Timeline for Subgroups of $\mathbb{R}^n$
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 19, 2023 at 9:14 | answer | added | Arun Wandakar | timeline score: 2 | |
| Jul 8, 2019 at 7:00 | comment | added | YCor | Concerning the (compact) space of subgroups of $\mathbf{R}^n$ with the Chabauty topology (that is, the Hausdorff topology on compact subsets of the 1-point compactification of $\mathbf{R}^n$), see Kloeckner's paper. | |
| Jul 8, 2019 at 6:54 | comment | added | YCor | @KConrad we have a classification modulo conjugation by $\mathrm{GL}_n(\mathbf{R})$, which is finer than a classification as topological groups, yet coincides here. Since it has not been said explicitly every closed subgroup is in the $\mathrm{GL}_n(\mathbf{R})$-orbit of $\mathbf{R}^a\times\mathbf{Z}^b\times\{0\}^{n-a-b}$ for a unique pair $(a,b)$ with $0\le a+b\le n$. It's not hard to prove (after passing to the spanned subspace and modding out the zero component, it boils down to proving that a discrete subgroup of $\mathbf{R}^n$ spanning the whole space is in the orbit of $\mathbf{Z}^n$). | |
| Jul 8, 2019 at 6:06 | comment | added | Todd Trimble | Abstractly (and ignoring topology), the groups $\mathbb{R}$ and $\mathbb{R}^n$ are isomorphic: both are $\mathbb{Q}$-vector spaces of dimension $2^{\aleph_0}$. A group is isomorphic to a subgroup of such iff it is torsionfree and of rank bounded by $2^{\aleph_0}$. But given what you are looking for, KConrad's suggestion makes a lot of sense. | |
| Jul 8, 2019 at 4:53 | comment | added | abx | In fact, the closed subgroups of $\Bbb{R}^n$ are easily classified, see for instance Bourbaki's General Topology VII, §1, Theorem 2. | |
| Jul 7, 2019 at 22:40 | comment | added | KConrad | A cleaner statement to aim for is to ask what the closed subgroups of $\mathbf R^n$ look like up to isomorphism (as a topological group). | |
| Jul 7, 2019 at 22:36 | comment | added | phdstud | I edited the question to be more precise on what I meant. | |
| Jul 7, 2019 at 22:35 | history | edited | phdstud | CC BY-SA 4.0 |
added 274 characters in body
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| Jul 7, 2019 at 22:34 | comment | added | Gabe Conant | The answers and comments for this question might be of interest. | |
| Jul 7, 2019 at 22:02 | comment | added | M. Winter | I am no expert, but I think this is hopeless. People are always on a hunt for nice lattices (for sphere packings etc.), and it would be much easier if they could be listed that easily. | |
| Jul 7, 2019 at 22:00 | review | First posts | |||
| Jul 7, 2019 at 22:53 | |||||
| Jul 7, 2019 at 21:59 | history | asked | phdstud | CC BY-SA 4.0 |