed (but again unproven) that there are only finitely many {r}-almost primes for any fixed {r} (the Fermat prime conjecture).)

The main tool that allows one to count almost primes in orbits is sieve theory. The reason for this lies in the simple observation that in order to ensure that an integer {n} of magnitude at most {x} is an {r}-almost prime, it suffices to guarantee that {n} is not divisible by any prime less than {x^{1/(r+1)}}. Thus, to create {r}-almost primes, one can start with the integers up to some large threshold {x} and remove (or “sieve out”) all the integers that are multiples of any prime {p} less than {x^{1/(r+1)}}. The difficulty is then to ensure that a sufficiently non-trivial quantity of integers remain after this process, for the purposes of finding points in the given set {V}.

The most basic sieve of this form is the sieve of Eratosthenes, which when combined with the inclusion-exclusion principle gives the Legendre sieve (or exact sieve), which gives an exact formula for quantities such as the number {\pi(x,z)} of natural numbers less than or equal to {x} that are not divisible by any prime less than or equal to a given threshold {z}. Unfortunately, when one tries to evaluate this formula, one encounters error terms which grow exponentially in {z}, rendering this sieve useful only for very small thresholds {z} (of logarithmic size in {x}). To improve the sieve level up to a small power of {x} such as {x^{1/(r+1)}}, one has to replace the exact sieve by upper bound sieves and lower bound sieves which only seek to obtain upper or lower bounds on quantities such as {\pi(x,z)}, but contain a polynomial number of terms rather than an exponential number. There are a variety of such sieves, with the two most common such sieves being combinatorial sieves (such as the beta sieve), based on various combinatorial truncations of the inclusion-exclusion formula, and the Selberg upper bound sieve, based on upper bounds that are the square of a divisor sum. (There is also the large sieve, which is somewhat different in nature and based on {L^2} almost orthogonality considerations, rather than on any actual sieving, to obtain upper bounds.) We will primarily work with a specific sieve in this notes, namely the beta sieve, and we will not attempt to optimise all the parameters of this sieve (which ultimately means that the almost primality parameter {r} in our results will be somewhat large). For a more detailed study of sieve theory, see the classic text of Halberstam and Richert, or the more recent texts of Iwaniec-Kowalski or of Friedlander-Iwaniec.

Very roughly speaking, the end result of sieve theory is that excepting some degenerate and “exponentially thin” settings (such as those associated with the Mersenne primes), all the orbits which are expected to have a large number of primes, can be proven to at least have a large number of {r}-almost primes for some finite {r}. (Unfortunately, there is a major obstruction, known as the parity problem, which prevents sieve theory from lowering {r} all the way to {1}; see this blog post for more discussion.) One formulation of this principle was established by Bourgain, Gamburd, and Sarnak:

Theorem 1 (Bourgain-Gamburd-Sarnak) Let {\Gamma} be a subgroup of {SL_2({\bf Z})} which is not virtually solvable. Let {f: {\bf Z}^4 \rightarrow {\bf Z}} be a polynomial with integer coefficients obeying the following primitivity condition: for any positive integer {q}, there exists {A \in \Gamma \subset {\bf Z}^4} such that {f(A)} is coprime to {q}. Then there exists an {r \geq 1} such that there are infinitely many {A \in \Gamma} with {f(A)} non-zero and {r}-almost prime.

This is not the strongest version of the Bourgain-Gamburd-Sarnak theorem, but it captures the general flavour of their results. Note that the theorem immediately implies an analogous result for orbits {\Gamma b \subset {\bf Z}^2}, in which {f} is now a polynomial from {{\bf Z}^2} to {{\bf Z}}, and one uses {f(Ab)} instead of {f(A)}. It is in fact conjectured that one can set {r=1} here, but this is well beyond current technology. For the purpose of reaching {r=1}, it is very natural to impose the primitivity condition, but as long as one is content with larger values of {r}, it is possible to relax the primitivity condition somewhat; see the paper of Bourgain, Gamburd, and Sarnak for more discussion.

By specialising to the polynomial {f: \begin{pmatrix} a & b \\ c & d \end{pmatrix} \rightarrow abcd}, we conclude as a corollary that as long as {\Gamma} is primitive in the sense that it contains matrices with all coefficients coprime to {q} for any given {q}, then {\Gamma} contains infinitely many matrices whose elements are all {r}-almost primes for some {r} depending only on {\Gamma}. For further applications of these sorts of results, for instance to Appolonian packings, see the paper of Bourgain, Gamburd, and Sarnak.

It turns out that to prove Theorem 1, the Cayley expansion results in {SL_2(F_p)} from the previous set of notes are not quite enough; one needs a more general Cayley expansion result in {SL_2({\bf Z}/q{\bf Z})} where {q} is square-free but not necessarily prime. The proof of this expansion result uses the same basic methods as in the {SL_2(F_p)} case, but is significantly more complicated technically, and we will only discuss it briefly here. As such, we do not give a complete proof of Theorem 1, but hopefully the portion of the argument presented here is still sufficient to give an impression of the ideas involved.

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In the last three notes, we discussed the Bourgain-Gamburd expansion machine and two of its three ingredients, namely quasirandomness and product theorems, leaving only the non-concentration ingredient to discuss. We can summarise the results of the last three notes, in the case of fields of prime order, as the following theorem.

Theorem 1 (Non-concentration implies expansion in {SL_d}) Let {p} be a prime, let {d \geq 1}, and let {S} be a symmetric set of elements in {G := SL_d(F_p)} of cardinality {|S|=k} not containing the identity. Write {\mu := \frac{1}{|S|} \sum_{s\in S}\delta_s}, and suppose that one has the non-concentration property

\displaystyle  \sup_{H < G}\mu^{(n)}(H) < |G|^{-\kappa} \ \ \ \ \ (1)

for some {\kappa>0} and some even integer {n \leq \Lambda \log |G|}. Then {Cay(G,S)} is a two-sided {\epsilon}-expander for some {\epsilon>0} depending only on {k, d, \kappa,\Lambda}.

Proof: From (1) we see that {\mu^{(n)}} is not supported in any proper subgroup {H} of {G}, which implies that {S} generates {G}. The claim now follows from the Bourgain-Gamburd expansion machine (Theorem 2 of Notes 4), the product theorem (Theorem 1 of Notes 5), and quasirandomness (Exercise 8 of Notes 3). \Box

Remark 1 The same argument also works if we replace {F_p} by the field {F_{p^j}} of order {p^j} for some bounded {j}. However, there is a difficulty in the regime when {j} is unbounded, because the quasirandomness property becomes too weak for the Bourgain-Gamburd expansion machine to be directly applicable. On theother hand, the above type of theorem was generalised to the setting of cyclic groups {{\bf Z}/q{\bf Z}} with {q} square-free by Varju, to arbitrary {q} by Bourgain and Varju, and to more general algebraic groups than {SL_d} and square-free {q} by Salehi Golsefidy and Varju. It may be that some modification of the proof techniques in these papers may also be able to handle the field case {F_{p^j}} with unbounded {j}.

It thus remains to construct tools that can establish the non-concentration property (1). The situation is particularly simple in {SL_2(F_p)}, as we have a good understanding of the subgroups of that group. Indeed, from Theorem 14 from Notes 5, we obtain the following corollary to Theorem 1:

Corollary 2 (Non-concentration implies expansion in {SL_2}) Let {p} be a prime, and let {S} be a symmetric set of elements in {G := SL_2(F_p)} of cardinality {|S|=k} not containing the identity. Write {\mu := \frac{1}{|S|} \sum_{s\in S}\delta_s}, and suppose that one has the non-concentration property

\displaystyle  \sup_{B}\mu^{(n)}(B) < |G|^{-\kappa} \ \ \ \ \ (2)

for some {\kappa>0} and some even integer {n \leq \Lambda \log |G|}, where {B} ranges over all Borel subgroups of {SL_2(\overline{F})}. Then, if {|G|} is sufficiently large depending on {k,\kappa,\Lambda}, {Cay(G,S)} is a two-sided {\epsilon}-expander for some {\epsilon>0} depending only on {k, \kappa,\Lambda}.

It turns out (2) can be verified in many cases by exploiting the solvable nature of the Borel subgroups {B}. We give two examples of this in these notes. The first result, due to Bourgain and Gamburd (with earlier partial results by Gamburd and by Shalom) generalises Selberg’s expander construction to the case when {S} generates a thin subgroup of {SL_2({\bf Z})}:

Theorem 3 (Expansion in thin subgroups) Let {S} be a symmetric subset of {SL_2({\bf Z})} not containing the identity, and suppose that the group {\langle S \rangle} generated by {S} is not virtually solvable. Then as {p} ranges over all sufficiently large primes, the Cayley graphs {Cay(SL_2(F_p), \pi_p(S))} form a two-sided expander family, where {\pi_p: SL_2({\bf Z}) \rightarrow SL_2(F_p)} is the usual projection.

Remark 2 One corollary of Theorem 3 (or of the non-concentration estimate (3) below) is that {\pi_p(S)} generates {SL_2(F_p)} for all sufficiently large {p}, if {\langle S \rangle} is not virtually solvable. This is a special case of a much more general result, known as the strong approximation theorem, although this is certainly not the most direct way to prove such a theorem. Conversely, the strong approximation property is used in generalisations of this result to higher rank groups than {SL_2}.

Exercise 1 In the converse direction, if {\langle S\rangle} is virtually solvable, show that for sufficiently large {p}, {\pi_p(S)} fails to generate {SL_2(F_p)}. (Hint: use Theorem 14 from Notes 5 to prevent {SL_2(F_p)} from having bounded index solvable subgroups.)

Exercise 2 (Lubotzsky’s 1-2-3 problem) Let {S := \{ \begin{pmatrix}1 & \pm 3 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix}1 & 0 \\ \pm 3 & 1 \end{pmatrix}}.

  • (i) Show that {S} generates a free subgroup of {SL_2({\bf Z})}. (Hint: use a ping-pong argument, as in Exercise 23 of Notes 2.)
  • (ii) Show that if {v, w} are two distinct elements of the sector {\{ (x,y) \in {\bf R}^2_+: x/2 < y < 2x \}}, then there os no element {g \in \langle S \rangle} for which {gv = w}. (Hint: this is another ping-pong argument.) Conclude that {\langle S \rangle} has infinite index in {SL_2({\bf Z})}. (Contrast this with the situation in which the {3} coefficients in {S} are replaced by {1} or {2}, in which case {\langle S \rangle} is either all of {SL_2({\bf Z})}, or a finite index subgroup, as demonstrated in Exercise 23 of Notes 2).
  • (iii) Show that {Cay(SL_2(F_p), \pi_p(S))} for sufficiently large primes {p} form a two-sided expander family.

Remark 3 Theorem 3 has been generalised to arbitrary linear groups, and with {F_p} replaced by {{\bf Z}/q{\bf Z}} for square-free {q}; see this paper of Salehi Golsefidy and Varju. In this more general setting, the condition of virtual solvability must be replaced by the condition that the connected component of the Zariski closure of {\langle S \rangle} is perfect. An effective version of Theorem 3 (with completely explicit constants) was recently obtained by Kowalski.

