In a couple of days I’ll be blogging for 4 years… and I’m in the process of resurrecting about 300 posts from a database-dump made in june. For example here’s my first post ever which is rather naive. This conversion program may last for a couple of weeks and I apologize for all unwanted pingbacks it will produce.

I’ll try to convert chunks of related posts in one go, so that I can at least give them correct self-references. Today’s work consisted in rewriting the posts of my virtual course, in march of this year, on dessins d’enfants and its connection to noncommutative geometry (a precursor of what Ive been blogging about recently). These posts were available through the PDF-archive but are from now on open to the internal search-function. Here are the internal links and a short description of their contents

• The best rejected proposal ever on Grothendieck’s programme and Belyi maps.
• Monsieur Mathieu on the dessins of and .
• The cartographers’ groups 1 on the free product structure of and related groups.
• The cartographers’ groups 2 continuation and a simpler proof due to Alperin.
• Anabelian geometry on sources of representations of the modular group.
• The noncommutative manifold of a Riemann surface a mental picture to get all finite dimensional representations of the coordinate ring of a curve.
• Noncommutative curves and their manifolds extension to twisted curves.
• Permutation representations of monodromy groups how dessins give such representations.

Besides, I’ve added a few scattered old posts, many more to follow…

Last time (possibly with help from the survival guide) we have seen that the universal map from the modular group to its profinite completion (limit over all finite index normal subgroups ) gives an embedding of the sets of (continuous) simple finite dimensional representations

and based on the example we would like the above embedding to be dense in some kind of noncommutative analogon of the Zariski topology on .

We use the Zariski topology on as in these two M-geometry posts¹. So, what’s this idea in this special case? Let be the vectorspace with basis the conjugacy classes of elements of (that is, the space of class functions). As explained here it is a consequence of the Artin-Procesi theorem that the linear functions separate finite dimensional (semi)simple representations of . That is we have an embedding

and we can define closed subsets of as subsets of simple representations on which a set of class-functions vanish. With this definition of Zariski topology it is immediately clear that the image of is dense. For, suppose it would be contained in a proper closed subset then there would be a class-function vanishing on all simples of so, in particular, there should be a bound on the number of simples of finite quotients which clearly is not the case (just look at the quotients ).

But then, the same holds if we replace ’simples of ‘ by ’simple components of permutation representations of ‘. This is the importance of Farey symbols to the representation problem of the modular group. They give us a manageable subset of simples which is nevertheless dense in the whole space. To utilize this a natural idea might be to ask what such a permutation representation can see of the modular group, or in geometric terms, what the tangent space is to in a permutation representation². We will call this the modular content of the permutation representation and to understand it we will have to compute the tangent quiver .

1. already, I regret terminology, I should have just called it noncommutative geometry [↩]
2. more precisely, in the ‘cluster’ of points making up the simple components of the representation representation [↩]

This is how my attention was drawn to what I have since termed anabelian algebraic geometry, whose starting point was exactly a study (limited for the moment to characteristic zero) of the action of absolute Galois groups (particularly the groups , where K is an extension of finite type of the prime field) on (profinite) geometric fundamental groups of algebraic varieties (defined over K), and more particularly (breaking 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 , there is the profinite compactification of the modular group , whose quotient by its centre 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).

The above text is taken from Alexander Grothendieck’s visionary text Sketch of a Programme. He was interested in the permutation representations of the modular group as they correspond via Belyi-maps and his own notion of dessins d’enfants to smooth projective curves defined over . One can now study the action of the absolute Galois group on these curves and their associated dessins. Because every permutation representation of factors over a finite quotient this gives an action of the absolute Galois group as automorphisms on the profinite compactification

where the limit is taken over all finite index normal subgroups . In this way one realizes the absolute Galois group as a subgroup of the outer automorphism group of the profinite group . As a profinite group is a compact topological group one should study its continuous finite dimensional representations which are precisely those factoring through a finite quotient. In the case of the simple continuous representations are precisely the components of the permutation representations of the modular group. So in a sense, anabelian geometry is the study of these continuous simples together wirth the action of the absolute Galois group on it.

In noncommutative geometry we are interested in a related representation theoretic problem. We would love to know the simple finite dimensional representations of the modular group as this would give us all simples of the three string braid group . So a natural question presents itself : how are these two ‘geometrical’ objects (anabelian) and (noncommutative) related and can we use one to get information about the other?

This is all rather vague so far, so let us work out a trivial case to get some intuition. Consider the profinite completion of the infinite Abelian group

As all simple representations of an Abelian group are one-dimensional and because all continuous ones factor through a finite quotient we see that in this case

is the set of all roots of unity. On the other hand, the simple representations of are also one-dimensional and are determined by the image of the generator so

Clearly we have an embedding and the roots of unity are even dense in the Zariski topology. This might look a bit strange at first because clearly all roots of unity lie on the unit circle which ’should be’ their closure in the complex plane, but that’s because we have a real-analytic intuition. Remember that the Zariski topology of is just the cofinite topology, so any closed set containing the infinitely many roots of unity should be the whole space!

Let me give a pedantic alternative proof of this (but one which makes it almost trivial that a similar result should be true for most profinite completions…). If is the generator of then the different conjugacy classes are precisely the singletons . Now suppose that there is a polynomial vanishing on all the continuous simples of then this means that the dimensions of the character-spaces of all finite quotients should be bounded by (for consider as the character of ), which is clearly absurd.

Hence, whenever we have a finitely generated group for which there is no bound on the number of irreducibles for finite quotients, then morally the continuous simple space for the profinite completion

should be dense in the Zariski topology on the noncommutative space of simple finite dimensional representations of . In particular, this should be the case for the modular group .

There is just one tiny problem : unlike the case of for which this space is an ordinary (ie. commutative) affine variety , what do we mean by the “Zariski topology” on the noncommutative space ? Next time we will clarify what this might be and show that indeed in this case the subset

will be a Zariski closed subset!

Here a list of saved pdf-files of previous NeverEndingBooks-posts on geometry in reverse chronological order.

(more…)

Last time we saw that a curve defined over gives rise to a permutation representation of or one of its subgroups (of index 2) or (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 , which is the inverse limit of finite groups
where N ranges over all normalsubgroups of finite index. These profinte completions are horrible beasts even for easy groups such as . Its profinite completion is

where the right hand side product of p-adic integers ranges over all prime numbers! The absolute Galois group acts on all curves defined over and hence (via the Belyi maps ans the corresponding monodromy permutation representation) there is an action of 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 of an algebraic curve defined over the prime field itself, it follows that the Galois group of over acts on the category of these maps in a natural way.
For instance, the operation of an automorphism on a spherical map given by the rational function above is obtained by applying to the coefficients of the polynomials P , Q. Here, then, is that mysterious group 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 , 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 of
where 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 termed anabelian algebraic geometry, whose starting point was exactly a study (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 , there is the profinite compactification of the modular group , 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 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 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 , 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 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 on its profinite geometric fundamental group), we can thus say that
all algebraic curves defined over number fields are contained in the profinite compactification and in the knowledge of a certain subgroup G of its group of outer automorphisms!

To study the absolute Galois group 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 where d is the number of edges in the dessin generated by the partitions and 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 though they are not conjugated as subgroups of .

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 there are other sources of semi-simple 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 and hence all their representations are also semi-simple -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 and that at least 12 of then are epimorphic images of .

Rational conformal field theories Another source of 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 -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…