Timeline for Geodesics in finite groups
Current License: CC BY-SA 3.0
27 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Mar 17, 2017 at 10:13 | history | edited | CommunityBot |
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| Jun 20, 2015 at 16:35 | vote | accept | Joonas Ilmavirta | ||
| Feb 10, 2015 at 10:11 | comment | added | Joonas Ilmavirta | @WłodzimierzHolsztyński, in Lie groups I'm interested in closed (=periodic) geodesics only, so I exclude all noncompact geodesics. On finite groups all geodesics are periodic but not so on Lie groups (of rank two or higher). | |
| Feb 10, 2015 at 5:41 | comment | added | Włodzimierz Holsztyński | Not all global geodesics of compact Lie groups are shifts of homomorphic images of $\ S^1.\ $ There are plenty of non-compact global geodesics. | |
| Feb 10, 2015 at 2:32 | answer | added | Yannick Voglaire | timeline score: 8 | |
| Oct 22, 2014 at 13:36 | history | edited | Joonas Ilmavirta | CC BY-SA 3.0 |
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| Oct 22, 2014 at 11:42 | comment | added | Richard Montgomery | Going one step further with the OPs analogy, it seems we ought to think of the p-Sylow subgroups as maximal tori, which fits well with the basic results of the two at theory: -any two are conjugate. I like this perspective! | |
| Oct 20, 2014 at 19:11 | comment | added | Joonas Ilmavirta | @მამუკაჯიბლაძე, that is true. Observing those automorphisms does actually help with the inverse problem. There are also finite cyclic groups of many sizes, whereas there is only one circle; finite groups cannot be scaled. The analogue is certainly not perfect (and cannot be), but I still find the analogue surprisingly strong. Or maybe the biggest surprise is in the observation that this thing seems not to have been studied. | |
| Oct 20, 2014 at 19:01 | comment | added | მამუკა ჯიბლაძე | One substantial difference is that the circle as a Lie group does not have any automorphisms except inversion, while finite cyclic groups have many. This in particular implies that your geodesics do not have well-defined direction. | |
| Oct 20, 2014 at 14:15 | history | edited | Joonas Ilmavirta | CC BY-SA 3.0 |
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| Oct 20, 2014 at 14:11 | comment | added | Joonas Ilmavirta | @AlexB., you are right. I had assumed it but had apparently not written it down. It's corrected now. I know that geodesics can be called cosets of cylcic subgroups (left and right translations give the same geodesics), but I have failed to find any useful material with this phrase. It especially seems that these things have not been studied from a "geodesic point of view". | |
| Oct 20, 2014 at 14:07 | history | edited | Joonas Ilmavirta | CC BY-SA 3.0 |
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| Oct 20, 2014 at 13:03 | comment | added | Alex B. | @Joonas You had better specify that $n>1$ then. By the way: there is of course a common name for these things, but it's cumbersome: left cosets of (non-trivial) cyclic subgroups. | |
| Oct 20, 2014 at 11:08 | comment | added | Joonas Ilmavirta | @AlexB., I only use nontrivial homomorphisms in my definition of geodesics. That is, a geodesic is a translate of a nontrivial cyclic subgroup. Without that restriction singletons would be geodesics and the second problem would be trivial. Of course maximality works as well but it's not necessary in this respect. | |
| Oct 20, 2014 at 10:22 | comment | added | Alex B. | Any singleton is a geodesic in your sense (if you don't impose maximality), which makes problem 2. trivial. | |
| Oct 20, 2014 at 8:46 | history | edited | Joonas Ilmavirta | CC BY-SA 3.0 |
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| Oct 20, 2014 at 1:56 | comment | added | Vincent | This seems obvious but to be on the safe side I ask you instead of changing it myself: shouldn't the first gamma be a phi? | |
| Oct 16, 2014 at 19:58 | history | edited | Joonas Ilmavirta | CC BY-SA 3.0 |
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| Oct 16, 2014 at 19:42 | comment | added | Joonas Ilmavirta | @PeterMichor, that is a very good point. Many thanks! You have convinced me to take a closer look at the setting with maximal cyclic subgroups and Lie subgroups. I am working on an abstract generalization or variation of something that has applications in the hope that it would become useful. Therefore finding reasonable definitions is a part of the problem. | |
| Oct 16, 2014 at 19:19 | comment | added | Peter Michor | Why do you think that each subgroup has to be a "Lie" subgroup; I would only take those which contain all "roots" of all elements also. Then they are again totally geodesic. There should be a place for an analog of a discrete subgroup of a Lie subgroup also. But of course your definition should be suitable for the applications you have in mind. | |
| Oct 16, 2014 at 16:55 | comment | added | Joonas Ilmavirta | @PeterMichor, if I only consider maximal cyclic subgroups, I lose the property that all subgroups are totally geodesic (which is true for Lie subgroups). I am hesitant to throw away this algebraic structure. But I do agree that maximal cyclic subgourps may correspond better to the mental image of usual geodesics. Actually, in my inverse problem it turns out that it suffices to consider minimal cyclic subgroups. But it would be interesting to consider the maximal variant as well. | |
| Oct 16, 2014 at 16:46 | history | edited | Joonas Ilmavirta | CC BY-SA 3.0 |
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| Oct 16, 2014 at 16:34 | comment | added | Peter Michor | I think your definition in the case of finite groups is not good enough: For a cyclic subgroup generated by $x$, also the subgroup generated by $x^2$ would be a geodesic, with half the number of elements. I would call geodesics only the translations of maximal cyclic subgroups. | |
| Oct 16, 2014 at 16:27 | comment | added | Joonas Ilmavirta | @Alan, I do not know of a general definition that covers both cases. The definition on Lie groups is based on the group $S^1$, whereas my definition on finite groups is based on finite cyclic groups, and this seems to make a big difference. If you can suggest a general definition, I would be happy to hear. | |
| Oct 16, 2014 at 16:22 | comment | added | Alan | I might be out of my league here, but isn't there a general defintion where both cases (finite and infinite groups) apply to? the finite group case seem to be a specific case to the infinite case, as in the infinite is the general case. I might be wrong, I haven't yet touched on this matter in Lie groups. | |
| Oct 16, 2014 at 16:09 | history | asked | Joonas Ilmavirta | CC BY-SA 3.0 |