The second example concerns Cayley graphs constructed using random elements of {SL_2(F_p)}.

Theorem 4 (Random generators expand) Let {p} be a prime, and let {x,y} be two elements of {SL_2(F_p)} chosen uniformly at random. Then with probability {1-o_{p \rightarrow \infty}(1)}, {Cay(SL_2(F_p), \{x,x^{-1},y,y^{-1}\})} is a two-sided {\epsilon}-expander for some absolute constant {\epsilon}.

Remark 4 As with Theorem 3, Theorem 4 has also been extended to a number of other groups, such as the Suzuki groups (in this paper of Breuillard, Green, and Tao), and more generally to finite simple groups of Lie type of bounded rank (in forthcoming work of Breuillard, Green, Guralnick, and Tao). There are a number of other constructions of expanding Cayley graphs in such groups (and in other interesting groups, such as the alternating groups) beyond those discussed in these notes; see this recent survey of Lubotzky for further discussion. It has been conjectured by Lubotzky and Weiss that any pair {x,y} of (say) {SL_2(F_p)} that generates the group, is a two-sided {\epsilon}-expander for an absolute constant {\epsilon}: in the case of {SL_2(F_p)}, this has been established for a density one set of primes by Breuillard and Gamburd.

— 1. Expansion in thin subgroups —

We now prove Theorem 3. The first observation is that the expansion property is monotone in the group {\langle S \rangle}:

Exercise 3 Let {S, S'} be symmetric subsets of {SL_2({\bf Z})} not containing the identity, such that {\langle S \rangle \subset \langle S' \rangle}. Suppose that {Cay(SL_2(F_p), \pi_p(S))} is a two-sided expander family for sufficiently large primes {p}. Show that {Cay(SL_2(F_p), \pi_p(S'))} is also a two-sided expander family.

As a consequence, Theorem 3 follows from the following two statments:

Theorem 5 (Tits alternative) Let {\Gamma \subset SL_2({\bf Z})} be a group. Then exactly one of the following statements holds:

  • (i) {\Gamma} is virtually solvable.
  • (ii) {\Gamma} contains a copy of the free group {F_2} of two generators as a subgroup.

Theorem 6 (Expansion in free groups) Let {x,y \in SL_2({\bf Z})} be generators of a free subgroup of {SL_2({\bf Z})}. Then as {p} ranges over all sufficiently large primes, the Cayley graphs {Cay(SL_2(F_p), \pi_p(\{x,y,x^{-1},y^{-1}\}))} form a two-sided expander family.

Theorem 5 is a special case of the famous Tits alternative, which among other things allows one to replace {SL_2({\bf Z})} by {GL_d(k)} for any {d \geq 1} and any field {k} of characteristic zero (and fields of positive characteristic are also allowed, if one adds the requirement that {\Gamma} be finitely generated). We will not prove the full Tits alternative here, but instead just give an ad hoc proof of the special case in Theorem 5 in the following exercise.

Exercise 4 Given any matrix {g \in SL_2({\bf Z})}, the singular values are {\|g\|_{op}} and {\|g\|_{op}^{-1}}, and we can apply the singular value decomposition to decompose

\displaystyle  g = u_1(g) \|g\|_{op} v_1^*(g) + u_2(g) \|g\|_{op}^{-1} v_2(g)^*

where {u_1(g),u_2(g)\in {\bf C}^2} and {v_1(g), v_2(g) \in {\bf C}^2} are orthonormal bases. (When {\|g\|_{op}>1}, these bases are uniquely determined up to phase rotation.) We let {\tilde u_1(g) \in {\bf CP}^1} be the projection of {u_1(g)} to the projective complex plane, and similarly define {\tilde v_2(g)}.

Let {\Gamma} be a subgroup of {SL_2({\bf Z})}. Call a pair {(u,v) \in {\bf CP}^1 \times {\bf CP}^1} a limit point of {\Gamma} if there exists a sequence {g_n \in \Gamma} with {\|g_n\|_{op} \rightarrow \infty} and {(\tilde u_1(g_n), \tilde v_2(g_n)) \rightarrow (u,v)}.

  • (i) Show that if {\Gamma} is infinite, then there is at least one limit point.
  • (ii) Show that if {(u,v)} is a limit point, then so is {(v,u)}.
  • (iii) Show that if there are two limit points {(u,v), (u',v')} with {\{u,v\} \cap \{u',v'\} = \emptyset}, then there exist {g,h \in \Gamma} that generate a free group. (Hint: Choose {(\tilde u_1(g), \tilde v_2(g))} close to {(u,v)} and {(\tilde u_1(h),\tilde v_2(h))} close to {(u',v')}, and consider the action of {g} and {h} on {{\bf CP}^1}, and specifically on small neighbourhoods of {u,v,u',v'}, and set up a ping-pong type situation.)
  • (iv) Show that if {g \in SL_2({\bf Z})} is hyperbolic (i.e. it has an eigenvalue greater than 1), with eigenvectors {u,v}, then the projectivisations {(\tilde u,\tilde v)} of {u,v} form a limit point. Similarly, if {g} is regular parabolic (i.e. it has an eigenvalue at 1, but is not the identity) with eigenvector {u}, show that {(\tilde u,\tilde bu)} is a limit point.
  • (v) Show that if {\Gamma} has no free subgroup of two generators, then all hyperbolic and regular parabolic elements of {\Gamma} have a common eigenvector. Conclude that all such elements lie in a solvable subgroup of {\Gamma}.
  • (vi) Show that if an element {g \in SL_2({\bf Z})} is neither hyperbolic nor regular parabolic, and is not a multiple of the identity, then {g} is conjugate to a rotation by {\pi/2} (in particular, {g^2=-1}).
  • (vii) Establish Theorem 5. (Hint: show that two square roots of {-1} in {SL_2({\bf Z})} cannot multiply to another square root of {-1}.)

Now we prove Theorem 6. Let {\Gamma} be a free subgroup of {SL_2({\bf Z})} generated by two generators {x,y}. Let {\mu := \frac{1}{4} (\delta_x +\delta_{x^{-1}} + \delta_y + \delta_{y^{-1}})} be the probability measure generating a random walk on {SL_2({\bf Z})}, thus {(\pi_p)_* \mu} is the corresponding generator on {SL_2(F_p)}. By Corollary 2, it thus suffices to show that

\displaystyle  \sup_{B}((\pi_p)_* \mu)^{(n)}(B) < p^{-\kappa} \ \ \ \ \ (3)

for all sufficiently large {p}, some absolute constant {\kappa>0}, and some even {n = O(\log p)} (depending on {p}, of course), where {B} ranges over Borel subgroups.

As {\pi_p} is a homomorphism, one has {((\pi_p)_* \mu)^{(n)}(B) = (\pi_p)_* (\mu^{(n)})(B) = \mu^{(n)}(\pi_p^{-1}(B))} and so it suffices to show that

\displaystyle  \sup_{B} \mu^{(n)}(\pi_p^{-1}(B)) < p^{-\kappa}.

To deal with the supremum here, we will use an argument of Bourgain and Gamburd, taking advantage of the fact that all Borel groups of {SL_2} obey a common group law, the point being that free groups such as {\Gamma} obey such laws only very rarely. More precisely, we use the fact that the Borel groups are solvable of derived length two; in particular we have

\displaystyle  [[a,b],[c,d]] = 1 \ \ \ \ \ (4)

for all {a,b,c,d \in B}. Now, {\mu^{(n)}} is supported on matrices in {SL_2({\bf Z})} whose coefficients have size {O(\exp(O(n)))} (where we allow the implied constants to depend on the choice of generators {x,y}), and so {(\pi_p)_*( \mu^{(n)} )} is supported on matrices in {SL_2(F_p)} whose coefficients also have size {O(\exp(O(n)))}. If {n} is less than a sufficiently small multiple of {\log p}, these coefficients are then less than {p^{1/10}} (say). As such, if {\tilde a,\tilde b,\tilde c,\tilde d \in SL_2({\bf Z})} lie in the support of {\mu^{(n)}} and their projections {a = \pi_p(\tilde a), \ldots, d = \pi_p(\tilde d)} obey the word law (4) in {SL_2(F_p)}, then the original matrices {\tilde a, \tilde b, \tilde c, \tilde d} obey the word law (4) in {SL_2({\bf Z})}. (This lifting of identities from the characteristic {p} setting of {SL_2(F_p)} to the characteristic {0} setting of {SL_2({\bf Z})} is a simple example of the “Lefschetz principle”.)

To summarise, if we let {E_{n,p,B}} be the set of all elements of {\pi_p^{-1}(B)} that lie in the support of {\mu^{(n)}}, then (4) holds for all {a,b,c,d \in E_{n,p,B}}. This severely limits the size of {E_{n,p,B}} to only be of polynomial size, rather than exponential size:

Proposition 7 Let {E} be a subset of the support of {\mu^{(n)}} (thus, {E} consists of words in {x,y,x^{-1},y^{-1}} of length {n}) such that the law (4) holds for all {a,b,c,d \in E}. Then {|E| \ll n^2}.

The proof of this proposition is laid out in the exercise below.

Exercise 5 Let {\Gamma} be a free group generated by two generators {x,y}. Let {B} be the set of all words of length at most {n} in {x,y,x^{-1},y^{-1}}.

  • (i) Show that if {a,b \in \Gamma} commute, then {a, b} lie in the same cyclic group, thus {a = c^i, b = c^j} for some {c \in \Gamma} and {i,j \in {\bf Z}}.
  • (ii) Show that if {a \in \Gamma}, there are at most {O(n)} elements of {B} that commute with {a}.
  • (iii) Show that if {a,c \in \Gamma}, there are at most {O(n)} elements {b} of {B} with {[a,b] = c}.
  • (iv) Prove Proposition 7.

Now we can conclude the proof of Theorem 3:

Exercise 6 Let {\Gamma} be a free group generated by two generators {x,y}.

  • (i) Show that {\| \mu^{(n)} \|_{\ell^\infty(\Gamma)} \ll c^n} for some absolute constant {0 < c<1}. (For much more precise information on {\mu^{(n)}}, see this paper of Kesten.)
  • (ii) Conclude the proof of Theorem 3.

— 2. Random generators expand —

We now prove Theorem 4. Let {{\bf F}_2} be the free group on two formal generators {a,b}, and let {\mu := \frac{1}{4}(\delta_a + \delta_b + \delta_{a^{-1}}+ \delta_{b^{-1}}} be the generator of the random walk. For any word {w \in {\bf F}_2} and any {x,y} in a group {G}, let {w(x,y) \in G} be the element of {G} formed by substituting {x,y} for {a,b} respectively in the word {w}; thus {w} can be viewed as a map {w: G \times G \rightarrow G} for any group {G}. Observe that if {w} is drawn randomly using the distribution {\mu^{(n)}}, and {x,y \in SL_2(F_p)}, then {w(x,y)} is distributed according to the law {\tilde \mu^{(n)}}, where {\tilde \mu := \frac{1}{4}(\delta_x + \delta_y + \delta_{x^{-1}}+ \delta_{y^{-1}})}. Applying Corollary 2, it suffices to show that whenever {p} is a large prime and {x,y} are chosen uniformly and independently at random from {SL_2(F_p)}, that with probability {1-o_{p \rightarrow \infty}(1)}, one has

\displaystyle  \sup_B {\bf P}_w ( w(x,y) \in B ) \leq p^{-\kappa} \ \ \ \ \ (5)

for some absolute constant {\kappa}, where {B} ranges over all Borel subgroups of {SL_2(\overline{F_p})} and {w} is drawn from the law {\mu^{(n)}} for some even natural number {n = O(\log p)}.

