Cyclic Extensions of Free Pro-p Groups

Let G be a pro-p group, which is a finite cyclic p-extension of a pro-p product of a free pro-p group F. We show that G can be described as the free pro-p product of the normalizers of a suitable collection of its cyclic subgroups of order p and some free pro-p group. From this we deduce that G can be realized as the fundamental group of a profinite connected graph of finite cyclic groups of bounded order. This result is thus the pro-p analogue of a special case of the well known characterization of virtually free groups as fundamental groups of graphs of groups, owing to Karrass, Pietrowski, Solitar, Cohen, and Scott. We conclude the paper by presenting a counterexample to a possible pro-p analogue of this result for noncyclic extensions of free pro-p groups, which we use in turn to construct a counterexample to pro-p versions of the Kurosh subgroup theorem.