Mild pro-p-groups and Galois groups of p-extensions of Q

摘要

In this paper we introduce a new class of finitely presented pro-p-groups G of cohomological dimension 2 called mild groups. If d(G), r(G) are respectively the minimal number of generators and relations of G, we give an infinite family of mild groups G with r(G)>= d(G) and d(G)>= 2 arbitrary. These groups can be constructed with G/G[G, G] finite, answering a question of Kuzmin. If G=G(S)(p) is the Galois group of the maximal p-extension of Q unramified outside a finite set of primes S and p not equal 2, we show that G is mild for a co-final class of sets S, even in the case p is not an element of S.