Let be an origami associated to a finite group generated by two elements and , and a subgroup of containing no nontrivial normal subgroup. Denoting by the normalizer of in , we have that the automorphism group is . For the sake of the exposition, we will denote .

In this language, we saw that the absolute homology group has a natural -invariant decomposition as

where has dimension and has codimension (i.e., dimension ).

Finally, we computed an explicit formula for the multiplicity in of a -irreducible -representation of character , and we used this formula to prove that (for any origami and any ).

So, this is it as far as the quick revision is concerned. Below the fold the reader will find my notes for J.-C. Yoccoz 3rd lecture: his main goal in it was to show how to use the information (derived in the previous two lectures) about -representations to deduce useful facts about -representations. As it turns out, a significant part of this classical and, in particular, the representation theory facts we’re going to use (without proof) can be found in the books of J.-P. Serre and W. Fulton and J. Harris. Nevertheless, this lecture will contain some new facts (from the forthcoming paper by J.-C. Yoccoz, D. Zmiaikou and C.M.) concerning the specific case of representations related to origamis.

Let be a field. We have that

Denote by the set of isomorphism classes of representations defined over and -irreducible.

A theorem of Brauer (see Theorem 24 in Serre’s book) says that if contains all th roots of unity where for all , then .

In particular, one has that where is the algebraic closure of .

Recall that the Galois group acts on , and is identified to the orbit space of this action.

Let be such an orbit and denote by be its character (by thinking of as a -representation). We can write

where is an integer known as Schur index.

Let be a -space where acts with character . We denote by the commuting algebra of in (i.e., the elements of commuting with the action of on ). In general is a field (not necessarily commutative) such that

where is the center of .

Now we consider the decomposition

into isotypical components (i.e., as -module for some ). Observe that the endomorphisms of as -module have the form

In the sequel, we’ll treat exclusively the case :

Standing Assumption. From now on, .

In this situation, given , we have three possibilities:

• is real, that is, , and (and takes only real values);
• is complex, that is, , , (and takes some complex non-real value);
• is quaternionic, that is, , , (and takes only real values).

(Here, designs Hamilton’s quaternions)

In the specific case of origamis, recall that the absolute homology group comes equipped with a (symplectic) intersection form

Since acts on by automorphisms, we have that is -invariant. Furthermore, since and are orthogonal (with respect to ), the restriction of to is non-degenerate.

Proposition. The subspaces introduced above are mutually orthogonal with respect to . In particular, the restriction of to each is non-degenerate.

Proof. Since is -invariant, it defines an isomorphism (of -modules) between and its dual. On the other hand, for each , we have that

where is the dual of . Therefore, this isomorphism (induced by ) preserves each isotypical component.

This proposition says that it makes sense to define

In the sequel, we will discuss the structure of in the three possible cases ( real, complex or quaternionic) in terms of adequate invariant bilinear forms. Below, the treatment of the latter objects in the quaternionic case will follow these notes here of Y. Le Cornulier.

- real-

The -module irreducible module of type can be equipped with a -invariant scalar product , and, by irreducibility, is unique up to multiplication by a positive scalar.

Proposition. The multiplicity of in is even, i.e., . Also, there exists an isomorphism of -modules such that

Consequently, for , by writing

we have that the map is an isomorphism between and the usual symplectic group .

Proof. Fix an isomorphism of -modules and define

Note that defines an isomorphism of -modules

On the other hand, by using the -invariant scalar product , we can identify with . Hence, it follows that induces a linear map ,

such that the matrix is antisymmetric and invertible. Thus, we have that is even, say and, furthermore, we can choose an isomorphism of the form such that can be written in the standard (symplectic) form in the statement of the proposition. Finally, the last claim of the proposition is a direct verification left as an exercise to the reader.

- complex (, )-

We fix an isomorphism and we equip with the corresponding structure of -vector space. In this way, is the underlying space of a -irreducible representation of , and we can equip with a -invariant (positive-definite) Hermitian product (unique up to multiplication by a positive scalar).

Proposition. There exists an isomorphism of -modules and a pair of integers with such that

Consequently, for , by writing

(), we have that the map is an isomorphism between and the unitary group of the Hermitian form

Proof. Fix an isomorphism of -modules and write

It is not hard to check that the bilinear forms on are -invariant. It follows that are linear combinations of the real and imaginary parts of the (positive-definite) Hermitian form . Thus, the fact that implies that

and, a fortiori, . Hence,

defines a -invariant non-degenerate Hermitian form on . In particular, we can write

where and , and, since is non-degenerate, we can find an automorphism of -modules and a pair of integers with such that verifies

Finally, the verification of the last claim of the proposition is straightforward (and again we leave as an exercise).

- quaternionic (, )-

We fix an isomorphism . Recall that given an element , its complex conjugate is , and its norm is given by the formula . Also, the complex conjugation satisfies .

Using the isomorphism we can render into a right -vector space by imposing

for and .

Definition. A Hermitian form on a right -vector space is a map verifying:

• (linearity on the 2nd factor);
• (“usual” Hermitian condition).

Observe that the two conditions above imply (i.e., is antilinear in the 1st factor).

Example. The standard Hermitian form on is .

In general, given an Hermitian form , we can write:

Observe that is symmetric while are antisymmetric. Moreover, they verify the relations

allowing to express in terms of for any . For later use, we will focus on antisymmetric forms, e.g., for sake of concreteness, because in the setting of origamis we will be interested in producing Hermitian forms from the symplectic intersection form.That being said, observe that satisfies

because , and hence is derived from by conjugation by whenever (i.e., is a purely imaginary quaternion with unit norm.

Conversely, given a bilinear antisymmetric verifying (1) above, then

is a Hermitian form.

Next, we observe that, by irreducibility, has an unique (up to multiplication by a positive real number) -invariant positive-definite Hermitian form whose components , , , form a basis of the space of -invariant -bilinear forms (and, in particular, the space of symmetric -invariant -bilinear forms has dimension 1, while the space of antisymmetric -invariant -bilinear forms has dimension 3).

At this point, we are ready to state the proposition whose proof will occupy complete today’s discussion.

Proposition. There exists an isomorphism of -modules and a pair of integers with such that the Hermitian form corresponding to by the formula (2) above satisfies

Consequently, for , by writing

,

, we have that is an isomorphism between and the unitary group of the Hermitian form

We will show this proposition with the aid of the following three lemmas:

Lemma 1. Let be a right -vector space and assume that is an isotypical -module of quaternionic type. Let be a -invariant antisymmetric non-degenerate -bilinear form. Then, there exists and such that

Proof of Lemma 1. Otherwise, for all , so that, by bilinearity,

i.e., . Since is antisymmetric, we deduce that

Since is -invariant, it follows that

for all . Since is non-degenerate, we get that for all , that is, the action of factors through the action of a subgroup whose elements have order two, i.e., , and for all . Since for all , we have that is Abelian and, in particular, it doesn’t have representations of quaternionic type. Of course, since the action of on factors through , we conclude that can’t be a representation of quaternionic type, a contradiction with our hypothesis.

Lemma 2. Under the assumptions of Lemma 1, we can write

where the ‘s are -invariant, irreducible, and mutually orthogonal with respect to .

Proof of Lemma 2. We proceed by induction on . For there is nothing to prove. For , we choose satisfying the conclusion of Lemma 1, and we define . We have that is -invariant and isotypical, and is -invariant. In particular, since is irreducible, we also have that is non-degenerate (otherwise its annihilator would be a non-trivial -invariant subspace of ). In particular, we can write

where is -orthogonal to , and hence -invariant of multiplicity .

Lemma 3. Let be a -invariant antisymmetric -bilinear form on . Then, there exists , such that

verifies

Proof of Lemma 3. For a -invariant antisymmetric -bilinear form , we define its adjunction by the formula

The adjunction has the following properties:

• because it commutes with the action of ;
• because is antisymmetric;
• ;
• for all ;
• if , then because and , i.e., is purely imaginary with unit norm, implies and hence , so that is purely imaginary with unit norm as well.

In particular, we have that is an anti-involution (i.e., and ) preserving the space of purely imaginary quaternions. Since an anti-involution can’t act by the identity on the space of purely imaginary quaternions (as is not commutative), it follows that there exists a purely imaginary such that

Also, recall that denotes the -component of the (unique up to multiplication by a positive real number) -invariant positive-definite Hermitian product on . We saw that satisfies (1) above, that is,

so that its adjunction verifies

(Also, similar formulas hold for and in the place of )

Now we come back to the bilinear form of the statement of the lemma. As we saw, there exists a purely imaginary with unit norm such that

On the other hand, the reader can check that the conclusion (3) of the lemma is equivalent to .

The adjunction of (when ) can be computed as follows.

so that . In particular, by choosing , such that , we obtain that

.

It follows that preserves the subspace generated by and . This leaves us with three possibilities:

• the restriction of to is a reflection;
• the restriction of to is ;
• the restriction of to is .

The case of reflection is easily excluded as follows: up to conjugation, we may assume that has the form

and this last map is not an anti-involution (as it is the conjugation by and thus a “true” involution).

The case of can also be excluded because it forces to the the complex conjugation, and this can’t be the adjunction of a non-trivial antisymmetric form (but only of a symmetric one). For instance, given an antisymmetric form , we can write . If the adjunction of is the complex conjugation, we would deduce

Here, in the last equality, we used our explicit knowledge of the adjunctions (see (4) above). Since is a basis, one has that , a contradiction (as implies that but is not the complex conjugation as we can see from (4) above).

It follows that and . By combining this with (5) above, we conclude that , and, as we already observed, this is equivalent to (3), so that the lemma is proved.

At this stage, we can complete the proof of the proposition (and today’s post). By Lemma 2, we can write

where are irreducible and mutually orthogonal with respect to the symplectic intersection form . Using this we can build an isomorphism such that

where are -invariant antisymmetric non-zero forms on .

By Lemma 3, we can choose such that the isomorphism

satisfies

where each verifies (3) above. In particular, we have that verifies (1) above and thus

is a -invariant Hermitian form. By uniqueness (up to real scalars), we have that

where .

Finally, by permuting coordinates and performing appropriate scaling, we can replace by an isomorphism such that the numbers become and, therefore,

for a pair of integers with . Finally, the verification that the map where

is an isomorphism between and the unitary group of the standard Hermitian form is left as an exercise to the reader.

Given an origami , we saw how to canonically describe it (through its monodromy group) by a finite group generated by two elements and , and a choice of subgroup such that , so that one has . Also, and act on the set of squares of by and . Furthermore, denoting by the normalizer of in , one has acting as , for .

Next, given a subfield , we decomposed and we reduced the problem of studying the action of on to understanding the action of on , and, to do so, we considered the set , and we proved that

as -module. In this way, since and , we were led to the study of the action of on . At this stage, we noticed that is the set of orbits of acting to the right on or equivalently the set of orbits of acting on by .

Nevertheless, we introduced the following notation: is the point of associated to , is the stabilizer of for the action of on , and is the order of the commutator .

Finally, if we denote by the smallest integer such that , we showed (in the very end of the previous post) the following lemma:

Lemma. .

After this quick revision, we will enter (below the fold) the content of Yoccoz’s 2nd lecture (on January 18, 2012).

Analogously to the definition of , we denote by the smallest integer such that . Observe that, by construction, divides , and divides . For later use, in the next proposition we collect a series of elementary facts:

Proposition. It holds:

1. is a cyclic group of order ;
2. is a cyclic group of order ;
3. the orbit of in has cardinality ;
4. ;
5. .

Proof. The first two items follow from the definitions of and . The third item follows from the second one (as the size of an orbit is the order of the group divided by the size of the stabilizer and this last quantity is given by item 2.). The fourth item is a matter of counting correctly the involved objects: , so that the natural “reflex” is to count through and . However, we should be slightly careful here because the representation is not unique. To solve this (easy) problem we observe that two representations verify , so that we have an unique representation once the condition , and hence the fourth item follows. Finally, the fifth item is a direct consequence of the third and fourth items.

Following the “general plan” mentioned in the previous post, we will dedicate today’s discussion to the study of the action of on in the case , i.e.,

Standing Assumption. In the sequel, .

Recall that , so that it suffices to understand the action of on each piece.

For this reason, it is natural to introduce the character of the representation of (or ) on , and the character of the representation of on .

By the fifth item of the proposition above,

On the other hand, since is the induced representation of the trivial representation of , one has

for . Here, is the characteristic function of .

Notation. We denote by the class determined by an element .

By the 2nd item of the proposition above,

Now, we consider a character of an irreducible representation (over ) of , and we denote by the multiplicity of in .

At this point, we observe that the action of on is “determined” by the knowledge of : indeed, morally speaking, the representation theory of finite groups (such as ) is “well-known” in the sense that irreducible characters are “determined”, at least when is a “classical” finite group (such as symmetric groups); therefore, the computation of permits (in principle) to calculate from character tables of finite groups.

In any event, recall that the multiplicity is given by

By using the formula of above, we can expand the right-hand side to get

where is the usual Kronecker’s delta.

Since is the character of a -representation, one has , and, hence,

It follows that

On the other hand, . By the second item of the proposition above, where , so that

Now, denoting by the degree of , we note that if are the eigenvalues of , then are the eigenvalues of . On the other hand, , so that and, hence,

In particular

where is the subspace of fixed vectors under the action of (a subspace of dimension , by definition).

