First Hurwitz triplet — Safekipedia Discoverer

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, which is the smallest possible group for this kind of surface, having a genus of 14. This means that each surface has a particular complexity measured by the number 14, which is a key number in this area of mathematics.

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

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

This discovery connects deep ideas from geometry and number theory, showing how numbers can explain patterns in shapes. It is an elegant example of how different areas of mathematics can come together to solve complex problems.

Arithmetic construction

The first Hurwitz triplet involves special mathematical objects called Hurwitz surfaces. These surfaces share the same symmetry group, which is the simplest possible for surfaces of a certain complexity. This connection arises from number theory.

Specifically, it relates to how the number 13 breaks down into smaller parts within a special number system. This breakdown helps create the three surfaces in the triplet, each formed by taking a certain mathematical "slice" of a geometric space called the hyperbolic plane.

Bound for systolic length and the systolic ratio

The Gauss–Bonnet theorem helps us understand the shape of certain surfaces. It connects the Euler characteristic of a surface — a number that describes its topology — to the total curvature across the surface. For surfaces with a specific type of curvature and a genus (number of "holes") of 14, the Euler characteristic is -26, and the total area of these surfaces can be calculated as 52π.

There is also a concept called the systole, which is the shortest distance across these special surfaces. For surfaces with genus 14, the lower bound for this distance is about 3.5187. This helps mathematicians study the geometry of these fascinating 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