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Abstract

A quasiplatonic surface is a compact Riemann surface, X, which admits a group of automorphisms, G, (called a quasiplatonic group) such that the quotient space, X/G, has genus 0 and the map pG:X-->X/G is branched over three points. For a given genus, by using computational methods we can determine all quasiplatonic groups which act on a quasiplatonic surface of that genus. Though in principal this method can be used to calculate all quasiplatonic groups, in practice it is unrealistic. Another approach is to fix a group and see what genera this group can act upon as a quasiplatonic group. In this paper we classify all Abelian groups which can act on a quasiplatonic surface. A partial classification has previously been supplied. This partial classification provides necessary but not sufficient conditions for a group to act upon a quasipatonic surface. The classification shown in the following provides both. We accomplish this by first looking at cyclic groups with orders of a single prime power. All other Abelian groups can be built up from this case, and so can the classification of the quasiplatonic surfaces upon which they act upon.

Author Bio

I am a native of the Portland area, and will begin a master㤼㸲s program in mathematics at Portland State University in the fall, and plan to go onto a doctoral studies following that. In addition to my studies at the University of Portland, I also attended an REU at North Carolina State University in the summer of 2005, and also the Park City Mathematics Institute in the summer of 2007. When I’m not doing math, I also enjoy running and reading science fiction.

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