The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis
Abstract.
We compute explicitly the action of the group of affine diffeomorphisms on the relative homology of two remarkable origamis discovered respectively by Forni (in genus 3) and ForniMatheus (in genus 4). We show that, in both cases, the action on the non trivial part of the homology is through finite groups. In particular, the action on some dimensional invariant subspace of the homology leaves invariant a root system of type. This provides as a byproduct a new proof of (slightly stronger versions of) the results of Forni and Matheus: the non trivial Lyapunov exponents of the KontsevichZorich cocycle for the Teichmüller disks of these two origamis are equal to zero.
Contents
1. Introduction
Our main objective is the explicit description of the action on homology of the affine group of the squaretiled translation surfaces constructed by Forni [7] and Forni Matheus [8], characterized by the total degeneracy of the socalled KontsevichZorich cocycle. Before going to the statements of our results, let us briefly recall some basic material about these notions.
1.1. The Teichmüller flow and the KontsevichZorich cocycle
Let be a closed oriented surface of genus , and be a finite subset of . We denote by the group of orientationpreserving homeomorphisms of which preserve , by the connected component of the identity in (i.e., is the subset of homeomorphisms in which are isotopic to the identity rel. ) and by the mapping class group of . When is empty, we just write , , for , , .
Definition 1.1.
A translation surface structure on is a complex structure on together with a nonzero holomorphic 1form w.r.t. this complex structure. The Teichmüller space (resp. the moduli space ) is the space of orbits for the natural action of (resp. of ) on the space of translation surface structures. We have thus .
The group acts naturally on by postcomposition on the charts defined by local primitives of the holomorphic form. The Teichmüller flow is the restriction of the action to the diagonal subgroup of on . For later reference, we recall some of the main structures of the Teichmüller space and the moduli space :

is stratified into analytic spaces obtained by fixing the multiplicities of the set of zeros of the holomorphic form (here ); the corresponding Teichmüller space is the space of orbits for the natural action of on the set of translation surface structures with prescribed zeroes in . One has ;

The total area function , is invariant so that the unit bundle and its strata are invariant (and, a fortiori, invariant);

the Teichmüller space has a locally affine structure modeled on the complex vector space : the local charts are given by the period map defined by integrating the holomorphic form against the homology classes in ;

