4
$\begingroup$

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?

$\endgroup$
  • 1
    $\begingroup$ 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. $\endgroup$ – PseudoNeo Oct 28 '15 at 13:59
1
$\begingroup$

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 $84(g-1)$.

$\endgroup$
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.