# What are the possible automorphism groups of Riemann surfaces of low genus?

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):

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.

• Just a comment: One reason linear groups arise naturally in this context is because of Serre's Theorem which says that if $G$ is a finite subgroup of $MCG(S)$ then the restriction to $G$ of the canonical homomorphism $MCG(S) \to Aut(H_1(S;A))$ is injective, as long as the coefficient group $A$ has no 2-torsion. If you just want a list of examples this comment is not particularly relevant, but if you want a proof then it might be more relevant. – Lee Mosher Sep 20 '15 at 17:13

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.

• Defining a subgroup of some $GL_n$ as generated by several explicit subgroups does not provide a very decipherable information about the group in general... – YCor Sep 21 '15 at 14:39

The canonical reference on the subject (though one I don't have in front of me) is:

Characters and Automorphism Groups of Compact Riemann Surfaces

Part of London Mathematical Society Lecture Note Series

AUTHOR: Thomas Breuer

DATE PUBLISHED: October 2000

FORMAT: Paperback

ISBN: 9780521798099

While there appears to be no online version, doubtlessly your library has it...

Update. There are two ways to interpret the question "Which 32-element groups show up as automorphism groups of Riemann surfaces of genus 3?" depending on whether the group is assumed to be the full automorphism group of the surface or only a subgroup of it. In my answer below I only assumed that the group was an automorphism group, not necessarily the full automorphism group. This may account for the discrepancy between my answer (there are two groups of order 32) and Dan Petersen's answer (there is a unique group of order 32). In any case, the paper by Broughton that I cite shows that the two groups of order 32 are not isomorphic since their centers are the two groups of order 4.

Another reference for the classification in genus 2 and 3 is:

S. Allen Broughton, Classifying finite group actions on surfaces of low genus, J. Pure Appl. Alg. v.69 #3 (1990) pp.233-270.

The point of view here is algebraic, reducing the problem to group theory. Table 4 on page 252 lists the 20 distinct actions in genus 2. These involve 18 different groups (the cyclic groups of order 2 and 6 each have two different actions). I think I once checked that only three of the actions are maximal, with groups of orders 10, 24, and 48, and all the other actions are obtained by restricting to subgroups of these three.

The actions in genus 3 are listed in Table 5, pages 254-255. There are 70 different group actions here, with 58 different groups (if I counted correctly). Just two of the actions involve groups of order 32. In both cases the group is a semidirect product of a group of order 2 and a group of order 16, the latter group being abelian in one case and nonabelian in the other. It doesn't look like it's too hard to track down the definitions of the groups in the paper. The table gives presentations for them.

I once gave a mini-course for undergraduates on this topic of symmetries of surfaces, mostly from a geometric/topological viewpoint. There are some pretty pictures here, though I couldn't locate a source giving pictures for all the examples in genus 2 and 3.

To answer your specific question, there is a unique curve of genus three with a 32-element automorphism group, the hyperelliptic curve
$$y^2 = x^8 - 1.$$ As for any hyperelliptic curve, the automorphism group fits in a central extension $$0 \to C_2 \to G \to D_{16} \to 1$$ where $C_2$ is the subgroup generated by the hyperelliptic involution and $D_{16}$ permutes the branch points of the hyperelliptic map, i.e. the solutions of $x^8-1=0$ on $\mathbf P^1$.

The group can be given a presentation $$\langle s, t \mid s^4, t^4, (st)^2, (s^{-1}t)^2\rangle.$$ I just learned this from a paper of Shaska and Wijesiri (a decidedly noncanonical reference).