Non-Abelian simple groups act with almost all signatures
The topological data of a finite group $G$ acting conformally on a compact Riemann surface is often encoded using a tuple of non-negative integers $(h;m_1,\ldots ,m_s)$ called its signature, where the $m_i$ are orders of non-trivial elements in the group. There are two easily verifiable arithmetic conditions on a tuple which are necessary for it to be a signature of some group action. We derive necessary and sufficient conditions on a group for the situation where all but finitely many tuples that satisfy these arithmetic conditions actually occur as the signature for an action of $G$ on some Riemann surface. As a consequence, we show that all non-abelian finite simple groups exhibit this property.