The Frattini module and p ′ - automorphisms of free pro - p groups

If a non-trivial subgroup A of the group of continuous automorphisms of a noncyclic free pro-p group F has finite order, not divisible by p, then the group of fixed points FixF (A) has infinite rank. The semi-direct product F>!A is the universal p-Frattini cover of a finite group G, and so is the projective limit of a sequence of finite groups starting with G, each a canonical group extension of its predecessor by the Frattini module. Examining appearances of the trivial simple module 1 in the Frattini module’s Jordan-Hölder series arose in investigations ([FK97], [BaFr02] and [Sem02]) of modular towers. The number of these appearances prevents FixF (A) from having finite rank. For any group A of automorphisms of a group Γ, the set of fixed points FixΓ(A) := {g ∈ Γ | α(g) = g, ∀α ∈ A} of Γ under the action of A is a subgroup of Γ. Nielsen [N21] and, for the infinite rank case, Schreier [Schr27] showed that any subgroup of a free discrete group will be free. Tate (cf. [Ser02, I.§4.2, Cor. 3a]) extended this to free pro-p groups. In light of this, it is natural to ask for a free group F , what is the rank of FixF (A)? When F is a free discrete group and A is finite, Dyer and Scott [DS75] demonstrated that FixF (A) is a free factor of F , i.e. F is a free product of FixF (A) and another free subgroup of F , thus bounding the rank of FixF (A) by that of F itself. That this bound would hold for A that are merely finitely generated was a conjecture attributed to Scott; this was proven first by Gersten [Ge87] and later, independently, by Bestvina and Handel [BH92] in a Communicated by 2000 Mathematics Subject Classification(s): 14G32, 20C20, 20D25, 20E05, 20E18, 20F14, 20F28. Supported by RIMS and Michael D. Fried, October 26 November 1, 2001. University of California, Irvine, Irvine, CA 92697-3875, USA E-mail address: [email protected]