Timeline for Pleated surfaces do not curl up too much
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Apr 16, 2012 at 7:51 | comment | added | Damiano Lupi | @Sam: I will read Thurston's notes! Thanks! @Misha: actually what I was about to do was trying to figure out how to "lift" uniform properness to universal covers. Thank you for the hint! | |
| Apr 16, 2012 at 1:11 | comment | added | Misha | @Damiano: One more remark: One can ask for uniform properness of the pleated map of the universal cover of $S$ to the hyperbolic space. You again get uniform bounds on uniform properness here, see Sam's and mine remarks on relative homotopy of paths in the surface. In my experience, uniform properness is most useful in the context of unbounded metric spaces, like the universal covers in this case. | |
| Apr 15, 2012 at 18:48 | comment | added | Sam Nead | @Misha - Of course, you are right: I didn't think to look in any references (all of my paper copies are at work). So... @Damiano, you should read Thurston's notes! Oh, and one remark about your comment: a pleating map $g: (S,\rho) \to N$ is generally not an inclusion. I'll leave it as an exercise (also answered in the notes) to give an example. | |
| Apr 15, 2012 at 16:34 | comment | added | Damiano Lupi | Thank you everyone for your help. Actually what I meant is what I clearly asked in the first part of my question, i.e. that the inclusion $(S,\rho)\hookrightarrow (N,d_N)$ is uniformly proper. In the last part of the question I wrote something very confused: what I was trying to point out is that the result of Minsky holds for loops while I needed a statement about distance between points, that is length of shortest paths between points. Anyway, your explanation about the diameter of $S$ solved my problems. Thank you very much and sorry about my inaccuracy. | |
| Apr 15, 2012 at 16:27 | vote | accept | Damiano Lupi | ||
| Apr 15, 2012 at 12:12 | comment | added | Misha | @Sam: The upper bound on diameter of a pleated surface is due to Thurston and could be found in his Notes (it is the same proof that you gave). Also, once you know an upper bound on diameter, then your version of the question (with the relative homotopy classes), follows from Minsky's result, there is no need to go through his proof. Lastly, I think, it's time for Damiano Lupi to re-read his post and explain what question he actually has meant to ask. (Otherwise, we might edit it to a point where the original problem, whatever it was, is unrecognizable.) | |
| Apr 15, 2012 at 9:00 | history | edited | Sam Nead | CC BY-SA 3.0 |
Added A to answer in correct place.
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| Apr 15, 2012 at 8:50 | history | edited | Sam Nead | CC BY-SA 3.0 |
Fixed last paragraph
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| Apr 15, 2012 at 8:44 | history | answered | Sam Nead | CC BY-SA 3.0 |