Timeline for Are nearby crossed homomorphisms from compact Lie groups crossed-conjugate?
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| Sep 16, 2021 at 10:46 | comment | added | Uri Bader | Ha, inspecting the proof allows me answering the question in the comments: assume the graphs of two nearby crossed homomorphisms are conjugated by $(\gamma,g)$. Then there exists $\gamma'\in \Gamma$ such that $(\gamma',e)\cdot (\gamma,g)^{-1}$ is in the graph of the second crossed homomorphism, so conjugation by $(\gamma',e)$ has the same effect as conjugation by $(\gamma,g)$. | |
| Sep 15, 2021 at 19:37 | comment | added | Uri Bader | Let me nevertheless try to explain further how I view this proof, and maybe convince you that it works... The idea is to identify the space of crossed homomorphisms with a certain space of subgroups of the semidirect product and exploit the fact that every $\Gamma\rtimes G$-orbit is open, but $\Gamma$-already acts transitively on each such orbit, as the stabilizer of each point contains the subgroup corresponds to this point which together with $\Gamma$ gives the full $\Gamma\rtimes G$. I hope that helps. Uri | |
| Sep 15, 2021 at 19:30 | comment | added | Uri Bader | @UrsSchreiber I rewrote my answer a bit and I hope it is clearer now. I thought that I have a more "constructive" argument and I intended to put it as another answer, but it didn't end up more transparent then this answer, so I abandoned this plan. | |
| Sep 15, 2021 at 19:27 | history | edited | Uri Bader | CC BY-SA 4.0 |
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| Sep 15, 2021 at 5:01 | comment | added | Urs Schreiber | Thanks. I do understand that you are claiming this, but I am trying to express that I am not following how you mean to arrive at that conclusion. I am still thinking that given a pair of nearby homomorphisms $G \to \Gamma \rtimes G$ which are the identity on $G$, the classical result only tells us that they are conjugate by some element $(\gamma, g) \in \Gamma \rtimes G$, while we need to show that we can find a conjugation by an element of the form $(\gamma', e)$. IF the action of G on Gamma is trivial on the center of G then it follows that $\gamma' = \gamma$ is the solution. But otherwise? | |
| Sep 14, 2021 at 18:32 | comment | added | Uri Bader | Sorry for not being clear. The above proof intends to show that the space of crossed homomorphisms modulo the action of $\Gamma$ is discrete. This shows that every two nearby crossed homomorphisms differ by a $\Gamma$-conjugation, but the proof does not provide the conjugating element. I am currently in the process of writing another answer which gives such an element in a more transparent way. | |
| Sep 14, 2021 at 18:28 | comment | added | Urs Schreiber | What's the proof of existence of this $\gamma$? | |
| Sep 14, 2021 at 16:39 | comment | added | Uri Bader | So you want a construction, not a proof of exitance? I can give a somewhat more constructive proof, I think. I will add it as another answer when I get the time. | |
| Sep 14, 2021 at 12:48 | comment | added | Urs Schreiber | If I hand you a pair of nearby homomorphisms $G \to \Gamma \rtimes G$ which are both the identity on $G$, how do you propose to construct an element $\gamma \in \Gamma$ such that conjugation with $(\gamma, e) \,\in\, \Gamma \rtimes G$ conjugates these two homomorphisms into each other? | |
| Sep 14, 2021 at 12:45 | comment | added | Uri Bader | @UrsSchreiber I am sorry, but I don't understand the last sentence of your last comment. | |
| Sep 14, 2021 at 8:00 | comment | added | Urs Schreiber | Your point seems to be (correct me if I am misunderstanding) that we can just as well identify the space of crossed homomorphisms $G \to \Gamma$ with that of actual homomorphisms $G \to \Gamma \rtimes G$ which are an inner automorphism on the $G$ factor, modulo that inner automorphism action. I certainly see that. But I admit to not yet following how that solves the problem: It seems to me that this goes to show that every conjugation in $\Gamma \rtimes G$ may be understood as taking a crossed hom. to a crossed hom. -- but we need this statement for any prescribed pair of nearbys. No? | |
| Sep 14, 2021 at 7:55 | comment | added | Urs Schreiber | Thanks for replying! Allow me to see if I am following the argument you propose: I was thinking of identifying the space of crossed homomorphisms with the subspace of actual homomorphisms $G \to \Gamma \rtimes G$ which are the identity on G. Given a pair of nearby such, the existing proof produces an element $(\gamma,g)$ which conjugates them into each other, and the problem seems to be to show that this element can be chosen to be of the form $(\gamma, e)$. Now, your point is, if I understand well, that (continued in next comment)... | |
| Sep 10, 2021 at 13:08 | history | answered | Uri Bader | CC BY-SA 4.0 |