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.