By putting these identities together, we obtain the following statement:

Proposition. The multiplicity of in is

Next, we observe that is the sum of copies of the regular representation of . Since the regular representation contains (all) irreducible representations with a multiplicity equal to its degree, we have that the multiplicity of in is .

In particular, the multiplicity of in is . By the previous proposition, we can write

Observe that the right-hand side of this identity depends only on the class : indeed, , , implies and, a fortiori, , i.e., the elements and are conjugated by an element . Therefore, we can rewrite

Now we proceed to rewrite this formula by using induced representations. More precisely, we think of the irreducible representation of as a representation of of on a vector space , and we denote by the induced representation of on , where are representants of the classes and . Denote by the permutation of determined by the action of , i.e., . In this notation, we have, by definition, .

Observe that is precisely the length of the cycle of containing , and the restriction of to is conjugated to the action of on . Thus, we get the following statement:

Theorem. The multiplicity is given by the formula

where is the dimension of the subspace of fixed by .

Even though this statement is a simple consequence of our discussion so far, we decided to call it a “theorem” because it has some interesting consequences. In fact, the rest of today’s post will be dedicated to the proof of the following “corollary” to this theorem:

Corollary 1. For any origami , and for any , one has .

We begin the discussion of the proof of the previous statement by showing the following result:

Corollary 2. The multiplicity is always greater than or equal to the multiplicity of the trivial representation.

Proof. By the theorem, . Since and for every , we conclude

so that the argument is complete.

Let us consider now the case of regular origamis (in the sense of D. Zmiaikou), i.e., , . In this context, the formulas for the multiplicities simplify to and .

Proposition. In the case of regular origamis:

• If then ;
• If then .

Proof. If , we have that is a homomorphism and hence as is a commutator. Therefore, in this case.

If , we have that , where denotes the representation with character . We claim that . Otherwise, , and hence the action of factorizes into the action of a commutative group (as and generate ). However, this would force (since commutative groups only have -dimensional irreducible representations), a contradiction proving our claim. Now, we observe that the claim and the fact that imply that has at least two eigenvalues , so that .

Partly motivated by this discussion, we introduce the following definition:

Definition. An origami is quasi-regular if the multiplicity of the trivial representation is zero.

Remark. By the previous proposition, any regular origami is quasi-regular (so that the nomenclature is coherent).

The following proposition allows to characterize quasi-regular origamis in terms of the commutator :

Proposition 1. is quasi-regular the normal subgroup generated by is contained in .

Proof. We know that . Thus, for all cycle acts trivially on .

Proposition 2. If is not quasi-regular, then for all .

Proof. By Corollary 2 above, it suffices to check that . Since and is not quasi-regular (i.e., ), we have that the permutation (associated to the action of ) is not the identity. On the other hand, since is a commutator, the permutation is even. In particular, is not a transposition, and, consequently, .

Before proceeding further, let us give an example of a quasi-regular but not regular origami.

Example. Let be a finite Heisenberg group. Observe that is generated by the two elements

We choose . Note that because

Hence, this data defines an origami . Since , is not a regular origami. On the other hand, the normalizer of is

is a normal subgroup (coinciding with the centralizer of ). Note also that

By Proposition 1, is a quasi-regular origami. However, this is not the most interesting example of quasi-regular origami (from the representation theory point of view) because is an Abelian group. In any case, this quasi-regular example shows two interesting features: is normal and . As we are going to see below, this is a “general phenomenon” for quasi-regular origamis.

Proposition. is quasi-regular if and only if is normal and is Abelian generated by two elements (i.e., is either cyclic group or a product of two cyclic groups).

Proof. If is normal and is Abelian, then the commutator is mapped into the identity under the natural map , and, by Proposition 1, is quasi-regular.

Conversely, if is quasi-regular, then . Thus, is generated by two elements satisfying inside , i.e., is an Abelian group (generated by two elements). Since is a subgroup of the Abelian group , it follows that is normal and is Abelian.

At this point, we are ready to complete today’s discussion by proving Corollary 1. We start by noticing that, by Proposition 2, it suffices to consider the case of quasi-regular origamis. The idea of the proof in this case is similar to the one in the case of regular origamis (see the proposition before Proposition 1), despite the fact that we need the following little trick.

Let be a quasi-regular origami and denote by and the orders of and in . Define and .

Note that is a complete system of representatives of . Define , for , , so that .

Observe that

and

so that is the product of , , .

In this language, the Theorem implies that

Proposition. If is a quasi-regular origami, then .

On the other hand, since is a commutator of elements of , we obtain

We have the following possibilities:

• for all : here either for all and, a fortiori, , or for some , so that (by the condition on the determinant);
• for at least two pairs of indices : in this situation, for and, a fortiori, .

In any event, we find that . This completes the proof of Corollary 1.

After following the first two lectures, I thought it could be a nice idea to try to make available the notes I’m taking for this course. So, I plan to write a series of posts whose titles will have the form “SPCS x” (where SPCS stands for“Surfaces à petits carreaux (suite)” and “x” stands for the number of the lecture). Of course, this goes without saying that any errors and/or mistakes are surely my sole fault: indeed, since the course is delivered in French, it may happen that I misinterpret some points.

Below the fold the reader will find my first set of notes, i.e., SPCS 1, corresponding to Yoccoz’s lecture on last January 11th.

-Introduction-

We start by recalling some distinct (but completely equivalent) points of view (discussed in last year’s course) on square-tiled surfaces / origamis.

1. Topological point of view

An origami is a finite (usually ramified) covering where is a connected surface and is not ramified on .

The squares are the connected components of . We denote by the set of squares.

A morphism between two origamis and is an application such that . An isomorphism is a homeomorphism such that .

2. Geometrical-Analytical point of view

We begin by reviewing the notion of translation structures. Let be a compact connected orientable surface genus , and , and , , .

A structure of translation surface on of type is a maximal atlas of local charts on verifying:

• the coordinate changes are locally given by translations;
• for each , there are neighborhoods of , of and a ramified covering of degree such that the injective restrictions of are local charts of (in particular, the “total angle” around is ).

Equivalently, we can say that a translation surface is a Riemann surface structure on together with a choice of a non-trivial holomorphic -form with a zero of order at . Indeed, this last definition can be connected to the previous one by noticing that the local primitives of provide local charts for an atlas as above.

Next, we recall that, given a translation surface structure, one has a period map , .

In this setting, an origami is a translation surface whose period map takes values in .

Remark (ambiguity on ). In the previous definition, the set was not explicitly chosen. In fact, by letting and , we will see that any choice of works.

In order to see that the geometrical-analytical definition of origami is equivalent to the topological one, we take , and we observe that (mod ) is a well-defined topological origami and, conversely, a topological origami defines a geometrical-analytical origami by taking the inverses of injective restrictions of as the local charts of an atlas with the desired properties.

3. Algebraic-Combinatorial point of view

An origami is a finite set equipped with two permutations generating a transitive action on . Given a topological origami , we set and , . In this way, the connectedness of corresponds to the transitivity of the action of on .

Conversely, given an algebraic-combinatorial origami, we see that , where is the equivalence relation and , is a topological origami.

A morphism between and is an application such that and . An isomorphism is a bijective morphism .

-Monodromy of origamis and description of origamis as quotients of groups-

This algebraic-combinatorial point of view of origamis was studied by D. Zmiaikou in his PhD thesis (under the supervision of J.-C. Yoccoz). In the sequel, we will follow closely some aspects of D. Zmiaikou’s thesis.

Definition (D. Zmiaikou). The monodromy group is the subgroup of the permutation group generated by and .

Convention. We will let acts on the right on , i.e., (because we want automorphisms of origamis to act on the left).

1. Ramification

The type is determined by the action of the commutator :

In fact, from the figure above one deduces that where is the ramification index of the left-down corner of the square . (Warning: the figure above is somewhat misleading as and are usually not the same square!).

2. Origamis and quotients of groups

Denote by and let be the elements corresponding to . Fix , and denote by the stabilizer of with respect to the (right-) action of on .

We identify , and , .

Note that is not an arbitrary subgroup of . Indeed, since the stabilizer of is , the transitivity faithfulness of the action implies that the intersection of the conjugates of is the trivial group . In other words, doesn’t contain non-trivial normal subgroups.

Conversely, given a finite group generated by two elements , and a subgroup containing no non-trivial normal subgroup, we obtain an origami by taking and , .

Of course, in this language, a change of basepoint corresponds to replace by one of its conjugates .

3. Automorphisms of an origami

By definition, an automorphism is a bijection such that and .

Denote by an element with . For every , one has (since is an automorphism). Thus, for every , , i.e., , i.e., belongs to the normalizer of in . In particular, for every , one has , i.e., the automorphisms of are given by the left action of the normalizer of in (see the convention above). So, by putting , we can identify

Definition (D.Zmiaikou). We say that is a regular origami if acts transitively on , or, equivalently, (and, a fortiori, ).

The general plan for the beginning of the course is to explain some results from a work (still in progress) by J.-C. Yoccoz, D. Zmiaikou and myself, where the following topics are studied:

• the action of on with ;
• consequences of the previous topic to the action of the affine group on and the Lyapunov exponents of the Kontsevich-Zorich cocycle;
• application of the previous topics to concrete examples.

In particular, the results we’re going to discuss below (and in the next two or three lectures) are part of a forthcoming paper by J-C. Y., D. Z., C. M.

-Homology of a origami as a -module-

Let be a subfield of . Given an origami , take and consider the first relative homology group

Recall that , , and . We denote by the -vector space of canonical basis , . Next, we take two copies of and we denote by , , the canonical basis of the first copy, and by , , the canonical basis of the second copy.

Proposition. We have an exact sequence of -modules

where , , and the map is explained in the picture below:

The relative homology cycles (with their canonical orientation) defining the map and (in red) the ``geometrical meaning'' of .

Proof. Let be the skeleton of . We have that gives rise to an exact sequence in homology:

where is the usual boundary operator.

Since is a -dimensional skeleton, and , and . Also, since is -dimensional, . By plugging this information into the previous exact sequence (and by reinterpreting the operators between the homology groups above), one can check that this exact sequence is precisely the one appearing in the conclusion of the proposition.

Corollary. One has as -module.

We recall that one can write a decomposition of -modules:

where is the -dimensional -module generated by and (hence it is isomorphic to ) and is the codimension (i.e., dimension ) given by the kernel of the map

that is, consists of homology classes projecting to zero in the torus.

On the other hand, the exact sequence in homology associated to is

By combining this with the previous corollary, we get

Corollary. as -module.

As we told above, our current goal is to understand the action of on . Of course, since the action of on is trivial, it suffices to understand the action of on , and, by the previous corollary, this amounts to study . Therefore, we’ll close today’s post with some preliminaries on the action of on .

Denote by the order of the commutator , and let be the subgroup generated by it. During the study of ramifications of origamis in terms of the commutator, we saw that the points of correspond to orbits of the action of on , or, equivalently, orbits of the action of on given by .

In this way, for each , we denote by the point of corresponding to the action of , and the stabilizer of for the action of on .

Lemma. .

Proof. Observe that

This completes the proof.

At this point, Jean-Christophe ended his lecture (as he was running out of time), and, so we also stop here for today. Next time, we will continue the study of the action of on .

Of course, the knowledge of Lyapunov exponents per se may not seem very exciting, but during the Christmas Workshop 2011 of Karlsruhe University, C. Weiss gave a talk (about his PhD thesis under the supervision of M. Möller) showing how this information about Lyapunov exponents can be put forward to exhibit new special curves (i.e., Kobayashi geodesics) inside Hilbert modular surfaces. As it turns out, special curves in Hilbert modular surfaces are very rigid objects, and before C. Weiss’ result, the list of previously known special curves was not very long: it contained Hirzebruch-Zagier cycles (a.k.a. Shimura curves or twisted diagonals) and twisted Teichmüller curves solely.

The goal of today’s post is to revisit the construction of Teichmüller curves through Prym eigenforms and to give a (very rough) sketch of proof of C. Weiss’ theorem.

-Prym varieties-

Let be a compact Riemann surface and let be a (non-trivial) holomorphic involution. The Prym variety is the Abelian variety

where (i.e., is the space of -anti-invariant holomorphic 1-forms on ) and (i.e., is the space of -anti-invariant cycles on ).

Standing Hypothesis. For the sake of today’s discussion, we will always assume that . In fact, this assumption is “motivated” by C. McMullen’s work (see Section 3 of his article) on Prym varieties associated to genus 2 Riemann surfaces.

Remark 1 Notice that , so

Remark 2 By Riemann-Hurwitz formula, , so that our standing hypothesis and the previous remark imply . Moreover, when , one has that is unramified.

-Real multiplication on Abelian varieties-

Let be an integer such that or modulo , and let

be the real quadratic orderof discriminant .

A polarized Abelian variety can be thought as , where is a lattice equipped with a symplectic pairing . In this way, the endomorphism ring of can be thought as

We say that is self-adjointif for all .

Definition 1 We say that a polarized Abelian variety of complex dimension admits a real multiplication by if there exists a representation such that

• (a) is self-adjoint for all ;
• (b) is a proper subring of , i.e., if for some and , then .

-Prym eigenforms-

Let be a Prym variety admitting real multiplication by . We say that is a Prym eigenform whenever .

