In a recent article (quickly discussed in a previous post of this blog), Giovanni Forni gave a criterion for the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle over the support of certain -invariant measures, and, after a kind invitation by Giovanni, I ended up contributing with an Appendix to Giovanni’s paper presenting some examples related to his main theorems.

As it turns out, while I was preparing the Appendix, some “scenes” were deleted from the final version of the article due to the usual limitations of space. Indeed, the following two remarks were omitted:

- (a) all -invariant measures supported on the stratum coming from (unramified) double covers of
*Veech surfaces*of have non-simple Lyapunov spectrum; - (b) all -invariant measures supported on the stratum coming from (unramified) double covers of
*square-tiled surfaces*of are*Lagrangian*(in the sense of Giovanni’s article).

Of course, since the proofs of these statements are (very) straightforward, they are not publication-quality, except (maybe) in this blog. So, I will use today’s post to present the (easy) proofs of (a) and (b).

–**Proof of (a)**–

Let be a Veech surface (of genus 2), and consider be a (unramified) double cover of , i.e., we have a (unramified) double cover and . In this situation, by Lemma A.1 in Giovanni’s article, we have that the Lyapunov spectrum of (the Kontsevich-Zorich cocycle over the -orbit of) is contained in the Lyapunov spectrum of . On the other hand, by a result of Matthew Bainbridge, we have that the (two) Lyapunov exponents of the (genus 2) Abelian differential is . It follows that the Lyapunov spectrum of the genus Abelian differential has the form for some .

Next, we observe that lives in the hyperelliptic locus of : for instance, it is not hard to show that the sheet exchange map (induced by the *unramified* double cover ) and a lift of the hyperelliptic involution of the genus (and hence hyperelliptic) surface can be lifted to generates a copy of Klein four group inside the automorphism group of (and this is sufficient to conclude hyperellipticity of ). In any case, see page 268 of H. Farkas and I. Kra’s book for *two* detailed proofs of the hyperellipticity of .

Once we know that (the -orbit of) lies in the hyperelliptic locus, we can apply Theorem 1.1 of this recent paper of Dawei Chen and Martin Möller to obtain that the sum of Lyapunov exponents of is , i.e., , that is, . Therefore, the Lyapunov spectrum of is , i.e., is a double Lyapunov exponent. This proves (a).

–**Proof of (b)**–

Recall that an unramified covering of a Riemann surface with group of deck transformations corresponds to a surjective morphism from the fundamental group of the Riemann surface to . In particular, the number of unramified double covers of a genus Riemann surface is the number of epimorphisms from its fundamental group to (and, as is Abelian, this is the same as the number of epimorphisms from its first homology group to ), that is, (and, moreover, it is possible to write algebraic equations for each of them, see this article of Y. Fuertes and G. González-Diez). In fact, if we denote by , , a “canonical” basis of the fundamental group of our Riemann surface, an epimorphism from the fundamental group to is determined by an assignment , such that is assigned to some or . Geometrically, such an epimorphism corresponds to taking two copies of our Riemann surface and declaring that the lift of a curve or corresponds to two disjoint copies (one in each copy of our Riemann surface) of the curve if we’ve assigned to it or one closed curve (with “twice the length of the original curve”) if we’ve assigned to it. In this language, it is easy to see why is the number of unramified double coverings: we can assign to , , in ways, but we have to exclude the trivial assignment ( for all ) because it corresponds to two disjoint copies of our initial Riemann surface (and hence it gives a *non-connected* cover [a case that we want to rule out]).

In the setting of item (b), we are looking at unramified double covers of a genus 2 Riemann surface (since ), that is, we have to deal with coverings of .

Remark 1Just to “double-check” the counting of unramified double coverings of , we recall that all such coverings are hyperelliptic genus surfaces (see the proof of item (a)). On the other hand, by a general result of H. Farkas, among the unramified double coverings of a genus Riemann surfaces, of them arehyperelliptic. In the case , and , so that the global scenario is coherent.

In the sequel, these coverings will be rendered *explicit*.

Let be a square-tiled surface. By a celebrated *classification result* of P. Hubert and S. Lelièvre (when is tiled by a prime number of squares) and C. McMullen (in general), it follows that the -orbit of *any* contains a –*shaped representative*:

In particular, given that item (b) concerns entire -orbits, the classification result above says that we can assume (without loss of generality) that is -shaped. In this case, the unramified double covers of are very easy to describe. In the -shaped origami in Figure 1 above, we have 4 pairs of sides identified by translations: as a representative of each pair, we have two bottom horizontal sides ( and ) connecting the double zero of , and two leftmost vertical sides ( and ) also connecting the double zero of . To each such pair, we assign (i.e., we consider an epimorphism to ). Now, we take two copies of and we glue the sides in a given pair with its “patner” in the *same* copy of if we’ve assigned to the pair, and with its “patner” in the *other* copy of if we’ve assigned to the pair. To aleviate the notation, given an unramified double cover of presented as two copies of with certain identifications of sides (determined by an surjective map ), we denote by and the two bottom horizontal sides in the first copy of , and by and the two bottom horizontal sides in the second copy of . Similarly, we introduce , , and . For instance, the figure below shows the location of these sides in the unramified double cover of determined by the map , , , and .

