# Grothendieck-Teichmuller conjecture

(1) In "Esquisse d'un programme", Grothendieck conjectures

Grothendieck-Teichmuller conjecture: the morphism $$G_{\mathbb{Q}} \longrightarrow Aut(\widehat{T})$$ is an isomorphism.

Here $G_{\mathbb{Q}}$ is the absolute Galois group and $\widehat{T}$ is the category whose

• objects are the profinite fundamental groupoids $T_{g,n}$ of the moduli stacks $\mathfrak{M}_{g,n}$ (restricted to certain basepoints, tangential basepoints or special automorphisms curves)

• morphisms are certain natural morphisms induced by natural operations like forgetting marked points and (auto-)gluing curves along such points.

(2) A parallel conjecture is that the morphism $$G_{\mathbb{Z}}^{\mathrm{mot}} \longrightarrow Aut(T_0^{\mathbb{mot}})$$ is an isomorphism. Here $G_{\mathbb{Z}}^{\mathrm{mot}}$ is the motivic Galois group of mixed Tate motives over $\mathbb{Z}$ and $T_0^{\mathbb{mot}}$ is the category of the mixed Tate fundamental groupoids of the $\mathfrak{M}_{0,n}$'s. The ($Gr^W$ of the) right hand side is known to be the Grothendieck-Teichmuller group $GRT$ so the conjecture translates as

Deligne-Drinfeld conjecture: the graded Lie algebra $\mathfrak{grt}_1$ is free with one generator for each odd degree $2k+1 > 1$.

Note that in both cases, the morphism $G \to Aut(T_{0,4})$ is known to be injective (by Belyi's theorem in the profinite case and by a recent theorem of F. Brown in the motivic case).

Another point in Grothendieck's Esquisse is his "Lego-Teichmuller conjecture" that the tower of Teichmuller groupoids $T$ is generated in dimension $dim(\mathfrak{M}_{g,n}) = 3g-3+n = 1$ with relations in dimension 2 (so we only get a finite number of these). This is also known. This is a purely topological fact that goes back to Moore-Seiberg and Drinfeld in the late 80's I think. In genus 0, it is obvious once stated in terms of a presentation of the little disc operad.

So the conjectures boils down to the fact that these natural geometric relations (forgetting marked points and gluing curved along them) between (motivic) fundamental groupoids of moduli spaces of curves characterize elements of the (motivic) Galois group. These are really amazing statements because they give a purely topological characterization of the Galois group a purely arithmetical object.

Question: Is there any evidence or intuition for these conjectures?

I think Grothendieck's intuition was that the study of varieties reduces to curves. This was his first approach to the Weil conjectures and his first test for the theory of the etale fundamental group. But this seems very vague to me. I'd be very happy to have more convincing evidence.

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Shouldn't one consider $\mathbb{Q}$-prounipotent fundamental groupoids ? –  DamienC May 6 '11 at 19:35
That's the same thing. The Betti realization of the mixed Tate groupoid of $M_{0,n}$ is the $\mathbb{Q}$-prounipotent completion of its topological fundamental groupoid (mainly because it is mixed Tate and a rational $K(\pi,1)$). –  YBL May 6 '11 at 19:46
"a purely topological characterization of the Galois group, a purely arithmetical object." - my guess is that Grothendieck's intuition was that to a large extent arithmetic "is" topology, which made him looking for e.g. a new foundation of homotopy theory, anabelian conjectures. –  Thomas Riepe May 8 '11 at 6:44
In some of his more discursive writing he talked about the first conjecture informally and if I remember well, did say why he thought it true. You may already have looked at his Esquisse and may not have found what I mention but if not do look at him in `chatty' mode as he was often interested in the process of clarifying why a result 'should' be true. –  Tim Porter Nov 24 '11 at 16:27