Alex Wright and I have just upload to ArXiv our paper Hodge-Teichmüller planes and finiteness results for Teichmüller curves.

As the title of the paper indicates, this article concerns finiteness results for certain classes of *Teichmüller curves* (i.e., closed -orbits in the moduli spaces of Abelian differentials). For example, one of the results in this paper is the finiteness of algebraically primitive Teichmüller curves in minimal strata of *prime* genus .

In fact, some parts of this paper were previously discussed in this blog here, and, for this reason, we will not make further comments on the contents of this article today.

Instead, let us take the opportunity to briefly discuss a “deleted scene” of this paper.

More concretely, at some stage of the paper, Alex and I need to know that there are Abelian differentials/translation surfaces with “rich monodromy” (whatever this means) in each connected component of every stratum of genus .

In this direction, we ensure the *existence* of such translation surfaces by an *inductive* argument going as follows.

Starting with an *hypothetical* connected component of genus containing *only* translation surfaces with “poor monodromy” ( orthogonal Hodge-Teichmüller planes in the notation of the paper), we follow and adpat some arguments of Kontsevich-Zorich paper on the classification of connected components of strata (cf. Section 5 of our paper) and we work a little bit with the Deligne-Mumford compactification of moduli spaces (cf. Section 6 of our paper) to show that the “poorness of monodromy” property passes down to *all* translations surfaces in both connected components and . Here, very roughly speaking, the basic idea is that, if we “degenerate” (e.g., pinch off a curve) in a careful way the translation surfaces in , we can decrease the genus *without* loosing the “poorness of monodromy” property, so that, if we keep degenerating, then we will find ourselves with plenty of genus translation surfaces with poor monodromy.

However, it is easy to see that the property that all translation surfaces in and have “poor monodromy” is *false*: indeed, in Section 4 of our paper, we exhibit *explicitely* two translation surfaces and with “rich monodromy”. Thus, the hypothetical connected component can *not* exist.

As it turns out, before thinking about this argument based on Kontsevich-Zorich classification of connected components and the features of Deligne-Mumford compactification, Alex and I thought of using the following simple-minded argument. One can check that the “richness of monodromy” property passes through finite branched coverings. Hence, it suffices to produce for each connected component an explicit finite branched cover of one of the translation surfaces or lying in to deduce that all connected components of all strata possess translation surfaces with “rich monodromy”.

Remark 1The strategy in the previous paragraph is very natural: for instance, after reading a preliminary version of our paper, Giovanni Forni asked us if we could not make the arguments in Sections 5 and 6 of our paper (related to Kontsevich-Zorich classification of connected components of strata and Deligne-Mumford compactification) more elementary by taking finite branched covers to produce explicit translation surfaces with rich monodromy on each connected component of every stratum. As we will see below, this does not quite work to fully recover the statements in Section 5 of our paper, but it allows to obtain at least part of our statements.

In particular, around November/December 2012, Alex and I started looking at finite branched covers of and “hitting” all connected components of all strata. Unfortunately, this strategy does not work as well as one could imagine: firstly, there are restrictions on the genera of strata we can reach using finite branched covers (thanks to Riemann-Hurwitz formula), and, secondly, it is not so easy to figure out in what connected component our finite branched cover lives.

Nevertheless, this elementary strategy permits to deduce the following proposition (allowing to deduce *partial* versions of the statements in Section 5 of our paper). Let and be the square-tiled surfaces with “rich monodromy” constructed in our paper (see also this post), i.e., the square-tiled surfaces below

associated to the pairs of permutations and , and and .

Remark 2Here, as it is usual in this theory, we are constructing square-tiled surfaces from a pair of permutations on elements by taking unit squares and gluing them (by translations) so that is the square to the right of the square and is the square on the top of the square .

Proposition 1For each odd, there exists a finite branched cover of of degree in the odd connected component of the minimal stratum (of Abelian differentials with a single zero of order ).

