Some Relations between Graph Theory Andriemann

In this paper, we present some new relationships between graph theory and the geometry of Riemann surfaces. For the most part, we present these relationships in terms of questions and open problems. Since these questions have arisen from a number of recent developments, we also describe these developments so as to provide a context for these questions. In fact, our main interest here is to present these new circles of ideas, and our hope is that presenting them in the form of questions will give a fuller picture of their potential than a mere exposition of the ideas themselves. The idea of thinking of a graph as some kind of combinatorial form of a Riemann surface, and of obtaining information about one in terms of the other, is not particularly new. Our own thinking on the subject was inspired by a desire to understand Selberg's famous 3=16 theorem in geometric terms; see 3], 4], 5], and 7]. Here, the crucial point is to model the behavior of a covering space on a k-regular graph, that is, a graph such that from each vertex emanate exactly k edges; see also 13], 14] for parallel ideas. What is somewhat surprising is that there are a number of diierent ways by which graph theory may enter into the study of Riemann surfaces. For instance, one has the theorem of Andreev-Koebe-Thurston 25], which associates to a combi-natorial triangulation of a topological surface a unique Riemann surface S together with a circle packing of the surface. According to 2], the countably many Riemann surfaces which arise in this way are dense in the moduli space. One also has an algebraic-geometric description which may be paraphrased as follows: according to a theorem of Belyi 1], those Riemann surfaces which are deened over some number eld | that is to say, which can be described as the solution set of a system of algebraic equations having coeecients in a xed number eld | are characterized as those Riemann surfaces which arise as branched covers of the Riemann sphere having branch locus consisting of three values in the sphere. Now, this branched covering is perfectly well described by a graph, so we again arrive at a situation where a dense set of Riemann surfaces is given by a graph-theoretic description. However, this description seems to be quite diierent from the previous one, and so the problem of reaching …