Let {B_n} denote the words in {{\bf F}_2} of length at most {n}. We may use the law (4) to obtain good bound on the supremum in (5) assuming a certain non-degeneracy property of the word evaluations {w(x,y)}:

Exercise 7 Let {n} be a natural number, and suppose that {x,y \in SL_2(F_p)} is such that {w(x,y) \neq 1} for {w \in B_{100n} \backslash \{1\}}. Show that

\displaystyle  \sup_B {\bf P}_w ( w(x,y) \in B ) \ll \exp(-cn)

for some absolute constant {c>0}, where {w} is drawn from the law {\mu^{(n)}}. (Hint: use (4) and the hypothesis to lift the problem up to {{\bf F}_2}, at which point one can use Proposition 7 and Exercise 6.)

In view of this exercise, it suffices to show that with probability {1-o_{p \rightarrow\infty}(1)}, one has {w(x,y) \neq 1} for all {w \in B_{100n} \backslash \{1\}} for some {n} comparable to a small multiple of {\log p}. As {B_{100n}} has {\exp(O(n))} elements, it thus suffices by the union bound to show that

\displaystyle  {\bf P}_{x,y}(w(x,y)=1) \leq p^{-\gamma} \ \ \ \ \ (6)

for some absolute constant {\gamma > 0}, and any {w \in {\bf F}_2 \backslash \{1\}} of length less than {c\log p} for some sufficiently small absolute constant {c>0}.

Let us now fix a non-identity word {w} of length {|w|} less than {c\log p}, and consider {w} as a function from {SL_2(k) \times SL_2(k)} to {SL_2(k)} for an arbitrary field {k}. We can identify {SL_2(k)} with the set {\{ (a,b,c,d)\in k^4: ad-bc=1\}}. A routine induction then shows that the expression {w((a,b,c,d),(a',b',c',d'))} is then a polynomial in the eight variables {a,b,c,d,a',b',c',d'} of degree {O(|w|)} and coefficients which are integers of size {O( \exp( O(|w|) ) )}. Let us then make the additional restriction to the case {a,a' \neq 0}, in which case we can write {d = \frac{bc+1}{a}} and {d' =\frac{b'c'+1}{a'}}. Then {w((a,b,c,d),(a',b',c',d'))} is now a rational function of {a,b,c,a',b',c'} whose numerator is a polynomial of degree {O(|w|)} and coefficients of size {O( \exp( O(|w|) ) )}, and the denominator is a monomial of {a,a'} of degree {O(|w|)}.

We then specialise this rational function to the field {k=F_p}. It is conceivable that when one does so, the rational function collapses to the constant polynomial {(1,0,0,1)}, thus {w((a,b,c,d),(a',b',c',d'))=1} for all {(a,b,c,d),(a',b',c',d') \in SL_2(F_p)} with {a,a' \neq 0}. (For instance, this would be the case if {w(x,y) = x^{|SL_2(F_p)|}}, by Lagrange’s theorem, if it were not for the fact that {|w|} is far too large here.) But suppose that this rational function does not collapse to the constant rational function. Applying the Schwarz-Zippel lemma (Exercise 23 from Notes 5), we then see that the set of pairs {(a,b,c,d),(a',b',c',d') \in SL_2(F_p)} with {a,a' \neq 0} and {w((a,b,c,d),(a',b',c',d'))=1} is at most {O( |w| p^5 )}; adding in the {a=0} and {a'=0} cases, one still obtains a bound of {O(|w|p^5)}, which is acceptable since {|SL_2(F_p)|^2 \sim p^6} and {|w| = O( \log p )}. Thus, the only remaining case to consider is when the rational function {w((a,b,c,d),(a',b',c',d'))} is identically {1} on {SL_2(F_p)} with {a,a' \neq 0}.

Now we perform another “Lefschetz principle” maneuvre to change the underlying field. Recall that the denominator of rational function {w((a,b,c,d),(a',b',c',d'))} is monomial in {a,a'}, and the numerator has coefficients of size {O(\exp(O(|w|)))}. If {|w|} is less than {c\log p} for a sufficiently small {p}, we conclude in particular (for {p} large enough) that the coefficients all have magnitude less than {p}. As such, the only way that this function can be identically {1} on {SL_2(F_p)} is if it is identically {1} on {SL_2(k)} for all {k} with {a,a' \neq 0}, and hence for {a=0} or {a'=0} also by taking Zariski closures.

On the other hand, we know that for some choices of {k}, e.g. {k={\bf R}}, {SL_2(k)} contains a copy {\Gamma} of the free group on two generators (see e.g. Exercise 23 of Notes 2). As such, it is not possible for any non-identity word {w} to be identically trivial on {SL_2(k) \times SL_2(k)}. Thus this case cannot actually occur, completing the proof of (6) and hence of Theorem 4.

Remark 5 We see from the above argument that the existence of subgroups {\Gamma} of an algebraic group with good “independence” properties – such as that of generating a free group – can be useful in studying the expansion properties of that algebraic group, even if the field of interest in the latter is distinct from that of the former. For more complicated algebraic groups than {SL_2}, in which laws such as (4) are not always available, it turns out to be useful to place further properties on the subgroup {\Gamma}, for instance by requiring that all non-abelian subgroups of that group be Zariski dense (a property which has been called strong density), as this turns out to be useful for preventing random walks from concentrating in proper algebraic subgroups. See this paper of Breuillard, Guralnick, Green and Tao for constructions of strongly dense free subgroups of algebraic groups and further discussion.

In the previous set of notes, we saw that one could derive expansion of Cayley graphs from three ingredients: non-concentration, product theorems, and quasirandomness. Quasirandomness was discussed in Notes 3. In the current set of notes, we discuss product theorems. Roughly speaking, these theorems assert that in certain circumstances, a finite subset {A} of a group {G} either exhibits expansion (in the sense that {A^3}, say, is significantly larger than {A}), or is somehow “close to” or “trapped” by a genuine group.

Theorem 1 (Product theorem in {SL_d(k)}) Let {d \geq 2}, let {k} be a finite field, and let {A} be a finite subset of {G := SL_d(k)}. Let {\epsilon >0} be sufficiently small depending on {d}. Then at least one of the following statements holds:

  • (Expansion) One has {|A^3| \geq |A|^{1+\epsilon}}.
  • (Close to {G}) One has {|A| \geq |G|^{1-O_d(\epsilon)}}.
  • (Trapping) {A} is contained in a proper subgroup of {G}.

We will prove this theorem (which was proven first in the {d=2,3} cases for fields {F} of prime order by Helfgott, and then for {d=2} and general {F} by Dinai, and finally to general {d} and {F} independently by Pyber-Szabo and by Breuillard-Green-Tao) later in this notes. A more qualitative version of this proposition was also previously obtained by Hrushovski. There are also generalisations of the product theorem of importance to number theory, in which the field {k} is replaced by a cyclic ring {{\bf Z}/q{\bf Z}} (with {q} not necessarily prime); this was achieved first for {d=2} and {q} square-free by Bourgain, Gamburd, and Sarnak, by Varju for general {d} and {q} square-free, and finally by this paper of Bourgain and Varju for arbitrary {d} and {q}.

Exercise 1 (Girth bound) Assuming Theorem 1, show that whenever {S} is a symmetric set of generators of {SL_d(k)} for some finite field {k} and some {d\geq 2}, then any element of {SL_d(k)} can be expressed as the product of {O_d( \log^{O_d(1)} |k| )} elements from {S}. (Equivalently, if we add the identity element to {S}, then {S^m = SL_d(k)} for some {m = O_d( \log^{O_d(1)} |k| )}.) This is a special case of a conjecture of Babai and Seress, who conjectured that the bound should hold uniformly for all finite simple groups (in particular, the implied constants here should not actually depend on {d}. The methods used to handle the {SL_d} case can handle other finite groups of Lie type of bounded rank, but at present we do not have bounds that are independent of the rank. On the other hand, a recent paper of Helfgott and Seress has almost resolved the conjecture for the permutation groups {A_n}.

A key tool to establish product theorems is an argument which is sometimes referred to as the pivot argument. To illustrate this argument, let us first discuss a much simpler (and older) theorem, essentially due to Freiman, which has a much weaker conclusion but is valid in any group {G}:

Theorem 2 (Baby product theorem) Let {G} be a group, and let {A} be a finite non-empty subset of {G}. Then one of the following statements hold:

  • (Expansion) One has {|A^{-1} A| \geq \frac{3}{2} |A|}.
  • (Close to a subgroup) {A} is contained in a left-coset of a group {H} with {|H| < \frac{3}{2} |A|}.

To prove this theorem, we suppose that the first conclusion does not hold, thus {|A^{-1} A| <\frac{3}{2} |A|}. Our task is then to place {A} inside the left-coset of a fairly small group {H}.

To do this, we take a group element {g \in G}, and consider the intersection {A\cap gA}. A priori, the size of this set could range from anywhere from {0} to {|A|}. However, we can use the hypothesis {|A^{-1} A| < \frac{3}{2} |A|} to obtain an important dichotomy, reminiscent of the classical fact that two cosets {gH, hH} of a subgroup {H} of {G} are either identical or disjoint:

Proposition 3 (Dichotomy) If {g \in G}, then exactly one of the following occurs:

  • (Non-involved case) {A \cap gA} is empty.
  • (Involved case) {|A \cap gA| > \frac{|A|}{2}}.

Proof: Suppose we are not in the pivot case, so that {A \cap gA} is non-empty. Let {a} be an element of {A \cap gA}, then {a} and {g^{-1} a} both lie in {A}. The sets {A^{-1} a} and {A^{-1} g^{-1} a} then both lie in {A^{-1} A}. As these sets have cardinality {|A|} and lie in {A^{-1}A}, which has cardinality less than {\frac{3}{2}|A|}, we conclude from the inclusion-exclusion formula that

\displaystyle |A^{-1} a \cap A^{-1} g^{-1} a| > \frac{|A|}{2}.

But the left-hand side is equal to {|A \cap gA|}, and the claim follows. \Box

The above proposition provides a clear separation between two types of elements {g \in G}: the “non-involved” elements, which have nothing to do with {A} (in the sense that {A \cap gA = \emptyset}, and the “involved” elements, which have a lot to do with {A} (in the sense that {|A \cap gA| > |A|/2}. The key point is that there is a significant “gap” between the non-involved and involved elements; there are no elements that are only “slightly involved”, in that {A} and {gA} intersect a little but not a lot. It is this gap that will allow us to upgrade approximate structure to exact structure. Namely,

Proposition 4 The set {H} of involved elements is a finite group, and is equal to {A A^{-1}}.

