Last time we saw

that a curve defined over $\overline{\mathbb{Q}} $ gives rise

to a permutation representation of $PSL_2(\mathbb{Z}) $ or one

of its subgroups $\Gamma_0(2) $ (of index 2) or

$\Gamma(2) $ (of index 6). As the corresponding

monodromy group is finite, this representation factors through a normal

subgroup of finite index, so it makes sense to look at the **profinite
completion** of $SL_2(\mathbb{Z}) $, which is the inverse limit

of finite

groups $\underset{\leftarrow}{lim}~SL_2(\mathbb{Z})/N $

where N ranges over all normalsubgroups of finite index. These

profinte completions are horrible beasts even for easy groups such as

$\mathbb{Z} $. Its profinite completion

is

$\underset{\leftarrow}{lim}~\mathbb{Z}/n\mathbb{Z} =

\prod_p \hat{\mathbb{Z}}_p $

where the right hand side

product of p-adic integers ranges over all prime numbers! The

_absolute Galois group_

$G=Gal(\overline{\mathbb{Q}}/\mathbb{Q}) $ acts on all curves

defined over $\overline{\mathbb{Q}} $ and hence (via the Belyi

maps ans the corresponding monodromy permutation representation) there

is an action of $G $ on the profinite completions of the

carthographic groups.

This is what Grothendieck calls **anabelian
algebraic geometry**

Returning to the general

case, since finite maps can be interpreted as coverings over

$\overline{\mathbb{Q}} $ of an algebraic curve defined over

the prime field $~\mathbb{Q} $ itself, it follows that the

Galois group $G $ of $\overline{\mathbb{Q}} $ over

$~\mathbb{Q} $ acts on the category of these maps in a

natural way.

For instance, the operation of an automorphism

$~\gamma \in G $ on a spherical map given by the rational

function above is obtained by applying $~\gamma $ to the

coefficients of the polynomials P , Q. Here, then, is that

mysterious group $G $ intervening as a transforming agent on

topologico- combinatorial forms of the most elementary possible

nature, leading us to ask questions like: are such and such oriented

maps ‚conjugate or: exactly which are the conjugates of a given

oriented map? (Visibly, there is only a finite number of these).

I considered some concrete cases (for coverings of low degree) by

various methods, J. Malgoire considered some others ‚ I doubt that

there is a uniform method for solving the problem by computer. My

reflection quickly took a more conceptual path, attempting to

apprehend the nature of this action of G.

One sees immediately

that roughly speaking, this action is expressed by a certain

outer action of G on the profinite com- pactification of the

oriented cartographic group $C_+^2 = \Gamma_0(2) $ , and this

action in its turn is deduced by passage to the quotient of the

canonical outer action of G on the profinite fundamental group

$\hat{\pi}_{0,3} $ of

$(U_{0,3})_{\overline{\mathbb{Q}}} $ where

$U_{0,3} $ denotes the typical curve of genus 0 over the

prime field Q, with three points re- moved.

This is how my

attention was drawn to what I have since termedanabelian, whose starting point was exactly a study

algebraic geometry

(limited for the moment to characteristic zero) of the action of

absolute Galois groups (particularly the groups Gal(K/K),

where K is an extension of finite type of the prime field) on

(profinite) geometric fundamental groups of algebraic varieties

(defined over K), and more particularly (break- ing with a

well-established tradition) fundamental groups which are very far

from abelian groups (and which for this reason I call

anabelian).

Among these groups, and very close to

the group $\hat{\pi}_{0,3} $ , there is the profinite

compactification of the modular group $Sl_2(\mathbb{Z}) $,

whose quotient by its centre ±1 contains the former as congruence

subgroup mod 2, and can also be interpreted as an oriented

cartographic group, namely the one classifying triangulated

oriented maps (i.e. those whose faces are all triangles or

monogons).

