# Teichmuller groupoids in Grothendieck's esquisse d'un programme

Grothendieck in his Esquisse d'un programme mentioned without any precise definition and construction that the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on the whole "tower" of the Teichmuller groupoids $\hat T_{g,\nu}$ (see p.5 in the above reference).

I do understand what does that mean in a very special case $g=0,\nu=4$.

What does it mean in general? What are these Teichmuller groupoids? Is there a more detailed exposition of the above general remark of Grothendieck?

ADD: If I understand correctly, $\hat T_{g,\nu}$ is very close to be a profinite completion of the fundamental group of the moduli space $M_{g,\nu}$ of smooth Riemann surfaces of genus $g$ with $\nu$ marked points (am I wrong?). Grothendieck claims that there are various natural morphisms between $\hat T_{g,\nu}$'s. Frankly I do not see any morphisms except $\hat T_{g,\nu}\to \hat T_{g,\mu}$ for $\mu<\nu$ induced by the map $M_{g,\nu}\to M_{g,\mu}$ which is just forgetting several marked points. Are there any other morphisms?

There is another type of important morphism between the (orbifold) fundamental groups of the moduli spaces $M_{g,\nu}\rightarrow M_{g',\nu'}$ that is considered in Grothendieck's tower. You can see this morphism in three different ways. One is directly on the surfaces of type $(g,\nu)$ and $(g',\nu')$ (of genus $g$ with $\nu$ boundary components, resp. genus $g'$ with $\nu'$ boundary components). This morphism exists if you can put a set of disjoint simple closed loops on the surface of type $(g',\nu')$ such that when you cut along them, you cut your surface into one piece of type $(g,\nu)$, or else into several pieces of which at least one is of type $(g,\nu)$. You can also think of including the smaller surface of type $(g,\nu)$ into the bigger one by gluing it to other smaller pieces along the edges of their boundary components, to form the bigger one of type $(g',\nu')$ (which is the image Grothendieck had in mind when he talked about Lego).

The second way to see this morphism is as a morphism of moduli spaces, where $M_{g,\nu}$ is mapped to a boundary component of the Deligne-Mumford compactification $\overline{M}_{g',\nu}$, in fact precisely the boundary component corresponding to taking the simple closed loops on the surface of type $(g',\nu')$ that "cut out" the one of type $(g,\nu)$ and shrinking them to length zero, so they become nodes.

The third way to view this same morphism is on the fundamental groups. This is pretty easy, since the (orbifold) fundamental group of $M_{g,\nu}$ is generated by Dehn twists along simple closed loops on the surface of type $(g,\nu)$, and these just map to the Dehn twists along the same simple closed loops when the $(g,\nu)$ surface is included in the $(g',\nu')$ one as above.

The Teichmüller tower can be considered to be the collection of all the fundamental groups of the $M_{g,\nu}$ linked by the point-erasing morphisms and by these. Or, as Grothendieck wanted, instead of fundamental groups, that depend on a certain choice of base point, you can replace the groups by more symmetric fundamental groupoids based at all "tangential base points" on the moduli spaces".

The automorphism group of the Teichmüller tower basically then consists of tuples $(\phi_{g,\nu})$ such that each $\phi_{g,\nu}$ is an automorphism of $\pi_1(M_{g,\nu})$ and the different $\phi_{g,\nu}$ in the same tuple commute with the homomorphisms of the tower.

• Belated welcome to MO! – David Roberts Oct 13 '17 at 5:35
• Many thanks! Is there a reference to these three specific constructions? Given my modest background on the subject, I would be happy to see more details. For example, in the second construction I am not sure how the map of $M_{g,\nu}$ to a boundary component of $\overline{M_{g',\nu}}$ induces a homomorphism $\pi_1(M_{g,\nu})\to \pi_1(M_{g',\nu})$. Probably to have such a map one should have that the natural inclusion map $M_{g',\nu}\to \overline{M_{g',\nu}}$ induces an isomorphism of the fundamental groups. Is it what you meant? – MKO Oct 13 '17 at 7:23
• Say you have a surface of type $(g',\nu')$ and you put one or more disjoint closed loops on it such that when you cut along them, you cut your surface into two pieces of types $(g_1,\nu_1)$ and $(g_2,\nu_2)$. Then the boundary component of $M_{g',\nu'}$ corresponding to curves with the disjoint closed loops you chose shrunk to nodes is isomorphic to the product of moduli spaces $M_{g1,\nu_1}\otimes M_{g_2,\nu_2}$. And more generally this holds for any number of pieces that you cut your original surfaces into. – Leila Schneps Oct 15 '17 at 4:12
• The map $\pi_1(M_{g,\nu})\rightarrow\pi_1(\overline{M}_{g,\nu})$ is trivial because the compactification is simply connected. It's not the same thing. A reference to the Deligne-Mumford compactification is the original Deligne-Mumford article, but to find info about the "subsurface-inclusion" morphisms see Hatcher-Lochak-Schneps in Crelle, 2000. – Leila Schneps Oct 15 '17 at 4:17
• PS. The tensor product above was meant to be just a product of spaces $M_{g_1,\nu_1}\times M_{g_2,\nu_2}$. – Leila Schneps Oct 15 '17 at 4:19