In Recoltes et Semailles, Grothendieck remarks that the theory of motives is related to anabelian geometry and Galois-Teichmuller theory. My understanding of these subjects is not very solid at this moment, but this is what I understand:


**Anabelian geometry** tries to ask how much information about a variety is contained in its etale fundamental group. In particular, there exist "anabelian varieties" which should be completely determined by the etale fundamental group (up to isomorphism). The determination of these anabelian varieties is currently ongoing.


**Galois-Teichmuller theory** tries to understand the absolute Galois group $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ in terms of the automorphisms of the "Teichmuller tower", which is constructed as follows. We begin with the moduli stacks of curves with genus $g$ and $\nu$ marked points. These moduli stacks $\mathcal{M}\_{g,\nu}$ have homomorphisms to each other, which correspond to "erasing" marked points and "gluing". The Teichmuller tower $\hat{T}\_{g,\nu}$ comes from the profinite fundamental groupoids of these moduli stacks.


The **theory of motives** is some sort of "universal cohomology theory" in the sense that any Weil cohomology theory (which is a functor from smooth projective varieties to graded algebras over a field) factors through it. This is obtained from some process of "linearization" of algebraic varieties (considering correspondences as morphisms, followed by the process of "passing to the pseudo-abelian envelope", and formally inverting the Lefschetz motive).


Related to the theory of motives is the concept of a **Tannakian category**, which provides a kind of higher-dimensional analogue of Galois theory. I think the category of motives is conjectured to be a Tannakian category, via Grothendieck's standard conjectures on algebraic cycles (please correct me if I am wrong about this).


So I'm guessing Tannakian categories might provide the link between the theory of motives and anabelian geometry and Galois-Teichmuller theory (which are both related to Galois theory) that Grothendieck was talking about in Recoltes et Semailles, but I'm not really sure. Either way, the ideas are still not very clear to me, and I'd like to understand the connections more explicitly.



How are motives related to anabelian geometry and Galois-Teichmuller theory?
Two clarifications:


For anabelian geometry, you should ask how much information about a variety is contained in the Galois action on its etale fundamental group. 


While it's true that the motivic Galois group is a higher-dimensional analogue of the Galois group, it also should be true that motives are "just" a special kind of Galois representation, i.e. under the Tate conjecture the $\ell$-adic realization functor should give a faithful functor from motives to $\ell$-adic Galois representations, so the category of motives is the category of Galois representations with some restrictions placed on the objects and morphisms.


Of course these restrictions are highly nontrivial. Only for the irreducible Galois representations do we have a good conjectural description of which ones come from motives, via the Fontaine-Mazur conjecture.


So we can see that all 3 of these relate to Galois actions - the first two to Galois actions on fundamental groups, and the last to Galois actions on $\ell$-adic vector spaces. However, Galois actions are used in different ways in the three. Thinking about the three concepts, you might be led to questions like these:


* Can we construct Galois representations from the Galois action on the fundamental group of a curve? (this would be the first step in relating motives to anabelian geometry)
* Do these Galois representations arise from motives? (this would be the second step)
* Are these motives related to the geometry of a curve? (seeking a deeper connection to anabelian geometry)
* Can the class of motives arising this way be used to construct or describe the motivic Galois group? (now we bring in Grothendieck-Teichmuller theory)


I think these questions at least touch on the beginning of what Grothendieck was thinking of.


Since Grothendieck, people have heavily studied these questions, primarily in the case of unipotent quotients of the fundamental group, starting with the paper of Deligne on the fundamental group of the projective line minus three points. I think it's fair to say that the answer to all these questions is yes, with the largest caveat for the last question - I believe we can understand certain very special quotients of the motivic Galois group this way, but I don't think anyone has a strategy to construct the whole thing. 


The story goes something like this:


Deligne looked at the maximal pro-$\ell$ quotient of the geometric fundamental group of the projective line minus three points. This is naturally an $\ell$-adic analytic group, and has a Lie algebra, which is an $\ell$-adic representation, and admits an action of the Galois group. This is supposed to be the $\ell$-adic realization of a motive, and Deligne worked to find the other realizations, including Hodge theory. This is a mixed motive, not a pure motive, so isn't constructed directly from linearizing algebraic varieties. 


All motives generated this way are mixed Tate motives, i.e extensions of powers of the Tate motive (the inverse of the Lefschetz motive), and are everywhere unramified. One can define the Tannakian category of everywhere unramified mixed Tate motives, and the Tannakian fundamental group is [known](http://www.ihes.fr/~brown/MTZ.pdf) to ask faithfully on the limit of these unipotent completions.


