Timeline for Series of discrete groups with a Lie group limit
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 21 at 22:14 | vote | accept | Fetchinson0234 | ||
| Aug 21 at 20:13 | history | became hot network question | |||
| Aug 21 at 13:12 | answer | added | Alexei Entin | timeline score: 11 | |
| Aug 21 at 13:04 | comment | added | Denis T | @Smiley Infinite symmetric group has no finite dimensional representations by Jordan's theorem: every finite subgroup of GL(N) is abelian-by-(finite of bounded order); and because all Lie groups are abelian-by-linear, there's no such embedding. | |
| Aug 21 at 13:00 | answer | added | Denis T | timeline score: 4 | |
| Aug 21 at 12:35 | comment | added | Sean Eberhard | The notion of limit of finitely generated groups was refined in the work of van den Dries and Wilkie. See sciencedirect.com/science/article/pii/0021869384902230. To apply this in the case of $S_N$ you would need to choose generators. | |
| Aug 21 at 12:33 | history | edited | Fetchinson0234 | CC BY-SA 4.0 |
added 17 characters in body
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| Aug 21 at 12:32 | comment | added | Sam Nead | You might be interested in the concept of “Gromov-Hausdorff” convergence of (pointed) metric spaces. This applies to Cayley graphs, so can be used to find limits of sequences of groups, equipped with generating sets and a “rescaling” function. | |
| Aug 21 at 12:25 | comment | added | Smiley1000 | I guess you can say that $\underset{\longrightarrow}{\operatorname{lim}} \mathbb{Z}_N \cong \mathbb{Q}/\mathbb{Z}$ embeds in $S^1$. In the same way, you can form the direct limit $\underset{\longrightarrow}{\operatorname{lim}} S_N$ which we might call $S_{(\infty)}$. I don't know whether this embeds in some Lie group. | |
| Aug 21 at 12:13 | history | asked | Fetchinson0234 | CC BY-SA 4.0 |