Platonic maps of low genus

In what ways can one tile a surface such that the tiling has a large measure of symmetry? This question lies at the basis of the research area with which this dissertation is concerned. To make the question more exact, we suppose we have a closed orientable surface with a connected finite graph with non-empty vertex set and non-empty edge set embedded into it, such that the complement of the graph consists of a union of two-dimensional disks. This puts a cell structure on the surface, with graph vertices, graph edges and disks being cells of dimension 0, 1, and 2, respectively. We can identify a disk together with its boundary as a polygon, where the edges on the disk boundary become sides of the polygon, and vertices on the boundary vertices of the polygon. A surface together with a cell division is called a map. A homeomorphism of the surface is called cellular with respect to a map if it sends cells to cells. The symmetry we demand of a map in this thesis is the existence of a group of cellular homeomorphisms that acts transitively on the set of oriented (0, 1)-flags of the graph, that is, on pairs of a vertex and an incident directed edge. This property entails that all disks have the same number of sides, when viewed as polygons, and all vertices of the graph have the same number of incident edges, but is in general stronger than these two conditions. We call a map on a surface satisfying our condition a platonic map. If there is also a cellular homeomorphism that preserves an oriented (0, 1)-flag but is not the identity, then we say the platonic map is reflexive, and call such a homeomorphism a reflection. Classical examples of reflexive platonic maps are the platonic solids (the tetrahedron, cube, octahedron, dodecahedron and icosahedron), whence their name. The platonic solids and their symmetry groups have long been of interest to mathematicians. The group of cellular homeomorphisms of a map M induces a subgroup of the mapping class group of the surface, which is the group formed by the equivalence classes of homeomorphisms with respect to isotopy. This subgroup is called the automorphism group of the map, Aut(M). One can in fact realize Aut(M) as an actual group of homeomorphisms of the surface. If the map is platonic, the subgroup Aut+(M) of orientation preserving map automorphisms can be generated by two elements R and S, which are primitive counterclockwise rotations around a polygon (disk) and an incident vertex, respectively. A presentation of Aut+(M) in R and S is called a standard map presentation ofM. A standard map presentation already contains all the information of a platonic map. A fascinating fact is that on each surface of genus g = 2 there are only a finite number of platonic maps. With the aid of group theory one can enumerate all possibile standard map presentations for a given genus, and this has indeed been done in [CD01] up to genus 15. This list has been the starting point of the investigation into platonic maps described in this thesis. We develop two fundamental tools to categorize and order platonic maps, namely polynomial families and diagonal maps. A polynomial family is a parametrized group-theoretic recipe for constructing an infinite series of platonic maps in a controlled way. Diagonal maps are the result of a standard construction applied, under certain conditions, to a map to yield a new map of the same genus. These tools, along with covering theory of platonic maps, are used throughout the thesis. We determine those reflexive platonic maps whose number of vertices is an odd prime. Also, we classify all reflexive platonic maps whose density is higher than 1/2 , with the stellar roles played by the tetrahedron and the Fermat maps. A remarkable property of a platonic map, a hidden gem lying dormant, is that it uniquely determines a compact Riemann surface on which the map can be realized by a graph with geodesic edges, and such that Aut(M) acts by isometries. Furthermore, there is a correspondence between compact Riemann surfaces and smooth complex algebraic curves. Both types of objects and the correspondence are of major importance in mathematics, and have guided a great deal of research, starting with Klein’s quartic curve. In general, this correspondence is not effective. For platonic maps, it is! We undertake the task to find algebraic curves for as many platonic maps as possible, so that other researchers may utilize them and study their properties further. We have succeeded for all platonic maps of genus at most 8, and for various other ones of higher genus, among which the members of the first Hurwitz triplet. In a separate chapter, we study the properties of this triplet and try to compute its Weierstrass points, answering a question by Kay Magaard and Helmut Volklein.