Galois action on mapping class groups

Abstract

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.