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I've had a desire to get some sort of handle on the Grothendeick-Teichmuller group for years now, but I've always felt that I could never find a source that was introductory and readable.

The obvious candidates are Schneps's: Grothendieck's "Long March through Galois theory"' andThe Grothendieck-Teichmuller group $\widehat{GT}$: a survey'. However I find them to be rather technical and off-putting for a person trying to understand this for the first time.

Do you know of an "easy" (by which I mean: well-motivated) introduction to the Grothendieck-Teichmuller group?

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Why is this CW? –  Martin Brandenburg May 6 '11 at 12:38
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V.G. Drinfeld: On quasi-Hopf algebras and on a group that is closely connected with $Gal(\overline{\mathbb{Q}}/\mathbb{Q}$), Algebra i Analiz, 2:4 (1990), 149–181

I am not saying it is easy to read but it is definitely well-motivated. By the way, the point of view on $\widehat{GT}$ might also be a bit different from what you expect.

In the very same spirit, but less technical, there is an excellent account of Drinfled's theory by Dror Bar-Natan: On Associators and the Grothendieck-Teichmuller Group I, Selecta Mathematica NS 4 (1998) 183-212 ( http://arxiv.org/abs/q-alg/9606021).


Let me now explain broadly how this works.

Consider the collection $\mathcal B_*=(\mathcal B_n)_n$ consisting of groupoids of parenthesised braids. It is an operad in groupoids (by using the so-called cabling operations).

A more algebroic way to look at it is to consider the free braided monoidal category with one generator $\bullet$, and consider the groupoid of isomorphisms in this category. Objects in this category are actually parenthesisations of $\bullet^{\otimes n}$... so if we fix $n$ we recover $\mathcal B_n$.

Now we would like to define the group $GT$ as the automorphism group of this operad in groupoids $\mathcal B_*$... the main problem is that it is rather small.

The main point is then, for any field $k$, to consider $k$-prounipotent completions $\mathcal B_n(k)$ of $\mathcal B_n$. Now $\widehat{GT}(k)$ is the automorphism group of the operad in $k$-groupoids $\mathcal B_*(k)$.

Using the version of Mac-Lane coherence Theorem for braided monoidal categories, one can prove that such automorphisms are determined by their action on $\mathcal B_3(k)$, and that the relations we have to impose are in $\mathcal B_3(k)$ and $\mathcal{B}_4(k)$.

The relation in $\mathcal B_3(k)$ is the hexagon axiom for braided monoidal categories and comes from the cabling operations from $\mathcal B_2(k)$ to $\mathcal B_3(k)$. The relation in $\mathcal B_4(k)$ is the pentagon axiom for braided monoidal categories and comes from the cabling operations from $\mathcal B_3(k)$ to $\mathcal B_4(k)$.

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That version of the coherence theorem being due to Joyal-Street... –  Todd Trimble May 6 '11 at 12:33
    
You are right. I should have written it. Sorry. –  DamienC May 6 '11 at 14:25
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"I am not saying it is easy to read" is quite un understatement. It is an incredibly dense article. You can't unbderstand it without background. Beside the wealth of rich ideas, one thing that makes it really hard to read is that everything is stated in terms of quasi-triangular quasi-hopf algebras when it is really not about those. They only occur though their category of representations. And these are braided monoidal categories i.e. reprsentations the operad of fundamental groupoids of the $F(\mathbb{C},n) = \{ (z_1\ldots,z_n) ~|~ z_i\neq z_j \}$. –  YBL May 15 '11 at 1:23
    
PS: see this answer mathoverflow.net/questions/39945/… for example. –  YBL May 15 '11 at 1:24
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See

MR2264540 (2007j:14028) Lochak, Pierre(F-PARIS6-IMJ); Schneps, Leila(F-PARIS6-IMJ) Open problems in Grothendieck-Teichmüller theory. Problems on mapping class groups and related topics, 165–186, Proc. Sympos. Pure Math., 74, Amer. Math. Soc., Providence, RI, 2006. 14G32 (18F30 20F34 57M07)

And anything else by Leila Schneps.

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e.g. Schnep's talks here: aimath.org/WWN/motivesdessins , this looks nice too: httP://www.phys.uu.nl/~3021009/topicsingalois/essay.pdf –  Thomas Riepe May 6 '11 at 4:41
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