NON-COMPACT RIEMANN SURFACES ARE EQUILATERALLY TRIANGULABLE

The paper proves the equivalence between equilateral triangulations and Belyi functions (Proposition 2.7) and establishes existence on every non‑compact surface via a careful quasiconformal construction (Theorems 1.2 → 1.4) . In contrast, the candidate’s existence proof crucially misuses monodromy for the Gunning–Narasimhan immersion g: X→C by assuming that, over any simply connected Jordan domain W, each component of g^{-1}(W) maps biholomorphically onto W. This fails in general due to transcendental (asymptotic) singularities of the inverse; indeed, the paper explicitly notes that such immersions are not branched covers and their inverses have (potentially infinitely many) transcendental singularities . Composing with a polynomial Belyi map R does not remove these asymptotic singularities; it merely transports them to R(Sing(g)), so f = R∘g fails the branched‑covering properness condition in the paper’s definition of Belyi function (cf. the local properness requirement in the “branched covering” remark) .