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 vertex groups being stabilizers of certain vertices of $T$ and edge groups (when non-trivial) being stabilizers of certain edges of $T$, 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 Kochloukova and Zalesskii (Math Z 267:109–128, 2011) 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 Kochloukova and Zalesskii (Math Z 267:109–128, 2011). Namely, we prove that a group $G$ from the class $\mathcal L$ has Euler–Poincaré 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 $\mathrm{def}(G)\ge 2$. 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.