Proof: It is clear that the identity element {1} is involved, and that if {g} is involved then so is {g^{-1}} (since {A \cap g^{-1} A = g^{-1}(A \cap gA)}. Now suppose that {g, h} are both involved. Then {A \cap gA} and {A\cap hA} have cardinality greater than {|A|/2} and are both subsets of {A}, and so have non-empty intersection. In particular, {gA \cap hA} is non-empty, and so {A \cap g^{-1} hA} is non-empty. By Proposition 3, this makes {g^{-1} h} involved. It is then clear that {H} is a group.

If {g \in A A^{-1}}, then {A \cap gA} is non-empty, and so from Proposition 3 {g} is involved. Conversely, if {g} is involved, then {g \in A A^{-1}}. Thus we have {H = A A^{-1}} as claimed. In particular, {H} is finite. \Box

Now we can quickly wrap up the proof of Theorem 2. By construction, {A \cap gA| > |A|/2} for all {g \in H},which by double counting shows that {|H| < 2|A|}. As {H = A A^{-1}}, we see that {A} is contained in a right coset {Hg} of {H}; setting {H' := g^{-1} H g}, we conclude that {A} is contained in a left coset {gH'} of {H'}. {H'} is a conjugate of {H}, and so {|H'| < 2|A|}. If {h \in H'}, then {A} and {Ah} both lie in {H'} and have cardinality {|A|}, so must overlap; and so {h \in A A^{-1}}. Thus {A A^{-1} = H'}, and so {|H'| < \frac{3}{2} |A|}, and Theorem 2 follows.

Exercise 2 Show that the constant {3/2} in Theorem 2 cannot be replaced by any larger constant.

Exercise 3 Let {A \subset G} be a finite non-empty set such that {|A^2| < 2|A|}. Show that {AA^{-1}=A^{-1} A}. (Hint: If {ab^{-1} \in A A^{-1}}, show that {ab^{-1} = c^{-1} d} for some {c,d \in A}.)

Exercise 4 Let {A \subset G} be a finite non-empty set such that {|A^2| < \frac{3}{2} |A|}. Show that there is a finite group {H} with {|H| < \frac{3}{2} |A|} and a group element {g \in G} such that {A \subset Hg \cap gH} and {H = A A^{-1}}.

Below the fold, we give further examples of the pivot argument in other group-like situations, including Theorem 2 and also the “sum-product theorem” of Bourgain-Katz-Tao and Bourgain-Glibichuk-Konyagin.

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Emmanuel Breuillard, Ben Green and I have just uploaded to the arXiv the short paper “A nilpotent Freiman dimension lemma“, submitted to the special volume of the European Journal of Combinatorics in honour of Yahya Ould Hamidoune.  This paper is a nonabelian (or more precisely, nilpotent) variant of the following additive combinatorics lemma of Freiman:

Freiman’s lemma.  Let A be a finite subset of a Euclidean space with |A+A| \leq K|A|.  Then A is contained in an affine subspace of dimension at most {}\lfloor K-1 \rfloor.

This can be viewed as a “cheap” version of the more well known theorem of Freiman that places sets of small doubling in a torsion-free abelian group inside a generalised arithmetic progression.  The advantage here is that the bound on the dimension is extremely explicit.

Our main result is

Theorem.  Let A be a finite subset of a simply-connected nilpotent Lie group G which is a K-approximate group (i.e. A is symmetric, contains the identity, and A \cdot A can be covered by up to K left translates of A.  Then A can be covered by at most K^{2+29K^9} left-translates of a closed connected Lie subgroup of dimension at most K^9.

We remark that our previous paper established a similar result, in which the dimension bound was improved to 6\log_2 K, but at the cost of worsening the covering number to O_K(1), and with a much more complicated proof  (91 pages instead of 8). Furthermore, the bound on O_K(1) is ineffective, due to the use of ultraproducts in the argument (though it is likely that some extremely lousy explicit bound could eventually be squeezed out of the argument by finitising everything).  Note that the step of the ambient nilpotent group G does not influence the final bounds in the theorem, although we do of course need this step to be finite.  A simple quotienting argument allows one to deduce a corollary of the above theorem in which the ambient group is assumed to be residually torsion-free nilpotent instead of being a simply connected nilpotent Lie group, but we omit the statement of this corollary here.

To motivate the proof of this theorem, let us first show a simple case of an argument of Gleason, which is very much in the spirit of Freiman’s lemma:

Gleason Lemma (special case).  Let A be a finite symmetric subset of a Euclidean space, and let 0 = H_0 \subset H_1 \subset \ldots \subset H_k be a sequence of subspaces in this space, such that the sets 2A \cap H_i are strictly increasing in i for i=0,\ldots,k.  Then |5A| \geq k|A|, where 5A = A+A+A+A+A.

Proof.    By hypothesis, for each i=1,\ldots,k, the projection B_i of 2A \cap H_i to H_i / H_{i-1} is non-trivial, finite, and symmetric.   In particular, since the vector space H_i/H_{i-1} is torsion-free, B_i+B_i is strictly larger than B_i.  Equivalently, one can find a_i in (2A \cap H_i) + (2A \cap H_i) that does not lie in (2A \cap H_i) + H_{i-1}; in particular, a_i \in 4A \cap H_i and a_i+A is disjoint from H_{i-1}+A.  As a consequence, the a_i+A are disjoint and lie in 5A, whence the claim. \Box

Note that by combining the contrapositive of this lemma with a greedy algorithm, one can show that any K-approximate group in a Euclidean space is contained in a subspace of dimension at most K^4, which is a weak version of Freiman’s lemma.

To extend the argument to the nilpotent setting we use the following idea.  Observe that any non-trivial genuine subgroup H of a nilpotent group G will contain at least one non-trivial central element; indeed, by intersecting H with the lower central series G = G_1 \geq G_2 \geq G_3 \geq \ldots of G, and considering the last intersection H \cap G_k which is non-trivial, one obtains the claim.  It turns out that one can adapt this argument to approximate groups, so that any sufficiently large K-approximate subgroup A of G will contain a non-trivial element that centralises a large fraction of A.  Passing to this large fraction and quotienting out the central element, we obtain a new approximate group.    If, after a bounded number of steps, this procedure gives an approximate group of bounded size, we are basically done.  If, however, the process continues, then by using some Lie group theory, one can find a long sequence H_0 \leq H_1 \leq \ldots \leq H_k of connected Lie subgroups of G, such that the sets A^2 \cap H_i are strictly increasing in i.   Using some Lie group theory and the hypotheses on G, one can deduce that the group \langle A^2 \cap H_{i+1}\rangle generated by A^2 \cap H_{i+1} is much larger than \langle A^2 \cap H_i \rangle, in the sense that the latter group has infinite index in the former.   It then turns out that the Gleason argument mentioned above can be adapted to this setting.

In the previous set of notes we saw how a representation-theoretic property of groups, namely Kazhdan’s property (T), could be used to demonstrate expansion in Cayley graphs. In this set of notes we discuss a different representation-theoretic property of groups, namely quasirandomness, which is also useful for demonstrating expansion in Cayley graphs, though in a somewhat different way to property (T). For instance, whereas property (T), being qualitative in nature, is only interesting for infinite groups such as {SL_d({\bf Z})} or {SL_d({\bf R})}, and only creates Cayley graphs after passing to a finite quotient, quasirandomness is a quantitative property which is directly applicable to finite groups, and is able to deduce expansion in a Cayley graph, provided that random walks in that graph are known to become sufficiently “flat” in a certain sense.

The definition of quasirandomness is easy enough to state:

Definition 1 (Quasirandom groups) Let {G} be a finite group, and let {D \geq 1}. We say that {G} is {D}-quasirandom if all non-trivial unitary representations {\rho: G \rightarrow U(H)} of {G} have dimension at least {D}. (Recall a representation is trivial if {\rho(g)} is the identity for all {g \in G}.)

Exercise 1 Let {G} be a finite group, and let {D \geq 1}. A unitary representation {\rho: G \rightarrow U(H)} is said to be irreducible if {H} has no {G}-invariant subspaces other than {\{0\}} and {H}. Show that {G} is {D}-quasirandom if and only if every non-trivial irreducible representation of {G} has dimension at least {D}.

Remark 1 The terminology “quasirandom group” was introduced explicitly (though with slightly different notational conventions) by Gowers in 2008 in his detailed study of the concept; the name arises because dense Cayley graphs in quasirandom groups are quasirandom graphs in the sense of Chung, Graham, and Wilson, as we shall see below. This property had already been used implicitly to construct expander graphs by Sarnak and Xue in 1991, and more recently by Gamburd in 2002 and by Bourgain and Gamburd in 2008. One can of course define quasirandomness for more general locally compact groups than the finite ones, but we will only need this concept in the finite case. (A paper of Kunze and Stein from 1960, for instance, exploits the quasirandomness properties of the locally compact group {SL_2({\bf R})} to obtain mixing estimates in that group.)

Quasirandomness behaves fairly well with respect to quotients and short exact sequences:

Exercise 2 Let {0 \rightarrow H \rightarrow G \rightarrow K \rightarrow 0} be a short exact sequence of finite groups {H,G,K}.

  • (i) If {G} is {D}-quasirandom, show that {K} is {D}-quasirandom also. (Equivalently: any quotient of a {D}-quasirandom finite group is again a {D}-quasirandom finite group.)
  • (ii) Conversely, if {H} and {K} are both {D}-quasirandom, show that {G} is {D}-quasirandom also. (In particular, the direct or semidirect product of two {D}-quasirandom finite groups is again a {D}-quasirandom finite group.)

Informally, we will call {G} quasirandom if it is {D}-quasirandom for some “large” {D}, though the precise meaning of “large” will depend on context. For applications to expansion in Cayley graphs, “large” will mean “{D \geq |G|^c} for some constant {c>0} independent of the size of {G}“, but other regimes of {D} are certainly of interest.

The way we have set things up, the trivial group {G = \{1\}} is infinitely quasirandom (i.e. it is {D}-quasirandom for every {D}). This is however a degenerate case and will not be discussed further here. In the non-trivial case, a finite group can only be quasirandom if it is large and has no large subgroups:

Exercise 3 Let {D \geq 1}, and let {G} be a finite {D}-quasirandom group.

  • (i) Show that if {G} is non-trivial, then {|G| \geq D+1}. (Hint: use the mean zero component {\tau\downharpoonright_{\ell^2(G)_0}} of the regular representation {\tau: G \rightarrow U(\ell^2(G))}.) In particular, non-trivial finite groups cannot be infinitely quasirandom.
  • (ii) Show that any proper subgroup {H} of {G} has index {[G:H] \geq D+1}. (Hint: use the mean zero component of the quasiregular representation.)

The following exercise shows that quasirandom groups have to be quite non-abelian, and in particular perfect:

Exercise 4 (Quasirandomness, abelianness, and perfection) Let {G} be a finite group.

