Properties of mapping class groups rolated to Gdois representations

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Let X be a non-singular algebraic curve over a field k of characteristic 0 which is obtained from a complete curve of genus g (【greater than or equal】 0) by removing n (【greater than or equal】 0) k-rational points (2-2g-n<0), and l be a prime number. The absolute Galois group of k acts naturally on the algebraic fundamental group π^<alg>_1 of X【cross product】k^^- (or pro-l fundamental group (the maximal pro-l quotient of π^<alg>_1)) so that we obtain a Galois representation.Let us consider the moduli space M_<g, n>/Q (Q : the rationals) of n-pointed complete curves of genus g and the universal family of curves over M_<g, n>. Then the algebraic fundamental group of M_<g, n> acts naturally on that of the general fiber so that we have a monodromy representation. (A foundation has been given by T.Oda.) This is the Galois representation in the case that the curve X is the universal curve, k being the function field of M_<g, n>. Let π_1(g, n) and Γ^n_g denote the fundamental group and the ma … More pping class group of a Riemann surface of genus g (【greater than or equal】 0) with n (【greater than or equal】 0) punctures respectively. Then the algebraic fundamental group of M_<g, n> 【cross product】 Q^^- and that of the general fiber are isomorphic to Γ^^<^>^n_g and π^^<^>_1 (g, n) respectively (^ : profinite completion). The natural action ρ_<g, n> of Γ^^<^>^n_g on π^^<^>_1 (g, n) is nothing but the (geometric part of) the monodromy representation. The group Γ^^<^>^n_g acts also on the pro-l fundamental group π^<(l)>_1(g, n), which is the pro-l completion of π_1(g, n), and we obtain a Galois representation ρ^<(l)>_<g, n>. In this research, we have investigated the kernels of the representations ρ_<g, n> and ρ^<(l)>_<g, n>. So far, the kernel of ρ^<(l)>_<g, n> has been known only in the case of g=0. In the case that l=2, by applying the method to prove the faithfulness of ρ_<1, 1>, the kernel of ρ^<(2)>_<1, 1> has been determined.On the other hand, whether the center of any open subgroup of Γ^^<^>^n_g is trivial or not is an open problem. (This is related to whether M_<g, n > is "anabelian" or not). We have shown that, if the representation ρ_<g, n> is faithful, then the center of any open subgroup of Γ^^<^>^<n+1>_g is trivial. Less