In this paper we study an action $D$ of the absolute Galois group $\Gamma =Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ on bicolored plane trees. In distinction with the similar action provided by the Grothendieck theory of “Dessins d’enfants” the action $D$ is induced by the action of $\Gamma$ on equivalence classes of conservative polynomials which are the simplest examples of postcritically finite rational functions. We establish some basic properties of the action $D$ and compare it with the Grothendieck action.

Dans cet article nous nous intéressons aux propriétés des composantes irréductibles associées à la géométrie réelle d’un dessin d’enfant. Plus précisément, nous étudions les composantes irréductibles de la courbe $\Gamma$ dont l’ensemble des points réels est l’image réciproque de ${\mathbf{P}}^1\left(\mathbf{R}\right)$ par une fonction de Belyi d’un dessin d’enfant.

We survey methods to compute three-point branched covers of the projective line, also known as Belyĭ maps. These methods include a direct approach, involving the solution of a system of polynomial equations, as well as complex analytic methods, modular forms methods, and $p$-adic methods. Along the way, we pose several questions and provide numerous examples.