Timeline for Mostow Rigidity Theorem and reconstruction from fundamental group
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 1, 2020 at 6:00 | comment | added | HJRW | The algorithm given by Toffee in their nice answer was described by Jason Manning in this 2002 G&T paper: arxiv.org/abs/math/0102154 . | |
| Apr 25, 2020 at 15:17 | vote | accept | Cameron Zwarich | ||
| Apr 25, 2020 at 0:22 | history | became hot network question | |||
| Apr 24, 2020 at 17:33 | comment | added | YCor | ... reading Toffee's answer, I understand that something is doable without reproving Thurston-Hamilton-Perelman... I understand (again, roughly) that if we know beforehand that hyperbolization exists, then it can be done in some kind of effective way, by "retrieving" the relevant representation $\Gamma\to\mathrm{PSL}_2(\mathbf{C})$. | |
| Apr 24, 2020 at 17:07 | comment | added | YCor | At a theoretical level the Mostow rigidity theorem precisely say that $\Gamma$ determines the manifold $X_\Gamma$ up to isometry. If I understand correctly, the question is to have a more "constructive" description of $\Gamma\mapsto X_\Gamma$? Maybe asking the manifold only should be more reasonable, because if you can output the metric easily, basically you reprove Thurston's hyperbolization conjecture (in this case in dim 3, from my very rough understanding the metric is output by the Hamilton-Perelman Ricci flow procedure). | |
| Apr 24, 2020 at 16:54 | answer | added | Toffee | timeline score: 11 | |
| Apr 24, 2020 at 16:49 | comment | added | Cameron Zwarich | I was thinking more generally about a mathematical construction rather than an effective algorithm, so any answer is fine. I don't have any particular application in mind; I was more thinking about why rigidity theorems are often stated in isomorphism form. | |
| Apr 24, 2020 at 16:37 | comment | added | YCor | What do you mean by reconstruct? Do you have an algorithmic question in mind? If so, how do you encode the group, and how do you encode the manifold? (a possible output is a triangulation but it doesn't say what the metric is) | |
| Apr 24, 2020 at 16:37 | comment | added | Moishe Kohan | Yes, but it's quite hard and is not algorithmic to the best of my knowledge. By the way, you should specify what you mean by "reconstruct." | |
| Apr 24, 2020 at 16:35 | history | edited | YCor |
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| Apr 24, 2020 at 16:20 | history | asked | Cameron Zwarich | CC BY-SA 4.0 |