In the sequel, we denote by the locus of Prym eigenforms inside the moduli space of Abelian differentials on genus Riemann surfaces. Also, given a list of non-zero natural numbers such that , we denote by the locus of Prym eigenforms whose list of orders of its zeroes coincides with .

Remark 3 In general, neither the holomorphic involution nor the representation are (uniquely) determined by the Prym eigenform . See, however, Theorem 5.1 of E. Lanneau and D.-M. Nguyen’s article for an uniqueness statement in genus .

The locus of Prym eigenforms is an important example in the dynamics of Teichmüller flow in view of the following theorem of C. McMullen:

Theorem 2 (McMullen) is a closed -invariant locus of .

Following C. McMullen, we call Weierstrass locus the locus of Prym eigenforms inside the minimal stratum of (this name is motivated by the fact that its construction is naturally related to Weierstrass points).

Remark 4 In the case of a Prym eigenform in the Weierstrass locus , its unique zero must be fixed by the holomorphic involution . On the other hand, by Remark 2, we know that and, moreover, doesn’t have fixed points when . Therefore, can occur only when .

By combining McMullen’s theorem above with a “dimension counting” argument, it is possible to show that

Corollary 3 (McMullen) For , is a finite union of Teichmüller curves. Moreover, each of these Teichmüller curves are primitive (i.e., they are not obtained by branching covers of low genera Teichmüller curves) unless is a square (i.e., unless the Teichmüller curve is associated to square-tiled surfaces, i.e., it is obtained by branched covers of the torus).

In the case of genus , C. McMullen used these Teichmüller curves to classify the closures of the orbits of the natural action on the moduli space of Abelian differentials with a single double zero (a Teichmüller-theoretical analog to Ratner’s theorems). See this article of K. Calta, and these articles of C. McMullen for more details.

-Special curves on Hilbert modular surfaces-

Consider the Hilbert modular surface . Here, is the Poincaré upper half-plane and is the lower half-plane (and the action of the subgroups of is through Moebius transformations).

A special curve or Kobayashi geodesic on is an algebraic curve that is totally geodesic with respect to the Kobayashi metric on .

A simple example of special curve on is the diagonal obtained by the composition of the map with the projection . In this article here, F. Hirzebruch and D. Zagier introduced the twisted diagonals obtained by the composition of the map with the projection , where and is its Galois conjugate. In the literature, twisted diagonals are known as Hirzebruch-Zagier cycles or Shimura curves.

More examples of special curves are given by the Teichmüller curves in genus 2 provided by McMullen’s Weierstrass loci . Roughly speaking, given , we can associated its Jacobian , where is the hyperelliptic involution of . By definition, whenever , one has that is a principally polarized Abelian variety with real multiplication by . Since the Hilbert modular surface parametrizes all principally polarized Abelian surfaces with real multiplication, and Teichmüller curves are Kobayashi geodesics, we can see naturally as a special curves on . Actually, C. McMullen showed that the embedding of on is given by the composition of a map of the form with the projection , where is holomorphic but it is not a Moebius transformation, i.e., is not given by some . In other words, the special curves induced by are not twisted diagonals. By following F. Hirzebruch and D. Zagier, C. Weiss considers in his PhD thesis (under the supervision of M. Möller) twisted versions of these Teichmüller curves on , i.e., he considers the composition of the map with the projection , where and is the holomorphic non-Moebius map described above. In this setting, C. Weiss showed that these twisted Teichmüller curves are special curves with finite area and he provides information on their stabilizer (inside or equivalently ).

In any event, the fact that special curves are totally geodesic with respect to Kobayashi metric indicates that they are very rigid objects. In particular, one could ask whether all special curves inside are twisted diagonals or twisted Teichmüller curves. The answer to this question is provided by the following result of C. Weiss:

Theorem 4 (C. Weiss) For every , the Teichmüller curves forming the Weierstrass loci and are neither twisted diagonals nor twisted Teichmüller curves.

The proof of this result depends on the precise knowledge of some Lyapunov exponents of the Kontsevich-Zorich cocycle over these Teichmüller curves. So, before entering into a sketch of proof of this theorem, let us quickly discuss Lyapunov exponents of Kontsevich-Zorich (KZ) cocycle of Teichmüller curves on the Weierstrass locus (for a brief review of Lyapunov exponents of KZ cocycle, please see these posts here).

Remark 5 Actually, the (genus ) case of can be treat without appealing to Lyapunov exponents: in fact, the Prym varieties arising from this case have a polarization of signature , so that they don’t lie in as it was defined in this post (as this last object parametrizes principally polarized Abelian surfaces). On the other hand, the (genus ) case of is non-trivial because the polarization has signature , i.e., it is a principal polarization, and hence the relevant Prym varieties lie on . In any event, we decided to include below an unified proof of the genus and cases just to show the role of played by Lyapunov exponents.

-Lyapunov spectrum of Teichmüller curves in the Weierstrass locus-

We begin with the following result of E. Lanneau and D.-M. Nguyen:

Theorem 5 The Teichmüller curves in always contain an Abelian differential whose (periodic) horizontal foliation has a decomposition into (maximal) cylinders accordingly to Model A+, Model A- or Model B below

The Teichmüller curves in always contain an Abelian differential wihose (periodic) horizontal foliation has a decomposition into (maximal) cylinders accordingly to Model A or Model B below

For a proof (and the original pictures!) of this theorem, see Proposition 3.2, Corollary 3.4 (for the case of ), and Appendix D (for the case of ) of E. Lanneau and D.-M. Nguyen’s article.

A consequence of this theorem is the fact that, in the language of this article of G. Forni, Teichmüller curves in the Weierstrass loci and have maximal homological dimension, i.e., these Teichmüller curves contain some Abelian differential whose (periodic) horizontal foliation has a decomposition into (maximal) cylinders so that their waist curves generates a Lagrangian subspace (with respect to the symplectic intersection form) in the absolute homology group. Indeed, let us check that the homological dimension is maximal in the cases of an Abelian differential in with a Model A+ cylinder decomposition and an Abelian differential in with a Model A cylinder decomposition (the other [analogous] cases left as an exercise to the reader).

Given a “Model A+” Abelian differential in , one can follow E. Lanneau and D.-M. Nguyen and draw the (closed) cycles below:

In this picture, the waist curves of horizontal cylinders are represented by the cycles , and . The subspace in absolute homology spanned by them has dimension (maximal) because is a canonical symplectic basis of homology (as one can see from the picture).

Similarly, given a “Model A” Abelian differential in , one can draw the (closed) cycles below

in order to see that the waist curves of horizontal cylinders (represented by and ) span a subspace of dimension (maximal) because is a canonical symplectic basis of homology.

Therefore, by G. Forni’s criterion, once we know that Teichmüller curves in and have maximal homological dimension, we have that the KZ cocycle over this Teichmüller curves is non-uniformly hyperbolic, i.e., none of the Lyapunov exponents of KZ cocycle over them is zero.

Actually, as it was mentioned in the introduction to this post, one can say more about these Lyapunov exponents due to the recent works of D. Chen and M. Möller, A. Eskin, M. Kontsevich and A. Zorich, and M. Möller.

In the (genus ) case of Teichmüller curves inside , we know that the sum of the three non-negative Lyapunov exponents is

because is a subset of the connected component of Abelian differentials in with odd spin structure (see Remark 1.2 of E. Lanneau and D.-M. Nguyen’s article or Lemma 2.1 of M. Möller’s article), and the sum of Lyapunov exponents of Teichmüller curves in this connected component is always as it was shown by D. Chen and M. Möller(with the aid of a beautiful formula derived by A. Eskin, M. Kontsevich and A. Zorich relating sum of exponents with slopes of divisors in moduli spaces). Moreover, M. Möller (see Section 5 of his recent paper) was able to show that the subbundle giving rise to the Prym variety over leads to a subbundle of the Hodge bundle (of dimension ) contributing with two explicit Lyapunov exponents: and . Therefore, given that the sum of the three exponents is , we conclude that the Lyapunov spectrum of Teichmüller curves in is

independently of the discriminant .

Remark 6 As a “side remark”, let me point out that one can “numerically test” these exponents by using (say) Proposition 4.2 of E. Lanneau and D.-M. Nguyen paper to build up explicit square-tiled surfaces (with “Model A+” cylinder decomposition) in whenever is a square, i.e., , by conveniently choosing parameters , and then using some programs by A. Zorich and V. Delecroix (available at SAGE after installation of sage-combinatpackage). I plan to make some comments on this program later (in future posts), but for now let me say that by testing a square-tiled surface (with squares) in with discriminant and choice of parameters , , , I got “numerical Lyapunov spectrum”:

In the (genus ) case of Teichmüller curves inside , we also know the sum of the four non-negative Lyapunov exponents is

because is a subset of the connected component of Abelian differentials in with even spin structure (see Proposition 2.2 of M. Möller’s paper), and the sum of Lyapunov exponents of Teichmüller curves in this connected component is always as it was shown by D. Chen and M. Möller. Moreover, M. Möller (see again Section 5 of his recent paper) proved that the subbundle of the Hodge bundle (of dimension ) giving rise to the Prym variety over contributes with two explicit Lyapunov exponents: and . In particular, the Lyapunov spectrum of Teichmüller curves in has the form

where .

Here, I don’t know how to determine the precise values of and , but this will not be important for the sequel.

Remark 7 Analogously to Remark 6 above, one can “numerically test” these exponents by using (say) Proposition D.2 of E. Lanneau and D.-M. Nguyen paperto build up explicit square-tiled surfaces (with “Model A” cylinder decomposition) in whenever is a square by conveniently choosing parameters , and then using SAGE. For instance, the parameters , , , and lead to a square-tiled surface in with squares and “numerical Lyapunov spectrum”

and the parameters , , , , lead to a square-tiled surface in with squares and “numerical Lyapunov spectrum”

Personally, I tend to believe that these numerical experiments “indicate” that and , but, as I said, the precise values of and are unknown.

In any case, we are ready to say a few words on the proof of C. Weiss’ theorem 4.

-Quick sketch of proof of Weiss’ theorem-

Let where is a Teichmüller curve. Recall that, by definition, is a Prym eigenform, i.e., and for every . Since (by our standing hypothesis), we can take such that and for all (where is, as usual, the Galois conjugate of ). In this way, we can attach a number to by computing the orbifold degree of the line bundle over (and normalizing it by the Euler characteristic of ). For a characterization of in terms of intersection theory (of some divisors of with ), see Sections 1 and 5 of M. Möller’s paper. As it turns out, it is possible to show that is one of the Lyapunov exponents of the Kontsevich-Zorich cocycle over .

In the case of (genus ) Teichmüller curves in , this exponent is known to be always , while in the cases (of genus and resp.) of Teichmüller curves in and resp., M. Möller (see again Section 5 of his paper) showed that and resp.

At this point, the plan for the proof of Weiss’ theorem is simple: we will use the Lyapunov exponent as an “invariant” to distinguish twisted diagonals, twisted Teichmüller curves and , .

We begin by distinguishing Teichmüller curves in and from twisted Teichmüller curves. Since twisted Teichmüller curves are obtained by twisting Teichmüller curves (of genus ) in , it is not hard to check (i.e., the twisting operation doesn’t change this exponent). On the other hand, we just saw that Teichmüller curves in , resp. satisfy , resp. , so that the claim follows.

Now, we complete the sketch by distinguishing Teichmüller curves in and from twisted diagonals. As we mentioned above, the quantity admits an interpretation in terms of intersecion theory inside the Hilbert modular surface , and, as a matter of fact, the definition of can be naturally extended to any special curve (Kobayashi geodesic) in . Moreover, a direct computation reveals that this quantity (a sort of “slope”) equals for the diagonal, and, since it can be checked that the twisting operation doesn’t change this quantity, one get that this quantity is also for twisted diagonals. Again, since for Teichmüller curves in and , we are done.

1. Triangle removal lemma

Most applications of Szemerédi’s regularity lemma deal with monotone problems, when throwing in more edges can only help. In these applications, one starts applying the original form of the regularity lemma to create a regular partition, then gets rid of all edges within the clusters of the partition, also the edges of non-regular pairs as well as those of regular pairs with small density. The leftover “pure” graph is much easier to handle and still contains most of the original edges. This is what happens in proving the triangle removal lemma.

The triangle removal lemma is the (intuitive, yet nontrivial) fact that if one has to delete at least edges of a graph with vertices to destroy all triangles in it, then the graph must contain at least triangles, where . If one only thinks naively, the conclusion is that the graph contains at least triangles, and the strength of the triangle removal lemma is that, instead of quadratic, the number of triangles is cubic. It was first proved by Ruzsa and Szemerédi in the paper Triple systems with no six points carrying three triangles, who also observed it implies Roth’s theorem, as we shall see in the next section.

Definition 1 Given , a graph is -far from being triangle free if one has to delete at least edges of to destroy all triangles in it.

In particular, every graph that is -far from being triangle-free has at least one triangle (indeed, at least of them).

Theorem 2 (Triangle removal lemma) For any , there is such that, whenever is -far from being triangle-free, then it contains at least triangles.

Proof: Let be an -far from being triangle-free graph and . We can assume by just taking sufficiently small so that

Consider the -regular partition given by Szemerédi’s regularity lemma. Let and the graph obtained from by deleting the following edges:

1. All edges incident in : there are at most edges.
2. All edges inside the clusters : the number of edges is at most

   

3. All edges that lie in irregular pairs: there are less than

   

   edges.

