American Journal of Mathematics

Let X be an algebraic curve of genus g, n-punctured, defined over a number field K. Then, the profinite or the pro-l completion of the topological fundamental group of X admits two actions: the action of the profinite completion of the mapping class group of the orientable surface of topological type (g, n) and the action of the absolute Galois group of K. This paper compares these two. In the profinite case, it is shown that the intersection of the images of these two actions is trivial if X is affine and its fundamental group is nonabelian. On the contrary, in the pro-l case, there are many curves such that the image of the Galois action contains the image of the mapping-class-group action. It is proved that the set of points corresponding to such curves is dense in the moduli space of (g, n)-curves.