Edray Goins gave a great colloquium today about his work on dessins d’enfants. And in this talk there was a picture that surprised me. It was one of the ones on the lower right of this poster. Here, I’ll put in a screen shot:

Let me tell you what you’re looking at. You are looking for elliptic curves E admitting a Belyi map f: E -> P^1, which is to say a map ramified only over 0,1, and infinity. For each such map, the blue graph is f^{-1}([0,1]), the preimage of the line segment joining o and 1 in P^1(R).

In four of these cases, the graph is piecewise linear! I didn’t know there were examples like this. Don’t know if this is easy, but: for which Belyi maps (of any genus, not just genus 1) is f^{-1}([0,1]) a union of geodesics?

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