Non-compact Riemann surfaces are equilaterally triangulable

TL;DR

This paper proves that all open Riemann surfaces can be constructed by gluing equilateral triangles in a locally finite way, showing they are Belyi surfaces and branched covers of the sphere over finitely many points.

Contribution

It establishes that every open Riemann surface admits a triangulation with equilateral triangles, linking geometric triangulations to algebraic properties of Belyi surfaces.

Findings

Every open Riemann surface can be triangulated with equilateral triangles.

All such surfaces are Belyi surfaces with a branched cover to the sphere.

Each surface is a finite-branching cover of the Riemann sphere.

Abstract

We show that every open Riemann surface can be obtained by glueing together a countable collection of equilateral triangles, in such a way that every vertex belongs to finitely many triangles. Equivalently, it is a _Belyi surface_: There exists a holomorphic branched covering to the Riemann sphere that is branched only over three values. It follows that every Riemann surface is a branched cover of the sphere, branched only over finitely many points.