Timeline for Continuous families of finite subgroups of a Lie group
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 12, 2015 at 23:02 | answer | added | Igor Belegradek | timeline score: 4 | |
| Feb 12, 2015 at 20:54 | comment | added | Charles Rezk | This question is pretty similar to (actually, is a special case of) this one. Some good answers are given there, though I'm not satisfied that I know a clean proof. | |
| Feb 12, 2015 at 18:39 | answer | added | Misha | timeline score: 8 | |
| Feb 12, 2015 at 11:10 | comment | added | Paul Taylor | It's a (fairly) clearly stated mathematical question asked on a site for such things, so it is not "off topic". | |
| Feb 12, 2015 at 10:17 | comment | added | YCor | Because it's off-topic here. | |
| Feb 12, 2015 at 10:14 | comment | added | Paul Taylor | @YCor, why not just post a full answer here? | |
| Feb 12, 2015 at 10:12 | comment | added | YCor | @PaulTaylor: I voted to move to stackexchange where I'd be happy to post an answer (but I'd need to find a reference for the finiteness result). | |
| Feb 12, 2015 at 10:07 | comment | added | YCor | In a general setting "continuous family" is not clear. You want a list of ambiguities? 1) Family indexed by what? a topological space? an interval? 2) does continuous mean locally injective? 3) Do you specify the group $F$ and look at representation, or do you consider the space of closed subgroups of $G$? 4) in the rep case, do you work in $Hom(F,G)$ or its quotient by conjugation, or its GIT quotient? In the case $F$ finite and $G$ compact Lie groups, all points of view are equivalent but it sort of comes after asking the basic questions. | |
| Feb 12, 2015 at 9:58 | comment | added | Paul Taylor | The notion of "continuous family" does have a clear meaning, so long as we know the relevant topologies. We need that on the set of finite, or better compact, subspaces of a compact Hausdorff space. This is provided by the Hausdorff metric or Vietoris topology. For the conjecture to be true, (the indexing space of) the family must also be connected. However, without worrying about unfamiliar topologies, a simpler result would suffice to answer the Question: given $g:[0,1]\to G$ with $g(t)^n=id$, are $g(0)$ & $g(1)$ conjugate? @YCor's Comment seems to solve this, so why not make it an Answer? | |
| Feb 12, 2015 at 7:47 | comment | added | YCor | it's still a bit unclear because you haven't defined "continuous family". Here's a precise statement which is hidden behind Misha's answer: Let $I$ be an interval with 0, $F$ a finite group, $G$ a compact Lie group. Let $(\pi_i)_{i\in I}$ be a continuous family of homomorphisms $F\to G$ (that is, $(i,h)\mapsto\pi_i(h)$ is continuous). Then there exists a continuous map $i\mapsto g_i$ from I to G such that $\pi_i(h)=g_i\pi_0(h)g^{−1}_i$ for all $h\in F$. Besides, up to conjugation in G, there are finitely many homomorphisms $F\to G$. | |
| Feb 12, 2015 at 4:46 | comment | added | Anon | @Ben-Webster, I am interested in what Ben Webster says - 'The space of representations of a finite group into a Lie group is a discrete union of conjugacy classes'. Is there a proof or a reference? I edited the question to make it clearer. | |
| Feb 12, 2015 at 4:44 | history | edited | Anon | CC BY-SA 3.0 |
Improved the statement.
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| Feb 11, 2015 at 19:01 | comment | added | Ben Webster♦ | @PaulTaylor Which Misha answered. The space of representations of a finite group into a Lie group is a discrete union of conjugacy classes, so any continuous map from a connected topological space lands inside one conjugacy class. | |
| Feb 11, 2015 at 18:08 | comment | added | Paul Taylor | @Misha, In $SO(3)$ for any integer $n$, consider the family of subgroups of rotations by multiples of $2\pi/n$, indexed by the axis of rotation. I agree that some thought needs to be put into what a "continuous family of finite subgroups" is, but this is a meaningful question. | |
| Feb 11, 2015 at 17:59 | review | Close votes | |||
| Feb 11, 2015 at 23:43 | |||||
| Feb 11, 2015 at 17:02 | comment | added | Misha | Such families do not exist as each finite group has only finitely many conjugacy classes of representations into a fixed target Lie group. | |
| Feb 11, 2015 at 16:29 | history | asked | Anon | CC BY-SA 3.0 |