Subgroup properties of pro-p extensions of centralizers

Abstract

We prove that a finitely generated pro-$p$ group acting on a pro-$p$ tree $T$ with procyclic edge stabilizers is the fundamental pro-$p$ group of a finite graph of pro-$p$ groups with edge and vertex groups being stabilizers of certain vertices and edges of $T$ respectively, in the following two situations: 1) the action is $n$-acylindrical, i.e., any non-identity element fixes not more than $n$ edges; 2) the group $G$ is generated by its vertex stabilizers. This theorem is applied to obtain several results about pro-$p$ groups from the class $\mathcal{L}$ defined and studied in [Math. Z. 267 (2011), 109-128] as pro-$p$ analogues of limit groups. We prove that every pro-$p$ group $G$ from the class $\mathcal{L}$ is the fundamental pro-$p$ group of a finite graph of pro-$p$ groups with infinite procyclic or trivial edge groups and finitely generated vertex groups; moreover, all non-abelian vertex groups are from the class $\mathcal{L}$ of lower level than $G$ with respect to the natural hierarchy. This allows us to give an affirmative answer to questions 9.1 and 9.3 in [Math. Z. 267 (2011), 109-128]. Namely, we prove that a group $G$ from the class $\mathcal{L}$ has Euler-Poincar\'e characteristic zero if and only if it is abelian, and if every abelian pro-$p$ subgroup of $G$ is procyclic and $G$ itself is not procyclic, then $def(G) \geq 2$. Moreover, we prove that $G$ satisfies the Greenberg-Stallings property and any finitely generated non-abelian subgroup of $G$ has finite index in its commensurator. We also show that all non-solvable Demushkin groups satisfy the Greenberg-Stallings property and each of their finitely generated non-trivial subgroups has finite index in its commensurator.