NON-COMPACT RIEMANN SURFACES ARE EQUILATERALLY TRIANGULABLE

Christopher B. Bishop; Lasse Rempe

NON-COMPACT RIEMANN SURFACES ARE EQUILATERALLY TRIANGULABLE

Christopher B. Bishop, Lasse Rempe

Pure Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We show that every open Riemann surface X 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, X is a Belyi surface: There exists a holomorphic branched covering f : X → Cˆ that is branched only over −1, 1 and ∞, and with no removable singularities at the boundary of X. It follows that every Riemann surface is a branched cover of the sphere, branched only over finitely many points.

Original language	English
Journal	Inventiones mathematicae
Publication status	Accepted/In press - 18 Sept 2025

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belyi_26_accepted_manuscript
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Licence: CC BY
Embargo ends: 19/09/26

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title = "NON-COMPACT RIEMANN SURFACES ARE EQUILATERALLY TRIANGULABLE",

abstract = "We show that every open Riemann surface X 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, X is a Belyi surface: There exists a holomorphic branched covering f : X → Cˆ that is branched only over −1, 1 and ∞, and with no removable singularities at the boundary of X. It follows that every Riemann surface is a branched cover of the sphere, branched only over finitely many points.",

author = "Bishop, \{Christopher B.\} and Lasse Rempe",

year = "2025",

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language = "English",

journal = "Inventiones mathematicae",

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AU - Bishop, Christopher B.

AU - Rempe, Lasse

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N2 - We show that every open Riemann surface X 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, X is a Belyi surface: There exists a holomorphic branched covering f : X → Cˆ that is branched only over −1, 1 and ∞, and with no removable singularities at the boundary of X. It follows that every Riemann surface is a branched cover of the sphere, branched only over finitely many points.

AB - We show that every open Riemann surface X 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, X is a Belyi surface: There exists a holomorphic branched covering f : X → Cˆ that is branched only over −1, 1 and ∞, and with no removable singularities at the boundary of X. It follows that every Riemann surface is a branched cover of the sphere, branched only over finitely many points.

M3 - Article

SN - 0020-9910

JO - Inventiones mathematicae

JF - Inventiones mathematicae

ER -

NON-COMPACT RIEMANN SURFACES ARE EQUILATERALLY TRIANGULABLE

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