Infinite generation of automorphism groups of free pro-p groups

Abstract

V. A. Roman~kov UDC 512.543.16:512.546.37 Introduction The main notions and facts pertaining to analytic and profinite groups are contained in [1] (see also [2-4]) and used in what follows without special references. The terms "subgroup" and "generators" are understood in the topological sense. We say that a group is infinitely generated if there is no finite set of generators for it. Let G be a profinite group. Denote by Aut G the group of all its topological automorphisms. The group Aut G is endowed with the naturally-defined topology whose base of identity neighborhoods is formed by congruence subgroups of the form F(N) = {~ e AutGt (G,7) _< N}, where N ranges over the set of all open normal subgroups of G. Observe that an automorphism 7 belongs to F(N) iff 7(N) = N and 3, induces the identity map on the quotient group G/N. With this topology, G is a Hausdorff topological group. The group Aut G is, in general, not profinite. Nevertheless, if G is finitely generated then Aut G is profinite. If G is a finitely generated pro-p group then every abstract automorphism 7 of G is continuous. Define the Frattini series of the group G by putting P,(C) = a, P2(a) = r = a,(a,a),..., P~+,(a) = r = P~(a),(P~(a),a),..., where the bar indicates closure. The group Aut (G, ep(G)) of all automorphisms of G which induce the identity map on the quotient group G/~(G) is a pro-p group; it is open and normal in Aut G and has finite index in it. A topological group G is called a p-adic analytic group if G is equipped with the structure of a p-adic analytic manifold so that the multiplication and the taking of inverses are analytic in G. A pro-p group Go is referred to as uniformly saturated if the following conditions are satisfied: (1) Go is finitely generated; (2) the quotient group