the Lebesgue measure on the Euclidean space induces an absolutely continuous invariant measure on such that the conditional measure induced on is invariant by the action (and hence invariant).
See Veech [16], [18], and the surveys of Yoccoz [20] and Zorich [23] for more details.
Once we get the existence of a natural invariant measure for the Teichmüller flow, it is natural to ask whether has finite mass and/or is ergodic with respect to the Teichmüller dynamics. In this direction, we have the following result:
Theorem 1.2 (Masur [14], Veech [16]).
The total volume of is finite and the Teichmüller flow is ergodic on each connected component of with respect to .
Remark 1.3.
The KontsevichZorich cocycle is the quotient of the trivial cocycle by the action of the mapping class group . As the action is symplectic, the Lyapunov exponents of with respect to any invariant ergodic probability are symmetric w.r.t. :
It turns out that the nonnegative exponents determine the Lyapunov spectrum of the Teichmüller flow^{1}^{1}1In fact, this is one of the motivation of the introduction of the KontsevichZorich cocycle.. Indeed, the Lyapunov exponents of the Teichmüller flow with respect to a invariant ergodic probability on are
See [21] and [11] for further details. On the other hand, Zorich and Kontsevich conjectured that the Lyapunov exponents of for the canonical absolutely continuous measure are all nonzero (i.e., nonuniform hyperbolicity) and distinct (i.e., all Lyapunov exponents have multiplicity 1). After the fundamental works of G. Forni [6] (showing the nonuniform hyperbolicity of ) and AvilaViana [1] (proving the simplicity of the Lyapunov spectrum), it follows that the ZorichKontsevich conjecture is true. In other words, the Lyapunov exponents of at a generic point are all nonzero and they have multiplicity 1.
1.2. The affine group of a translation surface
Let be a translation surface, i.e let be a nonzero holomorphic 1form w.r.t. some complex structure on . We denote as above by the set of zeros of .
Definition 1.4.
The affine group of is the group of orientation preserving homeomorphisms of which preserve and are given by affine maps in the charts defined by local primitives of . In these charts, the differential of an affine map is an element of . We obtain in this way a homomorphism from into . The automorphism group of is the kernel of this homomorphism.
Definition 1.5.
The image of this homomorphism is called the Veech group of and denoted by . It is a discrete subgroup of , equal to the stabilizer of for the action of on .
One has thus an exact sequence
(1.1) 
For a nice account on affine and Veech groups see the survey of Hubert and Schmidt [10].
In genus , the affine group injects into (and a fortiori into ): for elements with non trivial image in , this can be viewed from the period map (see Veech [19]); for elements in , this is a consequence of the Lefschetz fixed point theorem, as fixed points then have index 1.
Consider the natural surjective map:
between the relative and absolute cohomology groups of . We denote by the subspace of spanned by and and by its image in . The subspace (and therefore also ) is dimensional: this can be seen either as a standard fact from Hodge theory or more concretely from Veech’s zippered rectangles construction (see [17], [20], [23]). We denote by the orthogonal of with respect to the exterior product in . We have
because the form defines a nonzero element in .
The intersection form defines a nondegenerate pairing between the homology groups and , and also between and itself.
Let be the annihilator of and be the annihilator of . Both are codimension subspaces. We also introduce the orthogonal of with respect to the intersection form between and , and the orthogonal of for the intersection form on .
Observe that is the annihilator of , and also the image of under the map from to . Both and have dimension 2.
One has
These decompositions only depend on the image of the translation surface structure in Teichmüller space; they are constant along orbits, invariant under the action of , and covariant under the action of the mapping class groups. The same is true for the decompositions of the cohomology groups. In particular, the decomposition
is invariant under the KontsevichZorich cocycle . The subbundle correspond to the extreme exponents of .
Definition 1.6.
The restriction of the KontsevichZorich cocycle to the invariant subbundle is called the reduced KontsevichZorich cocycle and is denoted by .
Remark 1.7.
In general, doesn’t have a invariant supplement inside : see Appendix B.
1.3. Veech surfaces and squaretiled surfaces
Let be a translation surface.
Definition 1.8.
The stabilizer of this orbit in is exactly the affine group . We can thus view the reduced KontsevichZorich cocycle over this closed orbit as the quotient of the trivial cocycle
by the action of the affine group .
The two examples that we will consider belong to a special kind of Veech surfaces.
Definition 1.9.
is a squaretiled surface if the integral of over any path joining two zeros of belongs to . Equivalently, there exists a ramified covering unramified outside such that . Every squaretiled surface is a Veech surface. One says that the squaretiled surface is primitive if the relative periods of span the module . In this case the Veech group is a subgroup of of finite index (see [10], [23]).
In a squaretiled surface , the squares are the connected components of the inverse image , with as above. The set of squares of is finite and equipped with two onetoone self maps (for right) and (for up) which associate to a square the square to the right of it (resp. above it). The connectedness of the surface means that the group of permutations of generated by and acts transitively on . Conversely, a finite set , equipped with two onetoone maps and such that the group of permutations generated by and acts transitively on , defines a squaretiled surface. See [23].
For a squaretiled surface, it is easy to identify the factors and in the decomposition of the homology groups, and to see that in this case they are defined over .
Let be the inverse image of under the ramified covering . For each square , let be the homology class defined by a path in from the bottom left corner to the bottom right corner; let be the homology class defined by a path in from the bottom left corner to the upper left corner. Let (resp. ) be the sum over of the (resp. of the ). It is clear that both and belong to . Let (resp., ) be the class in obtained from (resp., ) by shifting each (resp., ) slightly upwards (resp., to the right).
Proposition 1.10.

The subspace (resp., ) is the kernel of the homomorphism from (resp., ) to (resp., ) induced by the ramified covering .

