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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 or , 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 be a finite group, and let . We say that is -quasirandom if all non-trivial unitary representations of have dimension at least . (Recall a representation is trivial if is the identity for all .)
Exercise 1 Let be a finite group, and let . A unitary representation is said to be irreducible if has no -invariant subspaces other than and . Show that is -quasirandom if and only if every non-trivial irreducible representation of has dimension at least .
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 to obtain mixing estimates in that group.)
Quasirandomness behaves fairly well with respect to quotients and short exact sequences:
- (i) If is -quasirandom, show that is -quasirandom also. (Equivalently: any quotient of a -quasirandom finite group is again a -quasirandom finite group.)
- (ii) Conversely, if and are both -quasirandom, show that is -quasirandom also. (In particular, the direct or semidirect product of two -quasirandom finite groups is again a -quasirandom finite group.)
Informally, we will call quasirandom if it is -quasirandom for some “large” , though the precise meaning of “large” will depend on context. For applications to expansion in Cayley graphs, “large” will mean “ for some constant independent of the size of “, but other regimes of are certainly of interest.
The way we have set things up, the trivial group is infinitely quasirandom (i.e. it is -quasirandom for every ). 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:
- (i) Show that if is non-trivial, then . (Hint: use the mean zero component of the regular representation .) In particular, non-trivial finite groups cannot be infinitely quasirandom.
- (ii) Show that any proper subgroup of has index . (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:
- (i) If is abelian and non-trivial, show that is not -quasirandom. (Hint: use Fourier analysis or the classification of finite abelian groups.)
- (ii) Show that is -quasirandom if and only if it is perfect, i.e. the commutator group is equal to . (Equivalently, is -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 be a finite -quasirandom group. Show that for any subgroup of , is -quasirandom, where is the index of in . (Hint: use induced representations.)
Now we give an example of a more quasirandom group.
This should be compared with the cardinality of the special linear group, which is easily computed to be .
Proof: We may of course take to be odd. Suppose for contradiction that we have a non-trivial representation on a unitary group of some dimension with . Set to be the group element
and suppose first that is non-trivial. Since , we have ; thus all the eigenvalues of are roots of unity. On the other hand, by conjugating by diagonal matrices in , we see that is conjugate to (and hence conjugate to ) whenever is a quadratic residue mod . As such, the eigenvalues of must be permuted by the operation for any quadratic residue mod . Since has at least one non-trivial eigenvalue, and there are distinct quadratic residues, we conclude that has at least distinct eigenvalues. But is a matrix with , a contradiction. Thus lies in the kernel of . By conjugation, we then see that this kernel contains all unipotent matrices. But these matrices generate (see exercise below), and so is trivial, a contradiction.
Exercise 6 Show that for any prime , the unipotent matrices
for ranging over generate as a group.
Exercise 7 Let be a finite group, and let . If is generated by a collection of -quasirandom subgroups, show that is itself -quasirandom.
Exercise 8 Show that is -quasirandom for any and any prime . (This is not sharp; the optimal bound here is , which follows from the results of Landazuri and Seitz.)
Remark 2 One can ask whether the bound in Lemma 2 is sharp, assuming of course that is odd. Noting that acts linearly on the plane , we see that it also acts projectively on the projective line , which has elements. Thus acts via the quasiregular representation on the -dimensional space , and also on the -dimensional subspace ; this latter representation (known as the Steinberg representation) is irreducible. This shows that the bound cannot be improved beyond . More generally, given any character , acts on the -dimensional space of functions that obey the twisted dilation invariance for all and ; these are known as the principal series representations. When 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 is the quadratic representation (thus taking values in while being non-trivial), the principal series representation splits into the direct sum of two -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 in a quadratic extension 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 , 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 be an odd prime. Show that for any , the alternating group is -quasirandom. (Hint: show that all cycles of order in are conjugate to each other in (and not just in ); in particular, a cycle is conjugate to its power for all . Also, as , is simple, and so the cycles of order 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 is -quasirandom for (but is only -quasirandom for due to icosahedral symmetry, and -quasirandom for due to lack of perfectness). Using Exercise 3 with the index subgroup , we see that the bound cannot be improved. Thus, (for large ) is not as quasirandom as the special linear groups (for large and 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 with the slightly larger symmetric group , then quasirandomness is destroyed (since , having the abelian quotient , is not perfect); indeed, is -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 , an alternating group , or is a finite simple group of Lie type such as . (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 is constructed from in characteristic .) In the case of finite simple groups of Lie type with bounded rank , it is known from the work of Landazuri and Seitz that such groups are -quasirandom for some 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 is -quasirandom for some and is sufficiently large depending on , then is a finite simple group of Lie type with rank . 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 (compare this with the property (T)-based methods in the previous notes, which could construct expander Cayley graphs in for any fixed ).
Last year, as part of my “open problem of the week” series (now long since on hiatus), I featured one of my favorite problems, namely that of establishing scarring for the Bunimovich stadium. I’m now happy to say that this problem has been solved (for generic stadiums, at least, and for phase space scarring rather than physical space scarring) by my old friend (and fellow Aussie), Andrew Hassell, in a recent preprint. Congrats Andrew!
Actually, the argument is beautifully simple and short (the paper is a mere 9 pages), though it of course uses the basic theory of eigenfunctions on domains, such as Weyl’s law, and I can give the gist of it here (suppressing all technical details).
This is a well known problem (see for instance this survey) in the area of “quantum chaos” or “quantum unique ergodicity”; I am attracted to it both for its simplicity of statement (which I will get to eventually), and also because it focuses on one of the key weaknesses in our current understanding of the Laplacian, namely is that it is difficult with the tools we know to distinguish between eigenfunctions (exact solutions to ) and quasimodes (approximate solutions to the same equation), unless one is willing to work with generic energy levels rather than specific energy levels.
The Bunimovich stadium is the name given to any planar domain consisting of a rectangle bounded at both ends by semicircles. Thus the stadium has two flat edges (which are traditionally drawn horizontally) and two round edges, as this picture from Wikipedia shows:
Despite the simple nature of this domain, the stadium enjoys some interesting classical and quantum dynamics. The classical dynamics, or billiard dynamics on is ergodic (as shown by Bunimovich) but not uniquely ergodic. In more detail: we say the dynamics is ergodic because a billiard ball with randomly chosen initial position and velocity (as depicted above) will, over time, be uniformly distributed across the billiard (as well as in the energy surface of the phase space of the billiard). On the other hand, we say that the dynamics is not uniquely ergodic because there do exist some exceptional choices of initial position and velocity for which one does not have uniform distribution, namely the vertical trajectories in which the billiard reflects orthogonally off of the two flat edges indefinitely.