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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 dðGÞ, rðGÞ are respectively the minimal number of generators and relations of G, we give an infinite family of mild groups G with rðGÞ f dðGÞ and dðGÞ f 2 arbitrary. These groups can be constructed with G=1⁄2G; G finite, answering a question of Kuzmin. If G 1⁄4 G S ðpÞ is the Galois group of the maximal pextension of Q unramified outside a finite set of primes S and p 3 2, we show that G is mild for a co-final class of sets S, even in the case p B S. ## Statement of results 1.1. Mild pro-p-groups. Let p be a prime number; for simplicity, we assume that p is odd. Let G 1⁄4 F =R be a finitely presented pro-p-group with F the free pro-p-group on x 1 ; . . . ; x m and R 1⁄4 ðr 1 ; . . . ; r d Þ the closed normal subgroup generated by nonidentity elements r 1 ; . . . ; r d A F p 1⁄2F ; F , where 1⁄2F ; F is the closed subgroup generated by the commutators 1⁄2x; y 1⁄4 x À1 y À1 xy. Let dðGÞ 1⁄4 dim F p H 1 ðG; Z=pZÞ and rðGÞ 1⁄4 dim F p H 2 ðG; Z=pZÞ. Then dðGÞ, rðGÞ are respectively the minimal number of generators and relations for G (cf. [25]). For the above

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Author

John Labute

Publisher

Walter de Gruyter GmbH

Published

2006

Language

EN

Field

Mathematics (Physical Sciences)