Simion Filip, Giovanni Forni and I have just upload to ArXiv our paper Quaternionic covers and monodromy of the Kontsevich-Zorich cocycle in orthogonal groups.
This article is motivated by Simion Filip’s recent work on the classification of possible monodromy groups for the Kontsevich-Zorich cocycle.
Very roughly speaking, the basic idea of this classification is the following. Consider the Kontsevich-Zorich cocycle on the Hodge bundle over the support of an ergodic -invariant probability measure on (a connected component of) a stratum of the moduli spaces of translation surfaces. Recall that, in a certain sense, the Kontsevich-Zorich cocycle is a sort of “foliated monodromy representation” obtained by using the Gauss-Manin connection on the Hodge bundle while essentially moving only along -orbits on moduli spaces of translation surfaces.
By extending a previous work of Martin Möller (for the Kontsevich-Zorich cocycle over Teichmüller curves), Simion Filip showed (in this paper here) that a version of the so-called Deligne’s semisimplicity theorem holds for the Kontsevich-Zorich cocycle: in plain terms, this means that the Kontsevich-Zorich cocycle can be completely decomposed into (-)irreducible pieces, and, furthermore, each piece respects the Hodge structure coming from the Hodge bundle. In other terms, the Kontsevich-Zorich cocycle is always diagonalizable by blocks and its restriction to each block is related to a variation of Hodge structures of weight .
The previous paragraph might seem abstract at first sight, but, as it turns out, it imposes geometrical constraints on the possible groups of matrices obtained by restriction of the Kontsevich-Zorich cocycle to an irreducible piece. More precisely, by exploiting the known tables (see § 3.2 of Filip’s paper) for monodromy representations coming from variations of Hodge structures of weight over quasiprojective varieties, Simion Filip classified (up to compact and finite-index factors) the possible Zariski closures of the groups of matrices associated to restrictions of the Kontsevich-Zorich cocycle to an irreducible piece. In particular, there are at most five types of possible Zariski closures for blocks of the Kontsevich-Zorich cocycle (cf. Theorems 1.1 and 1.2 in Simion Filip’s paper):
- (i) the symplectic group in its standard representation;
- (ii) the (generalized) unitary group in its standard representation;
- (iii) in an exterior power representation;
- (iv) the quaternionic orthogonal group (sometimes called , or ) of matrices on respecting a quaternionic structure and an Hermitian (complex) form of signature in its standard representation;
- (v) the indefinite orthogonal group in a spin representation.
Moreover, each of these items can be realized as an abstract variation of Hodge structures of weight over abstract curves and/or Abelian varieties.
Here, it is worth to stress out that Filip’s classification of the possible blocks of the Kontsevich-Zorich cocycle comes from a general study of variations of Hodge structures of weight . Thus, it is not clear whether all items above can actually be realized as a block of the Kontsevich-Zorich cocycle over the closure of some -orbit in the moduli spaces of translations surfaces.
In fact, it was previously known in the literature that (all groups listed in) the items (i) and (ii) appear as blocks of the Kontsevich-Zorich cocycle (over closures of -orbits of translation surfaces given by certain cyclic cover constructions). On the other hand, it is not obvious that the other 3 items occur in the context of the Kontsevich-Zorich cocycle, and, indeed, this realizability question was explicitly posed by Simion Filip in Question 5.5 of his paper (see also § B.2 in Appendix B of this recent paper of Delecroix-Zorich).
In our paper, Filip, Forni and I give a partial answer to this question by showing that the case of item (iv) is realizable as a block of the Kontsevich-Zorich cocycle.
Remark 1 Thanks to an exceptional isomorphism between the real Lie algebra in its standard representation and the second exterior power representation of the real Lie algebra , this also means that the case of of item (iii) is also realized.
Remark 2 We think that the examples constructed in this paper by Yoccoz, Zmiaikou and myself of regular origamis associated to the groups of Lie type might lead to the realizability of all groups in item (iv). In fact, what prevents Filip, Forni and I to show that this is the case is the absence of a systematic method to show that the natural candidates to blocks of the Kontsevich-Zorich cocycle over these examples are actually irreducible pieces.
In the remainder of this post, we will briefly explain our construction of an example of closed -orbit such that the Kontsevich-Zorich cocycle over this orbit has a block where it acts through a Zariski dense subgroup of (modulo compact and finite-index factors).
1. A quaternionic cover of a -shaped orgami
The starting point of our joint paper with Filip and Forni is the following. The group is related to quaternionic structures on vector spaces. In particular, it is natural to look for translation surfaces possessing an automorphism (symmetry) group admitting representations of quaternionic type.
Note that automorphism groups of translation surfaces (of genus ) are always finite (e.g., by Hurwitz’s automorphism theorem) and the simplest finite group with representations of quaternionic type is the quaternion group
where , , and .
Therefore, this indicates that we should look for translation surfaces whose group of automorphisms is isomorphic to . A concrete way of building such translation surfaces is to consider ramified covers of “simple translation surfaces” such that the group of deck transformations of is isomorphic to .
The first natural attempt is to take the flat torus, and define as the translation surface obtained as follows. We let , , be copies of the flat torus . Then, we glue by translation the rightmost vertical, resp. topmost horizontal side, of with the leftmost vertical, resp. bottommost horizontal side, of , resp. for each . In this way, we obtain a translation surface tiled by eight squares , , such that the natural projection is a ramified cover (branched only at the origin of ) whose group of automorphisms is isomorphic to (namely, an element acts by translating to for all ).
The translation surface constructed above is a square-tiled surface (origami) that we already met in this blog: it is the so-called Eierlegende Wollmilchsau.
