Non-compact Riemann surfaces are equilaterally triangulable
Complex Variables
2025-09-19 v3 Algebraic Geometry
Differential Geometry
Dynamical Systems
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.
Cite
@article{arxiv.2103.16702,
title = {Non-compact Riemann surfaces are equilaterally triangulable},
author = {Christopher J. Bishop and Lasse Rempe},
journal= {arXiv preprint arXiv:2103.16702},
year = {2025}
}
Comments
42 pages, 14 figures. Accepted manuscript; to appear in Inventiones Math. V3: The proof of Proposition 3.2 has been significantly expanded, with several new lemmas and figures added to provide further detail. Some additional clarifications were also made throughout the article