# Edray H Goins

# 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 \( F \).