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.
Aaron Wootton, Department of Mathematics, University of Portland email@example.com
"Classification of Quasplatonic Abelian Groups and Signatures,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 9
, Article 1.
Available at: http://scholar.rose-hulman.edu/rhumj/vol9/iss1/1