Monodromy Groups and Compositions of Belyi Maps

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 \phi, \, \beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C) such that the composition \Phi = \beta \circ \phi is also a Belyi map. (For example, a sufficient condition here is that \beta \bigl( \{ 0, \, 1, \, \infty \} \bigr) \subseteq \{ 0, \, 1, \, \infty \}.) I am interested in computing the monodromy group of the compositon \Phi. To this end, I will show the following.

Proposition. Say that \text{Mon}(\beta) \subseteq S_N and \text{Mon}(\phi) \subseteq S_M are the monodromy groups of \beta and \phi, respectively, as subgroups of the symmetric groups S_N and S_M, respectively. Then \text{Mon}(\Phi) \subseteq S_M \wr S_N is a subgroup of the wreath product S_M \wr S_N := {S_M}^N \rtimes S_N of the symmetric groups.

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System of Equations for Computing Shabat Polynomials

A few weeks ago, Dong Quan Ngoc Nguyen (University of Notre Dame) came to visit here at Purdue.  We spoke a little about the computer package Bertini (created by Daniel Bates, Jonathan Hauenstein, Andrew Sommese and Charles Wampler) and whether the homotopy continuation method can be used to compute Belyi maps and Shabat Polynomials.  I’ve been working on setting up a system of polynomial equations whose solutions give the coefficients of the Belyi maps, so it really comes down to actually finding the solutions to these equations.  The hope is that a polynomial homotopy continuation method will be much more efficient than say, using Groebner bases, to find all solutions!

Let me try and set up how this would work by working through some explicit examples.

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Monodromy Groups and Compositions of Shabat Polynomials

Last week, the great Naiomi Cameron visited me for a few days to discuss some new directions about Shabat polynomials.  I’ve been horrible about posting on this blog, so now that I’ve been motivated to work on Shabat polynomials again, I figured it’s time for me to write!

As considered in the 1994 paper by Georgii Borisovich Shabat and Alexander Zvonkin entitled Plane trees and algebraic numbers, the rational function

\displaystyle \beta(z) = - \dfrac {4}{531441} \, (z - 1) \, z^3 \, \bigl( 2 \, z^2 + 3 \, z + 9 \bigr)^3 \, \bigl( 8 \, z^4 + 28 \, z^3 + 126 \, z^2 + 189 \, z + 378 \bigr)

is a Shabat polynomial which happens to be the composition \beta = \phi \circ \Phi of two other Shabat polynomials.  The first has monodromy group G_\phi = Z_2 as a cyclic group, while the second has monodromy group G_\phi = A_7 as an alternating group.  The monodromy group of the composition has order |G_\beta| = 12 \, 700 \, 800 = |G_\phi| \cdot |G_\Phi|^2. Do we have G_\beta = Z_2 \ltimes \bigl( A_7 \times A_7 \bigr) as the wreath product of G_\Phi by G_\phi?

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“Jarden’s Property and Hurwitz Curves” by Robert A. Kucharczyk

Robert A. Kucharczyk has a new paper on the ArXiv entitled “Jarden’s Property and Hurwitz Curves”.
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“Enumeration of Grothendieck’s Dessins and KP Hierarchy” by Peter Zograf

Peter Zograf has a new paper on the ArXiv entitled “Enumeration of Grothendieck’s Dessins and KP Hierarchy”.
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“Generalized Onsager Algebras and Grothendieck’s Desssins d’Enfants” by Chernousov, Gille, and Pianzola

Vladimir Chernousov, Philippe Gille and Arturo Pianzola have a new paper on the ArXiv entitled “Generalized Onsager Algebras and Grothendieck’s Dessins d’Enfants”.
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“On computing Belyi maps” by Jeroen Sijsling and John Voight

Jeroen Sijsling and John Voight have a new paper on the ArXiv entitled “On computing Belyĭ Maps”.
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