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Galois action on mapping class groups. (English) Zbl 1406.14021
Summary: Let $$l$$ be a prime number. In the paper, we study the outer Galois action on the profinite and the relative pro-$$l$$ completions of mapping class groups of pointed orientable topological surfaces. In the profinite case, we prove that the outer Galois action is faithful. In the pro-$$l$$ case, we prove that the kernel of the outer Galois action has certain stability properties with respect to the genus and the number of punctures. Also, we prove a variant of the above results for arbitrary families of curves.

##### MSC:
 14H30 Coverings of curves, fundamental group 14H10 Families, moduli of curves (algebraic)
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