Also, there exists a finite branched cover of of degree in the hyperelliptic connected component of the minimal stratum (of Abelian differentials with a single zero of order ).

Finally, for each , there exists a finite branched cover of of degree in the even connected component of the minimal stratum (of Abelian differentials with a single zero of order ).

Remark 3In this statement, the nomenclature “hyperelliptic”, “even” and “odd” refers to the invariants introduced by Kontsevich and Zorich to distinguish between the connected components of strata. We will briefly review these notions below (as we will need them to prove Proposition 1.

We will dedicate the remainder of this post to outline the construction of the translation surfaces and satisfying the conclusions of Proposition 1. In particular, we will divide the discussion into three sections: in the next one, we will quickly review the invariants introduced by Kontsevich-Zorich to classify connected components of strata, and in the last two sections we will construct and respectively.

**1. Parity of spin structure and components of strata **

The results of Kontsevich-Zorich imply that any stratum (of Abelian differentials with zeroes of prescribed orders ) has at most three connected components

characterized by the following properties:

- consists
*entirely*of hyperelliptic translation surfaces; - , resp. , consists of translation surfaces whose
*parity of the spin structure*is even, resp. odd, (and they contain some non-hyperelliptic translation surface).

For convenience of the reader, we recall that the parity of the spin structure of a translation surface of genus can be computed as follows.

Firstly, one chooses a *canonical* (symplectic) basis of represented by paths avoiding the singularities of (i.e., zeroes of ).

Secondly, we define the index of a simple closed curved avoiding the singularities of as the degree of the Gauss map associated to (i.e., the degree of the map measuring the angle the tangents to make with the horizontal direction on [a notion that is well-defined outside the singularities]).

Finally, we form the quantity and we define the parity of the spin structure of as

Remark 4Of course, it is implicit in this definition that does not depend on the choice of the canonical basis . This fact is true and the reader is invited to consult Kontsevich-Zorich paper for more discussion on this (as well as a motivation for the nomenclature “parity of spin structure” for ).

In this notation, we say that has even, resp. odd, parity of spin structure if and only if , resp. .

**2. Explicit covers of **

Consider again the square-tiled surface given by the pair of permutations and .

For technical reason, it is desirable to “simplify” the geometry of in order to produce finite covers that are “simple” to handle.

More concretely, we will replace by the square-tiled surface given by the pair of permutations , in the –*orbit* of : indeed, where and .

Given an odd integer, let be the square-tiled surface associated to the pair of permutations

By defintion, is a degree cover of belonging to the stratum .

Remark 5The “shape” of the covering was “guessed” with the help of SAGE: in fact, we tried a few simple-minded finite coverings of (including for ) and we asked SAGE to determine their connected components; then, once we got the “correct” connected components (in minimal strata), we looked at the permutations corresponding these square-tiled surfaces and we found the “partner” leading to the expressions for the permutations and .

Let us now determine, for odd, the connected component of containing .

We think of it as horizontal cylinders determined by the permutation whose top and bottom boundaries are glued accordingly to the permutation . Using this geometrical representation, we can define the following cycles in .

- for each , let , , and be the homology classes of the vertical cycles within the horizontal cylinders connecting the middle of the bottom side of the squares , , and (resp.) to the top side of the squares , , and (resp.);
- for each , , let be the homology classes of the vertical cycles within the horizontal cylinders connecting the middles of the bottom side of the square to the top side of the squares ;
- let be the homology class of the concatenation of the vertical cycles within the horizontal cylinders connecting the middles of the bottom side of the square to the top side of the square for , the bottom side of the square to the left side of the square , and the right side of the square to the top side of the square ;
- is the horizontal cycle connecting the left vertical side of the square to the right vertical side of the square , and, for , , is the horizontal cycle connecting the left vertical side of the square to the right vertical side of the square .

