# Expressing Galois actions on fundamental groups explicitly

Let $X$ be some variety over $\mathbb{Q}$, and let $\pi_1(X\times_{\mathbb{Q}}\mathbb{C},x)$ denote its (topological) fundamental group. As is well known $Gal(\mathbb{Q})$ acts on this fundamental group. I was browsing old MSRI videos, and in the middle of one of them I saw an intriguing explicit description of this action:

http://www.msri.org/realvideo/ln/msri/1999/vonneumann/schneps/1/main/08.html

(you don't have to know anything from earlier in the talk to understand that page)

As it says there, there was also a talk by Ihara about this. I'm looking, however, for an explanation of this in a more systematic way, in a paper or a book. Do you know of a good reference for this?

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Just to clarify the question: I'm not sure what you mean by "(toplogical)," but Gal(Q) acts on the etale fundamental group, not the fundamental group of the complex manifold X(C). I assume this is why you gave the pi a hat. –  JSE Jul 25 '11 at 0:36
The OP may be emphasising that $\hat{\pi_1}$ is a topological group, not just a group, but maybe not... –  David Roberts Jul 25 '11 at 1:50
David Roberts was too kind, I was just being careless. I'll edit this. –  Makhalan Duff Jul 25 '11 at 1:57

I'm not precicely sure what you are looking for, but the following references I think are relevant.

The Grothendieck Theory of Dessins d'Enfants (London Mathematical Society Lecture Note Series) LMs 200

Geometric Galois Actions: Volume 1, Around Grothendieck's Esquisse d'un Programme LMS 242

Geometric Galois Actions: Volume 2, The Inverse Galois Problem, Moduli Spaces and Mapping Class Groups LMS 243

Leila Schnepps was very invloved with all three.

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