Properties of mapping class groups related to Galois representations

设X是特征为0的域k上的一条非奇异代数曲线，它是亏格g(&lt；大于或等于&gt；0)的完全曲线去掉n(&lt；大于或等于&gt；0)k-有理点(2-2g-n.&lt；0)而得到的，1是素数。K的绝对伽罗瓦群自然作用于X&lt；叉积&gt；k的代数基础群pi_1^&lt；alg&gt；(或PRO-1基本群(pi_1^&lt；alg&gt；的最大PRO-L商)，从而得到伽罗瓦表示。让我们考虑亏格G的n点完全曲线的模空间M_&lt；g，n&gt；/q(q为有理数)和M_&lt；g，n&gt；上的泛曲线族。然后，M&lt；g，n&gt；的代数基本群自然地作用于一般纤维的代数基本群，因此我们有一个单行表示。(T.Oda已经提供了一个基金会。)这是在曲线X是泛曲线的情况下的伽罗瓦表示，k是M&lt；g，n&gt；的函数域。在研究Galois表示时，群pi_1^&lt；alg&gt；(*pi_1^&lt；alg&gt；(g，n))的一个基本性质是它是无中心的(M.P.Anderson)。在这一研究中，我们用一个比较初等的方法证明了一个比这更强的性质：有限生成的自由幂零PRO-I群的辫子群可以看作近似于亏格0的PRO-L映射类群。在这一研究中，我们证明了这个群有一个独立于L的代数群结构。

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.