It is not hard to check that these cycles form a basis of . From this basis, we will compute the parity of the spin of using the orthogonalization procedure described in Appendix C of this paper of Zorich (a Gram-Schmidt orthogonalization process to produce canonical basis of homology modulo ). More precisely, we start with the cycles and . Then, by induction, we use the cycles , to successively render the cycles

into a canonical basis (where ). After performing this procedure, one has that the parity of the spin structure of is

where satisfies the following properties:

- whenever is a simple smooth closed curve in not passing through singular points whose Gauss map has degree ;
- is a quadratic form representing the intersection form in the sense that .

We affirm that for each odd. Indeed, we note that the cycles

of do not intersect the other cycles of defined above except for , and, moreover, they have trivial intersection with the cycle

satisfying . Thus, by replacing by and by applying the orthogonalization procedure to these two sets of and cycles in an independent way, we deduce that differs from by a term of the form

where , is an orthogonalization of the cycles

On the other hand, by a direct computation, one can check that

- , ,
- , ,
- , ,
- , ,
- ,

is an orthogonalization of the cycles

Moreover, for , and for , so that

as it was claimed.

It follows that . Since the square-tiled surfaces are not hyperelliptic, we deduce that:

Proposition 2For each odd, let be the square-tiled surface (defined above) covering the square-tiled in the -orbit of the square-tiled surface . Then, belongs to .

**3. Explicit covers of **

Consider now the square-tiled surface given by the pair of permutations and .

Once more, let us “simplify the geometry” by taking the square-tiled surface given by the pair of permutations , belongs to the -orbit of : indeed, where and .

Given an odd integer, let be the square-tiled surface associated to the pair of permutations

By defintion, is a degree cover of belonging to the stratum .

Remark 6Similarly to Remark 5, we “guessed” the shape of with the help of SAGE.

Now, let us determine, for odd, the connected component of containing . Again, we think of it as horizontal cylinders determined by the permutation whose top and bottom boundaries are glued accordingly to the permutation . Using this geometrical representation, we can define the following cycles in .

- for each , let , , and be the homology classes of the vertical cycles within the horizontal cylinders connecting the middle of the bottom side of the squares , , and (resp.) to the top side of the squares , , and (resp.);
- for each , , let be the homology classes of the vertical cycles within the horizontal cylinders connecting the middles of the bottom side of the square to the top side of the squares ;
- let be the homology class of the concatenation of the vertical cycles within the horizontal cylinders connecting the middles of the bottom side of the square to the top side of the square for , the bottom side of the square to the left side of the square , and the right side of the square to the top side of the square ;
- is the horizontal cycle connecting the left vertical side of the square to the right vertical side of the square , and, for , , is the horizontal cycle connecting the left vertical side of the square to the right vertical side of the square .

It is not hard to check that these cycles form a basis of . We affirm that for each odd. Indeed, we note that the cycles

of do not intersect the other cycles of defined above except for , and, moreover, they have trivial intersection with the cycle

satisfying . Thus, by replacing by and by applying the orthogonalization procedure to these two sets of and cycles in an independent way, we deduce that differs from by a term of the form

where , is an orthogonalization of the cycles

On the other hand, by a direct computation, one can check that, for ,

- , ,
- , ,
- , ,
- , ,
- ,

is an orthogonalization of the cycles

Moreover, for , one has for , and for , so that

as it was claimed.

Remark 7Contrary to the last section, we compared to instead of because the 20 cycles indicated above have a better geometrical behavior (for the orthogonalization process) than the corresponding 10 cycles (as a direct computation shows).

It follows that for , and for . Since and are hyperelliptic square-tiled surfaces of genus and resp., i.e., and , we deduce that : this last claim follows directly from the fact that the parity of the spin of a translation surface in is (where is the integer part of ), or by a direct calculation.

Because is not hyperelliptic for odd, we conclude that:

Proposition 3For each odd, let be the square-tiled surface (defined above) covering the square-tiled in the -orbit of the square-tiled surface . Then, and .

This completes the proof of Proposition 1 (as this proposition follows from Propositions 2 and 3 above).

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