Timeline for Recognizing a restriction from $SL_2(\mathbb{C})$ to $SL_2(\mathbb{Z})$
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| May 1, 2020 at 4:45 | comment | added | Ian Agol | By the way, the analogous question for $SL_n(\mathbb{Z})$, $n > 2$ follows from Margulis' superrigidity theorem. en.wikipedia.org/wiki/Superrigidity To clarify, do you want a characterization of when the representation is induced by a representation of $SL(2,\mathbb{C})$ for the standard embedding induced by $\mathbb{Z}\subset \mathbb{C}$? Or are other representations of $SL(2,\mathbb{Z})\to SL(2,\mathbb{C})$ allowed? | |
| Apr 13, 2019 at 6:10 | comment | added | Venkataramana | One "intrinsic" condition is that the Zariski closure in $GL_N(\mathbb C)$ iof the image of $SL(2,\mathbb Z)$ is $SL(2,\mathbb C)$ or $PGL(2,\mathbb C)$.. | |
| Mar 20, 2019 at 22:00 | comment | added | Nate | Indeed, and you can note I clarified that $E_{1,2}(1)$ acts unipotently. | |
| Mar 20, 2019 at 21:48 | comment | added | YCor | This is because usually $e_{ij}(x)$ is called elementary for arbitrary $x$, and then they're not conjugate in $SL_2(Z)$. For instance in $SL_2(Z)$, $e_{12}(1)$ and $e_{12}(2)$ are not conjugate. So the condition that the second one is mapped to a unipotent element doesn't imply that the first is mapped to a unipotent element. | |
| Mar 20, 2019 at 20:02 | history | edited | Nate | CC BY-SA 4.0 |
added 14 characters in body
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| Mar 20, 2019 at 20:02 | comment | added | Nate | Well by "elementary matrix" I meant the identity matrix with a single extra 1 somewhere off the diagonal. They are all conjugate, so there is no difference. I will try to clarify. | |
| Mar 20, 2019 at 19:53 | comment | added | YCor | "iff an elementary matrix acts unipotently": you mean "every" elementary matrix? | |
| Mar 20, 2019 at 19:51 | comment | added | Nate | Yes, finite dimensional ones. | |
| Mar 20, 2019 at 19:48 | comment | added | YCor | What kind of representations are you looking at? finite-dimensional ones? | |
| Mar 20, 2019 at 19:20 | history | asked | Nate | CC BY-SA 4.0 |