Where is a good place to start learning about the Grothendieck-Teichmuller group? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T18:58:03Z http://mathoverflow.net/feeds/question/64065 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/64065/where-is-a-good-place-to-start-learning-about-the-grothendieck-teichmuller-group Where is a good place to start learning about the Grothendieck-Teichmuller group? James D. Taylor 2011-05-05T23:35:34Z 2011-05-06T11:52:01Z <p>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.</p> <p>The obvious candidates are Schneps's: <code>Grothendieck's "Long March through Galois theory"' and</code>The 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.</p> <p>Do you know of an "easy" (by which I mean: well-motivated) introduction to the Grothendieck-Teichmuller group?</p> http://mathoverflow.net/questions/64065/where-is-a-good-place-to-start-learning-about-the-grothendieck-teichmuller-group/64070#64070 Answer by Igor Rivin for Where is a good place to start learning about the Grothendieck-Teichmuller group? Igor Rivin 2011-05-06T00:15:54Z 2011-05-06T00:15:54Z <p>See</p> <p>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) </p> <p>And anything else by Leila Schneps.</p> http://mathoverflow.net/questions/64065/where-is-a-good-place-to-start-learning-about-the-grothendieck-teichmuller-group/64082#64082 Answer by DamienC for Where is a good place to start learning about the Grothendieck-Teichmuller group? DamienC 2011-05-06T04:30:58Z 2011-05-06T05:19:19Z <p>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</p> <p>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. </p> <p>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 ( <a href="http://arxiv.org/abs/q-alg/9606021" rel="nofollow">http://arxiv.org/abs/q-alg/9606021</a>). </p> <hr> <p>Let me now explain broadly how this works. </p> <p>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). </p> <p>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 $\mathcal B_n$. </p> <p>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. </p> <p>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 $\mathcal B_*(k)$. </p> <p>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 $\mathcal{B}_4(k)$. </p> <p>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 $\mathcal B_4(k)$. </p>