4. All edges lying in a pair of clusters whose density is less than : their cardinality is at most

   

The number of deleted edges is less than and so contains a triangle. The three vertices of such triangle belong to three remaining distinct clusters, let us say . We’ll show that in fact these clusters support many triangles.

Call a vertex typical if it has at least adjacent vertices in and at least adjacent vertices in . As, by hypothesis,

whenever , have cardinality at least , there are more than typical vertices in . In fact, the number of vertices in with at least adjacent vertices in is greater than . If this were not the case, the subset of non-typical vertices would have more than elements and would satisfy

contradicting (1). As the same argument holds to , the number of typical vertices in is at least

Let be one of them and consider the vertices adjacent to .

Every edge between and generates a triangle. Observe that this number is at least

Summing this up in typical, has at least triangles. Because , this quantity is greater or equal to

1.1. Roth’s theorem

As an application of the triangle removal lemma, we prove Roth’s theorem proved in On certain sets of integers.

Theorem 3 (Roth) If has positive upper density, then it contains an arithmetic progression of length .

Proof: Assume that

Consider a graph in the following way:

1. , where have vertices labeled from 1 to each.
2. There is an edge from to iff .
3. There is an edge from to iff .
4. There is an edge from to iff .

Then form a triangle exists iff

that is, the triangles identify arithmetic progressions of length 3 in , including the trivial ones , . There are more than of these trivial triangles and they are all disjoint. This mere disjointness implies is -far from being triangle-free and so, by the triangle removal lemma, has at least triangles of which at least are non-trivial. The proof is complete by taking .

1.2. Corner’s theorem

This result was first proved by Ajtai and Szemerédi in Sets of lattice points that form no squares. The simpler proof we present here, using the triangle removal lemma, was obtained by Solymosi in Note on a generalization of Roth’s theorem. We point out, and leave the proof to the reader, that the corner’s theorem is a strengthening of Roth’s theorem.

Definition 4 A corner is an axis-aligned isosceles triangle of , that is, it is a set of three different elements of of the form

The corner’s theorem states that every set of positive density has a corner.

Theorem 5 (Corner’s theorem) For every , there exists such that any subset of with at least points has a corner.

Proof: We proceed as in the proof of Roth’s theorem. Let be a subset of with at least points and consider the graph defined by:

1. , where have vertices labeled by each.
2. There is an edge from to iff , that is, iff they are in the same horizontal line.
3. There is an edge from to iff , that is, iff they are in the same vertical line.
4. There is an edge from to iff , that is, iff they are in the same diagonal.

The triangles of correspond to the corners of , including the trivial ones . has more than of these trivial triangles and they are all disjoint, so that is -far from being triangle-free. By the triangle removal lemma, has at least triangles of which at least are non-trivial and give the required corner.

2. Graph removal lemma

The triangle removal lemma asserts that every graph for which it is necessary to throw a positive fraction of edges in order to destroy all triangles indeed has a positive fraction of triangles. The fact is that, as proved by Erdös, Frankl and Rödl in The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent, instead of fixing the triangle configuration, the result is also true for any fixed configuration. More formally, let be a finite graph with vertices and, analogously, consider the following

Definition 6 Given , a graph is -far from being -free if one has to delete at least edges of to destroy all copies of in it.

Theorem 7 (Graph removal lemma) For any , there is such that, whenever is -far from being -free, then it contains at least copies of .

The proof of this theorem is more intricate than that of the triangle removal lemma. Actually, it depends on the structure of the graph . If, for example, is a four-cycle, then the argument applied in the proof of the triangle removal lemma does not work, mainly because, once the “impure” edges are discarded, the copy of that remains may have two vertices in a same cluster. In other words, the connectivity properties of influence the distribution of the vertices along the clusters in a potential candidate for copy of in . As in the triangle removal lemma, this problem does not occur if is the complete graph in vertices. For this reason, the proof of the graph removal lemma will be accomplished in three parts:

Part 1. The establishment of the graph removal lemma for .

Part 2. We observe that, for a general , the application of the same idea in Part 1 only guarantees the existence of clusters, where is the chromatic number of .

Part 3. If we apply the same idea as in Part 1, allowing the choice of more than one vertex in a same cluster, we obtain the result for any .

As remarked above, Part 1 follows the same lines of the proof of the triangle removal lemma: we clean out the graph and the remaining copy of is supported in different clusters, which indeed contain many copies of . The construction of many copies is again accomplished by the typicality of most of the vertices, and is given by the following

Lemma 8 If is -regular and , then at least vertices of are adjacent to at least vertices of .

Proof: Let . Then

If , the -regularity guarantees that

thus contradicting (2).

Proof of the graph removal lemma for : Let be -far from being -free graph with , and consider the -regular partition given by Szemerédi’s regularity lemma. Let and the graph obtained from by deleting the following edges:

1. All edges incident in : there are at most edges.
2. All edges inside the clusters : the number of edges is at most

   

3. All edges that lie in irregular pairs: there are less than

   

   edges.

4. All edges lying in a pair of clusters whose density is less than : their cardinality is at most

   

The number of deleted edges is less than and so contains . The vertices of such belong to distinct remaining clusters, say . We’ll show that these clusters support many copies of . This is done in steps, the step consisting of choosing a vertex from in such a way that is adjacent to each of the previously chosen vertices . If there are ways of choosing , where is independent of , then contains at least

copies of and we’re done.

By Lemma 8, at least points in are joined to at least points of for each . Take one such point and denote by the subset of of all the adjacent vertices to , for each . We have

and hence any two of the clusters are -regular and have density at least . This concludes the first step.

We now proceed to step 2: again by Lemma 8, at least points in are joined to at least points of for each . Take one such point and denote by the subset of of all the adjacent vertices to , for each . We have

and hence any two of the clusters are -regular and have density at least .

Assuming, without loss of generality, that and , the above procedure can be repeated times. This concludes the proof.

Now let be an arbitrary graph. The chromatic number of is the smallest number of colors needed to paint the vertices of in such a way that no two adjacent vertices have the same color. Equivalently, it is the smallest for which is -partite, that is, for which one can divide the vertices of into disjoint subsets such that no two vertices on a same subset are adjacent. Let these subsets have cardinality and be the complete -partite graph whose subsets have cardinality . Obviously, contains and so the number of copies of in a given graph is at least the number of copies of .

Observe that if we apply the same idea as in Part 1 to a -far from being -free graph, the remaining copy of has vertices in at least clusters , and not necessarily in different clusters. This is not a problem: instead of choosing one vertex in each , we choose of them. If the same procedure works, each of these choices generates a copy of and thus of . This is how we proceed below.

Proof of the graph removal lemma: Apply the same argument as in Part 1 to obtain clusters such that any pair is -regular and has density at least . We can thus find points of which are joined to at least points of for each . Take one such point and denote by the set of all vertices of which are joined to , for each . Also, set (here is the difference: we don’t discard ). We have

for every and hence each pair among is -regular and has density at least .

Repeating this argument successively times in the first cluster, times in the second cluster, , times in the -th cluster, we construct vertices forming a copy of . This completes the proof.

We point out a recent proof of the graph removal lemma avoiding the use of Szemerédi’s regularity lemma has been obtained by Fox in the paper A new proof of the graph removal lemma. Although it does not use the regularity lemma, its idea is similar. Instead of using the mean square density given by the index

it uses a mean entropy density

where for and . Like in the regularity lemma, whenever a partition does not supports many copies of , Fox shows, using a Jensen defect inequality, that there is a refinement of the partition that increases the mean square entropy a fixed amount.

We finish this post mentioning another version of the graph removal lemma that counts the number of induced graphs. A subgraph of a graph is said to be induced if any pair of vertices of are adjacent if and only if they are adjacent in . For example, has an induced but does not have an induced four-cycle. This shows that induced graphs are harder to find, and actually that the excess of edges might prevent them to exist. In this setting, we have to consider a new definition of -far from being -free, in which one can remove or include edges.

Definition 9 Given , a graph is -unavoidable for if any graph that differs from in no more that edges has an induced copy of .

We thus have the graph removal lemma, proved by Alon, Fischer, Krivelevich and Szegedy in Efficient testing of large graphs.

Theorem 10 (Graph removal lemma for induced graphs) For any , there is such that, whenever is -unavoidable for , then it contains at least induced copies of .

We won’t prove it. Instead, we refer the reader to the original paper.

The content of this and the post Szemerédi’s regularity lemma can be downloaded in PDF format in my lecture notes available at my homepage.

In this first of two posts, we prove Szemerédi’s regularity lemma. The second post will give some applications of this lemma: the triangle removal lemma and Roth’s theorem. Some of the content has intersection with the Ergodic Ramsey Theory posts, whose interested reader may check here: ERT0, ERT1, ERT2, ERT3, ERT4, ERT5, ERT6, ERT7, ERT8, ERT9, ERT10, ERT11, ERT12, ERT13, ERT14, ERT15, ERT16.

1. Additive combinatorics

“Additive combinatorics is the theory of
counting additive structures in sets.“
T. Tao and V. Vu.

This theory has seen exciting developments and dramatic changes in direction in recent years, thanks to its connections with areas such as number theory, ergodic theory and graph theory. This section gives a brief historic introduction on the main results.

Van der Waerden’s theorem (see ERT6 for a topological dynamical proof), one of Kintchine’s Three Pearls of Number Theory, states that whenever the natural numbers are finitely partitioned (or, as it is customary to say, finitely colored), one of the cells of the partition contains arbitrarily long arithmetic progressions. In other words, the structure of the natural numbers can not be destroyed by partitions: arbitrarily large parts of persist inside some component of the partition. This result was first proved in and represents the first great result on additive combinatorics. Afterwards, in the mid-thirties, Erdös and Turán conjectured a density version of van der Waerden’s theorem. To present it, let us define what is the notion of density in the natural numbers.

Definition 1 Given a set , the upper density of is

If the limit exists, we say that has density, and denote it by . As pointed out by Erdös and Turán, having positive upper density is a notion of largeness and it is natural to ask if sets with this property have arbitrarily long arithmetic progressions. This quite recalcitrant question was only settled in by Szemerédi in the paper On sets of integers containing no elements in arithmetic progressions . Meanwhile, the first partial result was obtained by Roth in .

Theorem 2 (Roth) If has positive upper density, then it contains an arithmetic progression of length .

His proof relied on a Fourier-analytic argument of energy increment for functions: one decomposes a function as , where is good and is bad in a specific sense (this follows the same philosophy of Calderón-Zygmund’s theory on harmonic analysis). If the effect of is large, it is possible to break it into good and bad parts again and so on. In each step, the “energy” of increases a fixed amount. Being bounded, it must stop after a finite number of steps. At the end, controls the behavior of and for it the result is straightforward. See The remarkable effectiveness of ergodic theory in number theory for further details.

Sixteen years later, in the paper On sets of integers containing no four elements in arithmetic progression, Szemerédi  extended Roth’s theorem to

Theorem 3 (Szemerédi) If has positive upper density, then it contains an arithmetic progression of length .

Finally, in , Szemerédi settled the conjecture in its full generality.

Theorem 4 (Szemerédi) If has positive upper density, then it contains arbitrarily long arithmetic progression.

His proof required a complicated combinatorial argument and relied on a graph-theoretical result, known as Szemerédi’s regularity lemma, which turned out to be an important result in graph theory. It asserts, roughly speaking, that any graph can be decomposed into a relatively small number of disjoint subgraphs, most of which behave pseudo-randomly. This is the main topic of this post.

It is worth to mention Erdös also conjectured that if satisfies

then it contains arbitrarily long arithmetic progressions. This question is wide open: nobody knows even if contains arithmetic progressions of length . On the other hand, a remarkable result of Green and Tao states the conjecture for the particular case of the prime numbers.

Theorem 5 (Green and Tao) The prime numbers contain arbitrarily long arithmetic progressions.

2. Setting notation

is a graph, where is a finite set of vertices and is the set of edges, each of them joining two distinct elements of . For disjoint , is the number of edges between and and

is the density of the pair .

Definition 6 For and disjoint subsets , the pair is -regular if, for every and satisfying

we have

A partition of into pairwise disjoint sets in which is called the exceptional set is an equipartition if . We view the exceptional set as distinct parts, each consisting of a single vertex, and its role is purely technical: to make all other classes have exactly the same cardinality.

Definition 7 An equipartition is -regular if

1. ,
2. all but at most of the pairs are -regular.

The classes are called clusters or groups. Given two partitions of , we say refines if every cluster of is equal to the union of some clusters of .

3. Szemerédi’s regularity lemma

Szemerédi’s regularity lemma says that every graph with many vertices can be partitioned into a small number of clusters with the same cardinality, most of the pairs being -regular, and a few leftover edges. In my point of view, this result allows the decomposition of every graph with a sufficiently large number of vertices into many components uniformly (every component has the same number of vertices) in such a way the relation of the clusters is at the same time

uniform: the densities do not vary too much, and

randomic: even controlling the density, nothing can be said about the distribution of the edges.

As a toy model, let and consider the complete random graph with vertices in which every edge belongs to with probability . If are disjoint subsets of , the expected value of is , and the same happens for subsets , . Szemerédi’s regularity lemma says that, approximately, this is indeed the universal behavior.

