Given a Riemann surface $X$, we have some fairly standard methods for identifying which groups $G$ admit an effective and holomorphic action $G \times X \to X$. For instance, some fairly elementary arguments show that the only finite groups acting effectively and holomorphically on $\mathbb{P}^1$ are finite cyclic groups, $D_4, A_4, S_4$, and $A_5$. (See e.g. Miranda, Ch. III p. 80 for these arguments.) Another result is Hurwitz' theorem, which says that given a Riemann surface $X$, the only such finite groups must have order $\leq 84(g(X) - 1)$.

Now let's instead fix a group $G$. (Maybe you want $G$ to be a Lie group, maybe finite.) **What results tell us about the (non)existence of a Riemann surface $X$ on which $G$ acts effectively and holomorphically?** I'd be surprised to hear that given $G$, we can always construct such an $X$.

Unlike the problem of finding groups given $X$, I have not heard anything about this kind of problem while learning about Riemann surfaces.

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