  • (i) If {G} is abelian and non-trivial, show that {G} is not {2}-quasirandom. (Hint: use Fourier analysis or the classification of finite abelian groups.)
  • (ii) Show that {G} is {2}-quasirandom if and only if it is perfect, i.e. the commutator group {[G,G]} is equal to {G}. (Equivalently, {G} is {2}-quasirandom if and only if it has no non-trivial abelian quotients.)

Later on we shall see that there is a converse to the above two exercises; any non-trivial perfect finite group with no large subgroups will be quasirandom.

Exercise 5 Let {G} be a finite {D}-quasirandom group. Show that for any subgroup {G'} of {G}, {G'} is {D/[G:G']}-quasirandom, where {[G:G'] := |G|/|G'|} is the index of {G'} in {G}. (Hint: use induced representations.)

Now we give an example of a more quasirandom group.

Lemma 2 (Frobenius lemma) If {F_p} is a field of some prime order {p}, then {SL_2(F_p)} is {\frac{p-1}{2}}-quasirandom.

This should be compared with the cardinality {|SL_2(F_p)|} of the special linear group, which is easily computed to be {(p^2-1) \times p = p^3 - p}.

Proof: We may of course take {p} to be odd. Suppose for contradiction that we have a non-trivial representation {\rho: SL_2(F_p) \rightarrow U_d({\bf C})} on a unitary group of some dimension {d} with {d < \frac{p-1}{2}}. Set {a} to be the group element

\displaystyle  a := \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix},

and suppose first that {\rho(a)} is non-trivial. Since {a^p=1}, we have {\rho(a)^p=1}; thus all the eigenvalues of {\rho(a)} are {p^{th}} roots of unity. On the other hand, by conjugating {a} by diagonal matrices in {SL_2(F_p)}, we see that {a} is conjugate to {a^m} (and hence {\rho(a)} conjugate to {\rho(a)^m}) whenever {m} is a quadratic residue mod {p}. As such, the eigenvalues of {\rho(a)} must be permuted by the operation {x \mapsto x^m} for any quadratic residue mod {p}. Since {\rho(a)} has at least one non-trivial eigenvalue, and there are {\frac{p-1}{2}} distinct quadratic residues, we conclude that {\rho(a)} has at least {\frac{p-1}{2}} distinct eigenvalues. But {\rho(a)} is a {d \times d} matrix with {d < \frac{p-1}{2}}, a contradiction. Thus {a} lies in the kernel of {\rho}. By conjugation, we then see that this kernel contains all unipotent matrices. But these matrices generate {SL_2(F_p)} (see exercise below), and so {\rho} is trivial, a contradiction. \Box

Exercise 6 Show that for any prime {p}, the unipotent matrices

\displaystyle  \begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} 1 & 0 \\ t & 1 \end{pmatrix}

for {t} ranging over {F_p} generate {SL_2(F_p)} as a group.

Exercise 7 Let {G} be a finite group, and let {D \geq 1}. If {G} is generated by a collection {G_1,\ldots,G_k} of {D}-quasirandom subgroups, show that {G} is itself {D}-quasirandom.

Exercise 8 Show that {SL_d(F_p)} is {\frac{p-1}{2}}-quasirandom for any {d \geq 2} and any prime {p}. (This is not sharp; the optimal bound here is {\gg_d p^{d-1}}, which follows from the results of Landazuri and Seitz.)

As a corollary of the above results and Exercise 2, we see that the projective special linear group {PSL_d(F_p)} is also {\frac{p-1}{2}}-quasirandom.

Remark 2 One can ask whether the bound {\frac{p-1}{2}} in Lemma 2 is sharp, assuming of course that {p} is odd. Noting that {SL_2(F_p)} acts linearly on the plane {F_p^2}, we see that it also acts projectively on the projective line {PF_p^1 := (F_p^2 \backslash \{0\}) / F_p^\times}, which has {p+1} elements. Thus {SL_2(F_p)} acts via the quasiregular representation on the {p+1}-dimensional space {\ell^2(PF_p^1)}, and also on the {p}-dimensional subspace {\ell^2(PF_p^1)_0}; this latter representation (known as the Steinberg representation) is irreducible. This shows that the {\frac{p-1}{2}} bound cannot be improved beyond {p}. More generally, given any character {\chi: F_p^\times \rightarrow S^1}, {SL_2(F_p)} acts on the {p+1}-dimensional space {V_\chi} of functions {f \in \ell^2( F_p^2 \backslash \{0\} )} that obey the twisted dilation invariance {f(tx) = \chi(t) f(x)} for all {t \in F_p^\times} and {x \in F_p^2 \backslash \{0\}}; these are known as the principal series representations. When {\chi} is the trivial character, this is the quasiregular representation discussed earlier. For most other characters, this is an irreducible representation, but it turns out that when {\chi} is the quadratic representation (thus taking values in {\{-1,+1\}} while being non-trivial), the principal series representation splits into the direct sum of two {\frac{p+1}{2}}-dimensional representations, which comes very close to matching the bound in Lemma 2. There is a parallel series of representations to the principal series (known as the discrete series) which is more complicated to describe (roughly speaking, one has to embed {F_p} in a quadratic extension {F_{p^2}} and then use a rotated version of the above construction, to change a split torus into a non-split torus), but can generate irreducible representations of dimension {\frac{p-1}{2}}, showing that the bound in Lemma 2 is in fact exactly sharp. These constructions can be generalised to arbitrary finite groups of Lie type using Deligne-Luzstig theory, but this is beyond the scope of this course (and of my own knowledge in the subject).

Exercise 9 Let {p} be an odd prime. Show that for any {n \geq p+2}, the alternating group {A_n} is {p-1}-quasirandom. (Hint: show that all cycles of order {p} in {A_n} are conjugate to each other in {A_n} (and not just in {S_n}); in particular, a cycle is conjugate to its {j^{th}} power for all {j=1,\ldots,p-1}. Also, as {n \geq 5}, {A_n} is simple, and so the cycles of order {p} generate the entire group.)

Remark 3 By using more precise information on the representations of the alternating group (using the theory of Specht modules and Young tableaux), one can show the slightly sharper statement that {A_n} is {n-1}-quasirandom for {n \geq 6} (but is only {3}-quasirandom for {n=5} due to icosahedral symmetry, and {1}-quasirandom for {n \leq 4} due to lack of perfectness). Using Exercise 3 with the index {n} subgroup {A_{n-1}}, we see that the bound {n-1} cannot be improved. Thus, {A_n} (for large {n}) is not as quasirandom as the special linear groups {SL_d(F_p)} (for {p} large and {d} bounded), because in the latter case the quasirandomness is as strong as a power of the size of the group, whereas in the former case it is only logarithmic in size.

If one replaces the alternating group {A_n} with the slightly larger symmetric group {S_n}, then quasirandomness is destroyed (since {S_n}, having the abelian quotient {S_n/A_n}, is not perfect); indeed, {S_n} is {1}-quasirandom and no better.

Remark 4 Thanks to the monumental achievement of the classification of finite simple groups, we know that apart from a finite number (26, to be precise) of sporadic exceptions, all finite simple groups (up to isomorphism) are either a cyclic group {{\bf Z}/p{\bf Z}}, an alternating group {A_n}, or is a finite simple group of Lie type such as {PSL_d(F_p)}. (We will define the concept of a finite simple group of Lie type more precisely in later notes, but suffice to say for now that such groups are constructed from reductive algebraic groups, for instance {PSL_d(F_p)} is constructed from {SL_d} in characteristic {p}.) In the case of finite simple groups {G} of Lie type with bounded rank {r=O(1)}, it is known from the work of Landazuri and Seitz that such groups are {\gg |G|^c}-quasirandom for some {c>0} depending only on the rank. On the other hand, by the previous remark, the large alternating groups do not have this property, and one can show that the finite simple groups of Lie type with large rank also do not have this property. Thus, we see using the classification that if a finite simple group {G} is {|G|^c}-quasirandom for some {c>0} and {|G|} is sufficiently large depending on {c}, then {G} is a finite simple group of Lie type with rank {O_c(1)}. It would be of interest to see if there was an alternate way to establish this fact that did not rely on the classification, as it may lead to an alternate approach to proving the classification (or perhaps a weakened version thereof).

A key reason why quasirandomness is desirable for the purposes of demonstrating expansion is that quasirandom groups happen to be rapidly mixing at large scales, as we shall see below the fold. As such, quasirandomness is an important tool for demonstrating expansion in Cayley graphs, though because expansion is a phenomenon that must hold at all scales, one needs to supplement quasirandomness with some additional input that creates mixing at small or medium scales also before one can deduce expansion. As an example of this technique of combining quasirandomness with mixing at small and medium scales, we present a proof (due to Sarnak-Xue, and simplified by Gamburd) of a weak version of the famous “3/16 theorem” of Selberg on the least non-trivial eigenvalue of the Laplacian on a modular curve, which among other things can be used to construct a family of expander Cayley graphs in {SL_2({\bf Z}/N{\bf Z})} (compare this with the property (T)-based methods in the previous notes, which could construct expander Cayley graphs in {SL_d({\bf Z}/N{\bf Z})} for any fixed {d \geq 3}).

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In the previous set of notes we introduced the notion of expansion in arbitrary {k}-regular graphs. For the rest of the course, we will now focus attention primarily to a special type of {k}-regular graph, namely a Cayley graph.

Definition 1 (Cayley graph) Let {G = (G,\cdot)} be a group, and let {S} be a finite subset of {G}. We assume that {S} is symmetric (thus {s^{-1} \in S} whenever {s \in S}) and does not contain the identity {1} (this is to avoid loops). Then the (right-invariant) Cayley graph {Cay(G,S)} is defined to be the graph with vertex set {G} and edge set {\{ \{sx,x\}: x \in G, s \in S \}}, thus each vertex {x \in G} is connected to the {|S|} elements {sx} for {s \in S}, and so {Cay(G,S)} is a {|S|}-regular graph.

Example 1 The graph in Exercise 3 of Notes 1 is the Cayley graph on {{\bf Z}/N{\bf Z}} with generators {S = \{-1,+1\}}.

Remark 1 We call the above Cayley graphs right-invariant because every right translation {x\mapsto xg} on {G} is a graph automorphism of {Cay(G,S)}. This group of automorphisms acts transitively on the vertex set of the Cayley graph. One can thus view a Cayley graph as a homogeneous space of {G}, as it “looks the same” from every vertex. One could of course also consider left-invariant Cayley graphs, in which {x} is connected to {xs} rather than {sx}. However, the two such graphs are isomorphic using the inverse map {x \mapsto x^{-1}}, so we may without loss of generality restrict our attention throughout to left Cayley graphs.

Remark 2 For minor technical reasons, it will be convenient later on to allow {S} to contain the identity and to come with multiplicity (i.e. it will be a multiset rather than a set). If one does so, of course, the resulting Cayley graph will now contain some loops and multiple edges.

For the purposes of building expander families, we would of course want the underlying group {G} to be finite. However, it will be convenient at various times to “lift” a finite Cayley graph up to an infinite one, and so we permit {G} to be infinite in our definition of a Cayley graph.

