Grothendieck’s idea to study is via its action on geometric objects. We need two facts:
We have the following exact sequence where is a variety over a perfect field .
For , we have
From 1, the adjoint action of the arithmetic fundamental group preserves the geometric fundamental group , and it descends to a map . For example, if , then , and the action is the cyclotomic character.
If we consider and , then we have a group homomorphism where is the free group on two generators. The amazing thing is that this map is injective, first shown by Belyi (see this post for a proof of the original statement; the faithfulness of the action of the absolute Galois group can be shown by considering elliptic curves , see Theorem 2 of this article; For generalization of Belyi’s theorem, see here).
Drinfeld gives an explicit description of a subgroup of that conjectually is the image of the absolute Galois group. See Willwacher’s note Definition 1.2. The definition is as follows: Let be the two generators of , consists of all such that such that , satisfying
.
if and .
where is generator of .
Reference:
https://fr.wikipedia.org/wiki/Dessin_d%27enfant_(math%C3%A9matiques) (for explanation why dessin d’enfant corresponds to finite covers of .
https://mathoverflow.net/questions/1909/what-are-dessins-denfants
https://swc-math.github.io/aws/2005/05SchnepsNotes.pdf
https://drive.google.com/file/d/1M31uE5uReuX66pJ_KDsunz9mZAsTPsiE/view?usp=sharing