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 1 The 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 2 Here, 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 .
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 3 In 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.