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John Baez
John Baez
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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, andso 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 think it would be nicelike 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.

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, and 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 think it would be nice to 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.

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.

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John Baez
John Baez
  • 22.3k
  • 3
  • 85
  • 170

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, and 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 think it would be nice to 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.