First Hurwitz triplet

In the mathematical theory of Riemann surfaces, the first Hurwitz triplet is a special set of three different Hurwitz surfaces. These surfaces share the same automorphism group. This group is the smallest possible for this kind of surface, and the surfaces have a genus of 14. This number shows how complex these surfaces are.

There are two simpler cases. Genus 3 has one Hurwitz surface, called the Klein quartic. Genus 7 has another single example known as the Macbeath surface. The first Hurwitz triplet is more complex because it involves three surfaces together, all sharing the same symmetry.

The reason this triplet exists comes from number theory. In a special kind of number field, the prime number 13 breaks down into three smaller prime numbers. These smaller primes help create structures called congruence subgroups. These subgroups are linked to groups known as Fuchsian groups, which describe the geometry of the three surfaces in the triplet.

This discovery shows how ideas from geometry and number theory are connected. It is a nice example of how different parts of mathematics can work together to solve hard problems.

Arithmetic construction

The first Hurwitz triplet uses special objects called Hurwitz surfaces. These surfaces share the same symmetry group. This group is the simplest for surfaces of a certain complexity. This idea comes from number theory.

It relates to how the number 13 breaks down into smaller parts in a special number system. This breakdown helps create the three surfaces in the triplet. Each surface is made by taking a slice of a geometric space called the hyperbolic plane.

Bound for systolic length and the systolic ratio

The Gauss–Bonnet theorem helps us learn about the shapes of certain surfaces. It connects the Euler characteristic of a surface — a number that tells us about its shape — to the total curvature of the surface.

For surfaces with a special kind of curvature and 14 "holes," the Euler characteristic is -26. The total area of these surfaces can be found to be 52π.

There is also a concept called the systole. This is the shortest distance across these special surfaces. For surfaces with 14 "holes," the smallest possible distance is about 3.5187. This helps mathematicians understand the geometry of these interesting surfaces.

Ideal	3 − 2 η ⊲ O K {\displaystyle 3-2\eta \vartriangleleft O_{K}}
Systole	5.9039
Systolic Trace	− 4 η 2 − 8 η − 3 {\displaystyle -4\eta ^{2}-8\eta -3}
Systolic Ratio	0.2133
Number of Systolic Loops	91

Ideal	η + 3 ⊲ O K {\displaystyle \eta +3\vartriangleleft O_{K}}
Systole	6.3933
Systolic Trace	5 η 2 + 11 η + 3 {\displaystyle 5\eta ^{2}+11\eta +3}
Systolic Ratio	0.2502
Number of Systolic Loops	78

Ideal	2 η − 1 ⊲ O K {\displaystyle 2\eta -1\vartriangleleft O_{K}}
Systole	6.8879
Systolic Trace	− 7 η 2 − 14 η − 3 {\displaystyle -7\eta ^{2}-14\eta -3}
Systolic Ratio	0.2904
Number of Systolic Loops	364

First Hurwitz triplet