Theorem 8 (Szemerédi’s regularity lemma) For every and every integer , there exist integers and for which every graph with at least vertices has an -regular equipartition , where .

Note the importance of having an upper bound for the number of clusters. Otherwise, we could just take each of them to be a singleton.

The idea in the proof is similar to Roth’s approach. Start with an arbitrary partition of into disjoint classes of equal sizes. Proceed by showing that, as long as the partition is not -regular, it can be refined in a way to distribute the density deviation. This is done by introducing a bounded energy function that increases a fixed amount every time the refinement is made. After a finite number of steps, the resulting partition is -regular.

We now discuss what should be the energy function. The natural way of looking for it is to identify the obstruction for a pair to be -regular. This means there are subsets and such that , and

Consider the partitions and . The above inequality has the following probabilistic interpretation. Consider the random variable defined on the product by: let be a uniformly random element of and a uniformly random element of , let and be those members of the respective partitions for which and , and take

The expectation of is equal to

By assumption, deviates from at least whenever , and this event has probability

Then . Noting that the expectation of is

we conclude that

The fractions containing above represent the energy function we are looking for: given two disjoint subsets , define

For partitions , let

Definition 9 Given a partition of with exceptional set , the index of is

where the sum ranges over all unordered pairs of distinct parts of , with each vertex of forming a singleton part in its own.

Note that is a sum of terms of the form . The first good property it must have is boundedness.

Property 1. .

In fact, as ,

It is also monotone increasing with respect to refinements. This is the content of the next two properties.

Property 2. If are subsets of and are partitions of , respectively, then

This property follows easily from Cauchy-Schwarz inequality (the interested reader may check it in the survey Szemerédi’s regularity lemma and its applications in graph theory), but this analytical argument is not so clear. A soft way of proving it is to consider the probabilistic point of view, with the aid of the random variable . According to the above calculations,

and so, by Jensen’s inequality (which in this case is just Cauchy-Schwarz inequality),

Property 3. If refines , then

This is a direct consequence of Property 2 by breaking according to :

The next property grows the index of non -regular partitions and reflects the right choice of the energy function. In a few words, it says that

“The lack of uniformity implies energy increment”

and this idea permeates many results in recent developments in combinatorics, harmonic analysis, ergodic theory and others areas. Actually, all known proofs of Szemerédi’s theorem use this principle at some stage. To mention some of them:

1. the original proof of Roth considers good and bad parts of functions.
2. Furstenberg’s approach: every non-compact system has a weak mixing factor.
3. the Fourier-analytic proof of Gowers identifies arithmetic progressions via the nowadays called Gowers norms.
4. the construction of characteristic factors for multiple ergodic averages uses the Gowers-Host-Kra seminorms.

These two last results are still being developed to generate what is being called higher-order Fourier analysis. See this post of Terence Tao for a discussion about this topic. Going back to what matters, let’s prove the

Proposition 10 (Lack of uniformity implies energy increment 1) Suppose and are disjoint nonempty subsets of and the pair is not -regular. Then there are partitions of and of such that

Proof: The reader must convince himself that this is exactly relation (3). For those still not convinced, let’s do it again. Assume and are such that , and

Consider and . The evaluation of the variation will prove the proposition. On one hand, by the calculations in Property 2,

On the other, deviates from at least whenever , and this event has probability

Then which, together with (1), gives that

Proposition 11 (Lack of uniformity implies energy increment 2) Suppose and let be a non -regular equipartition of , where is the exceptional set. Then there exists a refinement of with the following properties:

1. is an equipartition of ,
2. ,
3. and
4. .

Proof: The idea is to apply the previous proposition to every non-regular pair. As there are at least of them, the index will increase the fixed amount. Let be the cardinality of every , . Saying that is not -regular means that, for at least pairs , , is not -regular. For each of these, let , be the partitions of , respectively, given by Proposition 10 and consider the smallest partition that refines and all , . By Proposition 10,

as . This proves that (and any of its refinements) satisfies (iv). The problem is that is not necessarily an equipartition. We adjust this by defining , splitting every part of arbitrarily into disjoint sets of size and throwing the remaining vertices of each part, if any, to the exceptional set. This new partition satisfies (i), (ii) and (iii), as we’ll verify below.

(i) is an equipartition by definition.

(ii) To get , every cluster of is divided in at most parts. After, every element of is divided in at most non-exceptional parts. This implies that

(iii) Each cluster of contributes with at most vertices to and so

Finally, we are able to prove the regularity lemma.

Proof: First, note that if the result is true for and , , then the result is also true for the pair . This allows us to assume that and is arbitrarily large.

Begin with an arbitrary partition of such that and . Apply Proposition 11 at most times to obtain an equipartition . Let be the largest number obtained by iterating the map at most times, starting from . Then has at most clusters. In addition, the cardinality of its exceptional set is bounded by

which is smaller than if is large. This concludes the proof.

There is a large literature about Szemerédi’s regularity lemma. We refer the reader to four references: my lecture notes available at my homepage, the book The probabilistic method of Alon and Spencer, the survey of Komlós and M. Simonovits and Tao’s perspective via random partitions. Merry Christmas!!

Acknowledgments. I would like to thank the organizers of the two conferences above (in particular, Corinna Ulcigrai and Gabriela Schmithüsen) for the invitation to deliver the talks at the origin of these notes.

-Some cyclic covers of the Riemann sphere -

We consider the family of curves (in the moduli space) described by the following algebraic equation

where are mutually distinct points of the Riemann sphere . The natural projection , , is a (ramified) covering and the reader can use the Riemann-Hurwitz formula to check that is a family of genus 10 curves.

The group of deck transformations of is naturally isomorphic to the cyclic group and it is generated by where . In other words, is a family of cyclic covers of the Riemann sphere. In particular, we can form the real, resp. complex Hodge bundle , resp. over the family whose fiber over a point is its first cohomology group, and we can decompose these bundles into a direct sum of eigenspaces of the action of in the first cohomology group of . For instance, in the case of the complex Hodge bundle, we observe that (because ), so that the eigenvalues of belong to the set of th roots of unity. Moreover, since the action of preserves the natural Hodge filtration (into holomorphic and anti-holomorphic 1-forms), we can write

where and are the eigenspaces associated to the eigenvalue of the action of on and , and is the eigenspace associated to the eigenvalue of the action of on the complex Hodge bundle .

We observe that the eigenspace associated to the eigenvalue is trivial: indeed, any non-trivial cohomology class in this eigenspace would project under into a non-trivial cohomology class of the Riemann sphere (whose first cohomology group is trivial).

Moreover, the reader can verify that the following list is a basis of holomorphic 1-forms of :

In particular, we see that for every . Furthermore, since is complex conjugate to , we conclude that , so that the dimensions of all pieces of the above decomposition of the complex Hodge bundle are determined.

This kind of discussion (of certain cyclic covers of ) was recently performed by C. McMullen (in this article here) in connection with a natural monodromy representation of braid groups in this context. For instance, let us fix an initial configuration of points and let us consider a continuous closed path of configurations of six points such that for each , are mutually distinct and (side remark: this last condition means that we find at time the same configuration we had at time as a set; of course, we could require that the configuration is the same as an ordered set; evidently, this alternative has its own interest as it leads to the pure braid group, but we will not consider it here). Informally, such a closed path corresponds to (continuously) move around an initial configuration of six points in a certain way and, after a while, we come back to the initial configuration. By definition, the braid group of configurations of six points is the group of isotopy classes of closed paths as above. Of course, any element of the braid group define a way of deforming complex structures in the moduli space via the map , and, again by definition, the variation of Hodge structures along such paths induces (with the aid of the so-called Gauss-Manin connection) a monodromy representation

of the braid group .

It is not hard to check that the action of commutes with the monodromy representation , so that the eigenspaces of can be used to diagonalize by blocks . In order to analyze the restrictions of on each of these eigenspaces, it is convenient to introduce the “intersection” form

for . In the literature, this Hermitian form is known as Hodge form. It is a Hermitian form with signature as its restriction to is positive-definite, while its restriction to is negative-definite. As a matter of fact, the Hodge form is preserved by Gauss-Manin connection, so that acts on by automorphisms preserving the Hodge form .

Now, we consider the restriction of to an eigenspace . Since , and , we see that acts on by automorphisms preserving the restriction of the Hodge form to it, that is, an Hermitian form of signature . In other words, we have that

where is the group of matrices preserving a Hermitian form of signature .

Of course, the representations and are complex conjugated, so we only need to understand them when and .

For , we have that acts by matrices, that is, acts by isometries (with respect to the positive definite Hermitian form of signature obtained by the restriction of the Hodge form to ). Moreover, the facts that and is “purely topological” (in the sense that it is defined as the eigenspace of the cohomological action of the automorphism ) imply that is a fixed part (rigid factor) of the Jacobian of . The definition of a fixed part is a complex torus (of positive dimension) such that there exists an isogeny , where is a family of curves and is independent of . In the case of , we can informally say that generates a rigid factor on the Jacobian of because the fact that a 1-form inside is holomorphic is independent on the Riemann surface structure (as and is independent of Riemann surface structures since it is “purely topological”).

For the other values of (i.e., and ), we do not have actions by isometries (nor rigid factors), but C. McMullen computed the image of monodromy representations (in particular ) by relating them to Artin systems, complex reflections (and sometimes to Burau representations). As a consequence of his computations, he was able to show that the action of on is irreducible for every .

Partly motivated by this, G. Forni, A. Zorich and I (see Appendix B of this preprint here) decided to “insert some dynamics” into C. McMullen’s discussion by attaching an Abelian differential to each Riemann surface in such a way that the resulting object in the moduli space of Abelian differentials is invariant under the natural action of on such moduli spaces (see this post here for a brief account on the natural -action on the moduli space of Abelian differentials).

More precisely, we attach to the Abelian differential . By checking at the ramification points (notice that there is no ramification at ), one see that has a zero of order over and zeroes of order over the five points . We encode this information by saying that (see this post here for an “explanation” of this notation). Observe that, by Riemann-Hurwitz formula, , i.e., (a fact that we already knew). In this way, we have that defines a locus of (the moduli space of genus 10 Abelian differentials with a zero of order and five double zeroes).

We claim that is invariant under the natural -action on . This can be proved by the following argument. By definition, is -anti-invariant, so that is -invariant, and hence can be projected into Riemann sphere . By direct verification, one sees that

that is, projects (under ) into a quadratic differential with a single zero of order (simple zero) and five poles of order (simple poles), i.e., . Now, by definitions, the natural -action on commutes with the covering map (as acts by post-composition with appropriate charts while the covering has to do with pre-compositionwith charts). By combining this with the fact that is -invariant (since this action don’t change the order of zeroes), we conclude that is also -invariant.

Remark 1 Just to “count dimensions”: has dimension because we need parameters to determine the complex structure of (as we can always normalize of them to be ), and parameter to determine the Abelian differential, and has dimension as well (by using period coordinates, see this post here).

Observe that we have an intermediate covering , , where

One has is an Abelian differential with a double zero over and no zeroes otherwise, i.e., . By the same argument above, defines a -invariant locus of dimension of . Since has dimension (again by using period coordinates, see this post here), we get that the locus is exactly . In other words, is a copy of (i.e., it is isomorphicto) inside the moduli space of Abelian differentials of genus 10.

Just to get a “feeling” of what the action on looks like, we notice the following facts. It is not hard to check that the flat structure associated to is described by the following octagon (whose opposite parallel sides are identified):

Flat structure associated to a .

Here, the vertices of this octagon are all identified to a single point corresponding to . Moreover, since are Weierstrass points of , one can organize the picture in such a way that the four points are located exactly at the middle points of the sides, and is located at the “symmetry center” of the octagon. See the picture above for an indication of the relative positions of (marked by a black dot) and (marked by crosses). In this way, we obtain a concrete description of where the action is reasonably easy to understand: given and denoting by the Abelian differential associated to the planar figure above, we define to be the Abelian differential associated to the object obtained by letting the matrix act on the planar figure above.

Moreover, since the locus is defined by Abelian differentials given by certain triple (ramified) covers of the Abelian differentials , one can check that the flat structure associated to are described by the following picture:

Flat structure associated to a .

Here, we glue the half-sides determined by the vertices (black dots) and the crosses of these five pentagons in a cyclic way, so that every time we positively cross the side of a pentagon indexed by , we move to the corresponding side on the pentagon indexed (mod ). For instance, in the figure above we illustrated the effect of going around the singularity point over .

In this language, the Teichmüller geodesic flow is the action of the diagonal subgroup of . From now on, we will restrict our attention to the unit area Abelian differentials inside our loci , and . It is not hard to check that the -action preserves the total area of Abelian differentials, and moreover, by the results of H. Masur and W. Veech, the action of the Teichmüller flow is ergodic in the subset of unit area elements of , (and, a fortiori, ) with respect to a natural (“Lebesgue-like”) -invariant probability (sometimes called Masur-Veech measure). Please see this post here for more information on this.

Remark 2 In fact, the works of K. Calta and C. McMullen allow to classify all -invariant probabilities supported on (somehow in the spirit of Ratner’s theorem). In principle, most of the discussion below could be extended to all of these probability measures, but, for sake of simplicity, we will stick to the Masur-Veech measure in the sequel.

