Lectures on Dessins d'Enfants

Monodromy Groups and Compositions of Belyi Maps

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A few weeks ago, I considered how the monodromy groups of Shabat polynomials change under composition by considering several examples.  I would like to explain a general phenomenon by considering the composition of Belyi maps on the sphere.

Say that we have two Belyi maps, namely such that the composition is also a Belyi map. (For example, a sufficient condition here is that .) I am interested in computing the monodromy group of the compositon . To this end, I will show the following.

Proposition. Say that and are the monodromy groups of and , respectively, as subgroups of the symmetric groups and , respectively. Then is a subgroup of the wreath product of the symmetric groups.


Monodromy Groups of Belyi Maps

First, let me review how to compute the monodromy group of a Belyi map of degree .

Let be a complex number in the “thrice punctured sphere” . We know that the fundamental group , the group of closed loops satisfying modulo equivalence, is the free group on two generators. We will choose these generators to be the closed loops explicitly given by and as loops around 0 and 1, respectively.

Now the Belyi map restricts to an -fold cover whose domain is the punctured sphere . Let be those complex numbers which form the preimages of . For each preimage , there are unique paths such that

For instance, open differentiating, one can deduce that such paths are the unique solutions to the ordinary differential equations

Since , we see that for some . In particular, there exist permutations such that and for all . Hence we have a group homomorphism

The set , called the monodromy group of , is that subgroup of which is the image of this group homomorphism.

Example #1

Consider the Belyi map given by of degree . We will show that is the Klein Vierergruppe.

Set and , so that is a 4-fold cover of punctured spheres. Choose , and write for some complex numbers and . Then in terms of the complex numbers

It is easy to check that the paths satisfy the differential equations

which have the exact solutions

In particular,

Hence while , so that the monodromy group is given by

Monodromy of Compositions of Belyi Maps

Now let’s consider two Belyi maps such that the composition is also a Belyi map. Denote and so that . Continuing notation as above, set

Hence is an -fold cover.

As before, we will choose the generators of the fundamental group to be the closed loops explicitly given by and . For each preimage , there are unique paths such that

As we know how to compute , we already know there exist such that

This means that we can find a -tuples and in such that

Hence we have a group homomorphism

The image of this group homomorphism is the monodromy group of . It is clear that there exists a subgroup such that we have a commutative diagram in the form

Example #2

For any positive integer , consider the Belyi map given by which is the composition of the two Belyi maps and . It is easy to check that and are both cyclic groups, while is a dihedral group. In particular, is a proper subgroup of the wreath product , also known as the generalized symmetric group.

Example #3

The Belyi map  is the composition of the Belyi maps and . One checks that is the wreath product of and .

(I would like to thank Michael Musty and Sam Schiavone for reading a previous blog post, then sending me an e-mail with this result!)

Example #4

The Belyi map  is the composition of the Belyi maps and . It is easy to check that and . Curiously, the monodromy group is a proper subgroup because it has order .  So what its the group ?

Question. Say that is a Belyi map which is the composition of and another Belyi map of degree . Since , we know that is a subgroup of the wreath product of and . Is it true that ?

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