Properties of mapping class groups related to Galois 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 1 be a prime number. The absolute Galois group of k acts naturally on the algebraic fundamental group pi_1^<alg> of X <cross product> k (or pro-1 fundamental group (the maximal pro-l quotient of pi_1^<alg>)) 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>. In this research, we have investigated the faithfulness of this monodromy representation and have shown that, in the case that g = 0, 1, it is faithful for all n.In investigating Galois representations, one of basic properties of the group pi_1^<alg> (*pi_1^<alg>(g, n) ) is that it is center free (M.P.Anderson). In this research we have shown, by a comparatively elementary way, a property which is stronger than this.The braid group of a finitely generated free nilpotent pro-I group can be regarded as approximating the genus 0 pro-l mapping class groups. In this research we have shown that this group has a structure an algebraic group which is independent of l.