On a triply periodic polyhedral surface whose vertices are Weierstrass points

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.