Finite groups generated by two elements are a rich source of examples of origamis (square-tiled surfaces). Indeed, given a finite group generated by and , we take a collection of unit squares indexed by the elements and we glue by translations the rightmost vertical, resp. topmost horizontal, side of with the leftmost vertical, resp. bottommost horizontal, side of , resp. , to obtain an origami naturally associated to the data of .

Such origamis were baptized *regular origamis* by David Zmiaikou in his PhD thesis and the study of the so-called Kontsevich-Zorich over regular (and quasi-regular) origamis was pursued in our joint article with Jean-Christophe Yoccoz.

In this short post, I would like discuss some informal questions posed by Jean-Christophe right after the completion of our joint paper with David Zmiaikou (in order to keep a trace of them in case someone [e.g., myself] wants to think about this matter in the future).

**1. Regular origamis and commutators **

The commutator determines the nature of the conical singularities of the origami : in fact, has exactly such singularities and the total angle around each of them singularities is .

Also, Yoccoz, Zmiaikou and I showed that the “potential blocks” of the Kontsevich-Zorich cocycle over the -orbit of are *completely* determined by the commutator (cf. Theorem 1.1 of our paper).

Given this scenario, Jean-Christophe asked whether the Lyapunov exponents of the Kontsevich-Zorich cocycle of were also completely determined from the knowledge of .

**2. Lyapunov exponents and commutators **

By the time we finished our joint paper, David Zmiaikou pointed out to me that the commutator is not a complete invariant for many algebraic question about : for example, Daniel Stork proved (among other things) that the pairs of permutations and have the same commutator but they generate *distinct* T-systems of the alternate group (cf. Section 2.6 of David Zmiaikou’s PhD thesis for more comments).

This remark led me to play with the origamis and in an attempt to answer Jean-Christophe’s question.

First, note that both of them have conical singularities and the total angle around each of them is . In particular, both and have genus .

On the other hand, a short computation with the aid of Sage (or a long calculation by hand) reveals that the -orbit of has cardinality and the -orbit of has cardinality (this is closely related to the comments in Section 5 of Stork’s paper).

Remark 1Recall that it is easy to algorithmically compute -orbits of origamis described by two permutations and of a finite collection of squares because is generated by and , and these matrices act on pairs of permutations by and (and the permutations and generate the same origami).

Moreover, this calculation also reveals that

- the -orbit of decomposes into four -orbits:
- two -orbits have size and all origamis in these orbits decompose into horizontal cylinders of width and height ;
- one -orbit has size and all origamis in this orbit decompose into horizontal cylinders of width and height ;
- one -orbit has size and all origamis in this orbit decompose into horizontal cylinders of width and height .

- the -orbit of decomposes into three -orbits:
- one -orbit contains a single origami decomposing into horizontal cylinders of width and height ;
- one -orbit has size and all origamis in this orbit decompose into horizontal cylinders of width and height ;
- one -orbit has size and all origamis in this orbit decompose into horizontal cylinders of width and height .

This information can be plugged into the Eskin-Kontsevich-Zorich formula to determine the sums and of the non-negative Lyapunov exponents of the Kontsevich-Zorich cocycle over the -orbits of and .

Indeed, if is an origami with conical singularities whose total angles around them are , , then Alex Eskin, Maxim Kontsevich and Anton Zorich showed that the sum of all non-negative Lyapunov exponents of the Kontsevich-Zorich cocycle over is

where is the decomposition of into horizontal cylinders and , resp. is the height, resp. width, of the horizontal cylinder .

In our setting, this formula gives

and

that is,

In particular, this allowed me to answer Jean-Christophe’s informal question: the knowledge of the commutator is *not* sufficient to determine the Lyapunov exponents.

**3. Lyapunov exponents and T-systems? **

Logically, this discussion led Jean-Christophe to ask a new question: how does the Lyapunov exponents of relate to algebraic invariants of ? For example, is the `Lyapunov exponent invariant’ equivalent to `T-systems invariant’ (or is the `Lyapunov exponent invariant’ a completely new invariant? [and, if so, can it be used to derive new interesting algebraic consequences for ?])

Remark 2André Kappes and Martin Möller used Lyapunov exponents in another (but not completely unrelated) setting as `new invariants’ to solve an algebraic problem (namely, classify commensurability classes of non-arithmetic lattices constructed by Pierre Deligne and George Mostow).

(Jean-Christophe and) I plan(ned) to come back to this question at some point in the future, but for now I will close this post here since I have nothing to say about this matter.

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