On a triply periodic polyhedral surface whose vertices are Weierstrass points

TL;DR

This paper constructs a specific triply periodic polyhedral surface whose vertices are Weierstrass points, resulting in a genus 3 Riemann surface with unique symmetry properties and explicit geometric structures.

Contribution

It provides a novel example of a triply periodic polyhedral surface with vertices as Weierstrass points, and analyzes its symmetry, hyperbolic, and translation structures.

Findings

Constructed a genus 3 Riemann surface from a triply periodic polyhedral surface.

Proved the surface is not hyperelliptic and is regular with transitive automorphism group.

Developed hyperbolic and translation structures compatible with the surface's conformal type.

Abstract

In this paper, we will construct an example of a closed Riemann surface $X$ that can be realized as a quotient of a triply periodic polyhedral surface $\Pi \subset \mathbb{R}^3$ where the Weierstrass points of $X$ coincide with the vertices of $\Pi.$ First we construct $\Pi$ by attaching Platonic solids in a periodic manner and consider the surface of this solid. Due to periodicity we can find a compact quotient of this surface, which has genus $g = 3.$ We claim that the resulting surface is not hyperelliptic and also that it is regular. By regular, we mean that the automorphism group of $X$ is transitive on flags. The symmetries of $X$ allow us to construct hyperbolic structures and various translation structures on $X$ that are compatible with its conformal type. The translation structures are the geometric representations of the holomorphic 1-forms of $X,$ which allow us to identify the Weierstrass points.