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Let G be a finitely generated infinite pro-p group acting on a pro-p treesuch that the restriction of the action to some open subgroup is free. Then weprove that G splits as a pro-p amalgamated product or as a pro-p HNN-extensionover an edge stabilizer.  Using this result we prove under certain conditions that free pro-p productswith procyclic amalgamation inherit from its free factors the property of each2-generated subgroup being free pro-p. This generalizes known pro-p results, aswell as some pro-p analogs of classical results in abstract combinatorial grouptheory.

Splitting theorems for pro-$p$ groups acting on pro-$p$ trees and 2-generated subgroups of free pro-$p$ products with procyclic amalgamations