Timeline for Invariant lifts of a closed curve on a surface of genus > 1
Current License: CC BY-SA 4.0
7 events
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| Oct 2, 2018 at 15:25 | comment | added | NWMT | @THeSilverDoe You don't need that much hyperbolic geometry though. By the way you may want to consider a simpler example, instead of a surface $S$ thake a graph $\Gamma$ with a single vertex, whose universal cover will be a regular tree. Things will more or less work out the same way. | |
| Sep 28, 2018 at 13:44 | comment | added | TheSilverDoe | Ok, thank you ! I think I begin to understand... but I have to learn a little bit of hyperbolic geometry. Thanks again ! | |
| Sep 28, 2018 at 0:46 | comment | added | NWMT | Hi, you have to think about how deck transformations move the basepoint $\hat b$. We already chose it to lie in the specific lift $\hat c_{\hat b}$. Any deck transformation that sends $\hat b$ into $\hat c_{\hat b}$ will also fix $\hat c_{\hat b}$ (having the effect of shifting the curve along itself). A family of examples of such deck transformations will come from taking powers of the $\pi_1$-image of your original loop $c$. Other deck transformation that move $\hat b$ off $\hat c_{\hat b}$ will necessarily give a different lift. | |
| Sep 26, 2018 at 12:55 | comment | added | TheSilverDoe | I understand that (intuitively), but how can you prove rigorously that it is impossible in the other cases ? Or more generally, how do you prove the statement "the images of all lifts of c correspond to distinct geodesics" ? Thank you ! | |
| Sep 26, 2018 at 12:48 | comment | added | NWMT | It is possible if the path along which the basepoint $b$ travels is $c$ (or some power thereof). | |
| Sep 26, 2018 at 10:10 | comment | added | TheSilverDoe | Thank you very much for this precise answer ! However, there is still a point that puzzles me : why "the images of all lifts of $c$ correspond to distinct geodesics" ? Intuitively, I would say the following : if two lifts $\widehat{c}$ and $\widehat{g}(\widehat{c})$ correspond to the same geodesic, then they are homotopic, in the sense that there is a homotopy H between $c$ and $c$ such that the path of a basepoint of $c$ under this homotopy is a non-trivial loop. And this seems to be impossible. Is that true ? How can you write it properly ? Thanks again ! | |
| Sep 26, 2018 at 0:32 | history | answered | NWMT | CC BY-SA 4.0 |