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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 ] 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  ≠ 2, we show that  G  is mild for a co-final class of sets  S , even in the case  p  ∉  S .

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