Timeline for Fenchel-Nielsen length-length coordinates on Teichmueller space?
Current License: CC BY-SA 4.0
20 events
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| Jan 6, 2022 at 17:05 | history | edited | JHM | CC BY-SA 4.0 |
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| Jan 5, 2022 at 22:05 | vote | accept | JHM | ||
| Jan 5, 2022 at 22:02 | history | edited | JHM | CC BY-SA 4.0 |
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| Jan 5, 2022 at 17:24 | history | edited | JHM | CC BY-SA 4.0 |
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| Jan 5, 2022 at 17:23 | comment | added | JHM | @AndyPutman yes i see your point. Another issue is that lengths are positive $\ell_a(g)>0$, so if you Nielsen twist around the red curves in the above figure, then the lengths of the green curves cannot distinguish betwen left and right Nielsen twists. | |
| Jan 5, 2022 at 16:41 | answer | added | Alex Nolte | timeline score: 1 | |
| Jan 5, 2022 at 16:03 | comment | added | Andy Putman | @JHM: If the sum of the lengths achieves a global minimum, then the lengths cannot be coordinates by Ian’s argument. Namely, if they did then by invariance of domain the length coordinates would be an open map to $\mathbb{R}^{6g-6}$, and thus their sum would also be an open map and hence have no local extrema. | |
| Jan 5, 2022 at 13:11 | comment | added | JHM | @AndyPutman Yes I agree that the sum of lengths of a collection of curves which fill the surface is proper. So sums of length functions do not provide coordinates on Teich following Agol's argument. But the possibility of length functions being coordinates still remains open from my POV. | |
| Jan 5, 2022 at 13:05 | comment | added | JHM | @TimothyBudd So if we fix the orientation of the surface, then we distinguish between $S$ and its "mirror" $S^{op}$. But do you think it's possible that there exists other hyperbolic structures on $S$ where the six curves have a prescribed length? | |
| Jan 5, 2022 at 9:48 | comment | added | Timothy Budd | @JHM: In your genus-2 example the lengths of the curves do not distinguish between a hyperbolic metric and its mirror image in the plane in which it is drawn, so I guess one should specify what exactly it means to "determine a hyperbolic metric". This is related to the example in my answer, that shows that the twist parameters can be determined up to a sign, which changes under reflection. | |
| Jan 5, 2022 at 2:31 | comment | added | Moishe Kohan | Actually, it is easy to see that if a (finite) collection of curves parameterizes then it must be filling. | |
| Jan 4, 2022 at 22:16 | history | edited | JHM | CC BY-SA 4.0 |
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| Jan 4, 2022 at 22:10 | history | edited | JHM | CC BY-SA 4.0 |
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| Jan 4, 2022 at 21:54 | comment | added | Andy Putman | @JHM: I think what's going on there is that the sum of the lengths is the proper function, and must achieve a minimum. The point is that if one curve gets too short, then any of the other curves that intersects it must get long. | |
| Jan 4, 2022 at 21:06 | comment | added | JHM | @AndyPutman. I dont think Prof Agol's answer in the above question is correct, despite it's being accepted by the OP. For example the claim that "If one has 6g−6 geodesics which parameterize, then they [the curves] must be filling" is not accurate. Rather it is the collection of curves which needs be filling, and not the individual curves themselves. Moreover lengths of curves $\ell_a$ are nowhere minimized in Teichmueller space, they are only minimized "at infinity". That is length functions are not proper on Teich (their sublevels are like huge open horoballs). | |
| Jan 4, 2022 at 19:27 | answer | added | Timothy Budd | timeline score: 1 | |
| Jan 4, 2022 at 17:17 | history | edited | JHM | CC BY-SA 4.0 |
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| Jan 4, 2022 at 16:18 | comment | added | Andy Putman | Your first question is answered here: mathoverflow.net/questions/243622/… | |
| Jan 4, 2022 at 15:47 | comment | added | abx | Nielsen, not Nielson. | |
| Jan 4, 2022 at 15:16 | history | asked | JHM | CC BY-SA 4.0 |