Generators for the automorphism groups of free metabelian pro-p-groups

Abstract

The automorphism groups of pro-finite groups attract attention of researchers. A number of inter- esting results and conjectures are contained in the articles by A. Lubotzky, D. Gildenhuys,. W. Herfort, and L. Ribes [1- 3]. At the conference on pro-finite groups in Oberwolfach (1990), A. Lubotzky for- mulated the problem: is the automorphism group of a pro-p-group of finite rank finitely generated? The generation is understood in the topological sense. Several well-known results on generators of the automorphism group of a finite rank free group in a nilpotent variety could be of value in describing generators of the automorphism group of a free pro-p-group. Different finite generating sets for the above-mentioned groups were given in [4-7]. Also, therein the principal obstacle to transferring finite generation to the case of pro-p-groups was delineated. The fact is that, for a free group of rank n _> 2 in a nilpotent Variety of class k > 3, the number of the automorphism group generators essentially depends on k in the descriptions available. In all probability this dependence is unavoidable. The present article is devoted to studying generators of the automorphism and IA-automorphism groups for free groups of finite rank in varieties of metabelian pro-p-groups. Preliminary information is collected in w Some profinite analogues of the Fox derivations are also defined there. They originated in the generalization, of the well-known Magnus embedding of a free metabelian group into a matrix group to the case of free metabelian pro-p-groups, that was obtained by V. N. Remeslennikov in [8]. In order to analyze the automorphism group of'a free metabelian pro-p-group, in w we present a profinite analogue of the well-known Bachmuth embedding [9] of the IA-automorphism group of a free metabelian group into the group of matrices over the ring of Laurent polynomials with integer coefficients. In our case, the group is represented by matrices over the ring of power series over p-adic integers. We also consider the case in which coefficients are chosen in a field with p elements. It is a free metabelian pro-p-group with the commutator subgroup of prime exponent that arises naturally in the situation in question. It turned out that the questions dealt with are related to the following: are the p-adic modules (or the modules over a field with p elements) that arise in studying the above series finitely generated (in the usual or generalized sense)? The key part of this paper, w is entirely devoted to the power series. Main results are duly stated and proved in w in the statements, p is prime. It is established that the IA-automorphism group of a free metabelian pro-p-group of rank n _> 9. with the commutator subgroup of prime exponent is infinitely generated (Theorem 1). This implies that the IA-automorphism groups of free metabelian groups and free pro-p-groups of rank n >_ 2 are infinitely generated (Corollary 1). Moreover, the number of generators of the IA-automorphism group of a free metabelian nilpotent group of class p and rank n _> 2 is proved to be infinitely increasing in k (Theorem 4). An analogous assertion holds without the assumption that the group is metabelian (Corollary 3). The automorphism group of a free rnetabelian pro-p-group of rank 2 with the commutator sub- group of prime exponent is finitely generated (Theorem 2). Its subgroup of automorphisms with determinant 1 (which is calculated according to the Bachmuth embedding) coincides with the inner Omsk. Translated from