Free products with amalgamation and HNN-extensions are two main constructions of combinatorial group theory. I shall discuss these two constructions in the category of pro-p groups, presenting results on splittings of pro-p groups as an amalgamated free pro-p product or a pro-p HNN-extension and relating them with pro-p version of Bass-Serre's theory of groups acting on trees. I shall also compare the pro-p results with similar results for abstract groups.
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