Timeline for Finitely generated subgroups of the absolute Galois group
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Feb 15, 2021 at 18:21 | comment | added | Arno Fehm | $\hat{F}_\omega$ is the absolute Galois group of any countable Hilbertian PAC field of characteristic zero by a result of Fried-Völklein, and there are plenty of these algebraic over $\mathbb{Q}$. See for example Jarden's Large normal extension of Hilbertian fields in Math. Z. 224, 1997. | |
| Feb 15, 2021 at 18:05 | comment | added | Carl-Fredrik Nyberg Brodda | @ArnoFehm That is interesting -- do you have a reference to $\hat{F}_\omega$ appearing as a subgroup? | |
| Feb 15, 2021 at 12:05 | comment | added | HJRW | There is work on the finitely generated subgroups of the profinite completions of free groups. For instance, all parafree groups can occur (see en.wikipedia.org/wiki/Parafree_group), and it is also known that Sela's limit groups all occur too. That said, classification seems a long way off. | |
| Feb 15, 2021 at 11:01 | comment | added | Arno Fehm | I'm not sure if this is really a question about the absolute Galois group of $\mathbb{Q}$. For example every such group is residually finite and hence Hopfian simply since it is a subgroup of a profinite group. Also, we know that every subgroup of the free profinite group $\hat{F}_\omega$ on countably many generators occurs - is it possible even there to say what they are? | |
| Feb 15, 2021 at 10:49 | history | asked | Carl-Fredrik Nyberg Brodda | CC BY-SA 4.0 |