One has
Moreover, the action of the affine group on is through the homomorphism from the affine group to and the standard action of on .
Proof.
The first part is an immediate consequence of the definitions of and .
Let be a class in . For the intersection form between and , we have
On the other hand, we have iff . This shows that and belong to . As this subspace is dimensional and , are linearly independent, we conclude that . It follows that . The last assertion of the proposition follows from a direct verification. ∎
We will denote by the kernel of the homomorphism induced by from to . We have . For and , we omit the coefficients to keep the notation simple: designates or (the context should remove the ambiguity); is the kernel of the homomorphism between the first absolute homology groups of and with real or rational coefficients.
1.4. Degenerate orbits
Veech has asked how “degenerate” the Lyapunov spectrum of can be along a nontypical orbit, for instance along a closed orbit.
This question was first answered by G. Forni [7] who exhibited a beautiful example of a squaretiled surface of genus such that the Lyapunov exponents of for the invariant measure supported on the orbit of verify .
Subsequently, Forni and Matheus [8] constructed a squaretiled surface of genus 4 such that the Lyapunov exponents (with respect to ) of the invariant measure supported on the orbit of verify .
More precisely, Forni’s example is the Riemann surface of genus 3
(1.2) 
equipped with the Abelian differential . This example was independently discovered, for different reasons, by Herrlich, Möller and Schmithüsen [9].
Similarly, Forni and Matheus’ example is the Riemann surface of genus 4
(1.3) 
equipped with the Abelian differential .
Remark 1.11.
Actually, these formulas for and are the description of entire closed orbits. The corresponding squaretiled surfaces (in the sense of the previous definition) belonging to these orbits are obtained by appropriate choices of the points . Also, we point out that these examples are particular cases of a more general family studied by I. Bouw and M. Möller [5].
Remark 1.12.
In the sequel, denotes the set of zeroes of and denotes the set of zeroes of . Note that and , so that is an Abelian differential in the stratum and is an Abelian differential in the stratum^{2}^{2}2This stratum has two connected components distinguished by the parity of the spin structure (see [12]). In particular, one can ask about the connected component of ForniMatheus’s surface. In the Appendix A below, we’ll use a squaretiled representation of this example to show that its spin structure is even. .
Remark 1.13.
An unpublished work of Martin Möller [15] indicates that such examples with totally degenerate KZ spectrum are very rare: they don’t exist in genus , Forni’s example is the unique totally degenerate orbit in genus 3 and the ForniMatheus example is the unique totally degenerate orbit in genus 4; also, the sole stratum in genus 5 possibly supporting a totally degenerate orbit is , although this isn’t probably the case (namely, Möller pursued a computer program search and it seems that the possible exceptional case can be ruled out).
Remark 1.14.
Forni’s version of Kontsevich formula for the sum of the Lyapunov exponents reveals the following interesting feature of the KontsevichZorich cocycle over a totally degenerate : it is isometric with respect to the Hodge norm on the cohomology on the orthogonal complement of the subspace associated to the exponents . For more details see [6] and [7]. Observe that this fact is far from trivial in general since the presence of zero Lyapunov exponents only indicates a subexponential (e.g., polynomial) divergence of the orbits (although in the specific case of the KZ cocycle, Forni manages to show that this “subexponential divergence” suffices to conclude there is no divergence at all).
1.5. Statement of the results
We start with . For this squaretiled surface, the Veech group is the full group and the automorphism group is the element quaternion group : see F. Herrlich and G. Schmithüsen [9] (and also Figure 1 below). We have tried to summarize the main conclusions of the computations of the next section.
Theorem 1.15.

One has a decomposition
into defined invariant subspaces. The action of on is through the group of permutations of the zeros of .

There exists a root system of type spanning which is invariant under the action of . The action of on is thus given by a homomorphism of to the automorphism group of . The image of this homomorphism is a subgroup of of order .

The inverse image in of the Weyl group is equal to the inverse image of the principal congruence subgroup by the canonical morphism from to . The morphism from to induced by is onto.

The intersection of the image of with is the subgroup of order formed by those elements of which preserve the intersection form on .

The intersection of the kernel of with the kernel of the action of on is sent isomorphically onto the principal congruence subgroup by the canonical morphism from to .
Remark 1.16.
Avila and Hubert communicated to the authors that they checked that the action of generators of on was through matrices of finite order. One of the referees also brought our attention to the PhD thesis of Oliver Bauer [2], who does some computations on the action of the affine group similar to ours.
We now consider . For this squaretiled surface, we will see that the Veech group is the full group and the automorphism group is the cyclic group .
Theorem 1.17.

One has decompositions
into defined invariant subspaces. The action of on is through the group of permutations of the zeros of .

