Mild pro-p-Galois groups of algebraic number fields

本研究旨在进一步发展John LaBant提出的弱PRO-p-群理论，并研究其在限制分支代数数域扩张的对偶性质中的应用。中心对象由Galois群G_S(P)给出，它是在有限位置集S外的代数数域的极大p-扩张。研究了弱PRO-P-群的群论性质，特别是它们与高阶Massey积的关系。在加拿大蒙特利尔麦吉尔大学的逗留将提供与John Labute教授合作的机会。研究项目的目的如下：1)用伽罗瓦表示给出形式G_S(P)的Pro-p-群的高阶Massey积和Milnor不变量的描述，并构造具有任意Zassenhaus不变的温和Pro-p-群的显例2)研究FAB-Pro-P-群，利用其相关的分次李代数的性质给出非解析FAB-群的纯代数或群论描述，研究了代数数域的Fab-扩张的应用3)回答了Serre关于上同调维大于2的单相关准p-群按p次幂分类为自由群的商的问题

This research project aims at further developing the theory of mild pro-p-groups as introduced by John Labute as well as studying their applications to duality properties of extensions of algebraic number fields with restricted ramification. The central object is given by the Galois group G_S(p) of the maximal p-extensions of an algebraic number field unramified outside a finite set of places S in the tame case. It is proposed to investigate group theoretical properties of mild pro-p-groups, in particular their relation to higher order Massey products. Furthermore, the objective is to give arithmetic realizations of these groups.The stay at McGill University, Montreal, Canada would provide the opportunity of a collaboration with Professor John Labute.The goals of the research project are the following:1) Give a description of higher order Massey products and Milnor invariants of pro-p-groups of the form G_S(p) in terms of Galois representations and construct explicit examples of mild pro-p-groups with arbitrary Zassenhaus invariant2) Study FAB pro-p-groups, give a purely algebraic or group theoretical description of non-analytic FAB-groups using properties of their associated graded Lie algebras, study applications to FAB-extensions of algebraic number fields3) Give an answer of Serre's question about the classification of one-relator pro-p-groups of cohomological dimension greater than 2 as quotients of free groups by p-th powers