Two weeks ago (October 29), Jean-Christophe Yoccoz, Artur Avila and I talked about the action on the moduli space of Abelian differentials and the spectral gap of the related unitary representations. The outcome of our conversation was the following result:
Since the short proof of this simple result depends on some facts from the theory of unitary representations (e.g., Ratner’s results about rates of mixing), we’ll divide this post into 5 sections:
- the first 4 sections covers some well-known useful facts about unitary representation such as Bargmann’s classification, examples of (regular) unitary representations coming from dynamical systems and Ratner’s estimates of rates of mixing; the basic reference for the facts stated in these sections is Ratner’s paper “The rate of mixing for geodesic and horocyclic flows” (besides the references therein).
- the last section contains the proof of theorem 1.
For the specialists, we can advance a few keywords of the proof of theorem 1: by a recent theorem of J. Ellenberg and D. McReynolds, given any finite index subgroup of the congruence subgroup , we can find a square-tiled surface whose Teichmüller curve is ; in particular, it suffices to find a subgroup such that the regular representation of on has complementary series; however, the existence of such subgroups can be easily derived from certain cyclic covering constructions plus a “reverse Ratner estimate” argument.
Remark 1 After we performed this cyclic covering construction, Pascal Hubert and Nicolas Bergeron informed us that this procedure was already known by A. Selberg. In fact, Selberg’s argument (see the subsection “Petites valeurs propres I. Critere geometrique d’existence” of this book project of N. Bergeron) and our argument are completely equivalent: Selberg uses that sufficiently small eigenvalues () of the Laplacian on lead to complementary series and we use that slow decay of correlations of the geodesic flow of lead to complementary series, which are two equivalent ways to state the same fact in view of Ratner’s paper.
Let be an unitary representation of , i.e., is a homomorphism from into the group of unitary transformations of the complex separable Hilbert space . We say that a vector is a -vector of if is . Recall that the subset of -vectors is dense in .
where is the exponential map (of matrices).
Exercise 1 Show that for any -vectors of and .
An important basis of is
Exercise 2 Show that , and . Furthermore, , and where is the Lie bracket of (i.e., is the commutator).
Furthermore, when the representation is irreducible, is a scalar multiple of the identity operator, i.e., for some and for any -vector of .
In general, as we’re going to see below, the spectrum of the Casimir operator is a fundamental object.
We introduce the following notation:
Note that satisfies the quadratic equation when .
Bargmann’s classification of irreducible unitary says that the eigenvalue of the Casimir operator has the form
where falls into one of the following three categories:
- Principal series: is purely imaginary, i.e., ;
- Complementary series: and is isomorphic to the representation , where belongs to the Hilbert space ;
- Discrete series: .
In other words, belongs to the principal series when , belongs to the complementary series when and belongs to the discrete series when for some natural number .
Note that, when (i.e., belongs to the complementary series), we have .
Given a dynamical system consisting of a action (on a certain space ) preserving some probability measure (), we have a naturally associated unitary representation on the Hilbert space of functions of the probability space . More concretely, we’ll be interested in the following two examples.
Hyperbolic surfaces of finite volume. It is well-known that is naturally identified with the unit cotangent bundle of the upper half-plane . Indeed, the quotient is diffeomorphic to via
Let be a lattice of , i.e., a discrete subgroup such that has finite volume with respect to the natural measure induced from the Haar measure of . In this situation, our previous identification shows that is naturally identified with the unit cotangent bundle of the hyperbolic surface of finite volume with respect to the natural measure .
Since the action of on and preserves the respective probability measures and (induced from the Haar measure of ), we obtain the following (regular) unitary representations:
Observe that is a subrepresentation of because the space can be identified with the subspace . Nevertheless, it is possible to show that the Casimir operator restricted to -vectors of coincides with the Laplacian on . Also, we have that a number belongs to the spectrum of (on ) if and only if belongs to the spectrum of on .
