ON AUTOMORPHISMS OF FREE PRO-p-GROUPS I

Let F be a (topologically) finitely generated free pro-p-group, and ß an automorphism of F . If p ^ 2 and the order of ß is 2 , then there is some basis of F such that ß either fixes or inverts its elements. If p does not divide the order of ß , then the subgroup of F of all elements fixed by ß is (topologically) infinitely generated; however this is not always the case if p divides the order of ß . Let p be a fixed prime number, and let F be a free pro-/?-group of finite rank. In this paper we study (continuous) automorphisms of F. The group of automorphisms Aut(F) of F is, in a natural way, a profinite group. In [9], Lubotzky gives global properties of the group Aut(F). Our interest here is rather more local; we describe properties of certain types of automorphisms. The group Aut(F) contains a pro-p-subgroup of finite index; hence the automorphisms of order prime to p must have finite order. In p / 2, we show that given an automorphism <p of order 2 of a finitely generated pro-p-group G (in particular, a free one), there is a minimal set of generators of G such that <p sends each of the generators in that set to itself or its inverse. As a consequence we can describe all the conjugacy classes of involutions of Aut(F) : they correspond bijectively to those of GL(« ,p), where ranki7 = n . In [4], Gersten proves that if a is an automorphism of an abstract free group of finite rank, then the elements of the group fixed by a form a subgroup of finite rank also. In contrast, in §3 we show that for a free pro-p-group F of finite rank, the equivalent result need not hold; in fact we prove that if the order of ß € Aut(F) is not divisible by p, and ß is not the identity, then the subgroup of the elements of F fixed by ß is necessarily infinitely generated (i.e. such a subgroup contains no dense subgroup which is finitely generated as an abstract group). This result depends strongly on the fact that the order of the automorphism does not involve the prime p . In fact, in §4 we Received by the editors November 17, 1988 and, in revised form, February 13, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 20E18; 20F28. ©1990 American Mathematical Society 0002-9939/90 $1.00+ $.25 per page