In view of the ergodicity of the Teichmüller flow, we can play the following game: starting with a “typical” (i.e., for -a.e. ), we can run the Teichmüller flow for a very long time until we come back very close to ; by completing the trajectory segment with a small path connecting to , we get a closed path ; then, we can look at the monodromy matrix on the (real and/or complex) Hodge bundle associated to all (homotopy classes of) paths obtained in this way. In the literature, this “monodromy representation over the Teichmüller flow” is known as the Kontsevich-Zorich (KZ) cocycle . For an introduction to the Kontsevich-Zorich cocycle the reader may consult this post here. Actually, we can think of the Kontsevich-Zorich cocycle as a part of the monodromy representation of the -action on the Hodge bundle (through parallel transport with respect to Gauss-Manin connection).

The main goal of today’s post is the study of the Lyapunov spectrum (that is, the collection of Lyapunov exponents) of the Kontsevich-Zorich cocycle on the Hodge bundle over as a prototypical example of the conjectural general behavior of KZ cocycle on the Hodge bundle of the support of any -invariant probability supported in any stratum of moduli space of Abelian differentials. Before proceeding, let me just do some propaganda on why one should care about Lyapunov exponents of KZ cocycle. A first reason (coming from Dynamics) is the fact that these exponents can be shown to govern deviations of Birkhoff sums (ergodic averages) of interval exchange transformations, translation flows and billiards on rational polygons essentially because the Teichmüller flow and KZ cocycle act as “renormalization dynamics” for these (zero entropy) systems. A second reason (coming from Statistical Mechanics) is the fact that these exponents were recently shown to govern the rates of diffusion on Ehrenfest “wind-tree” model of Lorenz gases. A third reason (coming from Algebraic Geometry) is the fact that the sum of Lyapunov exponents can be related to “orbifold degrees” of the determinant bundle of the Hodge bundle (by some formulas derived by M. Kontsevich, G. Forni, I. Bouw and M. Möller, and, more recently, A. Eskin, M. Kontsevich and A. Zorich). The reader is encouraged to consult A. Zorich’s survey, these posts here, and these articles here as some references for these three motivations for the study of Lyapunov exponents of KZ cocycle.

Remark 3 In fact, during the discussion below, we will consider exclusively the non-negative Lyapunov exponents of KZ cocycle: indeed, it is known that the Lyapunov spectrum of KZ cocycle is symmetric with respect to the origin (i.e., whenever is a Lyapunov exponent then is also a Lyapunov exponent) due to a certain “symplecticity” (see this post here for more details).

We begin with the remark that , and hence KZ cocycle, acts by monodromy (with respect to Gauss-Manin connection) on the Hodge bundle over . Therefore, the decomposition of in terms of eigenspaces of and the Hodge form are preserved by them. In particular, the restriction of to acts by matrices inside the group .

As a consequence, for and , acts on by isometries, and hence the Lyapunov exponents are all zero (as they measure exponential rates of growth). A nice information coming out of this is the continuity (and actually real-analyticity) of this part of the neutral Oseledets bundle (associated to the vectors with zero Lyapunov exponents): indeed, as we just saw, for or , one has , and, in general, depends continuously (actually analytically) on the base point . The reader should notice that this continuity of the neutral Oseledets bundle is somehow a “precious” information (since, in general, Oseledets theorem ensures only its measurability).

Remark 4 For some known examples (such as square-tiled cyclic covers, a family of cyclic covers of branched at points giving rise to interesting square-tiled surfaces, see these articles here for more information on them), it is possible to show continuity (and analyticity) of the neutral Osedelets bundle along the following lines. The variation of the Hodge form along cohomology classes is driven by the second fundamental form (Kodaira-Spencer map) of the Gauss-Manin connection on the Hodge bundle in the sense that we can write down variational formulas of the form

Thus, if the annihilator of is Teichmüller flow invariant, it is a natural candidate for the neutral Oseledets bundle . Furthermore, it is not hard to check that depends continuously (actually analytically) on the base point. Hence, the continuity (and analyticity) of follows whenever the equality is true. Actually, in the case of square-tiled cyclic covers, G. Forni, A. Zorich and I were capable (in Appendix A of this paper here) of verifying that the Teichmüller flow invariance of and the equality . Of course, this may lead one to conjecture that this maybe always true, but, as we are going to see below, this is not quite the case.

On the other hand, for , we have that is isomorphic to (as a -module) through the natural injective map . In particular, has the same exponents as the Kontsevich-Zorich cocycle on the Hodge bundle over . By the results of M. Bainbridge, in this particular (genus 2) situation, the (non-negative) Lyapunov exponents are and .

At this stage, it remains “only” to understand the (non-negative) Lyapunov exponents of the restriction of to for and . Actually, since is conjugated to , it suffices to discuss one of these cases, say . Here, we have the information that has monodromy . We claim that this is sufficient to conclude that has vanishing exponents at least. Indeed, this is a direct consequence of the following more general proposition about cocycles:

Proposition 1 Suppose that is a cocycle, i.e., a linear cocycle on a (measurable) complex vector bundle over an ergodic flow (with respect to a probability ) on the base of preserving a (measurable) family of Hermitian forms of signature . Assume that is a -integrable cocycle (with respect to some measurable family of norms on the fibers of ), so that the conditions of Oseledets theorem are met. Then, has vanishing exponents at least.

Remark 5 I’m sure that this proposition was known to experts in random products of matrices (such as A. Raugi and Y. Guivarch, and I. Goldscheid and G. Margulis) because its proof is completely elementary (as we’re going to see). However, I was unable to locate a precise reference where this appears for the first time.

The proof of this proposition has two ingredients. The first one is the fact that the stable and unstable Oseledets spaces (associated to negative and positive Lyapunov exponents of resp.) are “exiled” to the light-cone (null cone) of the Hermitian form :

Lemma 2 One has .

Proof: Take . On one hand, for every because the Hermitian form is preserved by (by hypothesis). On the other hand, the fact that implies that the norm of decays (exponentially fast) to zero as .

At this point, one is tempted to say that the exponential decay of implies as , so that one would have , that is, . But we should be a little bit careful (as we’re dealing with measurable families of Hermitian forms and norms). The formal argument goes as follows. Since our vector bundle is finite-dimensional, we can “compare” the (measurable) family of Hermitian form with the (measurable) family of norms in the sense that a Cauchy-Schwarz inequality is true up to a multiplicative factor maybe depending (measurably) on the base point , i.e., . By Luzin’s theorem, these comparisons can be made uniform on large compact sets (of almost full -measure), so that, by the ergodicity of the flow , we may assume is uniformly controlled by the norms of for a sequence of times going to infinity. In any event, the conclusion is that implies , and this completes the proof of the lemma.

The second ingredient is the following simple linear algebra lemma:

Lemma 3 Let be a complex vector subspace contained in the light-cone of a Hermitian form of signature . Then, .

Proof: We can always choose our coordinates so that . Without loss of generality, suppose that . Reasoning by contradiction, let’s assume that . Then, one could find linearly independent vectors . Using these vectors, we can define vectors () by temporarily “forgetting” the last coordinates of . Now, we consider a non-trivial linear combination of equal to :

where . It follows that the vector has its first coordinates equal to zero, i.e., On the other hand, since , one has , so that . In other words, we found a vanishing non-trivial linear combination of the linearly independent vectors , a contradiction.

Putting these two ingredients (lemmas) together, we find that . Since the total dimension of the fibers of is , we find that the neutral Oseledets bundle has dimension , so that the proof of Proposition 1 is complete.

Going back to our concrete example, we have that the restriction of to has 2 vanishing exponents (due to the monodromy ).

Actually, it is possible to show that the remaining two Lyapunov exponents are non-zero, and, by using a formula for the sum of Lyapunov exponents due to A. Eskin, M. Kontsevich and A. Zorich, and by computing the so-called Siegel-Veech constant of , one can determine their explicit value: and . The details of this computation will appear in a forthcoming article by G. Forni, A. Zorich and myself.

In any event, given the discussion in Remark 4 above, one can ask whether coincides with the annihilator of the second fundamental form of Gauss-Manin connection restricted to , and/or whether is continuous. In the forthcoming article by G. Forni, A. Zorich and myself (alluded to in the previous paragraph), we show that doesn’t coincide with the annihilator of , but this still leaves open the possibility that is continuous.

Remark 6 During an exposition at Rennes (in January 2011), Y. Guivarch asked whether still acts isometrically on (now that one can’t use variational formulas involving to deduce this property) or whether one has genuine subexponential growth in this subbundle. As it turns out, acts isometrically on by the following argument: one has that is outside the light-cone because the stable and unstable Oseledets subspaces have dimension (and corresponding exponents ), and so, if were non-trivial, we would get a subspace of dimension at least inside the light-cone of an Hermitian form of signature , a contradiction with the lemma above. In other words, the light-cone is a geometric mechanism of production of neutral Oseledets subbundles with isometric behavior genuinely different from the also geometric method of using the annihilator of the second fundamental form of Gauss-Manin connection of the Hodge bundle.

Heuristically, one strategy to “prove” that is not very smooth goes as follows: as it is indicated in this previous post here, the Lyapunov exponents of the Teichmüller flow can be deduced from the ones of the KZ cocycle by shifting them by ; in this way, it is possible to check that the smallest non-negative Lyapunov exponent of the Teichmüller flow is ; therefore, the generic points tend to be separated by Teichmüller flow by after time ; on the other hand, the largest Lyapunov exponent on the fiber is , so that the angle between the neutral Oseledets bundle over two generic points grows by after time ; hence, in general, one can’t expect the neutral Oseledets bundle to be better than Hölder continuous.

Of course, there are several details missing in this heuristic, and currently I don’t know how to render it into a formal argument. However, in a recent work still in progress, A. Avila, J.-C. Yoccoz and I prove (among other things) that is not continuous at all (and hence only measurable by Oseledets theorem). The next section contains a brief sketch of this proof of the non-continuity of .

-Coding of the Kontsevich-Zorich cocycle over -

The Teichmüller flow and the Kontsevich-Zorich cocycle over (connected components of) strata can be efficiently coded by means of the so-called Rauzy-Veech induction. Roughly speaking, given a (connected component of a) stratrum of Abelian differentials of genus , the Rauzy-Veech induction associates the following objects: a finite oriented graph , a finite collection of simplices (“Rauzy-Veech boxes”) and a finite number of copies of a Euclidean space over each vertex of , and, for each arrow of , a (expanding) projective map between (parts of) the simplices over the vertices connected by this arrow, and a matrix between the copies of over the vertices connected by this arrow. (I strongly recommend J.-C. Yoccoz’s survey for more details on the Rauzy-Veech induction).

In this language, the simplices (Rauzy-Veech boxes) over the vertices of this graph represent admissible paramaters determining translations surfaces (Abelian differentials on Riemann surfaces ) in , the (expanding) projective map between (parts of) the simplices (associated to vertices connected by a given arrow) correspond to the action of the Teichmüller flow on the parameter space (after running this flow for an adequate amount of time), and the matrices (attached to the arrows) on are the action of the Kontsevich-Zorich cocycle on the first cohomology group .

Among the main properties of the Rauzy-Veech induction, we can highlight the fact that it permits to “simulate” almost every (with respect to Masur-Veech measure) orbit of Teichmüller flow on on in the sense that these trajectories correspond to (certain) infinite paths on the graph . In order words, the Rauzy-Veech induction allows to code the Teichmüller flow as a subshift of a Markov shift on countably many symbols (as one can use loops on based on an arbitrarily fixed vertex as basic symbols / letters of the alphabet of our Markov subshift). Moreover, the KZ cocycle over these trajectories of Teichmüller flow can be computed by simply multiplying the matrices attached to the arrows one sees while following the corresponding infinite path on . Equivalently, we can think the KZ cocycle as a monoid of (countably many) matrices (as we can only multiply the matrices precisely when our oriented arrows can be concatened, but in principle we don’t dispose of the inverses of our matrices because we don’t have the right to “revert” the orientation of the arrows).

In the particular case of , the associated graph is depicted below:

Rauzy diagram associated to . The letters near the arrows are not important here, only the vertices (black dots) and the arrows between them.

Now, we observe that was defined by taking certain triple covers of Abelian differentials of , so that it is also possible to code the Teichmüller flow and KZ cocycle on by the same graph and the same simplices over its vertices, but by changing the matrices attached to the arrows: in the case of , these matrices acted on , but in the case of they act on and they contain the matrices of the case of as a block.

At this stage, one can prove non-continuity of as follows.

Firstly, one computes the restriction of KZ cocycle (or rather the matrices of the monoid) to on certain “elementary” loops and one checks that they have finite order. In particular, every time we can get the inverses of the matrices associated to these elementary loops by simply repeating these loops an appropriate number of times (namely, the order of the matrix minus 1). On the other hand, since these elementary loops are set up so that any infinite path (coding a Teichmüller flow orbit) is a concatenation of elementary loops, one conclude that the action (on ) of our monoid of matrices is through a group! In particular, given any loop (not necessarily an elementary one), we can find another loop such that the matrix attached to (i.e., the matrix obtained by multiplying the matrices attached to the arrows forming “in the order they show up” with respect to their natural orientation of ) is exactly the inverse of the matrix attached to .

Secondly, by computing with a pair of “sufficiently random” loops and , it is not hard to see that we can choose such that their attached matrices and have distinct and/or transverse central eigenspaces and (associated to eigenvalues of modulus ).

