Skipping the easy cases of genus 0 and 1, what groups can arise as the group of conformal transformations of a Riemann surfaces of genus, say, 2 or 3?
I'm frustrated because there are papers that supposedly answer this question (which are even open-access):
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Izumi Kuribayashi and Akikazu Kuribayashi, Automorphism groups of compact Riemann surfaces of genera three and four, Journal of Pure and Applied Algebra 65 (1990), 277-292
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Izumi Kuribayashi, On an algebraization of the Riemann-Hurwitz relation, Kodai Math. J. 7 (1984), 222-237.
Unfortunately, I don't understand the notation for groups used in this paper. There are lots of groups with names like $G(60,120)$ and $H(5 \times 40)$. These are actually specific subgroups of $\mathrm{GL(g,\mathbb{C})}$ where $g$ is the genus of the surface, obtained by looking at how automorphisms of a Riemann surface act on its space of holomorphic 1-forms. But I don't understand how they are defined, so I don't know how to answer questions like this:
- Which 32-element groups show up as automorphism groups of Riemann surfaces of genus 3?
This last question is the one I really want answered right now, but in general I would like to know more about automorphism groups of low-genus Riemann surfaces. I get the feeling that when such a group is reasonably big, it preserves a regular tiling of the surface by regular polygons, making the Riemann surface into the quotient of the hyperbolic plane by some Fuchsian group.
The classic example is of course Klein's quartic curve, the genus-3 surface tiled by 24 regular heptagons, whose automorphism group is $PSL(2,7)$, the largest allowed by the Hurwitz automorphism theorem.
Best Answer
I think that all of the notation is defined in the papers. For example, $G(60, 120)$ is defined in [Kuribayashi-Kuribayashi]. Proposition 2.3(d)(1) says $G(60, 120) = \langle G(5, 5 \times 2), L \rangle$, Proposition 2.1(d)(3) says $G(5, 5 \times 2) = \langle A(\zeta, \zeta^2, \zeta^3, \zeta^4) \rangle$, page 284 defines $A(a,b,c,d)$ and $L$ as explicit matrices, and $\zeta = \zeta_8 = e^{2\pi i/8}$ is defined on pages 285 and 279.