Uniform dessins of low genus

Abstract

It is known that every dessin (map or hypermap) corresponds to a finite index subgroup of a triangle group and can be embedded naturally into some Riemann surface [JSl, JS3]. A dessin is uniform if its (hyper)vertices all have the same valency, its (hyper)edges all have the same valency, and its (hyper)faces all have the same valency; uniform dessins correspond to torsion-free subgroups of triangle groups. By the theorems of Belyi [Bel] and Wolfart [Wol], a compact Riemann surface X is defined over the field of algebraic numbers Q if and only if X carries a dessin (also see [Gro]). In this thesis we study the uniform dessins of genus g < 3 and investigate their connections with algebraic curves, Belyi's Theorem, and the absolute Galois group Gal(Q/Q). An elliptic curve of modulus r can be uniformized by a finite index subgroup of a Euclidean triangle group if and only if r G Q(i) or r £ Q(p); these elliptic curves are said to have Euclidean Belyi uniformizations and naturally carry the uniform dessins of genus 1. Using results from number theory, it is proved that there are only five rational elliptic curves with Euclidean Belyi uniformizations. A classification of the genus 1 uniform maps is given which extends the notation for genus 1 regular maps found in [CMo]. Formulae are derived for the number of genus 1 uniform maps with a given number of vertices, and the refiexible maps are described. Belyi functions are computed in a number of cases, and arbitrarily large Galois orbits of genus 1 uniform dessins are constructed. The existence of two uniform maps of genus g > 1 lying on conformally equivalent Riemann surfaces is considered. This leads naturally to the study of arithmetic Fuchsian groups [Vi] and motivates the definitions of arithmetic and non-arithmetic maps. General results are proved for non-arithmetic maps, and specific examples are given in the arithmetic case.

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