We present the result that, under a certain condition, free pro-$p$ products with procyclic amalgamation inherit from its free factors the property of each 2-generator pro-$p$ subgroup being free pro-$p$. This generalizes known pro-$p$ results, as well as some pro-$p$ analogues of classical results in Combinatorial Group Theory. To present our theorem we discuss certain splitting theorems for pro-$p$ groups acting virtually freely on pro-$p$ trees; for instance, any infinite finitely generated pro-$p$ group acting on a pro-$p$ tree such that the restriction of the action to some open subgroup is free splits over an edge stabilizer either as an amalgamated free pro-$p$ product or as a pro-$p$ HNN-extension.
Based on the paper Splitting theorems for pro-p groups acting on pro-p trees with W. Herfort (TU Wien, Austria) and P. Zalesskii (UnB, Brazil).
Work partially supported by CAPES and CNPq.