Timeline for Change of length of curve when Fenchel-Nielsen length coordinate increase
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Jul 7, 2016 at 13:16 | comment | added | Cusp | @user331406 Actually I was reading this paper when I asked this question. I also guess that their will be no uniform behavior, it will depend on paths. | |
| Jul 7, 2016 at 9:03 | comment | added | user331406 | I have the feeling that the behavior of $l_\beta$ depends on the path you're looking at in Teichmüller space, even if you're only considering those along which $l_\alpha$ increases. There is a paper by Greg McShane and Hugo Parlier in which they show (section 8), that the sets where $l_\alpha > l_\beta$ and $l_\alpha < l_\beta$ are open and connected, and that their boundary (i.e the set where $l_\alpha = l_\beta$) is connected as well, so you have plenty of paths you can look at with different behavior of $l_\beta$. | |
| May 9, 2016 at 12:06 | history | edited | Cusp | CC BY-SA 3.0 |
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| May 9, 2016 at 10:37 | comment | added | Cusp | @JeanRaimbault 0 "I guess closer to what you want, would be to look at what happens in a continuous path in Teichmüller space along which lαlα increases", This is exactly what I want. | |
| May 9, 2016 at 9:47 | comment | added | Jean Raimbault | Another possibility, I guess closer to what you want, would be to look at what happens in a continuous path in Teichmüller space along which $l_\alpha$ increases. | |
| May 9, 2016 at 9:36 | comment | added | Jean Raimbault | The problem remains, because when you increase $l_\alpha$ you can always choose a faraway point in the mapping class group orbit of your hyperbolic structure where $l_\beta$ will be as large as you want. To exclude this you would have to ask something about the minimum of $\ell_\beta$ in (part of) the MCG orbit, I think. | |
| May 9, 2016 at 8:48 | comment | added | Cusp | @JeanRaimbault Now I understand your point. I have changed the question a little bit. I want the relation when $l_{\alpha}$ increases. Thanks for pointing that out. | |
| May 9, 2016 at 8:46 | history | edited | Cusp | CC BY-SA 3.0 |
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| May 9, 2016 at 8:12 | comment | added | Jean Raimbault | yes, and if what I said above holds then for any value of $l_\alpha$ there are points in Teichmüller space where $l_\beta$ is as large as you want it to be, in particular there cannot be an upper bound on $l_\beta$ depending only on $l_\alpha$. | |
| May 9, 2016 at 6:43 | comment | added | Cusp | My question was, what happens if we increase the length of "$\alpha$", not the other way around. | |
| May 9, 2016 at 6:14 | comment | added | Jean Raimbault | This seems unlikely to hold as stated. For example if you take a third simple curve which intersects $\beta$ but not $\alpha$ and do Dehn twists around it then you get a sequence of pairwise isometric hyperbolic metrics in Teichmüller space where $l_\alpha$ is (any) constant and $l_\beta$ goes to infinity. | |
| May 9, 2016 at 2:47 | history | asked | Cusp | CC BY-SA 3.0 |