Subgroup properties of pro- $$p$$ p extensions of centralizers

Abstract

We prove that a finitely generated pro-p group acting on a prop 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 L defined and studied in [16] as pro-p analogues of limit groups. We prove that every pro-p group G from the class 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 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 [16]. Namely, we prove that a group G from the class 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 def(G) ≥ 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.

Subgroups and homology of extensions of centralizers of pro-pgroups

“…The pro‐ p version of it is also valid with changing all the groups in the proof to pro‐ p groups. Applying this for and by Theorem A the rank gradient for pro‐ p groups is Then and by [, Prop. 3.4] H is abelian.…”

“…By restriction G acts on T but in general is not finite and hence applications of the pro‐ p version of Bass‐Serre theory for pro‐ p groups is difficult for this action. Still in Snopche, Zalesskii proved that T can be modified to a pro‐ p tree Γ with cofinite G ‐action. We shall use this several times in the paper and therefore state it in the form we need.…”

“…This property for abstract limit groups was proved in . The proof of [, Prop. 3.4] relies on developing a pro‐ p version of the Bass‐Serre theory in , see Theorem .…”

“…The study of the pro‐ p groups was continued by Snopce, Zalesskii in [, Prop. 3.4] where it was shown that a group from the class has Euler characteristic if and only if G is abelian.…”

“…The following result shows that for an open subgroup U of G , where is non‐abelian, the function is a monotonically increasing function of the index . Note that by [, Prop. 3.4] G has a non‐zero Euler characteristic and under the conditions of Theorem A, iii) asymptotically behaves as a non‐constant linear function.…”