Finite Extensions of Free Profinite Groups

From 1996 -- 2000 there was collaboration, supported by the ASF between P.A.Zalesskii (Belorussian Academy of Sciences, now UNB Brasilia), who meanwhile is a leading expert in the field of our project, and the author of this report. Allow me to provide an explanation of the project and its results dedicated rather to the Higher-Algebra-layman. Starting from something rather concrete, suppose somebody has a ball of wool, such that beginning and end are tied together, and he asks himself whether or not one can unravel it. A bit childish, my example, maybe to think about a computer board with conducting pathes which are not allowed to cross. One method, in order to mathematically decide the question of unravelling starts by considering space without wool. This is a connected topological space and one may consider pathes starting in a once forever fixed point and terminating there. Define a composition of any two such closed pathes just by first running through the first one, and then continue through the second one. Moreover two pathes will be consider to be equivalent, if one can deform them into each other without crossing the thread of wool. In this way the algebraic structure of a group arises, the fundamental group. In Austria around 1908 W.Wirtinger and O.Schreier investigated such groups. Knowing the algebraic structure of the group, one can decide wether or not one can unravel the ball of wool. Coming to the project. As known E.Galois 1832 successfully employed the notion of group to solving polynomial equations. He first showed how to construct from the field of rational numbers an algebraic extension by adjoining the roots of a polynomial equation. Doing so, one arrives at a complicated, but important extension field, the algebraic closure. The latter possesses a set of transformations into itself, which fixes every rational number. This is the infinite Galois group of the rationals, and its structure is far from being known. Its knowledge would lead to deeper insights even into approximation of certain real numbers. It is here, where our project yields an algebraic description of an important class of infinite Galois groups. Our groups possess a subgroup of finite p-power-index (where p is a fixed prime number) which is free pro-p,i.e., it has a system of generators without relations. Groups of this sort appear as the fundamental groups of certain spaces which have been glued from smaller pieces, as well as the Galois groups of extensions of so-called function fields, the most simple example consisting of all kind of fractions with numerator and denominator a polynomial. For function fields one can show the existence of a kind of Riemann surface such that its fundamental group appears embedded in a special way in the (ifinite) Galois group. A characterisation of fundamental groups of the above form has been provided (in even more general form) by H.Bass, J.P.Serre, U.Karras-A.Pietrowski-D.Solitar, P.Scott and D.Cohen. In our project, starting from work of O.V.Melnikov and P.A.Zalesskii the respective algebraic description for these infinite Galois groups has been established. Parts of the results have already taken into account in monographies on the subject.