The subspace is dimensional and the action of on it is through a homomorphism to the cyclic group (acting by rotations).

The subspace is dimensional and it splits over into two invariant subspaces of dimension 2.

There exists a root system of type spanning which is invariant under the action of . The action of on is thus given by a homomorphism of to the automorphism group of . The image of this homomorphism is a subgroup of of order .

The inverse image in of the Weyl group is sent isomorphically onto the Veech group by the canonical morphism from the affine group to the Veech group. The image of the morphism from to is the cyclic subgroup of index .

The intersection of the image of with is the subgroup of order , isomorphic to , formed by those elements of which preserve the intersection form on .

The kernel of is sent isomorphically onto the principal congruence subgroup by the canonical morphism from the affine group to the Veech group.
Actually, in Section 3, we study a family of squaretiled surfaces parametrized by an odd integer , the surface corresponding to . Some of the computations are valid for all , but the stronger statements only hold for .
In view of the observation following Definition 1.8 (in Subsection 1.3), an immediate consequence of the theorems is that, for the two squaretiled surfaces considered above, all Lyapunov exponents of the reduced KontsevichZorich cocycle are equal to zero.
Conversely (and less trivially), Möller has shown ([15]) that the action on for a totally degenerate squaretiled surface has to be through a finite group.
Acknowledgements
This research has been supported by the following institutions: the Collège de France, French ANR (grants 0863 petits diviseurs et resonances en géométrie, EDP et dynamique 0864 Dynamique dans l’espace de Teichmüller). We thank the Collège de France, IMPA (Rio de Janeiro), the MaxPlanck Institute für Mathematik in Bonn and the MittagLeffler Institute for their hospitality. We are also grateful to the referees for suggestions that greatly improved the presentation of the paper.
2. Proof of Theorem 1.15
This section is organized as follows. In Subsection 2.1, we recall the description of as a squaretiled surface. Also, the automorphism group is identified with the quaternion group. In order to understand the action of this group on the homology, we recall in Subsection 2.2 the list of irreducible representations of the quaternion group. In Subsection 2.3, we introduce generators for , compute the action of the quaternion group on homology and break the homology into invariant subspaces. Generators for the affine group are chosen in Subsection 2.4, and their action on homology are computed in Subsection 2.5. This allows to identify in Subsection 2.6 a subspace which complements in and is invariant under the action of the affine group. Finally, the action of the affine group on is analyzed in Subsection 2.7.
2.1. The squaretiled surface
We follow here F. Herrlich and G. Schmithüsen [9]. The set is identified with the quaternion group . We denote by the square corresponding to . The map (for right) is and the map (for up) is . The automorphism group is then canonically identified with , the element sending the square on the square . See Figure 1 below.
Here, it is shown the horizontal and vertical cylinder decompositions, and the right and top neighbors of each square, so that the (implicit) side identifications are easily deduced.
We will denote by the quotient of by its center . It is isomorphic to . We denote by the images of in .
For , the lower left corners of and correspond to the same point of . We will identify in this way with .
The map factors as
Here, the first map is a two fold covering ramified over the four points of order in , and may be viewed as the quotient map by the involution of , the four squares of being naturally labelled by .
2.2. Irreducible representations of
The group has 5 distinct irreducible representations, 4 onedimensional and one dimensional . The character table is
tr 
In the regular representation of in , the submodules associated to these representations are generated by



, , , 
2.3. Action of on
We consider the direct sum of two copies of . We denote by the canonical basis of the first copy and by the canonical basis of the second copy. We define a homomorphism from onto by sending on the homology class defined by the lower side of (oriented from left to right) and on the homology class defined by the left side of (oriented from upwards) (see Figure 2 below). The homomorphism is compatible with the actions of on (by the regular representation) and on (identifying with ).
The kernel of the homomorphism is the submodule of generated by the elements
(2.1) 
We have , hence has rank . Observe that, for we have .
Recall that is identified with . The boundary map is induced by
Let
We note that
We have and
(2.2) 
for . Similar statements hold for and . Let be the subspace of spanned by . The formulas imply the following lemma:
Lemma 2.1.
The subspace is invariant under the action of and complements in .
Proof.
The first part is clear. The second part follows from the fact that the images of under are linearly independent. ∎
Let
From Subsection 1.3, we have