# Dessins d'Enfants

For the past several years, I've been thinking about properties of Dessins d'Enfants. During June 4-8, 2012, I lead a research group on the subject at the Research Experiences for Undergraduate Faculty (REUF) at ICERM in Providence, Rhode Island. I've embarked on a project to realize each of the Archimedean Solids, , and as Dessins d'Enfant. Below you can find my lecture notes from the workshop and an explorer to play around with some Dessins.

 Introduction to Dessin d'Enfants (?? MB) Dessin Explorer (Mathematica Notebook) Animation featuring Monodromy Action (120 MB) Workshop Photos (Flickr Photostream) Lectures on Dessins d'Enfants Blog

## What is a Dessin d'Enfant?

Let $$X$$ be a compact, connected Riemann surface. It is well-known that $$X$$ is an algebraic variety, that is, $$X \simeq \left \{ P \in \mathbb P^n(\mathbb C) \, \bigl| \, F_1(P) = F_2(P) = \cdots = F_m(P) = 0 \right \}$$ in terms of a collection of homogeneous polynomials $$F_i$$ over $$\mathbb C$$ in $$(n+1)$$ variables $$x_j$$. Denote $$\mathcal O_X$$ as the ring of regular functions on $$X$$, that is, "polynomials" $$f, g: X \to \mathbb P^1(\mathbb C)$$; and denote $$\mathcal K_X$$ as its quotient field, that is, rational functions $$f/g: X \to \mathbb P^1(\mathbb C)$$. For example, if $$X \simeq \mathbb P^1(\mathbb C) = \mathbb C \cup \{ \infty \}$$, then $$\mathcal O_X \simeq \mathbb C[z]$$ consists of polynomials in one variable, while $$\mathcal K_X \simeq \mathbb C(z)$$ consists of rational functions in one variable. In particular, any rational map $$\beta: X \to \mathbb P^1(\mathbb C)$$ induces a map $$\beta^\ast: \mathbb C(z) \to \mathcal K_X$$ which sends $$J \mapsto J \circ \beta$$. The degree of such a map is the size of the group $$G = \text{Gal} \bigl( \mathcal K_X / \beta^\ast \, \mathbb C(z) \bigr)$$.

For each $$P \in X$$, let $$\mathcal O_P$$ be the localization of $$\mathcal O_X$$ at the kernel of the evaluation map $$\mathcal O_X \to \mathbb C$$ defined by $$f \mapsto f(P)$$. Let $$\mathfrak m_P$$ denote the maximal ideal of $$\mathcal O_P$$; we view this as the collection of rational maps $$\beta \in \mathcal K_X$$ which vanish at $$P$$. As shown by Weil and Belyĭ, the Riemann surface $$X$$ can be defined in terms of homogeneous polynomials $$F_i$$ over an algebraic closure $$\overline{\mathbb Q}$$ if and only if there exists a rational function $$\beta: X \to \mathbb P^1(\mathbb C)$$ such that $$\beta: \left \{ P \in X \, \bigl| \, \beta - \beta(P) \in {\mathfrak m_P}^2 \right \} \to \bigl \{ 0, \, 1, \, \infty \}$$. The difference $$\beta - \beta(P) \in \mathfrak m_P$$ for any $$P \in X$$ because the function vanishes at $$P$$; the condition $$\beta - \beta(P) \in {\mathfrak m_P}^2$$ means the derivative of the function vanishes as well. A rational function as above where these critical values are at most $$0$$, $$1$$, and $$\infty$$ is called a Belyĭ map.

Following Grothendieck, we associate a bipartite graph $$\Delta_\beta$$ to a Belyĭ map $$\beta: X \to \mathbb P^1(\mathbb C)$$ by denoting the "black" vertices as $$B = \beta^{-1}(0)$$, "white" vertices as $$W = \beta^{-1}(1)$$, midpoints of the faces as $$F = \beta^{-1}(\infty)$$, and edges as $$E = \beta^{-1}\bigl([0,1] \bigr)$$. This is a loopless, connected, bipartite graph, called a Dessin d'Enfant, which can be embedded on $$X$$ without crossings. The group $$G = \text{Gal} \bigl( \mathcal K_X / \beta^\ast \, \mathbb C(z) \bigr)$$ permutes the solutions $$P$$ to $$\beta(P) = z$$, and hence acts on the dessin $$\Delta_\beta$$. The hope is that in studying graphs $$\Delta_\beta$$ one can better understand quotients $$G$$ of the absolute Galois group $$\text{Gal} \bigl( \overline{\mathbb Q}/\mathbb Q \bigr)$$.

We are motivated by the following question: Given a loopless, connected, bipartite graph $$\Gamma$$ on a compact, connected Riemann surface $$X$$, when is $$\Gamma \simeq \Delta_\beta$$ the Dessin d'Enfant of a Belyĭ map $$\beta: X \to \mathbb P^1(\mathbb C)$$? Given such a loopless, connected, planar, bipartite graph $$\Gamma$$, we wish to use properties of the symmetry group $$G$$to construct a Belyĭ map $$\beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$$ such that $$\Gamma$$ arises as its Dessin d'Enfant.

