Subgroup properties of pro- $$p$$ 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–Poincare 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.