# 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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Why was this downvoted? These books are indeed the natural place to read about what Schneps was talking about. – JSE Jul 25 '11 at 0:34
I didn't downvote (or upvote), but I do have all three books out from the library, and I don't see anywhere where my question is discussed. – Makhalan Duff Jul 25 '11 at 1:56
I don't have the books in front of me, but one of them has an expository article by Oort which is very useful. And whichever article defines the Grothendieck-Teichmuller group will unavoidably give a very explicit description of (the little we really know about) the action of the Galois group on the etale fundamental group of P^1 - 0,1,infty. Maybe you should edit the post to make it more clear what your question is -- the three books above certainly serve as a good introduction to the story of Galois groups acting on fundamental groups. – JSE Jul 25 '11 at 2:05
I've read Oort's paper at some point. The point is that it is clear how the Galois action is defined, but what is not clear is its interpretation given in the link I put. – Makhalan Duff Jul 25 '11 at 2:22

Aha, now I think I have a better picture of what you're looking for. I would look at Matsumoto's notes from the Arizona Winter School program on Galois groups and fundamental groups:

http://math.arizona.edu/~swc/notes/files/05MatsumotoNotes.pdf

especially sections 2.2 and 4.1.

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