Sharpness of the Hurwitz 84(g-1) theorem.

There are usually courses at Mathcamp about surfaces; there should be courses about orbifolds! For instance, knowing that the smallest hyperbolic orbifold is the (2,3,7)-orbifold, having orbifold Euler characteristic , immediately gives that a closed hyperbolic surface of genus has no more than isometries (preserving orientation); this is “Hurwitz’ theorem.”

Just to show off this theorem, here is a cubical complex which is a surface with lots of symmetries (and the clever reader will recognize this as coming right out of Davis’ construction of aspherical manifolds): consider the -dimensional cube , and let be the edges around the origin, and be the square face containing the edges and . Define a subcomplex consisting of the squares and all squares in parallel to these. Now is a orientable surface (the link of any vertex is an -cycle, i.e., topologically an ). Don’t be fooled by the notation: has genus much larger than .

In fact, let’s calculate the genus. Every vertex of is contained in , and there are vertices in . Likewise, every edge in is contained in , and there are edges in . Finally, there are squares parallel to each of , so there are square faces in . Thus, and so is a surface of genus . This is a maybe a good exercise for someone first learning about Euler characteristic, but not especially interesting…

So here’s the punchline–or rather the punch-question–why is the genus growing exponentially in ? Because is very symmetric! And Hurwitz says to get so much symmetry, we need (linearly) as much genus. And we can find exponentially many symmetries of without any work. For starters, the group acts on by reflecting through hyperplanes, as does the group cyclically permuting the basis . If we want to be precise, let be the resulting group of order . Quotienting by these symmetries gives an orbifold , which one observes to be a square with cone points on each vertex (three with cone angle and one with cone angle ) and reflections in each of the four sides. Thus, the orbifold Euler characteristic of is , so the Euler characteristic of must be , just like we got before. One might argue that this method was “easier” than the previous method for calculating , but that misses the point—I (and probably everyone else) calculated the number of edges of by using a group action, if only implicitly.

The point is, even without doing any calculations or thinking very hard, the number of symmetries of is growing exponentially in , and therefore the genus must be growing exponentially in as well—the orbifold makes this reasoning precise.

There’s a lot of stuff left to be discovered about the number of automorphisms of genus surfaces. For instance, it’s known that the bound is attained for infinitely many genera, but there are also infinitely many genera for which it is not attained. Let be the maximal order of the automorphism group of a genus surface; Maclachlan and Accola proved (in 1968) that . This bound is sharp, too. There’s a beautiful paper [1] working out what happens in the arithmetic and non-arithmetic case. Anyway, what is known about the set of for which is attained? What is the asymptotic density of this set?