Abstract

Finitely generated virtually free pro-p groups are described. This generalizes Serre's result, stating that a torsion free virtually free pro-p group is free pro-p.

As a consequence of our main result certain finite subgroups and their conjugacy classes in the automorphism group of a finitely generated free pro-p group are classified.

Introduction

Let p be a prime number, and G a pro-p group containing an open free pro-p subgroup F. If G is torsion free, then, according to the celebrated theorem of Serre in [17], G itself is free pro-p.

The main objective of the announcement is to present a description of virtually free pro-p groups without the assumption of torsion freeness.

Theorem ASuppose G is a finitely generated pro-p group with an open free pro-p subgroup F. Then G is the fundamental pro-p group of a finite graph of finite p groups of order bounded by |G : F|

The theorem is the pro-p analog of the description of finitely generated virtually free discrete groups proved by Karrass, Pietrovski and Solitar [12]. In fact as a consequence we obtain that a finitely generated virtually free pro-p group is the pro-p completion of a virtually free discrete group. However, the discrete result is not used (and cannot be used) in the proof.

In the characterization of discrete virtually free groups Stallings theory of ends played a crucial role. One does not have such a powerful tool in the pro-p situation.