At this stage, the proof of the statement of item (b) will be complete as soon as we show that all possible unramified double coverings of are *Lagrangian* in the sense of Giovanni Forni, that is, for each such covering we can find a (periodic, i.e., rational) direction such that the genus Riemann surface (upstairs) decomposes into *cylinders* whose *waist curves* span a -dimensional subspace in homology. To simplify a little more our notation, we represent a map by the list of images of *in this order*.

Keeping this notation in mind, we begin by treating the cases where the horizontal direction of the unramified double covering is Lagrangian.

Remark 2We strongly encourage the reader to draw the coverings to check the geometrical facts claimed in the items below.

- : the waist curves of the 4 horizontal cylinders are homologous to , , and (see Figure … above); by direct inspection, we see that , and, moreover, the (non-trivial) homology classes and are
*distinct*(for instance, has intersection with the closed vertical curve connecting the middle of the sides in the first copy of , while doesn’t intersect this curve); therefore, , and span a -dimensional subspace in homology, that is, we have the Lagrangian property. - : the waist curves of the 4 horizontal cylinders are homologous to , , and , but this time (as one can see that from the first copy), and (as an intersection argument [similar to the previous item] with an adequate vertical path shows); thus, , , and span a -dimensional space in homology, and the Lagrangian property follows.
- : the waist curves of the 4 horizontal cylinders are homologous to , , and , but this time (as one can read from the first copy of ), and , and are linearly independent in homology (as an intersection argument shows); hence, this covering is also Lagrangian.
- , , and : the waist curves of the 3 horizontal cylinders are homologous to , and , and they are linearly independent in homology by appropriate intersection arguments in each case; thus, these covering are Lagrangian as well.
- , , and : the waist curves of the 3 horizontal cylinders are homologous to , and , and they are linearly independent in homology by appropriate intersection arguments in each case; thus, these covering are Lagrangian.

Next, we analyze the following three cases:

- , and : by looking the vertical direction of these coverings, or equivalently, by rotating by the associated square-tiled surfaces and looking the horizontal direction of the resulting origamis, we come back to the cases , and (resp.) treated above. So, the Lagrangian property for these coverings follows.

In resume, we showed that out of the unramified double coverings of are Lagrangian. However, the remaining case has only 2 horizontal cylinders (so that the span in homology of its waist curves has dimension ) and the situation doesn’t change after rotation by , that is, we still have a covering of type and hence cylinders only (in some sense this last case is “self-dual”).

To overcome this technical difficulty, the idea is to apply the parabolic element an appropriate number of times (i.e., apply for a convenient ) to the covering in order to change its “type”: since “twists” horizontal cylinders (it is a *Dehn twist*), one can hope to get back to one of the previous cases already treated after a certain amount of “twists”.

More precisely, we begin by noticing that the classification result of P. Hubert, S. Lelièvre and C. McMullen mentioned above implies that:

- any square-tiled surface tiled by an
*even*number of squares belongs to the -orbit of the -shaped origami in Figure 1 above with parameter ; - any square-tiled surface tiled by an
*odd*number of squares belongs to the -orbit of the -shaped origami in Figure 1 above with parameter or ;

In particular, we can assume that is an unramified double covering of type of a -shaped origami as above with parameter or . In these cases, we can number the squares of as follows:

In other words, by using a pair of permutations to code (one permutation giving the neighbor to the right of each square and another permutation giving the neighbor on the top of each square), one gets

in the case of parameter , and

in the case of parameter . In terms of a pair of permutations , the action of the parabolic element is .

Therefore, in the case , we get , where

, where

and, by induction, , where

Hence, one has is an unramified double covering of of type with

so that we come back to a previous considered case.

Similarly, in the case , we get , where

, where

and, by induction, , where

Hence, one has is an unramified double covering of of type with

so that we come back to a previous considered case. This completes the proof of item (b).

Remark 3Actually, the same kind of computation works with any parameter , but the notation becomes less transparent. So that’s why I decided to treat the cases and separately (since the classification result says that it suffices to treat these two cases).

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