and a bit further, on page

250

I would like to conclude this rapid outline

with a few words of commentary on the truly unimaginable richness

of a typical anabelian group such as $SL_2(\mathbb{Z}) $

doubtless the most remarkable discrete infinite group ever

encountered, which appears in a multiplicity of avatars (of which

certain have been briefly touched on in the present report), and which

from the point of view of Galois-Teichmuller theory can be

considered as the fundamental ‚building block‚ of the

Teichmuller tower

The element of the structure of

$Sl_2(\mathbb{Z}) $ which fascinates me above all is of course

the outer action of G on its profinite compactification. By

Bielyi’s theorem, taking the profinite compactifications of subgroups

of finite index of $Sl_2(\mathbb{Z}) $, and the induced

outer action (up to also passing to an open subgroup of G), we

essentially find the fundamental groups of all algebraic curves (not

necessarily compact) defined over number fields K, and the outer

action of $Gal(\overline{K}/K) $ on them at least it is

true that every such fundamental group appears as a quotient of one

of the first groups.

Taking the anabelian yoga

(which remains conjectural) into account, which says that an anabelian

algebraic curve over a number field K (finite extension of Q) is

known up to isomorphism when we know its mixed fundamental group (or

what comes to the same thing, the outer action of

$Gal(\overline{K}/K) $ on its profinite geometric

fundamental group), we can thus say that

all algebraic

curves defined over number fields are contained in the profinite

compactification $\widehat{SL_2(\mathbb{Z})} $ and in the

knowledge of a certain subgroup G of its group of outer

automorphisms!

To study the absolute

Galois group $Gal(\overline{\mathbb{\mathbb{Q}}}/\mathbb{Q}) $ one

investigates its action on dessins denfants. Each dessin will be part of

a finite family of dessins which form one orbit under the Galois action

and one needs to find invarians to see whether two dessins might belong

to the same orbit. Such invariants are called _Galois invariants_ and

quite a few of them are known.

Among these the easiest to compute

are

- the valency list of a dessin : that is the valencies of all

vertices of the same type in a dessin - the monodromy group of a dessin : the subgroup of the symmetric group $S_d $ where d is

the number of edges in the dessin generated by the partitions $\tau_0 $

and $\tau_1 $ For example, we have seen

before that the two

Mathieu-dessins

form a Galois orbit. As graphs (remeber we have to devide each

of the edges into two and the midpoints of these halfedges form one type

of vertex, the other type are the black vertices in the graphs) these

are isomorphic, but NOT as dessins as we have to take the embedding of

them on the curve into account. However, for both dessins the valency

lists are (white) : (2,2,2,2,2,2) and (black) :

(3,3,3,1,1,1) and one verifies that both monodromy groups are

isomorphic to the Mathieu simple group $M_{12} $ though they are

not conjugated as subgroups of $S_{12} $.

Recently, new

Galois invariants were obtained from physics. In Children’s drawings

from Seiberg-Witten curves

the authors argue that there is a close connection between Grothendiecks

programme of classifying dessins into Galois orbits and the physics

problem of classifying phases of N=1 gauge theories…

Apart

from curves defined over $\overline{\mathbb{Q}} $ there are

other sources of semi-simple $SL_2(\mathbb{Z}) $

representations. We will just mention two of them and may return to them

in more detail later in the course.

**Sporadic simple groups and
their representations** There are 26 exceptional finite simple groups

and as all of them are generated by two elements, there are epimorphisms

$\Gamma(2) \rightarrow S $ and hence all their representations

are also semi-simple $\Gamma(2) $-representations. In fact,

looking at the list of ‘standard generators’ of the sporadic

simples

(here the conjugacy classes of the generators follow the

notation of the Atlas project) we see that all but

possibly one are epimorphic images of $\Gamma_0(2) = C_2 \ast

C_{\infty} $ and that at least 12 of then are epimorphic images

of $PSL_2(\mathbb{Z}) = C_2 \ast

C_3 $.

**Rational conformal field theories** Another

source of $SL_2(\mathbb{Z}) $ representations is given by the

modular data associated to rational conformal field theories.

These

representations also factor through a quotient by a finite index normal

subgroup and are therefore again semi-simple

$SL_2(\mathbb{Z}) $-representations. For a readable

introduction to all of this see chapter 6 \”Modular group

representations throughout the realm\” of the

book Moonshine beyond the monster the bridge connecting algebra, modular forms and physics by Terry

Gannon. In fact, the whole book

is a good read. It introduces a completely new type of scientific text,

that of a neverending survey paper…