Pro-p groups acting on trees with finitely many maximal vertex stabilizers up to conjugation

Abstract

We prove that a finitely generated pro-p group G acting on a pro-p tree T splits as a free amalgamated pro-p product or a pro-p HNN-extension over an edge stabilizer. If G acts with finitely many vertex stabilizers up to conjugation, we show that it is the fundamental pro-p group of a finite graph of pro-p groups (\({\cal G},\Gamma \)) with edge and vertex groups being stabilizers of certain vertices and edges of T respectively. If edge stabilizers are procyclic, we give a bound on Г in terms of the minimal number of generators of G. We also give a criterion for a pro-p group G to be accessible in terms of the first cohomology H1(\(G,{\mathbb{F}_p}[[G]]\)).