Timeline for Is every automorphism of $\mathrm{Aut}^+(F_2)$ induced by conjugation inside $\mathrm{Aut}(F_2)$?
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Feb 18 at 21:53 | vote | accept | stupid_question_bot | ||
| Feb 17 at 18:41 | comment | added | stupid_question_bot | Another detail: One should rule out the possibility that the nontrivial coset of $Aut(F_2)/Aut^+(F_2)$ acts as an inner automorphism of $Aut^+(F_2)$. Since $Aut^+$ has trivial center, this would imply that the centralizer of $Aut^+$ inside $Aut$ has order 2, and makes the sequence $1\to Aut^+\to Aut\to C_2\to 1$ a split sequence with trivial action, hence a direct product. However, passing to the quotient by $Inn$, this would imply that $GL_2(\mathbb{Z})$ is a direct product of $SL_2(\mathbb{Z})$ with $C_2$, which is not true. Can you add this to your answer? Then I can accept. Thank you! | |
| Feb 17 at 18:29 | comment | added | stupid_question_bot | To add a little detail, a more precise reference to Charney-Crisp's claim is section 4 of the 1981 paper of Dyer-Grossman. | |
| Feb 16 at 21:25 | history | edited | LSpice | CC BY-SA 4.0 |
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| Feb 16 at 10:47 | history | answered | HJRW | CC BY-SA 4.0 |