## Is every finite group a quotient of the Grothendieck-Teichmuller group?

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The Grothendieck-Teichmuller conjecture asserts that the absolute Galois group $Gal(\mathbb{Q})$ is isomorphic to the Grothendieck-Teichmuller group. I was wondering, would this conjecture imply the Inverse Galois Problem? I.e. is every finite group a quotient of the Grothendieck-Teichmuller group?

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## 1 Answer

It's only a partial answer since this survey is already 5 years old, but it suggest that almost nothing is known about (non-abelian) finite quotients of $\widehat{GT}$ (question 1.7).

Edit: I should maybe recall what happen in the abelian case, and why it's encouraging: elements of $\widehat{GT}$ are pairs $(f,\lambda)$ where $f$ is in the derived subgroup of $\hat{F}_2$, and $\lambda \in \hat{\mathbb{Z}}^{\times}$, satisfying some complicated equations. It turns out that the set theoretic map $(f,\lambda) \mapsto \lambda$ induces a surjective group morphism $\widehat{GT}\rightarrow \hat{\mathbb{Z}}^{\times}$. And the good news is that the composite

$$G_{\mathbb{Q}} \hookrightarrow \widehat{GT} \rightarrow \hat{\mathbb{Z}}^{\times}$$

is nothing but the cyclotomic character.

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I have the feeling that their is very little known about $\widehat{GT}$ period! – Spice the Bird Jan 28 2012 at 3:38

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I too think this problem sounds extremely hard. To give some indication -- as far as I know (and I could certainly be wrong) -- the only reason we know the map GThat -> Zhat^* is surjective is that G_Q embeds in GThat! Without that, I don't think I even know how to show that the image is infinite! – JSE Jan 28 2012 at 22:19
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