Artur Avila, Jean-Christophe Yoccoz and I have just uploaded to ArXiv our paper Zorich conjecture for hyperelliptic Rauzy-Veech groups.
This is the first installment of a series of two articles on the Kontsevich-Zorich cocycle over certain -invariant loci in moduli spaces of translation surfaces obtained from cyclic cover constructions (inspired from the works of Veech and McMullen).
More precisely, the second paper of this series (still in preparation) studies the Kontsevich-Zorich cocycle over -orbits of certain cyclic covers of translation surfaces in hyperelliptic components called and in the literature. (The curious reader can find more explanations about this forthcoming paper in this old blog post here [cf. Remark 7].)
Of course, before studying the cyclic covers, we need to obtain some good description of the Kontsevich-Zorich cocycle on the hyperelliptic components and this is the purpose of the first article of the series.
Since the first paper of this series is not long, this post will just give a brief “reader’s guide” rather than entering into the technical details.
Remark 1 In the sequel, we will assume some familiarity with translation surfaces.
1. Rauzy-Veech groups and Zorich conjecture
The starting point of our article is a description of certain combinatorial objects — called hyperelliptic Rauzy diagrams — coding the dynamics of the Kontsevich-Zorich cocycle on and .
Remark 2 By the time that the first version of our article was written, we thought that we found a new description of these diagrams, but Pascal Hubert kindly pointed out to us that G. Rauzy was aware of it.
One important feature of this description of hyperelliptic Rauzy diagrams is the fact that we can order these diagrams by “complexity” in such a way that two consecutive diagrams can be related to each other by an inductive procedure.
The behavior of the Kontsevich-Zorich cocycle (with respect to Masur-Veech measures) is described in general by the Rauzy-Veech algorithm: roughly speaking, this algorithm is a natural way to attach matrices (acting on homology groups) to the arrows of Rauzy diagrams, and, in this language, the action of the Kontsevich-Zorich cocycle is just the multiplication of the matrices attached to concatenations of arrows of Rauzy diagrams.
In particular, the features of the Kontsevich-Zorich cocycle (with respect to Masur-Veech measures) can be derived from the study of the so-called Rauzy-Veech groups, i.e., the groups generated by the matrices attached to the arrows of a given Rauzy diagram. For example, the celebrated paper of Avila and Viana solving affirmatively a conjecture of Kontsevich and Zorich proves the simplicity of the Lyapunov exponents of the Kontsevich-Zorich cocycle by establishing (inductively) the pinching and twisting properties for Rauzy-Veech groups.
In our article, we exploit the “inductive” description of hyperelliptic Rauzy diagrams to compute the hyperelliptic Rauzy-Veech groups.
Remark 3 Our arguments for the computation of hyperelliptic Rauzy-Veech groups were inspired from the calculations of -blocks of the Kontsevich-Zorich cocycle over cyclic covers in the second paper of this series. In other words, we first developed some parts of the second paper before writing the first paper.
An interesting corollary of this computation is the fact that hyperelliptic Rauzy-Veech groups are explicit finite-index subgroups of the symplectic groups , so that they are Zariski dense in .
The Zariski density of (general) Rauzy-Veech groups in symplectic groups was conjectured by Zorich (see, e.g., Remark 6.12 in Avila-Viana paper) as a step towards the Kontsevich-Zorich conjecture established by Avila-Viana. Therefore, the previous paragraph means that Zorich conjecture is true for hyperelliptic Rauzy-Veech groups (and this justifies our choice for the title of our paper).
Here, it is worth to point out that Zorich conjecture asks more than what is needed to prove the Kontsevich-Zorich conjecture. Indeed, the Zariski denseness in symplectic groups imply the pinching and twisting properties of Avila-Viana, so that Zorich conjecture implies Kontsevich-Zorich conjecture. On the other hand, we saw in this previous blog post that a pinching and twisting group of symplectic matrices might not be Zariski dense: in other words, the techniques of Avila-Viana solve the Kontsevich-Zorich without addressing Zorich conjecture.
Thus, our proof of Zorich conjecture for hyperelliptic Rauzy-Veech groups gives an alternative proof of this particular case of Avila-Viana theorem.
2. Braid groups and A’Campo theorem
After a preliminary version of our article was complete, Martin Möller noticed some similarities between our characterization of hyperelliptic Rauzy-Veech groups and a result of A’Campo on the images of certain monodromy representations associated to hyperelliptic Riemann surfaces.
As it turns out, this is not a coincidence: we showed that the elements of the hyperelliptic Rauzy-Veech group associated to certain elementary loops on hyperelliptic Rauzy diagrams are induced by Dehn twists lifting the generators of a braid group; hence, this permits to recover our description of hyperelliptic Rauzy-Veech groups from A’Campo theorem.
Remark 4 In a certain sense, the previous paragraph is a sort of sanity test: the same groups found by A’Campo were rediscovered by us using different methods.
Closing this short post, let me point out that this relationship between loops in hyperelliptic Rauzy diagrams and Dehn twists in hyperelliptic surfaces reveals an interesting fact: the fundamental groups of the combinatorial model (hyperelliptic Rauzy diagram) coincides with the orbifold fundamental groups of and . In other words, the hyperelliptic Rauzy diagrams “see” the topology of objects ( and ) coded by them.