Timeline for Belyi functions on non-compact surfaces; or: Building Riemann surfaces from equilateral triangles
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Apr 2, 2021 at 18:13 | history | edited | Lasse Rempe | CC BY-SA 4.0 |
Added link to the preprint, as well as a reference to work of Bowers and Stephenson.
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| Dec 11, 2013 at 1:13 | vote | accept | Lasse Rempe | ||
| Oct 31, 2013 at 11:51 | answer | added | JHM | timeline score: 6 | |
| Jan 12, 2013 at 16:05 | answer | added | Loïc Teyssier | timeline score: 4 | |
| Jan 12, 2013 at 14:33 | history | edited | Lasse Rempe | CC BY-SA 3.0 |
typo fixed
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| Jan 11, 2013 at 14:18 | history | edited | Lasse Rempe | CC BY-SA 3.0 |
clarified "building from triangles" in the noncompact case, as pointed out by Misha
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| Jan 11, 2013 at 14:11 | comment | added | Lasse Rempe | @Misha, I think I see what you mean - for noncompact surfaces, being built from triangles is indeed (formally) weaker than having a Belyi function on it. To have a Belyi function, every corner should be adjacent to only finitely many triangles. I will add the question to clarify this. (Our theorem establishes the stronger property for all non-compact surfaces, and hence the weaker property also holds.) | |
| Jan 11, 2013 at 9:30 | comment | added | Lasse Rempe | Perhaps I should have clarified that, when we build the Riemann surface from infinitely many triangles, we only include those corner points that are contained in only finitely many triangles. Near these, we define a Riemann surface structure in the obvious way. (Near the others, it isn't at all clear how we would define a Riemann surface structure.) | |
| Jan 11, 2013 at 9:23 | comment | added | Lasse Rempe | I am not sure that I understand your comment correctly. Any holomorphic map is a "homeomorphism away from branch points". By "branched cover", I mean precisely the definition given in the second paragraph, which is stronger than both of your definitions. (The exponential map is a covering to its image on the plane, but it is not a Belyi function in my sense.) | |
| Jan 10, 2013 at 17:49 | comment | added | Misha | Not that it matters a lot, but when you say "branched cover", do you really mean that it is a topological covering (to its image) away from a certain discrete subset, or simply that it is a local homeomorphism away from branch points? This is unclear from your description. Note that, in general, gluing a surface from equilateral triangles will give you a developing map which is only a "branched cover" in the latter sense. In Belyi's case, both definitions agree because of properness. | |
| Jan 10, 2013 at 12:09 | history | asked | Lasse Rempe | CC BY-SA 3.0 |