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
17 events
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| Feb 18 at 21:53 | vote | accept | stupid_question_bot | ||
| Feb 16 at 14:09 | comment | added | Ian Agol | I think I realized my mistake. There is an elliptic involution of the twice punctured torus which exchanges the two punctures and is central in the mapping class group. So it acts trivially on $Aut(F_2)$. I’ll leave my comment up in case anyone gets confused about the same issue. | |
| Feb 16 at 10:47 | answer | added | HJRW | timeline score: 9 | |
| Feb 16 at 5:04 | comment | added | Ian Agol | I’m a bit puzzled here, because $Aut(F_2)$ is isomorphic to the subgroup of the mapping class group of the twice-punctured torus which preserves the two punctures by the Birman exact sequence. This is index two in the full mapping class group. But this seems to contradict the Dyer-Formanek result, so I must be mistaken, probably forgetting some technical assumptions. en.wikipedia.org/wiki/… | |
| Feb 15 at 10:20 | comment | added | HJRW | @DanielAsimov, RyanBudney: looking back over previous edits, the first version of the question was entirely unambiguous. | |
| Feb 14 at 21:05 | comment | added | stupid_question_bot | @DanielAsimov An automorphism of a group is pretty unambiguous IMO? IMO anyone writing "an automorphism of Aut(G)" to mean an automorphism of $G$ would be abusing notation, and the onus would be on them to make this clear. This is not the case in my question. | |
| Feb 14 at 21:03 | history | edited | stupid_question_bot | CC BY-SA 4.0 |
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| Feb 14 at 20:40 | comment | added | Ryan Budney | I agree with Daniel. The onus is on the question asker to write an unambiguous question, not on the audience to choose from among the ambiguity and answer whatever they prefer. | |
| Feb 14 at 20:11 | comment | added | Daniel Asimov | I'd much prefer that questions be written unambiguously than my having to decide whether one version or another makes a question trivial. | |
| Feb 14 at 18:41 | history | edited | LSpice | CC BY-SA 4.0 |
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| Feb 14 at 18:37 | comment | added | LSpice | @DanielAsimov, re, I think the question would become nonsensical, or at least trivial, if "automorphism of $\operatorname{Aut}^+(F_2)$" meant "element of $\operatorname{Aut}^+(F_2)$", so surely the only sensible reading is as "automorphism of $G$" in the usual sense, where $G$ happens to be $\operatorname{Aut}^+(F_2)$. | |
| Feb 14 at 17:57 | comment | added | Daniel Asimov | I am having trouble parsing what an "automorphism" of what is already an automorphism group (say Aut(G)) means. Does this mean an element of the automorphism group Aut(G), or does it mean an element of Aut(Aut(G)) ? | |
| Feb 14 at 17:41 | history | edited | stupid_question_bot | CC BY-SA 4.0 |
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| Feb 14 at 13:53 | comment | added | HenrikRüping | Just to rephrase what is already written in this question. This is equivalent to asking whether any automorphism of $Aut^+(F_2)$ can be extended to an automorphism of $Aut(F_2)$. | |
| Feb 14 at 7:30 | history | edited | YCor | CC BY-SA 4.0 |
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| Feb 14 at 1:57 | review | Close votes | |||
| Feb 15 at 1:32 | |||||
| Feb 13 at 22:23 | history | asked | stupid_question_bot | CC BY-SA 4.0 |