Timeline for Bisectors in symmetric spaces
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Nov 22, 2012 at 7:04 | comment | added | Misha | Selberg's bisectors are only defined for some types of symmetric spaces, most importantly, for the type A. Thus, if you have a symmetric space for type, say, C, you embed it in the one of type A and use bisectors coming from the ambient space. Computations with Riemannian bisectors are notoriously difficult even with modern computers. | |
| Nov 21, 2012 at 16:34 | comment | added | user28191 | Thanks. I'll think about this. I have a family of examples where I'm pretty sure the Riemannian bisectors will provide walls for a fundamental domain, but I don't know how to check this because I don't know how to explicitly describe them. Selberg's bisectors might work as well and look more computable. I'll think about it. But they look less likely to behave well with respect to maps between symmetric spaces, no? | |
| Nov 21, 2012 at 14:39 | comment | added | Misha | Selberg's construction was not used much, but it has advantages over the Riemannian one since equations for bisectors in his case are linear, you can also easily identify intersections of bisectors. You can also apply it for other symmetric spaces by embedding them isometric ally to the symmetric space for SL(n). If you are interested in algorithmic aspects of fundamental domains, Selberg's construction would work better than Riemannian one. I did not check this, but Selberg's construction might be related to classical reduction theory for lattices. | |
| Nov 20, 2012 at 14:04 | comment | added | user28191 | OK. I looked at Selberg's paper. I've never seen something like that before. I think that I'm still interested in the Riemannian metric, but this Selberg thing could be useful too. Does it appear commonly? Are there other more extended references? He talks about one case, SL(n), and doesn't say very much. | |
| Nov 19, 2012 at 10:31 | comment | added | user28191 |
Thanks. This is Selberg's paper On discontinuous groups...'? I'll see if I can get a hold of that. I'm not very worried if the bisectors are linear or not. I just want to build a fundamental domain for a group generated by some translations in the Siegel upper half-plane, where by translation I mean $Z \mapsto Z + B$`, $B$ a symmetric matrix.
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| Nov 19, 2012 at 7:09 | comment | added | Misha | In higher rank the most useful bisectors appear in the case of $SL(n)$ where instead of the Riemannian metric one uses a non -symmetric metric defined by Selberg in his 1960 paper. The key is that Selberg's bisectors are linear, unlike the Riemannian ones. | |
| Nov 18, 2012 at 19:55 | comment | added | user28191 | One might still be interested in constructing a nice fundamental domain for a discrete group, even if one already knows a priori whether the group is arithmetic. | |
| Nov 18, 2012 at 11:33 | comment | added | Benoît Kloeckner | One of the interest of bisectors is to prove that some non-arithmetic groups of isometries are nonetheless discrete by constructing fundamental domains for their action. Higher rank spaces are much more rigid, and all lattices are arithmetic. This does not mean that bissectors are useless, but their purpose seems less obvious, and it will probably be more difficult to find relevant literature. | |
| Nov 17, 2012 at 22:43 | answer | added | Joseph O'Rourke | timeline score: 3 | |
| Nov 17, 2012 at 22:06 | history | asked | user28191 | CC BY-SA 3.0 |