A virtually free pro-\(p\) need not be the fundamental group of a profinite graph of finite groups.
According to a celebrated result of \textit{J.-P.~Serre}, a torsion-free pro-\(p\) group that contains an open free pro-\(p\) group is itself free pro-\(p\) [Topology 3, 413-420 (1965; Zbl 0136.27402)]. The authors of the paper under review have extended this result to finitely generated pro-\(p\) groups [preprint (2007)]. The latter result is a pro-\(p\) analogue of a description of virtually free discrete groups, which does not depend on the cardinality of a set of generators. The goal of this note is to exhibit a countably generated pro-\(p\) group for which the corresponding result fails.
scientific article; zbMATH DE number 5675001
scientific article