Taking the approach in: http://www.msri.org/realvideo/ln/msri/1999/vonneumann/schneps/1/

I view the Grothendieck-Teichmuller conjecture as saying that $Gal(\mathbb{Q})$ is isomorphic to a well understood object. That is, it is isomorphic to $Out^*$ of the fundamental group of the Teichmuller lego.

This seems to indeed be informative about $Gal(\mathbb{Q})$! My question is whether the Grothendieck-Teichmuller philosophy has predictions about how $Gal(\mathbb{Q})$ acts on varieties defined over $\mathbb{Q}$ (for example $\mathbb{P}^1_{\mathbb{Q}}\smallsetminus ${$0,1,\infty$}). From the way that I formulated the conjecture, it is not obvious to me that it does; but I think I am missing the greater picture.

  • $\begingroup$ It seems to me that the statement should really be "this particular map from one group to the other is an isomorphism," and presumably this form of the conjecture should give more information about actions. $\endgroup$ – Qiaochu Yuan Nov 13 '11 at 0:22

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