Timeline for are the congruence subgroups $\Gamma(n)$ characteristic inside $\mathrm{SL}_2(\mathbb{Z})$?
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Jan 13, 2017 at 3:47 | history | edited | David E Speyer | CC BY-SA 3.0 |
added 95 characters in body
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| Jan 13, 2017 at 3:45 | comment | added | David E Speyer | @Amy You are right! I misread Theorem 2 of Hua and Reiner. Apparently, the word "inner" was in quotes because they didn't actually mean inner. | |
| Jan 13, 2017 at 3:00 | comment | added | stupid_question_bot | Sorry to resurrect this post, but isn't the outer automorphism group actually $\mathbb{Z}/2\times\mathbb{Z}/2$, generated by the $\epsilon$ you describe, and by conjugation by $[[-1,0],[0,1]]$ inside $\text{GL}_2(\mathbb{Z})$? Your answer is of course still valid, since $\Gamma(n)$ is normal inside $\text{GL}_2(\mathbb{Z})$ as well. | |
| Sep 27, 2016 at 16:37 | comment | added | Ben Wieland | ${SL_2(\mathbb Z)}$ is virtually free so $\widehat{SL_2(\mathbb Z)}$ is virtually free and thus center-free. | |
| Sep 27, 2016 at 14:06 | comment | added | David E Speyer | You are right, I don't know if $g$ lifts to a central element in $\widehat{SL_2(\mathbb{Z})}$. | |
| Sep 27, 2016 at 13:00 | comment | added | David E Speyer | Regarding your first comment: Correct. Regarding your second comment, I need to think about it. | |
| Sep 27, 2016 at 3:52 | comment | added | stupid_question_bot | Hmm, so your $X\mapsto g^{\alpha(X)}X$ only gives an automorphism of $SL_2(\widehat{\mathbb{Z}})$ right? I don't think this induces a continuous automorphism of $\widehat{SL_2(\mathbb{Z})}$ does it? For example, if you try to replace $g$ with a preimage in $\widehat{SL_2(\mathbb{Z})}$, it won't necessarily by central, right? | |
| Sep 27, 2016 at 3:45 | comment | added | stupid_question_bot | I think you left something out. You said "let $g\in SL_2(\widehat{\mathbb{Z}})$ be $\zeta Id$ in $SL_2(\mathbb{Z}_p)$ and 1;" Did you mean "...and 1 in $\prod_{q\ne p} SL_2(\mathbb{Z}_q)$"? | |
| Sep 27, 2016 at 3:30 | comment | added | David E Speyer | I thought about it a bit more, and the answer is no in more cases for the adelic case. For example, let $p$ be a prime $\equiv 1 \bmod 3$. We have a map $\alpha: \widehat{\Gamma(2p)} \to \mathbb{Z}/(3 \mathbb{Z})$ by $\Gamma(2p) \to SL_2(\mathbb{Z}/3) \cong A_4 \to \mathbb{Z}/(3 \mathbb{Z})$. Let $\zeta$ be a primitive cube root of $1$ in the $p$-adics and let $g \in SL_2(\widehat{\mathbb{Z}})$ be $\zeta \mathrm{Id}$ in $SL_2(\mathbb{Z}_p)$ and $1$; note that $g$ is central. So $X \mapsto g^{\alpha(X)} X$ gives an outer automorphism which doesn't preserve $\widehat{\Gamma(2p)}$. | |
| Sep 27, 2016 at 2:29 | vote | accept | stupid_question_bot | ||
| Sep 27, 2016 at 2:29 | comment | added | David E Speyer | Well, the $n$ odd part of the answer is the same, because the automorphism I give extends continuously to the adeles. For n even, I don't know if we get interesting new automorphisms. Dull jstor.org.proxy.lib.umich.edu/stable/2373578 gives some results on $Out(SL_2(R))$ for $R$ a general commutative ring, which might be useful, but I haven't unpacked them. | |
| Sep 27, 2016 at 1:52 | comment | added | stupid_question_bot | Hmm does this imply the same result when you look at the closure of these $\Gamma(n)$'s inside $\widehat{SL_2(\mathbb{Z})}$ and continuous automorphisms? | |
| Sep 27, 2016 at 1:13 | history | answered | David E Speyer | CC BY-SA 3.0 |