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I have read many times that it is crucial to understand the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) =: L$, in so far that some have stated (was it Richard Taylor ?) that this understanding is the main goal (or the very definition) of Number Theory (give or take). Anyway, in this question I am concerned with actions of this group. I know that there is, for instance, Grothendieck's "Dessins d'enfants" theory (although Deligne has stated that, at the point of writing and publishing his famous "projective line minus three points paper," the dessins had not helped that understanding very much at all). If one would want to define some action of $L$ on some space/set/geometry/etc., what would make that action interesting ?

Example given: I have never seen descriptions of $L$ on combinatorial geometries such as (axiomatic) projective planes, but (when) would such actions be interesting enough ? What are natural constraints on the geometry acted upon, so that we would have a nice action ?

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You should read the paper

I. Bauer, F. Catanese, F. Grunewald: Faithful actions of the absolute Galois group on connected components of moduli spaces, Invent. Math. 199, No. 3, 859-888 (2015). ZBL1318.14034.

Quoting from MathSciNet review:

The authors study the action of the absolute Galois group $\mathrm{Gal}(\bar{\mathbb{Q}}/ \mathbb{Q})$ on certain hyperelliptic curves, surfaces of general type and moduli spaces of surfaces of general type.

They prove a number of results on faithfulness of the action on moduli spaces and also provide a result on the existence of surfaces $X$ defined over number fields and Galois automorphisms $\sigma$ such that $X$ and its conjugate $X^{\sigma}$ (think of $X^{\sigma}$ as the variety cut out by the equations obtained from applying $\sigma$ to the coefficients of the equations of $X$ in some projective space) have nonisomorphic fundamental groups, in particular, fail to be diffeomorphic.

The existence of such varieties is classical and goes back to J.-P. Serre [C. R. Acad. Sci. Paris 258 (1964), 4194–4196; MR0166197]. The present paper even provides an infinite series of rather explicit examples.

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    $\begingroup$ @ Francesco Polizzi: why are these actions interesting for understanding the absolute Galois group (cf. my question) ? $\endgroup$
    – THC
    Commented Apr 3, 2018 at 18:44
  • $\begingroup$ @THC: If the action is faithful, it contains (in principle) all information about the absolute Galois group. Of course extracting anything of number-theoretic interest from such actions is no easy task... $\endgroup$
    – Daniel Litt
    Commented Apr 27, 2018 at 12:46
  • $\begingroup$ @DanielLitt: do you mean that a faithful action of the absolute Galois group on, say, the points of some axiomatic projective plane, could/would also be highly interesting ? $\endgroup$
    – THC
    Commented Apr 27, 2018 at 14:06
  • $\begingroup$ @THC: This is a matter of taste, but I'd say something like this isn't interesting unless one does something with it. But e.g. in principle the inverse Galois problem is about finding Galois actions on interesting objects... $\endgroup$
    – Daniel Litt
    Commented Apr 29, 2018 at 17:35
  • $\begingroup$ @DanielLitt: Could you give some examples of "doing something with the action" ? (I am not sure what you really mean, but it could be very helpful and relevant.) $\endgroup$
    – THC
    Commented Apr 30, 2018 at 10:59

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