A virtually free pro-p need not be the fundamental group of a profinite graph of finite groups

Splitting theorems for pro-p groups acting on pro-p trees

“…[ We end this section by recalling the example from [4] of an infinite countably generated pro-p group acting virtually freely on a pro-p tree that satisfies none of the conclusions of Theorem A. Example 3.9 Let A and B be groups of order 2 and H be a pro-2 HNN-extension with base group A × B, associated subgroups A and B, and stable letter t. Note that H admits an automorphism ϕ of order 2 that swaps A and B and inverts t. Moreover, the holomorph H ϕ is isomorphic to ((A × B) ϕ ) A (A × ϕt ), the free pro-2 product of the dihedral group of order 8 and the Klein four-group amalgamated along the cyclic group A.…”

“…Since the amalgamated free pro-2 product acts virtually freely on its standard pro-2 tree, the group G acts virtually freely on this pro-2 tree by restriction. The main result of Herfort and Zalesskii [4] shows that G does not split over a finite group as fundamental pro-2 group of a profinite graph of finite 2-groups. Its proof also shows that G does not decompose as an amalgamated free pro-2 product or as a pro-2 HNN-extension.…”

Virtually free pro-p groups

“…Set H 0 = Tor(G 0 ) and H = H 0 C. Since G 0 is virtually free pro-2, G and H are virtually free pro-2. The main result in [8] shows that H does not decompose as the fundamental pro-2 group of a profinite graph of finite 2-groups. It follows also from the proof in [8] that H does not split as a amalgamated free pro-2 product or a pro-2 HNN-extension over some finite subgroup.…”

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

“…(3) If every abelian pro-p subgroup of G is procyclic and G itself is not procyclic, then G has exponential subgroup growth; (4) There are only finitely many conjugacy classes of non-procyclic maximal abelian subgroups of G; (5) [Greenberg-Stallings Property] If H and K are finitely generated subgroups of G with the property that H ∩ K has finite index in both H and K, then H ∩ K has finite index in H, K ; (6) If H is a finitely generated subgroup of G, then H has a root in G; (7) If H is a finitely generated non-abelian subgroup of G, then |Comm G (H) :…”

“…This means that G can be described by taking iterated amalgamated free products and HNN extensions. The analogue of the structure theorem in the pro-p case does not hold in general [7]. Nevertheless, it was proved in [9] that every finitely generated infinite pro-p group that acts virtually freely on some pro-p tree D is isomorphic to the fundamental pro-p group of a finite graph of finite p-groups whose edge and vertex groups are isomorphic to the stabilizers of some edges and vertices of D.…”