Resumen de Subgroup properties of pro-pp extensions of centralizers

Ilir Snopce, Pavel A. Zalesskii

• We prove that a finitely generated pro- pp group acting on a pro- pp tree TT with procyclic edge stabilizers is the fundamental pro- pp group of a finite graph of pro- pp groups with vertex groups being stabilizers of certain vertices of TT and edge groups (when non-trivial) being stabilizers of certain edges of TT , in the following two situations: (1) the action is nn -acylindrical, i.e., any non-identity element fixes not more than nn edges; (2) the group GG is generated by its vertex stabilizers. This theorem is applied to obtain several results about pro- pp groups from the class LL defined and studied in Kochloukova and Zalesskii (Math Z 267:109–128, 2011) as pro- pp analogues of limit groups. We prove that every pro- pp group GG from the class LL is the fundamental pro- pp group of a finite graph of pro- pp groups with infinite procyclic or trivial edge groups and finitely generated vertex groups; moreover, all non-abelian vertex groups are from the class LL of lower level than GG 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 GG from the class LL has Euler–Poincaré characteristic zero if and only if it is abelian, and if every abelian pro- pp subgroup of GG is procyclic and GG itself is not procyclic, then def(G)≥2def(G)≥2 . Moreover, we prove that GG satisfies the Greenberg–Stallings property and any finitely generated non-abelian subgroup of GG has finite index in its commensurator.