Non-compact Riemann surfaces are equilaterally triangulable

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.

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