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We prove the pro‐  p   version of the Karras, Pietrowski, Solitar, Cohen and Scott result stating that a virtually free group acts on a tree with finite vertex stabilizers. If a virtually free pro‐  p   group   G   has finite centralizers of all non‐trivial torsion elements, a stronger statement is proved:   G   embeds into a free pro‐  p   product of a free pro‐  p   group and a finite   p  ‐group. Integral   p  ‐adic representation theory is used in the proof; it replaces the Stallings theory of ends in the pro‐  p   case.

Infinitely generated virtually free pro‐ p groups and p ‐adic representations