Consider the absolute Galois group $\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$. It seems to me that, in general, providing an "explicit" description of the elements of this group is a difficult task, see e.g. the related questions here and here. To make everything easier, I will assume we are working in ZFC.
I am interested in what is known about the (abstractly) finitely generated subgroups of $\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$. To be clear, I do not mean topologically finitely generated here, but rather just subgroups $H < \operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$ such that $H = \langle h_1, h_2, \dots, h_k \rangle$ for some elements $h_i \in \operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$. Of course, $\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$ itself is uncountable, so in particular not finitely generated in this sense.
The abstract properties of such finitely generated subgroups are of particular interest; to give some explicit questions:
- Are all finitely generated subgroups of $\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$ finitely presented?
- Do all such subgroups have finite cohomological dimension?
- Are all such subgroups residually finite and/or Hopfian?
but any similar type of abstract properties would also be interesting! Any "easy" explicit example of a finitely generated subgroup (other than $1$, or $C_2$ via conjugation!) of $\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$ would also be of interest -- especially if such an example is defined by (finitely many) generators and relations.
