  The topological data of a group action on a compact Riemann surface is often
encoded using a tuple $(h;m_1,\dots ,m_s)$ 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.
