Timeline for Proper discontinuity and existence of a fundamental domain
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Sep 22 at 1:30 | comment | added | Misha | @Learningmath: That's right, in the Riemannian setting. If you consider, say, groups acting on trees, then it is possible that every point has nontrivial stabilizer. | |
| Sep 12 at 16:21 | comment | added | Learning math | @Misha Thank you for your informative answer! I've a question. It seems to me that the unique Dirichlet Fundamental Domain described above may exist for almost all $x$ but not all $x?$ E.g. if we take a finite rotation group acting on $R^2,$ say the group of order $3$ where the generator rotates everything by $2\pi/3.$ In this case, it seems $0$ is fixed by every element, but it got no Dirichlet Fundamental Domain, although every other point in $R^2$ has one. Is it true, and if yes, is there such a characterization of points where the Dirichlet Fundamental Domain does not exist? | |
| Aug 20 at 16:22 | comment | added | Misha | @V.Rogov: Take a look at the revision of my note on proper actions posted on the arXiv, I tried to clean up the definitions. And, yes, in the simplicial setting one should take the interior of the closure to get a domain, it does not work otherwise. | |
| Aug 1 at 15:46 | comment | added | V. Rogov | I guess one should really replace $D$ with $(\overline{D})^{\circ}$ in this construction... | |
| Aug 1 at 13:57 | comment | added | V. Rogov | Thank you very much for a nice comment. I have a question on the simplicial construction in Seifert-Threlfall. How does one check that the resulting set is a domain? It seems, that if one takes the tetrahedral triangulation of a 2-sphere, one ends up with the complement of the vertices, which is not a domain. Maybe one should assume something on the triangulation and that $G$ is non-trivial to exclude pathologies like this? | |
| Feb 21 at 13:58 | comment | added | Learning math | @Misha Prof; Kapovich, I just came across the question and your answer. I'm wondering if there's any such similar theorem when a Lie group $G$ acts on a Riemannian manifold $M$ properly but not necessarily freely. Can we show the existence of a fundamental domain in this case? Someone else asked this on MO: mathoverflow.net/questions/251627/…. Thank you in advance! | |
| Jan 16, 2023 at 1:34 | history | edited | Misha | CC BY-SA 4.0 |
added 668 characters in body
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| Dec 10, 2016 at 4:06 | history | edited | Anton Petrunin | CC BY-SA 3.0 |
smooth smooth > smooth
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| Oct 10, 2016 at 5:41 | comment | added | Misha | @IliaSmilga: It seems that the Dirichlet domain method can be applied in greater generality of locally path-connected locally compact spaces as they appear to admit proper geodesic metrics (by work of Bing and Moise), provided that fixed point sets of finite order homeomorphisms have empty interior. | |
| Oct 10, 2016 at 5:06 | comment | added | Misha | @IliaSmilga: Not exactly, for smooth manifolds I assume diffeomorphisms and for PL manifolds I assume PL actions (for topological manifolds, I assume topological action, of course). You have to pick a category in which you are working (TOP, PL, or DIFF) and work with it. | |
| Oct 10, 2016 at 0:25 | comment | added | Ilia Smilga | I suppose you are also assuming that the action is by homeomorphisms, right? | |
| Oct 9, 2016 at 15:57 | vote | accept | Ilia Smilga | ||
| Oct 8, 2016 at 17:33 | history | edited | Misha | CC BY-SA 3.0 |
Added condition 4 and discussion of PL case.
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| Oct 8, 2016 at 15:15 | comment | added | Ilia Smilga | Nice proof :-) I was aware of the notion of Dirichlet fundamental domains but if the metric was not provided, it did not occur to me that I could construct one. | |
| Oct 8, 2016 at 14:05 | history | answered | Misha | CC BY-SA 3.0 |