Timeline for Number of conjugacy classes in GL(n,Z)
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Jul 9, 2013 at 19:30 | comment | added | JHM | @DerekHolt: do you have a reference for your claim on the largest finite subgroup? | |
| Sep 12, 2012 at 21:32 | comment | added | Ian Agol | The maximal groups in dimensions $\leq 9$ are analyzed in these papers: ams.org/mathscinet-getitem?mr=551303 | |
| Sep 11, 2012 at 21:30 | answer | added | Ralph | timeline score: 7 | |
| Sep 5, 2012 at 12:55 | comment | added | Ian Agol | The split Bieberbach groups are counted in 3D as 73 according to Wikipedia: en.wikipedia.org/wiki/Space_group These seem to be called arithmetic crystal classes, or symmorphic, at least in 3D. | |
| Sep 5, 2012 at 12:42 | comment | added | Ian Agol | I think the number of Bieberbach groups gives an upper bound. For each finite subgroup $G< GL_n(\mathbb{Z})$, one obtains a Bieberbach group $G \ltimes \mathbb{Z}^n$. Moreover, if two Bieberbach groups are obtained this way, then by Bieberbach's theorem, they are affinely equivalent. Since the $\mathbb{Z}^n$ subgroup is a maximal subgroup of translations, this means the affine map must send $\mathbb{Z}^n$ to $\mathbb{Z}^n$, so it lies in $GL_n(\mathbb{Z})$. However, this will be an overcount, since there are Bieberbach groups which are not of this form (e.g. Klein bottle group), ie not split. | |
| Sep 5, 2012 at 3:06 | comment | added | Tom Goodwillie | The number of conjugacy classes of finite subgroups of $GL_n(\mathbb Q)$ is a lower bound, and this number might be a little easier to think about. | |
| Sep 4, 2012 at 20:45 | history | edited | Gregor Samsa | CC BY-SA 3.0 |
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| Sep 4, 2012 at 20:42 | comment | added | Gregor Samsa | @David Speyer: You're right, it doesn't make sense to speak about big conjugacy classes. Since I was thinking about space group and their arithmetic equivalence for the last few days, things got mixed up in my head. Thanks for pointing that out. | |
| Sep 4, 2012 at 19:14 | comment | added | Derek Holt | The largest finite subgroup is the orthogonal group of order $2^nn!$, at least for all $n > 10$. These questions have been discussed on MO before, but I do not believe that any asymptotic results on the number of classes are known. | |
| Sep 4, 2012 at 15:29 | comment | added | Will Sawin | If I counted right $GL_1$ has $2$ and $GL_2$ has $9$. The number should increase fairly rapidly with $n$. | |
| Sep 4, 2012 at 14:19 | comment | added | David E Speyer | I am confused by question (2). These conjugacy classes are infinite, so how are you measuring their size? | |
| Sep 4, 2012 at 12:59 | history | edited | Gregor Samsa | CC BY-SA 3.0 |
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| Sep 4, 2012 at 12:38 | history | asked | Gregor Samsa | CC BY-SA 3.0 |