A Note on the Quasi Simple Filtration of Profinite Groups

We illustrate some intuitions of Poénaru (of more than 60 years ago) in connection with the property of quasi simple filtration, which was formulated later by Brick and Mihalik for finitely presented groups. This property adapts certain topological decompositions, which are proper to Low Dimensional Topology and Riemannian Geometry, to the context of Algebraic Topology, mostly to fundamental groups and universal coverings. We sketch some steps in the evolution of the property of quasi simple filtration, showing that there are significant margins of generalization to algebraic fundamental groups in theory of profinite groups.