Dessin D'enfant - Regular Maps and Triangle Groups

Regular Maps and Triangle Groups

The five Platonic solids – the regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron – viewed as two-dimensional surfaces, have the property that any flag (a triple of a vertex, edge, and face that all meet each other) can be taken to any other flag by a symmetry of the surface. More generally, a map embedded on a surface with the same property, that any flag can be transformed to any other flag by a symmetry, is called a regular map.

If a regular map is used to generate a clean dessin, and the resulting dessin is used to generate a triangulated Riemann surface, then the edges of the triangles lie along lines of symmetry of the surface, and the reflections across those lines generate a symmetry group called a triangle group, for which the triangles form the fundamental domains. For example, the figure shows the set of triangles generated in this way starting from a regular dodecahedron. When the regular map lies on a surface whose genus is greater than one, the universal cover of the surface is the hyperbolic plane, and the triangle group in the hyperbolic plane formed from the lifted triangulation is a (cocompact) Fuchsian group representing a discrete set of isometries of the hyperbolic plane. In this case, the starting surface is the quotient of the hyperbolic plane by a finite index subgroup Γ in this group.

Conversely, given a Riemann surface that is a quotient of a (2,3,n) tiling (a tiling of the sphere, Euclidean plane, or hyperbolic plane by triangles with angles π/2, π/3, and π/n), the associated dessin is the Cayley graph given by the order two and order three generators of the group, or equivalently, the tiling of the same surface n-gons meeting three per vertex. Vertices of this tiling give black dots of the dessin, centers of edges give white dots, and centers of faces give the points over infinity.