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.

The main purpose of this article is to study pro-p groups with quadratic Fp-cohomology algebra, i.e. H • -quadratic pro-p groups. Prime examples of such groups are the maximal Galois pro-p groups of fields containing a primitive root of unity of order p.We show that the amalgamated free product and HNN-extension of H • -quadratic pro-p groups is H • -quadratic, under certain necessary conditions. Moreover, we introduce and investigate a new family of pro-p groups that yields many new examples of H • -quadratic groups: p-RAAGs. These examples generalise right angled Artin groups in the category of pro-p groups. Finally, we explore "Tits alternative behaviour" of H • -quadratic pro-p groups.

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Given a torsion-free p-adic analytic pro-p group G with dim(G) < p, we show that the selfsimilar actions of G on regular rooted trees can be studied through the virtual endomorphisms of the associated Z p -Lie lattice. We explicitly classify 3-dimensional unsolvable Z p -Lie lattices for p odd, and study their virtual endomorphisms. Together with Lazard's correspondence, this allows us to classify 3-dimensional unsolvable torsion-free p-adic analytic pro-p groups for p 5, and to determine which of them admit a faithful self-similar action on a p-ary tree. In particular, we show that no open subgroup of SL 1 1 (∆ p ) admits such an action. On the other hand, we prove that all the open subgroups of SL △ 2 (Z p ) admit faithful self-similar actions on regular rooted trees. *

Let p be a prime. Uniform pro-p groups play a central role in the theory of p-adic Lie groups. Indeed, a topological group admits the structure of a p-adic Lie group if and only if it contains an open pro-p subgroup which is uniform. Furthermore, uniform pro-p groups naturally correspond to powerful