In this way, the periodic orbits (pseudo-Anosov orbits) of the Teichmüller flow coded by the infinite paths and obtained by infinite concatenation of the loops and have distinct and/or transverse neutral Oseldets bundle, but this is no contradiction to continuity since the base points of these periodic orbits are not very close. However, we can use and to produce a contradiction as follows. Let a large integer. Since our monoid acts by a group, we can find a loop such that the matrix attached to it is . It follows that the matrix attached to the loop is , i.e., . Therefore, the infinite paths and produce periodic orbits whose neutral Oseledets bundle still are and (and hence, distinct and/or transverse), but this time their basepoints are arbitrarily close (as ) because the first “symbols” (loops) of the paths coding them are equal (to ).

Remark 7 Actually, this argument is part of more general considerations (in the forthcoming paper by A. Avila, J.-C. Yoccoz and myself) on certain cyclic covers obtained by taking copies of a regular polygon with sides, and cyclically gluing the sides of these polygons in such a way that their middle points become ramification points: indeed, corresponds to the case and of this construction.

Remark 8 It is interesting to notice that the real version of Kontsevich-Zorich cocycle over on is a irreducible symplectic cocycle with non-continuous neutral Oseledets bundle. In principle, this irreducibility at the real level makes it difficult to see the presence of zero exponents, so that the passage to its complex version (where we can decompose it as a sum of two complex conjugated monodromy representations by matrices in and ) reveals a “hidden truth” not immediately detectable from the real point of view (thus confirming the famous quotation of J. Hadamard: “the shortest route between two truths in the real domain passes through the complex domain”). I believe this example has some independent interest because, to the best of my knowledge, most examples of symplectic cocycles and/or diffeomorphisms exhibiting some zero Lyapunov exponents usually have smooth neutral Oseldets bundle due to some sort of “invariance principle” (see this article of A. Avila and M. Viana for some illustrations of this).

Closing today’s post, we state in the next (final) section two “optimistic guesses” (part of a forthcoming paper by G. Forni, A. Zorich and myself) on the features of the KZ cocycle over the support of general -invariant probabilities. Notice that we call these “optimistic guesses” instead of “conjectures” because we think they’re shared (to some extent) by others working with Lyapunov exponents of KZ cocycle (and so it would be unfair to state them as “our” conjectures).

-Two optimistic guesses-

Optimistic Guess 1. Let be a -invariant probability in some connected component of a stratrum of Abelian differentials and denote by its support. Then, there exists a finite (ramified) cover such that (the lift of) the Hodge bundle over can be decomposed into a direct sum of continuous

where , are distinct -irreducible representations admiting Hodge filtrations , such that , , , are complex vector spaces (taking into account the multiplicities of the irreducible factors , ), and is the tautological bundle , , , . Moreover, this decomposition is unique and it can’t be further refined after passing to any further finite cover.

Remark 9 When is the (unique) -invariant probability supported on a Teichmüller cover (i.e., a closed -orbit), the Optimistic Guess 1 is a consequence of Deligne’s semisimplicity theorem.

Optimistic Guess 2. In the setting of Optimistic Guess 1, denote by

and

Then, the Lyapunov spectrum of the KZ cocycle on is simple, i.e.,

and the Lyapunov spectrum of the KZ cocycle on is “as simple as possible”, i.e.,

Remark 10 This “guess” is based on the general philosophy (supported by works as the ones of A. Raugi and Y. Guivarch, and I. Goldscheid and G. Margulis) that, after reducing our cocycle to irreducible pieces, if the cocycle restricted to such a piece is “sufficiently generic” inside a certain Lie group of matrices , then the Lyapunov spectrum on this piece should look like the “Lyapunov spectrum” (i.e., collection of the logarithms of the norms of eigenvalues) of the “generic” matrix of . For instance, since a generic matrix inside the group has spectrum

where , the above guess essentially claims that, once one reduces the KZ cocycle to irreducible pieces, its Lyapunov spectrum on each piece must be as generic as possible.

Remark 11 Notice that the previous guess doesn’t make any attempt to compare Lyapunov exponents within distinct irreducible factors: indeed, in general non-isomorphic representations may lead to the same exponent by “pure chance” (as it happens in the case of certain genus Abelian differentials associated to the “wind-tree model”).

Firstly, let me say that we mostly consider this article as a survey. Indeed, a large portion of this article is dedicated to revisit some variational formulas in G. Forni’s paper using the perspective of Differential Geometry: more precisely, we re-interpret Forni’s variational formulas for the growth of Hodge norms of vectors and isotropic subspaces of the Hodge bundle in terms of features of the second fundamental form (a.k.a. Kodaira-Spencer map) and the curvature form of the Gauss-Manin connection on the Hodge bundle.

Then, we use this point of view to detect the minimal set of assumptions on a subbundle of the (real) Hodge bundle under which a “Kontsevich-Zorich-Forni like formula” for the sum of Lyapunov exponents holds: whenever is a -equivariant and Hodge-star invariant, the sum of the non-negative Lyapunov exponents related to is described by the average of the eigenvalues of a certain “curvature form” (see Corollary 3.5).

Furthermore, also by using this point of view, we deduce a new result (Theorem 3) relating the neutral Oseledets bundle of the Kontsevich-Zorich cocycle (i.e., the Oseledets subspace associated to the zero Lyapunov exponents) and the annihilator of the second fundamental form of the Gauss-Manin connection. In particular, we show that, if is Teichmuller-flow invariant then , and, if is -invariant, then , and hence and coincide whenever they are -invariant, i.e., the annihilator of the second fundamental form is a natural candidate for the neutral Oseledets bundles of the Kontsevich-Zorich cocycle (at least under the appropriate invariance assumptions). An interesting corollary of this (and the fact that the infinitesimal variation of the Hodge norm along the Kontsevich-Zorich cocycle is measured by the second funamental form, i.e., , see Lemma 2.4) is the fact that the Kontsevich-Zorich cocycle acts by isometries (with respect to the Hodge norm) along whenever this subbundle is -invariant.

Finally, we “test” this new result against two classes of examples (presented in Appendix A and B). In the first class of examples, namely, square-tiled cyclic covers, we verify that both and are -invariant. As a consequence, we derive that (see Theorem 7). In other words, the neutral Oseledets bundle (responsible for the zero exponents of the Kontsevich-Zorich cocycle) has a natural geometric explanation: it is the annihilator of the second fundamental form.  Some nice consequences of this fact are that the Kontsevich-Zorich cocycle acts by isometries along , and is continuous (and actually real-analytic) in the case of square-tiled cyclic covers: indeed, this is so because the second fundamental form is a continuous (actually, real-analytic) object. However, as we announce  in Appendix B (leaving the details for a forthcoming paper), the neutral Oseledets bundle doesn’t coincide with in general! In fact, based on some constructions of C. McMullen, we exhibit an example where is not -invariant (and hence ) despite the fact that and have the same rank! In any event, even though the annihilator of the second fundamental form is not responsible for the zero exponents in this example, the mechanism for the existence of is not very complicated: essentially, we are dealing with a cocycle of matrices preserving an indefinite non-degenerate Hermitian form (i.e., we have a cocycle of matrices where is the signature of the invariant Hermitian form), and a simple (linear-algebra) argument allows to prove the existence of zero exponents in this context. Also, let me point out that in this example the Kontsevich-Zorich cocycle also acts by isometries, but the conceptual reason behind this is different from the square-tiled cyclic covers case! Of course, we will come back to this issue later in this blog (most likely when the promised forthcoming paper comes out).

Closing this post, let me make two points. The first one is that, besides the “applications” given in the Appendices to this survey article, we feel that this point of view (of using the second fundamental form to understand the Kontsevich-Zorich cocycle) might be helpful in other contexts (and that’s what motivated us to write down this survey). For instance, Alex Eskin communicated to us that the discussion in our survey is useful when trying to derive certain semisimplicity statements of the “algebraic hulls” (in the sense of Zimmer) of the Kontsevich-Zorich cocycle (needed in his work with Maryam Mirzakhani on classification of -invariant measures in moduli spaces). The second point is that, while in the case of square-tiled cyclic covers we showed that the neutral Oseledets bundle is continuous by comparison with the annihilator of the second fundamental form, the same kind of reasoning can’t be applied to the second class of examples (in Appendix B of our article), and so it is natural to ask about the regularity of in this situation. Here, in a work (in progress) by A. Avila, J.-C. Yoccoz and myself, we are able to show (among other things) that is not continuous at all, so that it is only measurable at best. In particular, this gives an example of a symplectic cocycle whose neutral Oseledets bundle is not continuous (in contrast with “most” examples in the literature where zero Lyapunov exponents “usually” are associated to continuous subbundles).  Evidently, I also plan to come more on this work in progress in due time, but for now I think that’s all I have to say on zero exponents of the Kontsevich-Zorich cocycle!

• this theorem has some roots in Physics (more precisely, Statistical Mechanics) and
• its proof has a lot to do with recent advances in the study of the Lyapunov exponents of the Kontsevich-Zorich cocycle (a subject that I’m particularly interested in).

So, while I was preparing these slides here (written in Portuguese), I thought that it could be a nice idea to publish a sort of “extended version” of the slides in this blog. The outcome of this is the text below the fold. I hope you’ll enjoy your reading as much as I enjoyed preparing these notes!

-Ehrenfest wind-tree model-

In 1912, Paul and Tatiana Ehrenfest introduced in their article “Begriffliche Grundlagen der statistischen Auffassung in der Mechanik” (a review on Statiscal Mechanics later translated as The conceptual Foundations of the Statistical Approach in Mechanics) the following model (called “wind-tree model”) of Lorenz gases: one considers a particle moving in a billiard table obtained by removing from the plane a certain number of rectangular obstacles.

In 1980, J. Hardy and J. Weber studied a periodic version of the wind-tree model where rectangular obstacles of dimensions , , are disposed along the integral lattice (i.e., the centers of these rectangles are exactly the points of the plane with integral coordinates).

The figures below (extracted from Vincent Delecroix’s Ph.D. thesis) show some examples of billiard trajectories in periodic wind-tree models.

Fig. 1. Periodic wind-tree model with parameters and .

The meaning of the colors in this figure is the following. In the left side of this picture, our trajectory starts with angle (with the horizontal direction), and the initial piece is colored in blue. Then, it collides with a rectangle, and the new angle is , and the corresponding piece of trajectory is colored in yellow. After that, we have a collision leading to a green piece of trajectory with angle , and finally a collision leading to a red piece of trajectory with angle .

Fig. 2. Piece of trajectory in a periodic wind-tree model drew with a little Python script written by Vincent Delecroix.

The script used to produce the figure above is available at Vincent Delecroix’s webpage.

Fig. 3. Two orbits with distinct dynamical behaviors in a periodic wind-tree model.

Denote by the translation flow in a periodic wind-tree model in the direction starting at . In their article (quoted above), J. Hardy and J. Weber showed that the periodic wind-tree model exhibits a diffusion of (i.e., for a.e. , one has ) along special directions , , corresponding to generalized diagonals (e.g., when , is such a direction). Roughly speaking, the proof of the diffusion result of J. Hardy and J. Weber is based on the observation that the translation flow along generalized diagonals of periodic wind-tree models can be interpreted as a particular type of skew-product over a rotation of the circle (and this kind of dynamical system can be reasonably analyzed with direct calculations). Moreover, it was implicitly conjectured that one should expect “abnormal diffusion” along typical directions , that is,

Here, the “justification” of the word “abnormal” comes by comparison with Brownian motion and/or central limit theorem: once we know that the diffusion is “sublinear” (maybe after removing the “average”), the “natural” scaling factor to converge (maybe to a normal distribution) is , i.e., a “normal” diffusion would correspond to

Remark 1 In the case of translation flows, a theorem of S. Kerckhoff, H. Masur and J. Smillie ensures that the diffusion is “always” sublinear, i.e.,

for almost every and .

In this direction, the following (very recent) theorem of V. Delecroix, P. Hubert and S. Lelièvre confirms the “abnormal” diffusion conjecture:

Theorem 1 (Delecroix, Hubert, Lelièvre (2011)) For the periodic wind-tree model, one has

in the following cases:

• (a) for a.e. , , and any with infinite -trajectory;
• (b) for , for a.e. , and any with infinite -trajectory;
• (c) for , , with and , for a.e. , and any with infinite -trajectory.

Our main goal here is the presentation of a sketch of proof of this theorem. In particular, we will see how the value emerges in the result above: in a nutshell, it is a Lyapunov exponent of the Kontsevich-Zorich cocycle over a -invariant locus of genus 5 Riemann surfaces.

Remark 2 In fact, Vincent Delecroix found recently that the speed of diffusion may be faster than 2/3 for other wind-tree models (namely, -periodic wind-tree models where is an appropriately chosen lattice in ). We will say a few words on this by the end of this post.

The organization of the subsequent discussion is as follows. In the next section, we briefly recall the so-called Katok-Zemlyakov construction of translation surfaces associated to rational billiards. Then, in a subsequent section, we reduce the study of diffusion speed to the study of algebraic intersections of pieces of trajectories of with certain -valued cocycles. Afterwards, a section will be dedicated to relate these algebraic intersections with the Lyapunov exponents of the Kontsevich-Zorich cocycle (by following the works of A. Zorich and G. Forni). At this point, the sketch of proof of Delecroix-Hubert-Lelièvre theorem will be completed by computing the relevant Lyapunov exponents (and deriving the value ). Finally, the last section will consist in a few words on Remark 2 above.

