# Applications of the Hurwitz Theorem on Number of Automorphisms?

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?

• It's not really an application, but Hurwitz's theorem proves that not all points on a Riemann surface of genus $g \geq 2$ are equivalent (of course it's a very weak consequence of the finiteness of $\mathrm{Aut}(S)$). This is a massive difference between (real) differential geometry and complex geometry, and it's the first example of this phenomenon (on Riemann surfaces of genus $0$ or $1$, all the points are equivalent). One of the proofs of the theorem actually goes through the study of some particular points on the surface, the Weierstraß points. – PseudoNeo Oct 28 '15 at 13:59

You might know the geometric reformulation of the Hurwitz theorem, which says that there exists a unique compact, connected hyperbolic 2-orbifold $P_{237}$ of minimum area, namely the $(2,3,7)$ triangle reflection orbifold. The connection between the version you state and the geometric version is that the quotient space $S / \text{Aut}(S)$ is a compact, connected, hpyerbolic 2-orbifold, and the quotient map $S \mapsto S / \text{Aut}(S)$ is an orbifold covering map of degree equal to $|\text{Aut}(S)|$, so $$\text{Area}(P_{237}) \le \text{Area}(S) \, / \, |\text{Aut}(S)|$$ $$|\text{Aut}(S)| \le \frac{\text{Area}(S)}{\text{Area}(P_{237})}$$ Those areas can be computed using the Gauss-Bonnet formula, which yields the Hurwitz theorem.

This geometric version has application to the classification of Fuchsian groups up to isomorphism. For instance: there are only finitely many isomorphism classes of Fuchsian groups $K$ such that $\pi_1(S)$ is isomorphic to a finite index subgroup of $K$, because the index $[K:\pi_1(S)]$ is bounded by the same constant $84(g-1)$.

One such application that I managed to find is the following:

Define the $\textit{genus}$ of a graph to be the minimal genus among compact Riemann surfaces that the graph embeds in (i.e. planar)

Define the $\textit{genus of a group}$ $G$ to be the minimum genus among its Cayley graphs.

Then Tucker used the Hurwitz Automorphisms Theorem to prove these striking theorems - in fact, one's intuition is probably that they would be false, particularly the last one:

There are only finitely many groups of a given genus $g \geq 2$.

If a group has a Cayley graph ebmedding in the Klein bottle, it has one embedding in the torus.

There exists only one group $G$ of genus 2. It has order 96.

For a reference see pp. 305 and 316 in Gross and Tucker's Topological Graph Theory, Dover 2012 Edition.