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Description

Let $K$ be a field and let $overline K$ be its algebraic closure, i.e. the field formed by adding to $K$ all roots of polynomials defined over $K$. The absolute Galois group $G_K={ m Aut}_K(overline K)$ is defined to be the group of automorphisms of $overline K$ which fix $K$. But $G_K$ acts on many kinds of objects associated to $K$, not just algebraic numbers. $G_K$ is a profinite group which is sometimes extremely mysterious, and never more so than when $K$ is the field ${sf Q}$ of rational numbers, the case considered in this talk. Our approach to $G_{sf Q}$ is to consider its action on two kinds of topological objects:

* {it dessins d'enfants}, which are cellular graphs embedded in topological surfaces, and

* {it diffeomorphisms} of topological surfaces (modulo those isotopic to the identity).

In studying the Galois action on dessins d'enfants, we ask ourselves above all the following question: {it can we find Galois invariants of dessins of a topological nature ? and can we find enough to characterize the Galois action?} In studying the action on diffeomorphisms, we obtain a parametrization of each element $sigmain G_{sf Q}$ as a pair $(lambda,f)$ with $lambda$ in the profinite completion of ${sf Z}$ and $f$ in the profinite free group $F_2$ on two generators. Here the main question is: {it can we find properties of pairs $(lambda,f)$ ensuring that they come from an element of $G_{sf Q}$?} In some sense, these questions come down to asking if the language of topology is rich enough to use for the study of arithmetic.