-Katok-Zemlyakov construction-

Given a rational polygon , that is, a polygon whose inner angles are all rational multiples of , it is not hard to see that the billiard flow in is equivalent to a translation flow in a translation surface (see this link here for definitions and more details on these notions) obtained by reflecting the sides of the polygon every time a trajectory hits the boundary of . In other words, instead of changing of direction by the reflection law in the billiard , we keep the same direction but reflect the polygon in order to be able to continue the trajectory. Below, the reader can see a figure (extracted from slides of a talk of Corinna Ulcigrai) where this unfolding process is illustrated in the case of a square:

In the literature, this unfolding procedure was introduced by R. Fox and R. Keshner (in 1936) and rediscovered (independently) by A. Katok and A. Zemlyakov (in 1975), and it is popularly called Katok-Zemlyakov construction.

In the particular case of the figure above, we get a torus as the resulting translation surface of the Katok-Zemlyakov construction applied to the billiard in the unit square.

We observe that the rationality condition above was imposed on the polygon only to get that the associated translation surface is compact (as the group generated by the reflections along the sides of is finite when the inner angles belong to ). However, the Katok-Zemlyakov construction can also be applied to the non-compact billiard table of the -periodic wind-tree model with parameters , but the resulting translation surface is not easy to visualize.

We overcome this visualization difficulty as follows. The figure below shows a fundamental domain in :

Fig. 4. Fundamental domain in (see Delecroix's thesis)

Here, the meaning of the vectors and near the boundaries of the fundamental domain is the following. We cover with countably many copies of indexed by in the natural way. Then, any billiard trajectory in may be coded by a billiard trajectory in and a sequence of integers determining the copies of the trajectory is passing by. In other words, every time a trajectory in hits the boundary, we use the corresponding vector (one of the vectors and ) to indicate that we’re changing from a copy of of index to the copy of of index . Using technical jargon, this means that we can see the billiard on as a skew-product of the billiard on by a -valued cocycle.

In this context, we can apply the Katok-Zemlyakov construction to . The resulting translation surface is indicated below.

Fig. 5. Translation surface (see Delecroix's thesis)

It follows that the translation surface obtained from the Katok-Zemlyakov construction applied to is a -cover of (with the -cocycle indicated in the picture above).

Remark 3 A “more natural” fundamental domain for the billiard in is showed in the left side of the picture below:

Fig. 6. Two fundamental domains in (see Delecroix's thesis)

  However, from the dynamical point of view, these domains are completely equivalent.

Concerning the (flat) geometry of the translation surface , one can check that it has distinct conic singularities with total angle , that is, the natural Abelian differential on has 4 double zeros, i.e., . By Gauss-Bonnet theorem, is a Riemann surface of genus 5.

In resume, the billiard flow in direction on is dynamically equivalent to a skew-product given by a -cocycle over the translation flow in direction on the genus 5 translation surface .

-Speed of diffusion and algebraic intersections of cycles-

The speed of diffusion of (with ) can be studied as follows.

In the genus 5 surface , denote by the segment of orbit and consider the closed segment obtained from by “closing” its endpoints and with a segment of bounded length.

Next, we introduce the cycle

where , , are the cycles indicated in Figure 5.

Since , it makes sense to consider the algebraic intersection . In this notation, we have the following fact:

Lemma 2 It holds . In particular,

Proof: The second statement follows from the first one, and the latter follows by definitions and the fact that the fundamental domain has diameter .

From this lemma, we get:

In other words, the study of speed of diffusion is equivalent to the study of algebraic intersections of certain cycles in .

Remark 4 The algebraic intersections correspond to “special Birkhoff sums”: for sufficiently small times and it stays like that until hits the “boundary” of , say at a time ; at this point, for sufficiently close to , where is a vector of the form or (depending on the boundary cycle we landed), and it stays like that until hits again the “boundary” of , say at a time ; at this point, for sufficiently close to , where is a vector of the form or (depending on the boundary cycle we landed); and so on ad infinitum.

-Algebraic intersections, Teichmüller flow and Kontsevich-Zorich cocycle-

Following the works of Anton Zorich (1994 and 1997), and Giovanni Forni (1997 and 2002), we will try to understand the algebraic intersections by means of the so-called Kontsevich-Zorich cocycle over the Teichmüller flow.

Roughly speaking, the basic idea is: is a very long and complicated segment (of length ) in , so that a direct study might be complicated; in order to overcome this difficulty, we make shorter, by first rotating (if necessary) so that one can suppose that , and then applying the diagonal matrix to ; since is a vertical segment of length on , by applying to , one gets a vertical segment of length on .

In particular, by taking (i.e., ), we have that is a vertical segment of length . The picture below shows the effect of on a translation surface formed by 4 “zippered rectangles”:

Fig. 7. Action of on a translation surface.

In the literature, the action of on translation surfaces is called Teichmüller flow. In this language, we see that the Teichmüller flow can be used to shorten pieces of trajectories of the translation flow in translation surfaces, or in other words,

The Teichmüller flow is a nice renormalization procedure on translation flows.

Of course, there is “price” to pay: our unit length segment doesn’t live anymore in the surface , but in the distorted surface . At this point, we have the impression that the use of Teichmüller flow only pushes the difficulty from one place to another, but this is not quite true: it is known that the Teichmüller flow has nice recurrence properties in the moduli space of translation surfaces, that is, for infinitely many times , is “essentially” the same surface as in the moduli space, i.e., and are very similar surfaces up to some cutting and pasting operations. The figure below (extracted from A. Zorich’s survey) illustrates some “cutting and pasting operations” in a surface derived from a translation surface by applying the Teichmüller flow:

Fig. 8. Actions of the Teichmüller flow and the mapping class group.

The cutting and pasting operations on a surface of genus correspond to the action of the so-called mapping class-group of isotopy classes of orientation-preserving diffeomorphisms of : indeed, these cutting and pasting operations are simply changes on the “names” of homology classes in given by the natural (symplectic) action on homology of certain diffeomorphisms.

Going back to our initial problem of studying algebraic intersections , we saw that the action of on allows us to think of ( and also) (to some extent) as a segment of length for in a surface essentially “equal” to (at least for some infinite sequence of times ). However, since the equality in the moduli space of translation surfaces only means that is equal to up to some element of the mapping-class group, one may change into another cycle when using to “cut and paste” into .

In other words, because the algebraic intersection is preserved by the natural action on homology of the diffeomorphism , one has

Remark 5 Strictly speaking, the previous equality is not exactly true: while the Teichmüller flow makes shorter, since is obtained from by “closing” its endpoints with arbitrarysegments of bounded length, this introduces some little technical difficulties. However, it is possible to show that the potential discrepancies introduced by these “closing” procedures don’t affect the study of speed of diffusion, that is, it holds

Here, we used the “time change” .

In the literature, the family of linear operators (matrices) on the first homology group induced by the action of the elements of the mapping-class group used to convert into a surface “essentially” equal to is called the Kontsevich-Zorich cocycle over the Teichmüller flow. Keeping this in mind, one sees that the expression measures the (exponential) rate of growth of , i.e., the study of speed of diffusion is equivalent to the study of the exponential rate of growth of the Kontsevich-Zorich cocycle applied to the particular homology cycle .

The reader used with Ergodic Theory (and, in particular, Oseledets theorem) recognizes that the computation of is equivalent to determining the Lyapunov exponents of the Kontsevich-Zorich cocycle.

In resume, the problem of studying algebraic intersections is equivalent to the question of determining the Lyapunov exponents of the Kontsevich-Zorich (KZ) cocycle. Again, this seems to lead us nowhere, but this is not quite true: Lyapunov exponents of KZ cocycle received a lot of attention recently, and, as we’re going to see in the next section, we can compute them in several situations (specially in the presence of symmetric surfaces such as ).

-End of “proof” of Delecroix-Hubert-Lelièvre theorem-

As one can check in Figure 5, has a group of symmetries isomorphic to the Klein group . This group of symmetries is generated by a natural “horizontal translation” and a natural “vertical translation” such that (i.e., and are involutions). In particular, we can define , resp. , the subspaces of , resp. , invariant homology classes, and , resp. , the subspaces of , resp. , anti-invariant homology classes. Using these subspaces, one can introduce

By definition, we have . Moreover, since it is possible to verify that the action on homology of and commutes with the Kontsevich-Zorich cocycle (“ and acts by pre-composition with translation charts while KZ cocycle acts by post-composition with translation charts”), the KZ cocycle preserves the susbpaces , , and, hence, can be diagonalized by blocks. Therefore, Lyapunov exponents of are the ones coming from the blocks , , and . Moreover, since the symplectic intersection form is preserved, the Lyapunov spectrum on each block is symmetric: if is a Lyapunov exponent, then is also a Lyapunov exponent. Thus, the Lyapunov spectrum on a block is determined by the list of non-negative Lyapunov exponents. In the sequel, when talking about Lyapunov exponents, we’ll care exclusively about the non-negativeones.

We start with the block . By definition, the quotient is (shown in Figure 4). The translation surface has an unique conical singularity with total angle , that is, (i.e., the natural Abelian differential on has a single zero of order 2). Moreover, by definition, is canonically identified with the homology group , and, under this identification, corresponds to the Kontsevich-Zorich cocycle associated to the -orbit of .

By a celebrated Ratner-like classification theorem of closures of -orbits of genus 2 translation surfaces by K. Calta and C. McMullen (2003, 2005 and 2007), one has is:

• (a) dense in for a.e. ;
• (b) closed (Veech non-arithmetic) for , , but , ;
• (c) closed (Veech arithmetic) for .

In particular, the statement of Delecroix-Hubert-Lelièvre theorem is justified by this classification result.

Moreover, in the case of , the Lyapunov exponents of the Kontsevich-Zorich cocycle were computed by M. Bainbridge and, more recently, by A. Eskin, M. Kontsevich and A. Zorich (using a far-reaching formula to appear in a work still in preparation). The outcome of their work is the fact that the Kontsevich-Zorich cocycle on , or equivalently , has Lyapunov exponents and .

Next, we analyze the blocks and . The surfaces and are hyperelliptic Riemann surfaces of genus in the odd spin connected component of translation surfaces with two double zeroes (see this post here and references therein for more details on connected components of strata ), that is, and belong to the hyperelliptic locus of . In this case, by the aforementioned work of A. Eskin, M. Kontsevich and A. Zorich (see also this recent paper of D. Chen and M. Möller), one gets that the sum of the 3 non-negative Lyapunov exponents of KZ cocycle over these surfaces is . On the other hand, by definition, , resp. , is canonically identified to , resp. , and , resp. , corresponds to the KZ cocycle over , resp. .

In particular, the sum of the non-negative Lyapunov exponents of , resp. is . However, we already saw that contributes with the Lyapunov exponents and , so that , resp. , contributes with a Lyapunov exponent .

Finally, we analyze the block . We consider the surface . It belongs to the hyperelliptic connected component of the stratum genus 3 translation surfaces with two double zeroes. By the work of Eskin, Kontsevich and Zorich (see also the article of Chen and Möller quoted above), the sum of the 3 non-negative Lyapunov exponents of the KZ cocycle is . Here, is canonically identified with , so that contributes with a exponent (as already contributes with and ).

Therefore, the sketch of proof of Delecroix-Hubert-Lelièvre theorem will be complete as soon as we determine the relevant Lyapunov exponent controlling the growth of . By direct inspection, one sees that decomposes as a sum of and , so that the relevant exponents are (associated to and ). Moreover, it can be shown that doesn’t belong to the stable Oseledets space of the KZ cocycle (i.e., the subspace associated to the negative exponents) because is a integer cycle (so that its size can’t go to zero as ). In other words, has size about as , so that

and the argument is “complete”.

-Faster diffusion in other periodic wind-tree models-

Closing today’s post, we will say a few words on a nice argument found by V. Delecroix to construct -periodic wind-tree models () with a diffusion faster than .

The basic idea is that -periodic wind-tree models are less symmetric than their -periodic counterpart: instead of having a Klein group of symmetries, one keeps only one of the involutions , , let’s say for sake of concreteness. In this case, the quotient of by is still a genus surface, but we can’t compute all exponents individually because we don’t dispose of a nice genus 2 surface (i.e., we lose the bundle). Nevertheless, it is possible to prove that the sum of the two relevant Lyapunov exponents controlling is (recall that it was also before), and, actually, has size about as (i.e., it is who controls the rate of growth of ). However, by setting up the parameters and the lattice properly (in particular, , that is, is a square-tiled surface), one can apply a recent criterion of simplicity of Lyapunov exponents of the KZ cocycle over square-tiled surfaces developed by M. Möller, J.-C. Yoccoz and myself (see this preprint here for a statement of this criterion and its application in a very concrete case). In a certain sense, this criterion is a version for square-tiled surfaces of a simplicity result of A. Avila and M. Viana (see these papers here and here) for the KZ cocycle with respect to the Masur-Veech measure, and it is likely that we will discuss this issue in a future post. In any case, it turns out that V. Delecroix was able to check this simplicity criterion in the case of appropriately chosen and , so that one gets in this particular situation. Of course, since ,one has , and the desired result (of diffusion faster than ) follows.

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