Unfortunately, the Eierlegende Wollmilchsau is not a good example for our purposes. Indeed, it is known that the Kontsevich-Zorich cocycle over the -orbit of the Eierlegende Wollmilchsau acts through a finite group of matrices (see, e.g., this paper here). In particular, this provides no meaningful information from the point of view of realizing the items in Filip’s list of possible monodromy groups because in his list one always ignores compact and/or finite-index factors.
This indicates that we should look for other translation surfaces than the flat torus.
In this direction, Filip, Forni and I took to be the simplest -shaped square-tiled surface in genus described in this picture here
where any two sides with the same labels are identified by translation.
Next, we take copies , , of this -shaped square-tiled surface , and we glue by translations the corresponding vertical, resp. horizontal, sides of and , resp. . Alternatively, we label the sides of as indicated in the figure below (where is called )
and we glue by translations the pairs of sides with the same labels.
In this way, we obtain a translation surface (called in our joint paper with Filip and Forni) such that the natural projection is a ramified cover branched only at the unique conical singularity of . Also, the automorphism group of is isomorphic to and each acts on by translating each to for all .
A direct inspection reveals that is a genus surface with four conical singularities whose cone angles are . In this setting, the Kontsevich-Zorich cocycle (over ) is simply the action on of the group of affine homeomorphisms of .
Similarly to the investigation of Delecroix, Hubert and Lelièvre of the so-called wind-tree models, the translation surface has a rich group of symmetries allowing us to decompose the Kontsevich-Zorich cocycle.
More precisely, by taking the quotient of by the center of its automorphism group , we obtain a translation surface of genus with four conical singularities whose cone angles are . Moreover, by taking the quotient of by the subgroups , and of its automorphism group , we obtain three genus surfaces , and each having two conical singularities whose cone angles are . In summary, we have intermediate covers , and for .
Using these intermediate covers together with the fact that has a finite-index subgroup whose elements commute with the automorphisms of (i.e., up to finite-index, the Kontsevich-Zorich cocycle commutes with the action of on ), we can determine the natural candidates for blocks of the Kontsevich-Zorich cocycle over , namely,
where is the subspace generated by comes from (is isomorphic to ), comes from for each , and is the symplectic orthogonal of the direct sum of the other subspaces.
These subspaces have the structure of –modules, and, by a quick comparison with the character table of , one can show that , , , and (resp.) are the isotypical components of the trivial, -kernel, -kernel, -kernel and the unique four-dimensional faithful irreducible representation of (resp.): for example, is the isotypical component of because acts as on and the character of in is while the other characters in take the value .
Furthermore, is -dimensional because and have genera and (so that and have dimensions and ), and is the symplectic orthogonal of the symplectic subspace . Hence, as a -module.
Note that acts via symplectic automorphisms of the -module (because the actions of and the automorphism group on commute), and carries a quaternionic structure. In particular, we are almost in position to apply Filip’s classification results to determine the group of matrices through which acts on .
Indeed, if we have that acts irreducibly on , then Filip’s list of possible groups says that acts through a (virtually Zariski dense) subgroup of (because preserves a quaternionic structure on ).
However, there is no reason for the action of the affine homeomorphisms on an isotypical component of the automorphism group to be irreducible in general (as far as I know). Nevertheless, the semisimplicity theorems of Möller and Filip mentioned in the introduction tells us that can split into irreducible pieces in one of the following three ways:
- (a) is irreducible, i.e., it does not decompose further;
- (b) where and are irreducible pieces;
- (c) where , , are irreducible pieces isomorphic to .
By applying Filip’s classification to each of these items, we find that (up to compact and finite-index factors) there are just three cases:
- (a’) if is -irreducible, then acts through a Zariski-dense subgroup of ;
- (b’) if with and irreducible pieces, then acts through a subgroup of ;
- (c’) if with , , irreducible pieces isomorphic to , then acts through a subgroup of .
We claim that the situations (b’) and (c’) can’t occur, so that we are in situation (a’).
We start by ruling out the case (c’). In this situation, the nature of would force all Lyapunov exponents of on to vanish. On the other hand, the formulas of Eskin-Kontsevich-Zorich for the sum of non-negative Lyapunov exponents for the square-tiled surfaces , , , and (together with the facts that , ) allows to show that
where is the sum of non-negative Lyapunov exponents of on . This means that , and, thus, there must be some non-zero Lyapunov exponent of in . In particular, we can not be in situation (c’).
Remark 3 At this point, we have that we are in situation (a’) or (b’). Hence, we already have at this stage that the Kontsevich-Zorich cocycle over has a irreducible piece where it acts through a Zariski dense subgroup of or . Of course, this suffices to deduce that we can realize a non-trivial case ( or ) of item (iv) in Filip’s list.
Let us now close this post by sketching the computation (done in Section 6 of our joint paper with Filip and Forni) permitting to rule out the situation in (b’).
The basic idea is very simple: if we had a decomposition with and , then the sole possibility for the subspace is to be the central subspace of any matrix (of the action on ) of with “simple spectrum” in the quaternionic sense (i.e., the matrix has an unstable [modulus ] eigenvalue, a central [modulus ] eigenvalue, and an stable [modulus ] eigenvalue, all of them with multiplicity four). Therefore, we can contradict the existence of once we exhibit two matrices of with “simple spectrum” whose central spaces are distinct.
Here, we do not have an abstract method to produce two matrices with the properties above, so that we are obliged to compute by hands some matrices of . As the reader can imagine, this calculation is straightforward but somewhat tedious, and, for this reason, we are not going to repeat them here: instead, we refer the curious reader to Section 6 of our joint paper with Filip and Forni for the details.