Moduli spaces of Abelian differentials. An interesting space philosophically related to the hyperbolic surfaces of finite volumes are the moduli spaces of Abelian differentials: we consider the space of Riemann surfaces of genus equipped with Abelian (holomorphic) 1-forms of unit area modulo the equivalence relation given by biholomorphisms of Riemann surfaces respecting the Abelian differentials. By writing , we can make act naturally on by
The case of is particularly clear: it is well-known that is isomorphic to the unit cotangent bundle of the modular curve. In this nice situation, the action has a natural absolutely continuous (w.r.t. Haar measure) invariant probability , so that we have a natural unitary representation on .
After the works of H. Masur and W. Veech, we know that the general case has some similarities with the genus 1 situation, in the sense that, after stratifying by listing the multiplicities of the zeroes of and taking connected components of these strata, this action has an absolutely continuous (with respect to a natural “Lesbegue” class induced by the period map) invariant probability . In particular, we get also an unitary representation on .
Once we have introduced two examples (coming from Dynamical Systems) of unitary representations, what are the possible series (in the sense of Bargmann classification) appearing in the decomposition of our representation into its irreducible factors.
In the case of hyperbolic surfaces of finite volume, we understand precisely the global picture: the possible irreducible factors are described by the rates of mixing of the geodesic flow on our hyperbolic surface.
More precisely, let
be the 1-parameter subgroup of diagonal matrices of . It is not hard to check that the geodesic flow on a hyperbolic surface of finite volume is identified with the action of the diagonal subgroup on .
Ratner showed that the Bargmann’s series of the irreducible factors of the regular representation of on can be deduced from the rates of mixing of the geodesic flow along a certain class of observables. In order to keep the exposition as elementary as possible, we will state a very particular case of Ratner’s results (referring the reader to Ratner’s paper for more general statements). We define equipped with the usual inner product . In the sequel, we denote by
the intersection of the spectrum of the Laplacian with the open interval ,
with the convention when ) and
We remember the reader that the subset detects the presence of complementary series in the decomposition of into irreducible representations. Also, since is a lattice, it is possible to show that is finite and, a fortiori, . Because essentially measures the distance between zero and the first eigenvalue of on , it is natural to call the spectral gap.
- when ;
- when , and is not an eigenvalue of the Casimir operator ;
- otherwise, i.e., when and either or is an eigenvalue of the Casimir operator .
Here is a constant such that is uniformly bounded when varies on compact subsets of .
In other words, Ratner’s theorem relates the (exponential) rate of mixing of the geodesic flow with the spectral gap: indeed, the quantity roughly measures how fast the geodesic flow mixes different places of phase space (actually, this is more clearly seen when and are characteristic functions of Borelian sets), so that Ratner’s result says that the exponential rate of mixing of is an explicit function of the spectral gap of .
In the case of moduli spaces of Abelian differentials, our knowledge is less complete than the previous situation: as far as I know, the sole result about the “spectral gap” of the representation on the space of zero-mean -functions (wrt to the natural measure ) on a connected component of the moduli space is:
Theorem 3 (A. Avila, S. Gouezel, J.C. Yoccoz) The unitary representation has spectral gap in the sense that it is isolated from the trivial representation, i.e., there exists some such that all irreducible factors of in the complementary series are isomorphic to the representation with .
In the proof of this result, Avila, Gouezel and Yoccoz proves firstly that the Teichmüller geodesic flow (i.e., the action of the diagonal subgroup on the moduli space of Abelian differentials is exponentially mixing (indeed this is the main result of their paper) and they use a reverse Ratner estimate to derive the previous result from the exponential mixing.
Observe that, generally speaking, the result of Avila, Gouezel and Yoccoz says that doesn’t contain all possible irreducible representations of the complementary series, but it is doesn’t give any hint about quantitative estimates of the “spectral gap”, i.e., how small can be in general. In fact, at the present moment, it seems that the only situation where one can say something more precise is the case of the moduli space :
Theorem 4 (Selberg/Ratner) The representation has no irreducible factor in the complementary series and it holds .
In fact, using the notation of Ratner’s theorem, Selberg proved that . Since we already saw that , the first part of the theorem is a direct consequence of Selberg’s result, while the second part is a direct consequence of Ratner’s result.
