# On what kind of objects do the Galois groups act?

I am neither number theorist nor algebraic geometer. I am wondering whether Galois groups of number fields (say the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$) act on objects which are not related a priori to the number theory.

I am aware of two such situations of rather different nature:

(1) Grothendieck's dessins d'enfants: $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on certain graphs on 2-dimensional surfaces.

(2) $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on the profinite completion of the topological $K$-theory (of sufficiently nice spaces, e.g. finite $CW$-complexes).

As far as I understand (am I wrong?) the most important and best studied examples of actions of Galois groups are actions on $l$-adic cohomology of varieties over number fields. But this is not what I am looking for: number fields appear in the formulation of the problem from the vary beginning.

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There are some nice examples in knot theory and quantum algebra.

If $X$ is an algebraic variety over $\mathbb{Q}$, there is a canonical outer action $G_{\mathbb{Q}} \rightarrow Out(\hat{\pi}_1(X(\mathbb{C})))$ (which also exists for other fields than $\mathbb{Q}$ using the so-called algebraic fundamental group but I don't want to say something wrong about that). Roughly speaking, this is because finite covering of $X$ can be defined over $\bar{\mathbb{Q}}$, together with the relation between (regular) finite covering and finite quotients of the fundamental group. This action thus has the same origine as for dessin d'enfants.

Of particular interest is the study of this action in the case $X$ is the moduli space of algebraic curves of genus $g$ with $n$ marked points. This was suggested in Grothendieck's esquisse, and leads to the so-called Grothendieck-Teichmuller theory which gives a rather explicit description of a group which actually contains $G_{\mathbb{Q}}$. (see Cartographic group and flat stringy connection or Where is a good place to start learning about the Grothendieck-Teichmuller group?)

Actually, there are several flavours of the Grothendieck-Teichmuller group: a profinite one $\widehat{GT}$ which does contain $G_{\mathbb{Q}}$ and a group $GT(k)$ defined for every field $k$. A deep result of Drinfeld assert that this latter group is in some sense a universal automorphism group of braided monoidal categories. Indeed, it acts on the set of Drinfeld associator with coefficients in $k$.

Now, there is also a morphism

$G_{\mathbb{Q}}\rightarrow GT(\mathbb{Q}_{\ell})$

for every prime number $\ell$. Hence the absolute galois group acts on each kind of object in which associators come up (assuming that it is in a situation where one can work over $\mathbb{Q}_{\ell}$). It leads to, I think, quite surprizing examples like action on finite type invariants of knots and links, on quantization functor of Lie bialgebras, and several other constructions arising in deformation/quantization theory as they are often related to Drinfeld associators.

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The examples of knots and quantum algebra are new for me. Are they directly related to dessins d'enfants or the construction of the action of $G_{\mathbb{Q}}$ is really different? (Even if such a relation does exist, I can imagine that by itself it should be very surprising and far from obvious.) –  semyon alesker Jul 6 '11 at 13:32
So far I know, dessin d'enfants encodes finite covering of $P^1(\mathbb{C}-0,1,\infty$, which is isomorphic to the moduli space $M_{0,4}$ of Riemann spheres whith 4 marked points. On the other hand, as explained in the first link I pointed out, the Grothendieck-Teichmuller group is the automorphism group of the "tower" of fundamental groups of the $M_{0,n}$ equipped with natural geometric operations like adding or erasing points. So they have the same origin and in both case the faithfulness follows from Belyi's theorem. –  Adrien Jul 6 '11 at 14:19
Now these fundamental groups are close to braid group, which explain partially the relation with knots. But the precise role that it plays in braid theory, quantum algebra and so on really comes from the beautiful work of Drinfeld. –  Adrien Jul 6 '11 at 14:20
Is there a good place to read about it? –  semyon alesker Jul 6 '11 at 14:48
Well, you can check the link above wehre somebody asked a similar question, and the nice answer of Damien. From the algebraic perspective, you can read the paper of Bar Natan "On associators and the Grothendieck-Teichmuller group". For the relation with the absolute Galois group, there are a lot of surveyx on the webpage of Leila Schneps : math.jussieu.fr/~leila/articles.html –  Adrien Jul 6 '11 at 15:51

The work of Dennis Sullivan (see his ICM 1970 address for a summary) is a veritable source of such unexpected Galois actions.

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Thank you. Actually the example (2) I mentioned is a special case of the Sullivan's work you referred to. In fact I learned it from Sullivan's book "Geometric topology: localization, periodicity and Galois symmetry" which is probably a detailed exposition of that note. –  semyon alesker Jul 6 '11 at 13:16

This is not exactly an incarnation of the question you asked, in the sense that is not so much an action of a Galois group but rather an action whose existence is governed by a Galois group of number-theoretic origin, but it seems likely to be of interest.

Let $K$ be a number field, and let $K^{(1)}$ be the maximal unramified abelian extension of $K$. The Galois group of $K^{(1)}/K$ is a subquotient of Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$) which is isomorphic to the class group of $K$. Note that by Minhyong Kim's answer here, we can characterize this subquotient purely Galois-theoretically. Several authors have discovered surprising links between the arithmetic of number fields and actions of groups on spheres. In particular, when $K$ is the real cyclotomic field $K_m=\mathbb{Q}(\zeta_m+\zeta_m^{-1})$, the class group appears to govern the free actions of binary dihedral groups on spheres $S^n$ with $n\equiv 3\pmod{4}$. Let me loosely quote/paraphrase from Lang's "Units and Class Groups in Number Theory and Algebraic Geometry" (bolding mine):

C. T. C. Wall has already shown to depend in part on the 2-primary component of the ideal class group in real cyclotomic fields $K_m^+$ for suitable $m$...Using the algebraic background of a paper of Wall, applied to the surgery exact sequence, Thomas gives examples for the binary dihedral group $D_{4p}$ of order $4p$ operating freely on $S^{4k-1}$ with $k\geq 2$, when the order of $[(K_p^+)^{( 1)}:K_p^+]$ is odd.

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Furthermore, according to Thomas, there exist free actions by $D_{4p}$ which can be topologically distinguished only by an invariant in the 2-primary part of the ideal class group of $K_p^+$.

Perhaps needless to say, the study of these degrees $[(K_p^+)^{( 1)}:K_p^+]$, even their 2-part, is of tremendous interest in algebraic number theory (Vandiver's conjecture, etc.), so the link to actions on spheres is surprising.

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