We will also sometimes consider a generalisation of a Cayley graph, known as a Schreier graph:

Definition 2 (Schreier graph) Let {G} be a finite group that acts (on the left) on a space {X}, thus there is a map {(g,x) \mapsto gx} from {G \times X} to {X} such that {1x = x} and {(gh)x = g(hx)} for all {g,h \in G} and {x \in X}. Let {S} be a symmetric subset of {G} which acts freely on {X} in the sense that {sx \neq x} for all {s \in S} and {x \in X}, and {sx \neq s'x} for all distinct {s,s' \in S} and {x \in X}. Then the Schreier graph {Sch(X,S)} is defined to be the graph with vertex set {X} and edge set {\{ \{sx,x\}: x \in X, s \in S \}}.

Example 2 Every Cayley graph {Cay(G,S)} is also a Schreier graph {Sch(G,S)}, using the obvious left-action of {G} on itself. The {k}-regular graphs formed from {l} permutations {\pi_1,\ldots,\pi_l \in S_n} that were studied in the previous set of notes is also a Schreier graph provided that {\pi_i(v) \neq v, \pi_i^{-1}(v), \pi_j(v)} for all distinct {1 \leq i,j \leq l}, with the underlying group being the permutation group {S_n} (which acts on the vertex set {X = \{1,\ldots,n\}} in the obvious manner), and {S := \{\pi_1,\ldots,\pi_l,\pi_1^{-1},\ldots,\pi_l^{-1}\}}.

Exercise 1 If {k} is an even integer, show that every {k}-regular graph is a Schreier graph involving a set {S} of generators of cardinality {k/2}. (Hint: first show that every {k}-regular graph can be decomposed into {k/2} unions of cycles, each of which partition the vertex set, then use the previous example.

We return now to Cayley graphs. It is easy to characterise qualitative expansion properties of Cayley graphs:

Exercise 2 (Qualitative expansion) Let {Cay(G,S)} be a finite Cayley graph.

  • (i) Show that {Cay(G,S)} is a one-sided {\epsilon}-expander for {G} for some {\epsilon>0} if and only if {S} generates {G}.
  • (ii) Show that {Cay(G,S)} is a two-sided {\epsilon}-expander for {G} for some {\epsilon>0} if and only if {S} generates {G}, and furthermore {S} intersects each index {2} subgroup of {G}.

We will however be interested in more quantitative expansion properties, in which the expansion constant {\epsilon} is independent of the size of the Cayley graph, so that one can construct non-trivial expander families {Cay(G_n,S_n)} of Cayley graphs.

One can analyse the expansion of Cayley graphs in a number of ways. For instance, by taking the edge expansion viewpoint, one can study Cayley graphs combinatorially, using the product set operation

\displaystyle  A \cdot B := \{ab: a \in A, b \in B \}

of subsets of {G}.

Exercise 3 (Combinatorial description of expansion) Let {Cay(G_n,S_n)} be a family of finite {k}-regular Cayley graphs. Show that {Cay(G_n,S_n)} is a one-sided expander family if and only if there is a constant {c>0} independent of {n} such that {|E_n \cup E_n S_n| \geq (1+c) |E_n|} for all sufficiently large {n} and all subsets {E_n} of {G_n} with {|E_n| \leq |G_n|/2}.

One can also give a combinatorial description of two-sided expansion, but it is more complicated and we will not use it here.

Exercise 4 (Abelian groups do not expand) Let {Cay(G_n,S_n)} be a family of finite {k}-regular Cayley graphs, with the {G_n} all abelian, and the {S_n} generating {G_n}. Show that {Cay(G_n,S_n)} are a one-sided expander family if and only if the Cayley graphs have bounded cardinality (i.e. {\sup_n |G_n| < \infty}). (Hint: assume for contradiction that {Cay(G_n,S_n)} is a one-sided expander family with {|G_n| \rightarrow \infty}, and show by two different arguments that {\sup_n |S_n^m|} grows at least exponentially in {m} and also at most polynomially in {m}, giving the desired contradiction.)

The left-invariant nature of Cayley graphs also suggests that such graphs can be profitably analysed using some sort of Fourier analysis; as the underlying symmetry group is not necessarily abelian, one should use the Fourier analysis of non-abelian groups, which is better known as (unitary) representation theory. The Fourier-analytic nature of Cayley graphs can be highlighted by recalling the operation of convolution of two functions {f, g \in \ell^2(G)}, defined by the formula

\displaystyle  f * g(x) := \sum_{y \in G} f(y) g(y^{-1} x) = \sum_{y \in G} f(x y^{-1}) g(y).

This convolution operation is bilinear and associative (at least when one imposes a suitable decay condition on the functions, such as compact support), but is not commutative unless {G} is abelian. (If one is more algebraically minded, one can also identify {\ell^2(G)} (when {G} is finite, at least) with the group algebra {{\bf C} G}, in which case convolution is simply the multiplication operation in this algebra.) The adjacency operator {A} on a Cayley graph {Cay(G,S)} can then be viewed as a convolution

\displaystyle  Af = |S| \mu * f,

where {\mu} is the probability density

\displaystyle  \mu := \frac{1}{|S|} \sum_{s \in S} \delta_s \ \ \ \ \ (1)

where {\delta_s} is the Kronecker delta function on {s}. Using the spectral definition of expansion, we thus see that {Cay(G,S)} is a one-sided expander if and only if

\displaystyle  \langle f, \mu*f \rangle \leq (1-\epsilon) \|f\|_{\ell^2(G)} \ \ \ \ \ (2)

whenever {f \in \ell^2(G)} is orthogonal to the constant function {1}, and is a two-sided expander if

\displaystyle  \| \mu*f \|_{\ell^2(G)} \leq (1-\epsilon) \|f\|_{\ell^2(G)} \ \ \ \ \ (3)

whenever {f \in \ell^2(G)} is orthogonal to the constant function {1}.

We remark that the above spectral definition of expansion can be easily extended to symmetric sets {S} which contain the identity or have multiplicity (i.e. are multisets). (We retain symmetry, though, in order to keep the operation of convolution by {\mu} self-adjoint.) In particular, one can say (with some slight abuse of notation) that a set of elements {s_1,\ldots,s_l} of {G} (possibly with repetition, and possibly with some elements equalling the identity) generates a one-sided or two-sided {\epsilon}-expander if the associated symmetric probability density

\displaystyle  \mu := \frac{1}{2l} \sum_{i=1}^l \delta_{s_i} + \delta_{s_i^{-1}}

obeys either (2) or (3).

We saw in the last set of notes that expansion can be characterised in terms of random walks. One can of course specialise this characterisation to the Cayley graph case:

Exercise 5 (Random walk description of expansion) Let {Cay(G_n,S_n)} be a family of finite {k}-regular Cayley graphs, and let {\mu_n} be the associated probability density functions. Let {A > 1/2} be a constant.

  • Show that the {Cay(G_n,S_n)} are a two-sided expander family if and only if there exists a {C>0} such that for all sufficiently large {n}, one has {\| \mu_n^{*m} - \frac{1}{|G_n|} \|_{\ell^2(G_n)} \leq \frac{1}{|G_n|^A}} for some {m \leq C \log |G_n|}, where {\mu_n^{*m} := \mu_n * \ldots * \mu_n} denotes the convolution of {m} copies of {\mu_n}.
  • Show that the {Cay(G_n,S_n)} are a one-sided expander family if and only if there exists a {C>0} such that for all sufficiently large {n}, one has {\| (\frac{1}{2} \delta_1 + \frac{1}{2} \mu_n)^{*m} - \frac{1}{|G_n|} \|_{\ell^2(G_n)} \leq \frac{1}{|G_n|^A}} for some {m \leq C \log |G_n|}.

In this set of notes, we will connect expansion of Cayley graphs to an important property of certain infinite groups, known as Kazhdan’s property (T) (or property (T) for short). In 1973, Margulis exploited this property to create the first known explicit and deterministic examples of expanding Cayley graphs. As it turns out, property (T) is somewhat overpowered for this purpose; in particular, we now know that there are many families of Cayley graphs for which the associated infinite group does not obey property (T) (or weaker variants of this property, such as property {\tau}). In later notes we will therefore turn to other methods of creating Cayley graphs that do not rely on property (T). Nevertheless, property (T) is of substantial intrinsic interest, and also has many connections to other parts of mathematics than the theory of expander graphs, so it is worth spending some time to discuss it here.

The material here is based in part on this recent text on property (T) by Bekka, de la Harpe, and Valette (available online here).

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Let {G = (G,+)} be a finite additive group. A tiling pair is a pair of non-empty subsets {A, B} such that every element of {G} can be written in exactly one way as a sum of an element {a} of {A} and an element {b} of {B}, in which case we write {G = A \oplus B}. The sets {A, B} are then called tiles, with {B} being a complementary tile to {A} and vice versa. For instance, every subgroup {H} of {G} is a tile, as one can pick one representative from each coset of {H} to form the complementary tile. Conversely, any set formed by taking one representative from each coset of {H} is also a tile.

Tiles can be quite complicated, particularly when the group {G} is “high-dimensional”. We will therefore restrict to the simple case of a cyclic group {G = {\bf Z}/N{\bf Z}}, and restrict even further to the special case when the modulus {N} is square-free. Here, the situation should be much simpler. In particular, we have the following conjecture of Coven and Meyerowitz, which asserts that the previous construction of a tile is, in fact, the only such construction:

Conjecture 1 (Coven-Meyerowitz conjecture, square-free case) Let {N} be square-free, and let {A} be a tile of {{\bf Z}/N{\bf Z}}. Then there exists a subgroup {H} of {{\bf Z}/N{\bf Z}} such that {A} consists of a single representative from each coset of {H}.

Note that in the square-free case, every subgroup {H} of {{\bf Z}/N{\bf Z}} has a complementary subgroup {H^\perp} (thus {{\bf Z}/N{\bf Z} = H \oplus H^\perp}). In particular, {H} consists of a single representative from each coset of {H^\perp}, and so the examples of subgroups of {{\bf Z}/N{\bf Z}} are covered by the above conjecture in the square-free case.

In the non-square free case, the above assertion is not true; for instance, if {p} is a prime, then the multiples of {p} in {{\bf Z}/p^2{\bf Z}} are a tile, but cannot be formed from taking a single representative from all the cosets of a given subgroup. There is a more general conjecture of Coven and Meyerowitz to handle this more general case, although it is more difficult to state:

Conjecture 2 (Coven-Meyerowitz conjecture, general case) Let {N} be a natural number, and let {A} be a tile of {{\bf Z}/N{\bf Z}}. Then there exists a set {S_A} of prime powers with {|A| = \prod_{p^j \in S_A} p} such that the Fourier transform

\displaystyle  \hat 1_A(k) := \sum_{n \in A} e^{2\pi i kn / N}

vanishes whenever {k} is a non-zero element of {{\bf Z}/N{\bf Z}} whose order is the product of elements of {S_A} that are powers of distinct primes. Equivalently, the generating polynomial {\sum_{n \in A} x^n} is divisible by the cyclotomic polynomials {\phi_m} whenever {m} is the product of elements of {S_A} that are powers of distinct primes.

