(1) In "Esquisse d'un programme", Grothendieck conjectures
Grothendieck-Teichmüller 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-Teichmüller 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-Teichmüller conjecture" that the tower of Teichmüller 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.