## Summary of REUF4 Results

Here is a summary of the main results we found during the 2012 summer program.
• Every Belyĭ map $$\beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$$ of degree $$\deg(\beta) = 1$$ is in the form $\beta(z) = \dfrac {a \, z + b}{c \, z + d} \qquad \text{where} \qquad a \, d - b \, c \neq 0.$
• Up to fractional linear transformation, every Belyĭ map $$\beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$$ of degree $$\deg(\beta) = 2$$ is in the form $\beta(z) = \left( \dfrac {a \, z + b}{c \, z + d} \right)^2 \qquad \text{where} \qquad a \, d - b \, c \neq 0.$
• Consider four distinct complex numbers $$z^{(-1)}$$, $$z^{(0)}$$, $$z^{(+1)}$$, and $$z^{(\infty)}$$ with cross-ratio $$\bigl( z^{(-1)}, \, z^{(0)}; \, z^{(+1)}, \, z^{(\infty)} \bigr) = -1$$ and define the rational function $\beta(z) = \left[ \dfrac {2 \, \bigl( z^{(0)} - z^{(1)} \bigr) \, \bigl( z^{(\infty)} - z^{(1)} \bigr) \, \bigl( z - z^{(0)} \bigr) \, \bigl( z - z^{(\infty)} \bigr)}{ \bigl( z^{(0)} - z^{(1)} \bigr)^2 \, \bigl(z - z^{(\infty)} \bigr)^2 + \bigl( z^{(\infty)} - z^{(1)} \bigr)^2 \, \bigl(z - z^{(0)} \bigr)^2} \right]^2.$ Then $$\beta(z)$$ is a Belyĭ map whose associated Dessin d'Enfant $$K_{2,2}$$ has vertices $$B = \bigl \{ z^{(0)}, \, z^{(\infty)} \bigr \}$$ and $$W = \bigl \{ z^{(-1)}, \, z^{(+1)} \bigr \}$$.
• Every planar complete bipartite graph $$K_{m,n}$$ be realized as the Dessin d'Enfant of some Belyĭ map, namely either $$\beta(z) = z^n$$ or $$\beta(z) = 4 \, z^n/(z^n + 1)^2$$.
• Every path graph be realized as the Dessin d'Enfant of some Belyĭ map, namely $$\beta(z) = \bigl( 1 + \cos \, ( n \, \arccos z) \bigr)/2$$.
• Every bipartite cycle graph be realized as the Dessin d'Enfant of some Belyĭ map, namely $$\beta(z) = (z^n + 1)^2/ (4 \, z^n)$$.
• The Möbius Transformations $$r(z) = (z-1)/z$$ and $$s(z) = z/(z-1)$$ generate a subgroup of $$\text{Aut} \bigl( \mathbb P^1(\mathbb C) \bigr)$$ isomorphic to $$S_3 = \left \langle r, \, s \, \bigl | \, r^3 = s^2 = (s \, r)^2 = 1 \right \rangle$$.
• Let $$\phi(z)$$ be a rational function. The composition $$\phi \circ \beta$$ is a Belyĭ map for every Belyĭ map $$\beta$$ if and only if $$\phi$$ is a Belyĭ map which maps the set $$\bigl \{ (0:1), \, (1:1), \, (1:0) \bigr \}$$ to itself.
• Let $$\Gamma = \bigl( B \cup W, \, E \bigr)$$ be the Dessin d'Enfant associated to a Belyĭ map $$\beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$$. For each $$\gamma(z) \in S_3$$, let $$\Gamma_\gamma = \bigl( B_\gamma \cup W_\gamma, \, E_\gamma \bigr)$$ be the Dessin d'Enfant associated to the composition $$\gamma^{-1} \circ \beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$$. That is, $B_\gamma = \beta^{-1}\bigl( \gamma(0) \bigr), \qquad W_\gamma = \beta^{-1} \bigl( \gamma(1) \bigr), \qquad \text{and} \qquad E_\gamma = \beta^{-1} \bigl( \gamma([0,1]) \bigr).$
• If $$\gamma = 1$$, then $$\Gamma_1 = \Gamma$$ is the original Dessin d'Enfant.
• If $$\gamma = s$$, then $$\Gamma_s$$ can be obtained from $$\Gamma$$ by interchanging the white vertices $$W$$ with the midpoints of the faces $$F$$.
• If $$\gamma = s \, r$$, then $$\Gamma_s$$ can be obtained from $$\Gamma$$ by interchanging the black vertices $$B$$ with the white vertices $$W$$. In other words, $$\Gamma_s$$ is the dual graph to $$\Gamma$$.
• If $$\gamma = r \, s$$, then $$\Gamma_s$$ can be obtained from $$\Gamma$$ by interchanging the black vertices $$B$$ with the midpoints of the faces $$F$$.
• If $$\gamma = r$$, then $$\Gamma_r$$ can be obtained from $$\Gamma$$ by cyclically rotating the black vertices $$B$$ to the midpoints of the faces $$F$$ to the white vertices $$W$$.
• If $$\gamma = r^2$$, then $$\Gamma_{r^2}$$ can be obtained from $$\Gamma$$ by cyclically rotating the black vertices $$B$$ to the white vertices $$W$$ to the midpoints of the faces $$F$$.

## Dessin Explorer

We decided we needed to visualize these graphs, so we wrote code which would do just that. Here is a Mathematica notebook which will visualize Dessins either in the complex plane or on the Riemann sphere.