It can be shown (with a modest amount of effort) that Conjecture 2 implies Conjecture 1, but we will not do so here, focusing instead exclusively on the square-free case for simplicity.

It was observed by Laba that Conjecture 2 is connected to the following well-known conjecture of Fuglede:

Conjecture 3 (One-dimensional Fuglede conjecture, tiling to spectral direction) Let {E} be a compact subset of {{\bf R}} of positive measure which is a tile (thus {{\bf R} = E \oplus \Lambda} for some set {\Lambda \subset {\bf R}}). Then {L^2(E)} (with Lebesgue measure) has a spectrum, that is to say an orthogonal set of plane waves {x \mapsto e^{2\pi i \xi x}}.

Indeed, it was shown by Laba that Conjecture 2 implies Conjecture 3 in the case when {E} is the finite union of unit intervals. Actually, thanks to the more recent work of Farkas, Matolcsi, and Mora we know that Conjecture 2 in fact implies the universal spectrum conjecture of Lagarias and Wang, which in turn was known to imply Conjecture 3 in full generality. (On the other hand, the conjecture fails in four and higher dimensions; see the papers of Kolountzakis-Matolcsi and of Farkas-Revesz.)

Given the simple statement of Conjecture 1, it is perhaps somewhat surprising that it remains open, even in simple cases such as when {N} is the product of just four primes. One reason for this is that some plausible strengthenings of this conjecture (such as the Tijdeman-Sands conjecture) are known to be false, as we will review below. On the other hand, as we shall see, tiling sets have a lot of combinatorial structure, and in principle one should be able to work out a lot of special cases of the conjecture. Given the combinatorial nature of this problem, it may well be quite suitable for a polymath project in fact, if there is sufficient interest.

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In the last set of notes, we obtained the following structural theorem concerning approximate groups:

Theorem 1 Let {A} be a finite {K}-approximate group. Then there exists a coset nilprogression {P} of rank and step {O_K(1)} contained in {A^4}, such that {A} is covered by {O_K(1)} left-translates of {P} (and hence also by {O_K(1)} right-translates of {P}).

Remark 1 Under some mild additional hypotheses (e.g. if the dimensions of {P} are sufficiently large, or if {P} is placed in a certain “normal form”, details of which may be found in this paper), a coset nilprogression {P} of rank and step {O_K(1)} will be an {O_K(1)}-approximate group, thus giving a partial converse to Theorem 1. (It is not quite a full converse though, even if one works qualitatively and forgets how the constants depend on {K}: if {A} is covered by a bounded number of left- and right-translates {gP, Pg} of {P}, one needs the group elements {g} to “approximately normalise” {P} in some sense if one wants to then conclude that {A} is an approximate group.) The mild hypotheses alluded to above can be enforced in the statement of the theorem, but we will not discuss this technicality here, and refer the reader to the above-mentioned paper for details.

By placing the coset nilprogression in a virtually nilpotent group, we have the following corollary in the global case:

Corollary 2 Let {A} be a finite {K}-approximate group in an ambient group {G}. Then {A} is covered by {O_K(1)} left cosets of a virtually nilpotent subgroup {G'} of {G}.

In this final set of notes, we give some applications of the above results. The first application is to replace “{K}-approximate group” by “sets of bounded doubling”:

Proposition 3 Let {A} be a finite non-empty subset of a (global) group {G} such that {|A^2| \leq K |A|}. Then there exists a coset nilprogression {P} of rank and step {O_K(1)} and cardinality {|P| \gg_K |A|} such that {A} can be covered by {O_K(1)} left-translates of {P}, and also by {O_K(1)} right-translates of {P}.

We will also establish (a strengthening of) a well-known theorem of Gromov on groups of polynomial growth, as promised back in Notes 0, as well as a variant result (of a type known as a “generalised Margulis lemma”) controlling the almost stabilisers of discrete actions of isometries.

The material here is largely drawn from my recent paper with Emmanuel Breuillard and Ben Green.

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A common theme in mathematical analysis (particularly in analysis of a “geometric” or “statistical” flavour) is the interplay between “macroscopic” and “microscopic” scales. These terms are somewhat vague and imprecise, and their interpretation depends on the context and also on one’s choice of normalisations, but if one uses a “macroscopic” normalisation, “macroscopic” scales correspond to scales that are comparable to unit size (i.e. bounded above and below by absolute constants), while “microscopic” scales are much smaller, being the minimal scale at which nontrivial behaviour occurs. (Other normalisations are possible, e.g. making the microscopic scale a unit scale, and letting the macroscopic scale go off to infinity; for instance, such a normalisation is often used, at least initially, in the study of groups of polynomial growth. However, for the theory of approximate groups, a macroscopic scale normalisation is more convenient.)

One can also consider “mesoscopic” scales which are intermediate between microscopic and macroscopic scales, or large-scale behaviour at scales that go off to infinity (and in particular are larger than the macroscopic range of scales), although the behaviour of these scales will not be the main focus of this post. Finally, one can divide the macroscopic scales into “local” macroscopic scales (less than {\epsilon} for some small but fixed {\epsilon>0}) and “global” macroscopic scales (scales that are allowed to be larger than a given large absolute constant {C}). For instance, given a finite approximate group {A}:

  • Sets such as {A^m} for some fixed {m} (e.g. {A^{10}}) can be considered to be sets at a global macroscopic scale. Sending {m} to infinity, one enters the large-scale regime.
  • Sets such as the sets {S} that appear in the Sanders lemma from the previous set of notes (thus {S^m \subset A^4} for some fixed {m}, e.g. {m=100}) can be considered to be sets at a local macroscopic scale. Sending {m} to infinity, one enters the mesoscopic regime.
  • The non-identity element {u} of {A} that is “closest” to the identity in some suitable metric (cf. the proof of Jordan’s theorem from Notes 0) would be an element associated to the microscopic scale. The orbit {u, u^2, u^3, \ldots} starts out at microscopic scales, and (assuming some suitable “escape” axioms) will pass through mesoscopic scales and finally entering the macroscopic regime. (Beyond this point, the orbit may exhibit a variety of behaviours, such as periodically returning back to the smaller scales, diverging off to ever larger scales, or filling out a dense subset of some macroscopic set; the escape axioms we will use do not exclude any of these possibilities.)

For comparison, in the theory of locally compact groups, properties about small neighbourhoods of the identity (e.g. local compactness, or the NSS property) would be properties at the local macroscopic scale, whereas the space {L(G)} of one-parameter subgroups can be interpreted as an object at the microscopic scale. The exponential map then provides a bridge connecting the microscopic and macroscopic scales.

We return now to approximate groups. The macroscopic structure of these objects is well described by the Hrushovski Lie model theorem from the previous set of notes, which informally asserts that the macroscopic structure of an (ultra) approximate group can be modeled by a Lie group. This is already an important piece of information about general approximate groups, but it does not directly reveal the full structure of such approximate groups, because these Lie models are unable to see the microscopic behaviour of these approximate groups.

To illustrate this, let us review one of the examples of a Lie model of an ultra approximate group, namely Exercise 28 from Notes 7. In this example one studied a “nilbox” from a Heisenberg group, which we rewrite here in slightly different notation. Specifically, let {G} be the Heisenberg group

\displaystyle  G := \{ (a,b,c): a,b,c \in {\bf Z} \}

with group law

\displaystyle  (a,b,c) \ast (a',b',c') := (a+a', b+b', c+c'+ab') \ \ \ \ \ (1)

and let {A = \prod_{n \rightarrow \alpha} A_n}, where {A_n \subset G} is the box

\displaystyle  A_n := \{ (a,b,c) \in G: |a|, |b| \leq n; |c| \leq n^{10} \};

thus {A} is the nonstandard box

\displaystyle  A := \{ (a,b,c) \in {}^* G: |a|, |b| \leq N; |c| \leq N^{10} \}

where {N := \lim_{n \rightarrow \alpha} n}. As the above exercise establishes, {A \cup A^{-1}} is an ultra approximate group with a Lie model {\pi: \langle A \rangle \rightarrow {\bf R}^3} given by the formula

\displaystyle  \pi( a, b, c ) := ( \hbox{st} \frac{a}{N}, \hbox{st} \frac{b}{N}, \hbox{st} \frac{c}{N^{10}} )

for {a,b = O(N)} and {c = O(N^{10})}. Note how the nonabelian nature of {G} (arising from the {ab'} term in the group law (1)) has been lost in the model {{\bf R}^3}, because the effect of that nonabelian term on {\frac{c}{N^{10}}} is only {O(\frac{N^2}{N^8})} which is infinitesimal and thus does not contribute to the standard part. In particular, if we replace {G} with the abelian group {G' := \{(a,b,c): a,b,c \in {\bf Z} \}} with the additive group law

\displaystyle  (a,b,c) \ast' (a',b',c') := (a+a',b+b',c+c')

and let {A'} and {\pi'} be defined exactly as with {A} and {\pi}, but placed inside the group structure of {G'} rather than {G}, then {A \cup A^{-1}} and {A' \cup (A')^{-1}} are essentially “indistinguishable” as far as their models by {{\bf R}^3} are concerned, even though the latter approximate group is abelian and the former is not. The problem is that the nonabelian-ness in the former example is so microscopic that it falls entirely inside the kernel of {\pi} and is thus not detected at all by the model.

The problem of not being able to “see” the microscopic structure of a group (or approximate group) also was a key difficulty in the theory surrounding Hilbert’s fifth problem that was discussed in previous notes. A key tool in being able to resolve such structure was to build left-invariant metrics {d} (or equivalently, norms {\| \|}) on one’s group, which obeyed useful “Gleason axioms” such as the commutator axiom

\displaystyle  \| [g,h] \| \ll \|g\| \|h\| \ \ \ \ \ (2)

for sufficiently small {g,h}, or the escape axiom

\displaystyle  \| g^n \| \gg |n| \|g\| \ \ \ \ \ (3)

when {|n| \|g\|} was sufficiently small. Such axioms have important and non-trivial content even in the microscopic regime where {g} or {h} are extremely close to the identity. For instance, in the proof of Jordan’s theorem from Notes 0, which showed that any finite unitary group {G} was boundedly virtually abelian, a key step was to apply the commutator axiom (2) (for the distance to the identity in operator norm) to the most “microscopic” element of {G}, or more precisely a non-identity element of {G} of minimal norm. The key point was that this microscopic element was virtually central in {G}, and as such it restricted much of {G} to a lower-dimensional subgroup of the unitary group, at which point one could argue using an induction-on-dimension argument. As we shall see, a similar argument can be used to place “virtually nilpotent” structure on finite approximate groups. For instance, in the Heisenberg-type approximate groups {A \cup A^{-1}} and {A' \cup (A')^{-1}} discussed earlier, the element {(0,0,1)} will be “closest to the origin” in a suitable sense to be defined later, and is centralised by both approximate groups; quotienting out (the orbit of) that central element and iterating the process two more times, we shall see that one can express both {A \cup A^{-1}} and {A'\cup (A')^{-1}} as a tower of central cyclic extensions, which in particular establishes the nilpotency of both groups.