In view of the previous theorem, it is natural to make the following conjecture:
Conjecture (J.-C. Yoccoz) The representations don’t have complementary series.
While we are not going to discuss this conjecture here, we see that the goal of the theorem 1 is to show that this conjecture is false if we replace the invariant natural measure by other invariant measures supported on smaller loci.
After this long revision of the basic facts around unitary -representations, we focus on the proof of theorem 1. Before starting the argument, let us briefly recall the definition of Teichmüller curves of square-tiled surfaces and the definition of the associated representation. Firstly, we remind that a square-tiled surface is a Riemann surface obtained by gluing the sides of a finite collection of unit squares of the plane so that a left side (resp., bottom side) of one square is always glued with a right side (resp., top) of another square together with the Abelian differential induced by the quotient of under these identifications. It is known that square-tiled surfaces are dense in the moduli space of Abelian differentials (because is square-tiled iff the periods of are rational) and the -orbit of any square-tiled surface is a nice closed submanifold of which can be identified with for an appropriate choice of a finite-index subgroup of (in the literature is called the Veech group of our square-tiled surface). Furthermore, the Teichmüller geodesic flow is naturally identified with the geodesic flow on . In other words, the Teichmüller flow of the -orbit of a square-tiled surface corresponds to the geodesic flow of where is a finite-index subgroup of (hence is a lattice of ). In the converse direction, J. Ellenberg and D. McReynolds recently proved that:
Theorem 5 (Ellenberg and McReynolds) Any finite-index subgroup of the congruence subgroup gives rise to a Teichmüller curve of a square-tiled surface.
For a nice introduction to the Teichmüller flow and square-tiled surfaces see this survey of A. Zorich. In any case, the fact that the -orbits of square-tiled surfaces are identified to the unit cotangent bundle of permit to introduce the regular unitary -representation on the space of zero-mean -functions (wrt the natural measure on ) of .
In this notation, the theorem 1 says that there are square-tiled surfaces such that the representation associated to its -orbit has irreducible factors in the complementary series.
In view of Ellenberg and McReynolds theorem, it suffices to find a finite-index subgroup such that has complementary series. As we promised, this will be achieved by a cyclic covering procedure. Firstly, we fix a congruence subgroup such that the corresponding modular curve has genus , e.g., . Next, we fix a homotopically non-trivial closed geodesic of after the compactification of its cusps and we perform a cyclic covering of (i.e., we choose a subgroup ) of high degree such that a lift of satisfies . Now, we select two small open balls and of area whose respective centers are located at two points of belonging to very far apart fundamental domains of the cyclic covering , so that the distance between the centers of and is . Define , . Take and the zero mean parts of and .
We claim that has complementary series, i.e., (i.e., ) for a sufficiently large . Actually, we will show a little bit more: is arbitrarily close to for large (i.e., the spectral gap of can be made arbitrarily small).
Indeed, suppose that there exists some such that for every . By Ratner’s theorem 2, it follows that
for any . On the other hand, since the distance between the centers of and is , the support of is disjoint from the image of the support of under the geodesic flow for a time . Thus, , and, a fortiori,
Putting these two estimates together and using the facts that and , we derive the inequality
In particular, , a contradiction for a sufficiently large .
Remark 2 As we pointed out in the introduction, the algebraic part of the proof of theorem 1 was already known by Selberg: in fact, as pointed out to us by N. Bergeron and P. Hubert, the same cyclic covering construction giving arbitrarily small first eigenvalue of (i.e., arbitrarily small spectral gap) was found by Selberg and the reader can find an exposition of this argument in the subsection 3.10.1 of Bergeron’s book project. In particular, although the “difference” between Selberg argument and the previous one is the fact that the former uses the first eigenvalue of the Laplacian while we use the dynamical properties of the geodesic flow (more precisely the rates of mixing) and a reverse Ratner estimate, it is clear that both arguments are essentially the same.
Remark 3 We note that isn’t a congruence subgroup (i.e., it doesn’t contain for any choice of ) for large because Selberg proved that the first eigenvalue of any congruence subgroup is uniformly bounded from below by (and hence there is no complementary series for the hyperbolic surfaces related to congruence subgroups).