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.

中文翻译：

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非阿贝尔单群作用于几乎所有的签名

紧致黎曼曲面上群作用的拓扑数据通常使用称为其签名的元组 $(h;m_1,\dots,m_s)$ 进行编码。一个元组有两个容易验证的算术条件，它是某个组动作的签名所必需的。在下文中，当这些算术条件实际上足以成为满足它们的除有限多个元组之外的所有元组的签名时，我们在群 $G$ 上推导出充分必要条件。因此，我们证明所有非阿贝尔有限单群都表现出这个性质。