P-projective groups and pro-p trees

Combining results of J-P. Serre and L. Ribes and P. Zalesskii one concludes that a profinite group G acting without inversion of edges and with pro-p' stabilizers on a pro-p tree T must be p-projective (Thm.~B). It is shown that a finitely generated, p-projective virtual pro-p group G has such an action on a pro-p tree T (Thm.~C). However, not every such profinite group G can act on a locally-finite pro-p tree without inversion of edges, with finite vertex stabilizers and with finitely many orbits (Thm.~E). This fact is deduced from the existence of p-irrational, finitely generated, p-projective, virtual pro-p groups (Prop.~D) using the theory of p-Lefschetz numbers