So I am giving a talk in which we'll prove the semi-famous Hurwitz Theorem:
Let $S$ be a compact Riemann surface of genus $g \geq 2$. Then $|Aut(S)| \leq 84(g-1)$, the group of holomorphic automorphisms on $S$.
My understanding is that this theorem is just a curiosity; I know of only one application of this theorem - to planar embeddings of graphs in surfaces.
Are there any other applications, if even a bit bizarre? Maybe in some dynamics of moduli spaces?