Non-Abelian Simple Groups Act with Almost All Signatures

Abstract

The topological data of a group action on a compact Riemann surface is often encoded using a tuple (h; m 1 , . . . , ms) called its signature. There are two easily verifiable arithmetic conditions on a tuple necessary for it to be a signature of some group action. In the following, we derive necessary and sufficient conditions on a group G for when these arithmetic conditions are in fact sufficient to be a signature for all but finitely many tuples that satisfy them. As a consequence, we show that all non-Abelian finite simple groups exhibit this property.