The escape axiom (3) is a particularly important axiom in connecting the microscopic structure of a group {G} to its macroscopic structure; for instance, as shown in Notes 2, this axiom (in conjunction with the closely related commutator axiom) tends to imply dilation estimates such as {d( g^n, h^n ) \sim n d(g,h)} that allow one to understand the microscopic geometry of points {g,h} close to the identity in terms of the (local) macroscopic geometry of points {g^n, h^n} that are significantly further away from the identity.

It is thus of interest to build some notion of a norm (or left-invariant metrics) on an approximate group {A} that obeys the escape and commutator axioms (while being non-degenerate enough to adequately capture the geometry of {A} in some sense), in a fashion analogous to the Gleason metrics that played such a key role in the theory of Hilbert’s fifth problem. It is tempting to use the Lie model theorem to do this, since Lie groups certainly come with Gleason metrics. However, if one does this, one ends up, roughly speaking, with a norm on {A} that only obeys the escape and commutator estimates macroscopically; roughly speaking, this means that one has a macroscopic commutator inequality

\displaystyle  \| [g,h] \| \ll \|g\| \|h\| + o(1)

and a macroscopic escape property

\displaystyle  \| g^n \| \gg |n| \|g\| - o(|n|)

but such axioms are too weak for analysis at the microscopic scale, and in particular in establishing centrality of the element closest to the identity.

Another way to proceed is to build a norm that is specifically designed to obey the crucial escape property. Given an approximate group {A} in a group {G}, and an element {g} of {G}, we can define the escape norm {\|g\|_{e,A}} of {g} by the formula

\displaystyle  \| g \|_{e,A} := \inf \{ \frac{1}{n+1}: n \in {\bf N}: g, g^2, \ldots, g^n \in A \}.

Thus, {\|g\|_{e,A}} equals {1} if {g} lies outside of {A}, equals {1/2} if {g} lies in {A} but {g^2} lies outside of {A}, and so forth. Such norms had already appeared in Notes 4, in the context of analysing NSS groups.

As it turns out, this expression will obey an escape axiom, as long as we place some additional hypotheses on {A} which we will present shortly. However, it need not actually be a norm; in particular, the triangle inequality

\displaystyle  \|gh\|_{e,A} \leq \|g\|_{e,A} + \|h\|_{e,A}

is not necessarily true. Fortunately, it turns out that by a (slightly more complicated) version of the Gleason machinery from Notes 4 we can establish a usable substitute for this inequality, namely the quasi-triangle inequality

\displaystyle  \|g_1 \ldots g_k \|_{e,A} \leq C (\|g_1\|_{e,A} + \ldots + \|g_k\|_{e,A}),

where {C} is a constant independent of {k}. As we shall see, these estimates can then be used to obtain a commutator estimate (2).

However, to do all this, it is not enough for {A} to be an approximate group; it must obey two additional “trapping” axioms that improve the properties of the escape norm. We formalise these axioms (somewhat arbitrarily) as follows:

Definition 1 (Strong approximate group) Let {K \geq 1}. A strong {K}-approximate group is a finite {K}-approximate group {A} in a group {G} with a symmetric subset {S} obeying the following axioms:

  • ({S} small) One has

    \displaystyle  (S^{A^4})^{1000K^3} \subset A. \ \ \ \ \ (4)

  • (First trapping condition) If {g, g^2, \ldots, g^{1000} \in A^{100}}, then {g \in A}.
  • (Second trapping condition) If {g, g^2, \ldots, g^{10^6 K^3} \in A}, then {g \in S}.

An ultra strong {K}-approximate group is an ultraproduct {A = \prod_{n \rightarrow \alpha} A_n} of strong {K}-approximate groups.

The first trapping condition can be rewritten as

\displaystyle  \|g\|_{e,A} \leq 1000 \|g\|_{e,A^{100}}

and the second trapping condition can similarly be rewritten as

\displaystyle  \|g\|_{e,S} \leq 10^6 K^3 \|g\|_{e,A}.

This makes the escape norms of {A, A^{100}}, and {S} comparable to each other, which will be needed for a number of reasons (and in particular to close a certain bootstrap argument properly). Compare this with equation (12) from Notes 4, which used the NSS hypothesis to obtain similar conclusions. Thus, one can view the strong approximate group axioms as being a sort of proxy for the NSS property.

Example 1 Let {N} be a large natural number. Then the interval {A = [-N,N]} in the integers is a {2}-approximate group, which is also a strong {2}-approximate group (setting {S = [10^{-6} N, 10^{-6} N]}, for instance). On the other hand, if one places {A} in {{\bf Z}/5N{\bf Z}} rather than in the integers, then the first trapping condition is lost and one is no longer a strong {2}-approximate group. Also, if one remains in the integers, but deletes a few elements from {A}, e.g. deleting {\pm \lfloor 10^{-10} N\rfloor} from {A}), then one is still a {O(1)}-approximate group, but is no longer a strong {O(1)}-approximate group, again because the first trapping condition is lost.

A key consequence of the Hrushovski Lie model theorem is that it allows one to replace approximate groups by strong approximate groups:

Exercise 1 (Finding strong approximate groups)

  • (i) Let {A} be an ultra approximate group with a good Lie model {\pi: \langle A \rangle \rightarrow L}, and let {B} be a symmetric convex body (i.e. a convex open bounded subset) in the Lie algebra {{\mathfrak l}}. Show that if {r>0} is a sufficiently small standard number, then there exists a strong ultra approximate group {A'} with

    \displaystyle  \pi^{-1}(\exp(rB)) \subset A' \subset \pi^{-1}(\exp(1.1 rB)) \subset A,

    and with {A} can be covered by finitely many left translates of {A'}. Furthermore, {\pi} is also a good model for {A'}.

  • (ii) If {A} is a finite {K}-approximate group, show that there is a strong {O_K(1)}-approximate group {A'} inside {A^4} with the property that {A} can be covered by {O_K(1)} left translates of {A'}. (Hint: use (i), Hrushovski’s Lie model theorem, and a compactness and contradiction argument.)

The need to compare the strong approximate group to an exponentiated small ball {\exp(rB)} will be convenient later, as it allows one to easily use the geometry of {L} to track various aspects of the strong approximate group.

As mentioned previously, strong approximate groups exhibit some of the features of NSS locally compact groups. In Notes 4, we saw that the escape norm for NSS locally compact groups was comparable to a Gleason metric. The following theorem is an analogue of that result:

Theorem 2 (Gleason lemma) Let {A} be a strong {K}-approximate group in a group {G}.

  • (Symmetry) For any {g \in G}, one has {\|g^{-1}\|_{e,A} = \|g\|_{e,A}}.
  • (Conjugacy bound) For any {g, h \in A^{10}}, one has {\|g^h\|_{e,A} \ll \|g\|_{e,A}}.
  • (Triangle inequality) For any {g_1,\ldots,g_k \in G}, one has {\|g_1 \ldots g_k \|_{e,A} \ll_K (\|g_1\|_{e,A} + \ldots + \|g_k\|_{e,A})}.
  • (Escape property) One has {\|g^n\|_{e,A} \gg |n| \|g\|_{e,A}} whenever {|n| \|g\|_{e,A} < 1}.
  • (Commutator inequality) For any {g,h \in A^{10}}, one has {\| [g,h] \|_{e,A} \ll_K \|g\|_{e,A} \|h\|_{e,A}}.

The proof of this theorem will occupy a large part of the current set of notes. We then aim to use this theorem to classify strong approximate groups. The basic strategy (temporarily ignoring a key technical issue) follows the Bieberbach-Frobenius proof of Jordan’s theorem, as given in Notes 0, is as follows.

  1. Start with an (ultra) strong approximate group {A}.
  2. From the Gleason lemma, the elements with zero escape norm form a normal subgroup of {A}. Quotient these elements out. Show that all non-identity elements will have positive escape norm.
  3. Find the non-identity element {g_1} in (the quotient of) {A} of minimal escape norm. Use the commutator estimate (assuming it is inherited by the quotient) to show that {g_1} will centralise (most of) this quotient. In particular, the orbit {\langle g_1 \rangle} is (essentially) a central subgroup of {\langle A \rangle}.
  4. Quotient this orbit out; then find the next non-identity element {g_2} in this new quotient of {A}. Again, show that {\langle g_2 \rangle} is essentially a central subgroup of this quotient.
  5. Repeat this process until {A} becomes entirely trivial. Undoing all the quotients, this should demonstrate that {\langle A \rangle} is virtually nilpotent, and that {A} is essentially a coset nilprogression.

There are two main technical issues to resolve to make this strategy work. The first is to show that the iterative step in the argument terminates in finite time. This we do by returning to the Lie model theorem. It turns out that each time one quotients out by an orbit of an element that escapes, the dimension of the Lie model drops by at least one. This will ensure termination of the argument in finite time.

The other technical issue is that while the quotienting out all the elements of zero escape norm eliminates all “torsion” from {A} (in the sense that the quotient of {A} has no non-trivial elements of zero escape norm), further quotienting operations can inadvertently re-introduce such torsion. This torsion can be re-eradicated by further quotienting, but the price one pays for this is that the final structural description of {\langle A \rangle} is no longer as strong as “virtually nilpotent”, but is instead a more complicated tower alternating between (ultra) finite extensions and central extensions.

Example 2 Consider the strong {O(1)}-approximate group

\displaystyle  A := \{ a N^{10} + 5 b: |a| \leq N; |b| \leq N^2 \}

in the integers, where {N} is a large natural number not divisible by {5}. As {{\bf Z}} is torsion-free, all non-zero elements of {A} have positive escape norm, and the nonzero element of minimal escape norm here is {g=5} (or {g=-5}). But if one quotients by {\langle g \rangle}, {A} projects down to {{\bf Z}/5{\bf Z}}, which now has torsion (and all elements in this quotient have zero escape norm). Thus torsion has been re-introduced by the quotienting operation. (A related observation is that the intersection of {A} with {\langle g \rangle = 5{\bf Z}} is not a simple progression, but is a more complicated object, namely a generalised arithmetic progression of rank two.)

To deal with this issue, we will not quotient out by the entire cyclic group {\langle g \rangle = \{g^n: n \in {\bf Z} \}} generated by the element {g} of minimal escape norm, but rather by an arithmetic progression {P = \{g^n: |n| \leq N\}}, where {N} is a natural number comparable to the reciprocal {1/\|g\|_{e,A}} of the escape norm, as this will be enough to cut the dimension of the Lie model down by one without introducing any further torsion. Of course, this cannot be done in the category of global groups, since the arithmetic progression {P} will not, in general, be a group. However, it is still a local group, and it turns out that there is an analogue of the quotient space construction in local groups. This fixes the problem, but at a cost: in order to make the inductive portion of the argument work smoothly, it is now more natural to place the entire argument inside the category of local groups rather than global groups, even though the primary interest in approximate groups {A} is in the global case when {A} lies inside a global group. This necessitates some technical modification to some of the preceding discussion (for instance, the Gleason-Yamabe theorem must be replaced by the local version of this theorem, due to Goldbring); details can be found in this recent paper of Emmanuel Breuillard, Ben Green, and myself, but will only be sketched here.

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