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

 February 23, 2007 mathematics

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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 -1/84 , immediately gives that a closed hyperbolic surface of genus g has no more than 84(g-1) isometries (preserving orientation); this is “Hurwitz’ 84(g-1) 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 n -dimensional cube I^n , and let e_1, \ldots, e_n be the n edges around the origin, and e_i e_j be the square face containing the edges e_i and e_j . Define a subcomplex \Sigma^2_n \subset I^n consisting of the squares e_1 e_2, e_2 e_3, \ldots, e_{n-1} e_n, e_n e_1 and all squares in I^n parallel to these. Now \Sigma^2_n is a orientable surface (the link of any vertex is an n -cycle, i.e., topologically an S^1 ). Don’t be fooled by the notation: \Sigma^2_n has genus much larger than n .

In fact, let’s calculate the genus. Every vertex of I^n is contained in \Sigma^2_n , and there are 2^n vertices in I^n . Likewise, every edge in I^n is contained in \Sigma^2_n , and there are n 2^{n-1} edges in I^n . Finally, there are 2^{n-2} squares parallel to each of e_i e_j , so there are n 2^{n-2} square faces in \Sigma^2_n . Thus, \chi(\Sigma^2_n) = 2^n - n 2^{n-1} + n 2^{n-2} = 2^{n} - n \cdot 2^{n-2}, and so \Sigma^2_n is a surface of genus g = 1 + 2^{n-3} (n - 4) . 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 n ? Because \Sigma^2_n 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 \Sigma^2_n without any work. For starters, the group (\mathbb{Z}/2\mathbb{Z})^n acts on \Sigma^2_n by reflecting through hyperplanes, as does the group \mathbb{Z}/n\mathbb{Z} cyclically permuting the basis e_1, \ldots, e_n . If we want to be precise, let G_n be the resulting group of order n 2^{n} . Quotienting \Sigma^2_n by these symmetries gives an orbifold \Sigma^2_n / G_n , which one observes to be a square with cone points on each vertex (three with cone angle \pi/2 and one with cone angle 2 \pi/n ) and reflections in each of the four sides. Thus, the orbifold Euler characteristic of \Sigma^2_n / G_n is 3/4 + 1/n - 4/2 + 1 = 1/n - 1/4 , so the Euler characteristic of \Sigma^2_n must be (1/n - 1/4) \cdot n \cdot 2^{n} = 2^{n} - n \cdot 2^{n-2} , just like we got before. One might argue that this method was “easier” than the previous method for calculating \chi(\Sigma^2_n) , but that misses the point—I (and probably everyone else) calculated the number of edges of I^n 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 \Sigma^2_n is growing exponentially in n , and therefore the genus must be growing exponentially in n 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 g surfaces. For instance, it’s known that the 84(g-1) bound is attained for infinitely many genera, but there are also infinitely many genera for which it is not attained. Let N(g) be the maximal order of the automorphism group of a genus g surface; Maclachlan and Accola proved (in 1968) that N(g) \geq 8(g+1) . 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 g for which 84(g-1) is attained? What is the asymptotic density of this set?

[1] M. Belolipetsky, G.A. Jones, A bound for the number of automorphisms of an arithmetic Riemann surface, Math. Proc. Cambridge Philos. Soc. 138 (2005) 289–299. https://doi.org/10.1017/S0305004104008035.